diff --git a/-dFQT4oBgHgl3EQfKDXj/content/tmp_files/2301.13259v1.pdf.txt b/-dFQT4oBgHgl3EQfKDXj/content/tmp_files/2301.13259v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0bde215f1a6bf6411b4cab4e0467a30f7deb7fba --- /dev/null +++ b/-dFQT4oBgHgl3EQfKDXj/content/tmp_files/2301.13259v1.pdf.txt @@ -0,0 +1,2598 @@ +arXiv:2301.13259v1 [stat.ME] 30 Jan 2023 +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +Abstract. In this article, we define extensions of copula-based dependence measures for data with +arbitrary distributions, in the non-serial case, i.e., for independent and identically distributed random +vectors, as well as in serial case, i.e., for time series. These dependence measures are covariances with +respect to a multilinear copula associated with the data. We also consider multivariate extensions based +on M¨obius transforms. We find the asymptotic distributions of the statistics under the hypothesis of +independence or randomness and under contiguous alternatives. This enables us to find out locally +most powerful test statistics for some alternatives, whatever the margins. Numerical experiments are +performed for combinations of these statistics to assess the finite sample performance. +1. Introduction +In many cases, simple measures of dependence like Kendall’s tau and Spearman’s rho, perform as well +as more complex statistics like Cram´er-von Mises statistics based on empirical processes, and are gener- +ally much faster to compute. However, tests based of such measures are not always consistent. Neverthe- +less, tests of independence or randomness based on copulas should always be performed. Here, we are in- +terested in copula-based dependence measures for a sample of iid observations Xi = (Xi1, . . . , Xid) ∼ H, +with vector of margins F = (F1, . . . , Fd), d ≥ 2, called the non-serial case, as well as for the serial case, +i.e., for stationary time series Y1, . . . , Yn with common cumulative distribution function (cdf) F, where +one defines the random vectors Xt = (Yt, . . . , Yt+1−d); hereafter, the series Y is extended in a circular +way by setting Yt+n = Yt for all t ∈ Z. +In the bivariate case, when the margins are continuous, most copula-based dependence measures are +theoretically defined as the correlation ̺K(C) = cor +� +K−1 +1 (Ui1), K−1 +2 (Ui2) +� +, since by continuity of the +margins, Ui = (Ui1, Ui2) = F(Xi) ∼ C, for a unique copula C. Here K = (K1, K2) is a given vector +of cdfs, with mean µ1, µ2, and variances σ2 +1, σ2 +2. The value under independence is clearly 0. Next, by +Key words and phrases. Independence; randomness; multilinear copula; Spearman’s rho, van der Waerden’s coefficient; +Savages’s coefficient. +Funding in partial support of this work was provided by the Fonds qu´eb´ecois de la recherche en sant´e and the Natural +Sciences and Engineering Research Council of Canada. +1 + +2 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +Hoeffding’s identity (Hoeffding, 1940), +(1) +̺K(C) = γK(C) +σ1σ2 += +1 +σ1σ2 +� +R2 [C {K1(x1), K2(x2)} − K1(x1)K2(x2)] dx1dx2, +since +� +K−1 +1 (U1), K−1 +2 (U2) +� +has joint cdf C ◦ K. For example, suppose that (U1, U2) ∼ C. Then, taking +K1 = K2 = D, where D is the cdf of the uniform distribution over (0, 1), one obtains Spearman’s +correlation ρS(C) = 12E(U1U2) − 3. +The case K1 = K2 = Φ, where Φ is the cdf of the standard +Gaussian distribution yields the van der Waerden coefficient ρvdw(C) = E +� +Φ−1(U1)Φ−1(U2) +� +, while if +K1 = K2 is the cdf of a Bernoulli(1/2), one gets Blomqvist’s coefficient 4C(1/2, 1/2) − 1. Note that by +definition, when K1 = K2, ̺K(C+) = 1 for the complete dependence, where C+(u1, u2) = min(u1, u2) +is the Fr´echet-Hoeffding upper bound. However, when K1 ̸= K2, the covariance γK1,K2(C) must be +divided by E +� +K−1 +1 (U1)K−1 +2 (U1) +� +− µ1µ2 to give 1 for complete dependence. Blest’s coefficient (Blest, +2000, Genest and Plante, 2003) can be seen as such an example if one considers a natural modification. +In fact, Blest’s coefficient has been originally defined as the covariance between (1 − U1)2 and U2. An +obvious modification is obtained by taking K1(u) = u1/2, u ∈ [0, 1], K2 = D, so the modified coefficient +is 12E(U 2 +1 U2)−2, normalised to give 1 for complete dependence. Genest and Plante (2003) also proposed +a symmetrised Blest’s coefficient, which, in our general setting, amounts to defining +γ∗ +K1,K2(C) = γK1,K2(C) + γK2,K1(C) +2 +. +Not all copula-based dependence measures are defined by a covariance (up to a constant), a well-known +example being Kendall’s tau, defined by τ(C) = 4 +� +(0,1)2 C(u1, u2)dC(u1, u2) − 1 = 4E{C(U1, u2)} − 1. +However, Kendall’s tau and Spearman’s rho have an equivalent limiting distribution, even under a +sequence of contiguous alternatives. +Estimating the dependence measures defined previously is relatively straightforward. In the continu- +ous case, the copula is replaced by the empirical copula +ˆCn(u1, u2) = n−1 � +i=1 +I +� +n +n + 1Fn1(Xi1) ≤ u +� +I +� +n +n + 1Fn2(Xi2) ≤ u2 +� +, +u1, u2 ∈ [0, 1]. +In fact, γK(C) can be estimated by γK(Cn), and according to Genest and R´emillard (2004), one has +γK +� +ˆCn +� += +� +R2 [Cn {K1(x1), K2(x2)} − K1(x1)K2(x2)] dx1dx2. +Asymptotic limits and their representations are easier to work with the latter representation, being a +linear functional of the empirical process �Cn(u1, u2) = n1/2 � +ˆCn(u1, u2) − u1u2 +� +. These dependence +measures and their estimation work well when the margins are continuous. However, for applications, + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +3 +there is a need to consider arbitrary distributions, i.e., when at least of the margins is not continuous. +In this case, since there are ties, one might be tempted to replace the ranks by the mid-ranks. However, +the asymptotic distribution might not be simple enough and it makes sense to base the dependence +measures on copulas. +The main problem here is that the copula is not unique. If X ∼ H, there are infinitely many copulas +satisfying Sklar’s equation H = C ◦ F (Sklar, 1959). To construct solutions for this equation, for any +copula C, take V ∼ C independent of X ∼ H and set U = ψF(X, V), where Uj = ψFj(Xj, Vj) = +Fj(Xj−) + Vj∆Fj(Xi), with Fj(x−) = P(Xj < x) and ∆Fj(x) = Fj(x) − Fj(x−) = P(Xj = x), +j ∈ {1, . . ., d}. It is known (Ferguson, 1967, R¨uschendorf, 1981, Neˇslehov´a, 2007, Brockwell, 2007) that +for any j ∈ {1, . . . , d}, Uj ∼ U(0, 1), and the joint cdf CC of U is a copula satisfying Sklar’s equation. +In addition, there is one interesting copula C✠ in this family, the so-called multilinear copula, obtained +by taking C = Π, the independence copula, i.e., Π(u) = �d +j=1 D(uj). One interesting property of C✠ is +that if H(x) = �d +j=1 Fj(xj), then CC = Π if and only if CC = C✠. As a by-product, taking the empirical +joint cdf Hn with the vector of margins Fn = (Fn1, Fn2), one obtains the empirical multilinear copula +�C✠ +n , for which an explicit expression will be given in the next section. Note that contrary to �Cn, �C✠ +n is a +genuine copula, so all dependence measures presented before can be computed with C✠ and its empirical +counterpart �C✠ +n . This is the approach that we propose here. Note that the asymptotic behaviour of +the associated versions of Kendall’s tau and Spearman’s rho has been studied in Genest et al. (2014), +and tests of independence based on �C✠ +n were proposed in Genest et al. (2019), while in the serial case, +tests of randomness based on the serial version �C✠,s +n +have been studied in Nasri (2022), as well as the +asymptotic behaviour of the serial versions of Kendall’s tau and Spearman’s rho. +The main aim of this article is to define bivariate and multivariate extensions of the dependence +measures when the margins are arbitrary, to find explicit expressions of the measures, and to study +their asymptotic behaviour. We will also look at the asymptotic distribution of the test statistics under +a sequence of contiguous alternatives to be able to suggest locally powerful tests for given dependence +models, in the same spirit as Genest and Verret (2005) did in the bivariate case for continuous margins. +To this end, we also present a new representation of the multilinear copulas in the serial and non-serial +cases that enables us to perform calculations more easily. Note that in both Genest et al. (2019) and +Nasri (2022), the main focus was on using Cram´er-von Mises statistics of related multilinear processes, +which is not done here. + +4 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +In Section 2, we recall the definitions and properties of multilinear copulas in a serial setting (Nasri, +2022) and non-serial setting Genest et al. (2019), together with their associated M¨obius transforms. +Next, in Section 3, we define the serial and non-serial versions of dependence measures, providing explicit +formulas that are easy to implement, and we study their asymptotic behaviour under the null hypothesis +of independence or randomness. Multivariate extensions similar to those defined in Genest and R´emillard +(2004) and Genest et al. (2014) will also be studied. Next, in Section 4, we will study the asymptotic +behaviour of the proposed dependence measures under a sequence of contiguous alternatives, using the +results of Genest et al. (2019) and Nasri (2022). This will enable us to find the locally most powerful +tests based of the proposed dependence measures. We will also discuss how to combine the proposed +dependence measures. Finally, numerical experiments will be performed in Section 5 to assess the power +of the tests for finite samples. +2. Multilinear copulas and associated empirical processes +From now on, we consider the following two settings: the non-serial case and the serial case. In the +non-serial case, we have independent and identically distributed (iid) random vectors U1, . . . , Un ∼ C, +for a given copula C, and the observations are Xi = F−1(Ui), i ∈ {1, . . . , n}. In the serial setting, we +have a series of random variables U1, . . . , Un ∼ U(0, 1), and the observed time series is Yt = F −1(Ut), +t ∈ {1, . . . , n}, where (U1, . . . , Un) is d-Markov process with copula C, meaning that the distribution of +(Ut, . . . , Ut+1−d) is C, C has density c, and the joint density of (U1, . . . , Un), evaluated at (u1, . . . , un), +is given by +(2) +n +� +t=d+1 +c(ut, ut−1, . . . , ut+1−d) +cd−1(ut−1, . . . , ut+1−d), +where cd−1(ud, . . . , u2) = +� 1 +0 +c(ud, . . . , u2, s)ds. We can now define the multilinear copula. For any +j ∈ {1, . . . , d}, set JFj(xj, uj) = E +� +ψFj(Xj, uj)|Xj = xj +� += P +� +Fj(xj−) + Vj∆Fj(xj) ≤ uj +� +. +Then, +JFj(xj, uj) = + + + +I{Fj(xj) ≤ uj}, +if ∆Fj(xj) = 0, +D +� +uj−Fj(xj−) +∆Fj (xj) +� +, +if ∆Fj(xj) > 0. +, where D is the cdf of U ∼ U(0, 1). Note that +when ∆Fj(xj) > 0, JFj(xj, uj) = 0 if uj ≤ Fj(xj−), JFj(xj, uj) = 1 if uj ≥ Fj(xj), and JFj(xj, uj) = +uj − Fj(xj−) +∆Fj(xj) +if Fj(xj−) ≤ uj ≤ Fj(xj). Using properties of conditional expectations, one obtains +(3) +C✠(u) = E + + + +d +� +j=1 +JFj(Xj, uj) + + + , +u ∈ [0, 1]d. + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +5 +As a result, +(4) +�C✠ +n (u) = n−1 +n +� +i=1 +d +� +j=1 +JFnj(Xij, uj) = n−1 +n +� +i=1 +d +� +j=1 +D +�uj − Fnj(Xij−) +∆Fnj(Xij) +� +, +u ∈ [0, 1]d. +This new expression is different from what appears in the literature, e.g., Genest et al. (2017, 2019), but +it is easier to manipulate for our purposes. In fact, ˆC✠ +n was previously defined by +ˆC✠ +n (u) = n−1 +n +� +i=1 +d +� +j=1 +� +λFnj(uj)I{Xij ≤ F −1 +nj (uj)} + {1 − λFnj(uj)}I{Xij < F −1 +nj (uj)} +� +, +where, for any cdf G and u ∈ (0, 1), λG(u) = + + + + + +u − G +� +G−1(u)− +� +∆G {G−1(u)} +, +∆G +� +G−1(u) +� +> 0, +1, +otherwise. +. Next, the +empirical serial multilinear copula, first defined and studied in Nasri (2022), can also be written as +(5) +�C✠,s +n +(u) = n−1 +n +� +t=1 +d +� +j=1 +D +�uj − Fn(Yt+1−j−) +∆Fn(Yt+1−j) +� +, +u ∈ [0, 1]d, +where Fn(y) = n−1 +n +� +t=1 +{Yt+1−j ≤ y}, y ∈ R. +Using the circular construction, it follows that for +any j ∈ {1, . . . , d}, Fn(y) = n−1 +n +� +t=1 +{Yt+1−j ≤ y}. Further define the empirical multilinear processes +�C✠ +n = n1/2 � +�C✠ +n − Π +� +and �C✠,s +n += n1/2 � +�C✠,s +n +− Π +� +. Next, let Nd be the set of all subsets A of {1, . . ., d} +with card (A) = |A| > 1, and let Sd be the set of all elements A of Nd with A ∋ 1. It has been shown, +e.g., Genest and R´emillard (2004), Ghoudi and R´emillard (2018), Genest et al. (2019), Nasri (2022), +that M¨obius transforms of empirical processes have nice asymptotic properties for tests of independence +or tests of randomness. To this end, define +(6) +G✠ +A,n(u) = MA +� +�C✠ +n +� +(u) = n−1/2 +n +� +i=1 +� +j∈A +� +D +�uj − Fnj(Xij−) +∆Fnj(Xij) +� +− uj +� +, +A ∈ Nd, +and +(7) +G✠,s +A,n(u) = MA +� +�C✠,s +n +� +(u) = n−1/2 +n +� +t=1 +� +j∈A +� +D +�uj − Fn(Yt+1−j−) +∆Fn(Yt+1−j) +� +− uj +� +, +A ∈ Sd, +where the M¨obius transform MA is defined in Appendix A. Next, for any s, t ∈ [0, 1], and any cdf G, set +(8) ΓG(s, t) = s∧t−st− +� +x:∆G(x)>0 +I{G(x−) ≤ s∧t ≤ s∨t ≤ G(x)}{(s ∧ t) − G(x−)} {G(x) − s ∨ s)} +∆G(x) +. +The main findings of Genest et al. (2019) and Nasri (2022) that we need can be summarised as follows: + +6 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +Theorem 1. Under the null hypothesis of independence, +� +G✠ +A,n : A ∈ Nd +� +converge jointly in ℓ∞ � +(0, 1)d� +to independent Gaussian processes +� +G✠ +A : A ∈ Nd +� +, where E +� +G✠ +A(u)G✠ +A(v) +� += +� +j∈A +ΓFj(uj, vj). Fur- +thermore, under the null hypothesis of randomness, +� +G✠,s +A,n : A ∈ Sd +� +converge jointly in ℓ∞ � +(0, 1)d� +to +independent Gaussian processes +� +G✠,s +A +: A ∈ Sd +� +, where E +� +G✠,s +A (u)G✠,s +A (v) +� += +� +j∈A +ΓF (uj, vj). +Remark 1. The formulas for the covariances in Theorem 1 follows from (D.6) and (8) in Nasri (2022). +One can check that for any s, t ∈ [0, 1], ΓG(s, t) ≥ 0 with equality if and only if s ∧ t = 0 or s ∨ t = +1. It is interesting to note that for sets A of size 2, G✠ +A,n and G✠,s +A,n are empirical multilinear copula +processes. In fact, for A = {j, k} ∈ Nd, j < k, G✠ +A(u1, u2) = n1/2 � +�C✠ +A (u1, u2) − u1u2 +� +, where �C✠ +A is +the multilinear copula for the pairs (Xij, Xik), i ∈ {1, . . ., n}. Similarly, for any A = {1, 1 + ℓ} ∈ Sd, +G✠,s +A (u1, u2) = n1/2 � +�C✠,s +A +(u1, u2) − u1u2 +� +, where C✠,s +A +is the multilinear copula for the pairs (Yt, Yt−ℓ), +t ∈ {1, . . . , n}. +3. Dependence measures for arbitrary distributions +From now on, let K = (K1, . . . , Kd) be a vector of margins with mean µj and variance σ2 +j , j ∈ +{1, . . . , d}, and define Lj(u) = +� u +0 +K−1 +j +(v)dv. Next, for any j ∈ {1, . . ., d}, and any cdf G, define +Kj,G(x) = +� 1 +0 +K−1 +j +{G(x−) + s∆G(x)} ds. +Then Kj,G(x) = K−1 +j +{G(x)}, if G is continuous at x, and Kj,G(x) = LKj{G(x)} − LKj{G(x−)} +∆G(x) +, if G +is not continuous at x. The extension of the covariance measures are defined the following way: In the +non-serial case, for any A ∈ Nd, set +γK,A +� +�C✠ +n +� += n−1/2(−1)|A| +� +RA +� +G✠ +A,n {K(x)} dx, +while in the serial case, for any A ∈ Sd, set +γK,A +� +�C✠,s +n +� += n−1/2(−1)|A| +� +RA +� +G✠,s +A,n {K(x)} dx. +It then follows from Proposition 2 in Appendix A that for any A ∈ Nd, in the non-serial case, +(9) +γK,A +� +�C✠ +n +� += n−1 +n +� +i=1 +� +j∈A +�Lj{Fnj(Xij)} − Lj{Fnj(Xij−)} +∆Fnj(Xij) +− µj +� +, + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +7 +while in the serial case, for any A ∈ Sd, +(10) +γK,A +� +�C✠,s +n +� += n−1 +n +� +t=1 +� +j∈A +�Lj{Fn(Yt+1−j)} − Lj{Fn(Yt+1−j−)} +∆Fn(Yt+1−j) +− µj +� +. +Example 1. For Spearman’s rho, Kj ≡ D, so Lj(u) = u2 +2 . For van der Waerden’s coefficient, Kj ≡ Φ, +so Lj = −φ ◦ Φ−1, µj = 0. For Savage’s coefficient, Kj(x) ≡ 1 − e−x, x ≥ 0, so Lj(u) = u − u log u, +µj = 1, with the convention that 0 log 0 = 0. Finally, for the modified Blest’s coefficient in the bivariate +case, K−1 +1 (u) = u2, u ∈ [0, 1], K2 = D. As a result, one gets the following formula for 12γK1,K2 +� +�C✠ +n +� +: +2n−1 +n +� +i=1 +� +F 2 +nj(Xij−) + Fnj(Xij−)Fnj(Xij) + F 2 +nj(Xij) − 1 +� +{Fn2(Xi2−) + Fn2(Xi2) − 1} . +Remark 2. For continuous margins, ∆Fnj(Xij) = n−1 a.s., so +Lj{Fnj(Xij)} − Lj{Fnj(Xij−)} +∆Fnj(Xij) +≈ K−1 +j +� +n +n + 1Fnj(Xij) +� +. +Note that in general, n−1 �n +i=1 K−1 +j +� +n +n+1Fnj(Xij) +� +̸= µj, while +n−1 +n +� +i=1 +Lj{Fnj(Xij)} − Lj{Fnj(Xij−)} +∆Fnj(Xij) += µj, +j ∈ {1, . . . , d}. +This shows that even for continuous margins, one should use formulas (9)–(10) based on the multilinear +copulas, since we do not need to work with the normalised +n +n+1Fnj(Xij). +The following result is an immediate consequence of Theorem 1, the continuous mapping theorem, +together with representations (9) and (10). When K−1 +j +is unbounded, one can use the same technique as +in the corresponding proofs in Genest and R´emillard (2004), meaning that one integrates GA,n{K(x)} +on large compact sets and show that the remainder can be made arbitrarily small, since K−1 +j +is square +integrable by hypothesis. The covariance formulas follows from (D.6)-(D.7) in Nasri (2022). +Corollary 1. Under the hypothesis of independence, +� +n1/2γK,A,n +� +�C✠ +n +� +: A ∈ Nd +� +converge jointly to +independent Gaussian random variables with variance ς2 +K,F,A = +� +j∈A +ς2 +Kj,Fj, where for any cdf G, +(11) +ς2 +Kj,G = +� +{Kj,G{G(x)} − µ}2 dG(x) = +� +R2 ΓG{Kj(x), Kj(y)}dxdy, +j ∈ {1, . . . , d}. +Furthermore, under the hypothesis of randomness, +� +n1/2γK,A,n +� +�C✠,s +n +� +: A ∈ Sd +� +converge jointly to +independent Gaussian random variables with variance ς2 +K,F,A = +� +j∈A +ς2 +Kj,F . + +8 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +Remark 3. It follows from Genest and R´emillard (2004) that σ2 +j = +� +R2{Kj(x ∧ y) − Kj(x)Kj(y)}dxdy. +Finally, ς2 +Kj,Fj = var +� +Kj,Fj(Xj) +� +, if Xj ∼ Fj, j ∈ {1, . . . , d}. +The next result is fundamental for applications since it shows how to normalised the statistics to +standard Gaussian distributions in the limit. Its proof is given in Appendix B.1. +Lemma 1. In the non-serial case, +s2 +Kj,Fnj = n−1 +n +� +i=1 +�Lj{Fnj(Xij)} − Lj{Fnj(Xij−)} +∆Fnj(Xij) +− µj +�2 +P r +−→ ς2 +Kj,Fj, +j ∈ {1, . . . , d}, +and in the serial case, +s2 +Kj,Fn = n−1 +n +� +t=1 +�Lj{Fn(Yt)} − Lj{Fn(Yt−)} +∆Fn(Yt) +− µj +�2 +P r +−→ ς2 +Kj,F , +j ∈ {1, . . . , d}. +4. Asymptotic behaviour along contiguous alternatives and local power +In this section, we consider contiguous alternatives of the form Cθn for the non-serial case as well as +the serial case described by (2), where Cθ0 = Π, for some θ0, and θn = θ0 + n−1/2δ. It is assumed that +the copula family Cθ is smooth enough, namely that the Conditions 1–2 in Genest et al. (2019) are met. +More precisely, assume that Cθ has a continuous density cθ which is square integrable, continuously +differentiable in a neighbourhood of θ0, with ˙c = ∇θcθ(u)|θ=θ0, (u) ∈ (0, 1)d, ˙C(u) = +� +(0,u] +˙c(s)ds, and +lim +n→∞ +� +(0,1)d[n1/2[{cθn(u)}1/2 − 1] − δ ˙c(u)/2]2du = 0. +Before stating the limiting distribution under the sequence of contiguous alternatives Cθn, for any A ∈ +Nd, set qA = MA( ˙C). It follows from Lemma 2, stated in the Appendix, and proven in Nasri (2022), +that in the non-serial case, MF(qA) = MA ◦MF +� +˙C +� += MA +� +˙C✠� +, while in the serial case, MF ⊗d(qA) = +MA ◦ MF ⊗d +� +˙C +� += MA +� +˙C✠,s� +. Under the previous conditions, the following results were obtained by +Genest et al. (2019) in the non-serial case, and by Nasri (2022) in the serial case. +Theorem 2. Under the sequence of contiguous alternatives Cθn, in the non-serial case, the processes +G✠ +A,n, A ∈ Nd, converge jointly in ℓ∞ � +(0, 1)d� +to G✠ +A + δMA( ˙C✠). Furthermore, in the serial case, the +processes G✠,s +A,n, A ∈ Sd, converge jointly in ℓ∞ � +(0, 1)d� +to G✠,s +A ++ δMA( ˙C✠). +Remark 4. Nasri (2022) also considered Poisson contiguous alternatives with conditional mean λt,n = +λ0 + δn−1/2Yt−1. In this case, for any A ∈ Sd, the processes G✠,s +A,n converge jointly in ℓ∞ � +(0, 1)d� +to + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +9 +G✠,s +A ++ +δ +λ0 I{A = {1, 2}}MF(f)(u1)MF (f)(u2), where f(u) = {LF (u) − λ0u}, and F is the cdf of the +Poisson with parameter λ0. +As a corollary, we obtain the asymptotic behaviour of the proposed dependence measures under the +sequence of contiguous alternatives Cθn. +Corollary 2. Under the sequence of contiguous alternatives Cθn, in the non-serial case, the random +variables n1/2γK,A +� +�C✠ +n +� +, A ∈ Nd, converge jointly to independent Gaussian random variables with +mean δ ˙γK,A +� +C✠� +and variance ς2 +K,F,A, where +(12) +˙γK,A +� +C✠� += +� +˙cA(u) +� +j∈A +� +Kj,Fj ◦ F −1 +j +(uj) − µj +� +du, +and CA is the copula restricted to components Uj with j ∈ A. Furthermore, in the serial case, the +random variables n1/2γK,A +� +�C✠,s +n +� +, A ∈ Sd, converge jointly to independent Gaussian random variables +with mean δ ˙γK,A +� +C✠,s� +and variance ς2 +K,F,A, where +(13) +˙γK,A +� +C✠,s� += +� +˙cA(u) +� +j∈A +� +Kj,F ◦ F −1(uj) − µj +� +du. +Note that if the margin Fj is continuous, Kj,Fj ◦ F −1 +j +(uj) = K−1 +j +(uj). In particular, in the serial case, +if the margin F is continuous, then ˙γK,A +� +C✠,s� += +� +˙cA(u) +� +j∈A +� +K−1 +j +(uj) − µj +� +du. +Remark 5. Since MA +� +C✠ +θ +� +(u) = Eθ + +� +j∈A +� +JFj(Xj, uj) − uj +� + +, Proposition 2 in Appendix A yields +γK,A +� +C✠ +θ +� += +(−1)|A| +� +RA Eθ + +� +j∈A +� +JFj{Xj, Kj(xj)} − Kj(xj) +� + + dx = Eθ + +� +j∈A +{Kj(Xj) − µj} + + , +so ˙γK,F,A = ∂θ γK,A +� +C✠ +θ +��� +θ=θ0. As a result, one obtains formulas (12) and (13). In particular, if +˙cA = � +j∈A Jj(uj), then in the non-serial case, for any A ∈ Nd, +(14) +˙γK,A +� +C✠� += +� +j∈A +� 1 +0 +� +Kj ◦ F −1 +j +(uj) − µj +� +Jj(uj)duj = +� +j∈A +cov +� +Kj ◦ F−1 +j +(U), Jj(U) +� +, +where U ∼ U(0, 1), while in the serial case, for any A ∈ Sd, +(15) +˙γK,A +� +C✠,s� += +� +j∈A +cov +� +Kj ◦ F−1(U), Jj(U) +� +. + +10 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +4.1. Applications for local power. First note that for many copula families satisfying the smooth- +ness conditions listed at the beginning of the section, one has ˙cA(u) = � +j∈A J(uj), and Jj is often a +quantile function. In this case, choosing K−1 +j += Jj would make sense in order to have a non-zero mean, +and hence having more local power by maximising formulas (14)–(15). This is what was proposed in +Genest and Verret (2005) in the bivariate case, where the margins were assumed to be continuous. In +fact, the next proposition shows that this choice is also optimal for any margins. The proof of the +following result is given in Appendix B.2. +Proposition 1. Suppose that ˙cA(u) ∝ � +j∈A G−1 +j (uj), u ∈ (0, 1)d, where G = (G1, . . . , Gd) is a vector +of margins with means (˜µ1, . . . , ˜µd) and variances +� +˜σ2 +1, . . . , ˜σ2 +d +� +, and assume U ∼ U(0, 1). +Then, in +the non-serial case cov +� +Kj ◦ F−1 +j +(U), G−1 +j +(U) +� += cov {Kj(Xj), Gj(Xj)}, j ∈ {1, . . . , d}, where Xj ∼ Fj, +so ˙γK,A +� +C✠� += � +j∈A cov {Kj(Xj), Gj(Xj)}. In particular, if K = G, then ˙γK,A +� +C✠� += ς2 +K,F,A. In +the serial case, cov +� +Kj ◦ F−1(U), G−1 +j +(U) +� += cov {Kj(X), Gj(X)}, j ∈ {1, . . . , d}, where X ∼ F, so +˙γK,A +� +C✠,s� += � +j∈A cov {Kj(X), Gj(X)}. In particular, if K = G, then ˙γK,A +� +C✠,s� += ς2 +K,F,A. +Remark 6. Under the assumptions of Proposition 1, it follows from Proposition 3 in Genest and Verret +(2005) that the ARE between the test based on K, and G is given by +� +j∈A +cor2 {Kj(Xj), Gj(Xj)} in the non- +serial case, and the ARE is +� +j∈A +cor2 {Kj(X), Gj(X)} in the serial case. This shows that whenever ˙cA(u) ∝ +� +j∈A G−1 +j (uj), the ARE is maximised by taking K = G. Moreover, this result is independent of the +margins, although the solution might not be unique. This is the case for example for a Bernoulli margin +in the serial case. In fact, for any A ∈ Sd, nr2 +A,n = +Z2 +A,n +{pn(1−pn)}|A| , where pn = n−1 +n +� +t=1 +{Yt = 1}, and +ZA,n = n−1/2 +n +� +t=1 +� +j∈A +[I{Yt+1−j = 1} − pn]. In the non-serial case, if all margins are Bernoulli, pnj = +n−1 +n +� +i=1 +{Xij = 1}, and ZA,n = n−1/2 +n +� +i=1 +� +j∈A +[I{Xij = 1} − pnj], A ∈ Nd, then nr2 +A,n = +Z2 +A,n +� +j∈A pnj(1−pnj). +We will now look at some well-known copula families. The following expressions for ˙C or ˙c come from +Genest et al. (2007). +Example 2. If Cθ is the equicorrelated Gaussian copula, then ˙cA(u) = +� +B⊂A,|B|=2 +� +j∈B +Φ−1(uj). It follows +from (14)–(15) that in the non-serial case, ˙γK,A +� +C✠� += I{|A| = 2} +� +j∈A +cov +� +Kj ◦ F−1 +j +(U), Φ−1(U) +� +, +while in the serial case, ˙γK,A +� +C✠,s� += I{|A| = 2} +� +j∈A +cov +� +Kj ◦ F−1(U), Φ−1(U) +� +. As a result, van der +Waerden’s coefficients should be locally the most powerful when restricted to pairs, i.e., when |A| = 2. + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +11 +For the Farlie-Gumbel-Morgensten’s copula family, ˙cA(u) = I{A = {1, . . ., d}} +d +� +j=1 +(1 − 2uj). It follows +that in the non-serial case, ˙γK,A +� +C✠� += 2d(−1)dI{A = {1, . . . , d}} +d +� +j=1 +cov +� +Kj ◦ F−1 +j +(U), U +� +, and in the +serial case, ˙γK,A +� +C✠,s� += 2d(−1)dI{A = {1, . . ., d}} +d +� +j=1 +cov +� +Kj ◦ F−1(U), U +� +, so Spearman’s rho with +A = {1, . . . , d} should be locally the most powerful. +For Claytons’s copula family, ˙cA(u) = +� +B⊂A,|B|=2 +� +j∈B +(1 + log uj). In the non-serial case, one gets +˙γK,A +� +C✠� += I{|A| = 2} +� +j∈A +cov +� +Kj ◦ F−1 +j +(U), log U +� +, and in the serial case, ˙γK,A +� +C✠,s� += I{|A| = +2} +� +j∈A +cov +� +Kj ◦ F−1(U), log U +� +. As a result, Savage’s coefficients for pairs should be locally the most +powerful. +Finally, for Frank’s copula family, ˙cA(u) = |A| − 1 +2 ++ 2|A|−1 � +j∈A +uj − +� +j∈A +uj, and it then follows from +formulas (12)–(13) that in the non-serial case, ˙γK,A +� +C✠� += 2|A|−1 � +j∈A cov +� +Kj ◦ F−1 +j +(U), U +� +, and in +the serial case, ˙γK,A +� +C✠,s� += 2|A|−1 � +j∈A cov +� +Kj ◦ F−1(U), U +� +. So even if ˙cA is not a product, ˙γK,A +can be computed. As a result, Spearman’s rho for all sets should be locally the most powerful. The good +performance of combination of Spearman’s rho for pairs was confirmed in numerical experiments in the +serial case for Frank’s family; see, e.g., Nasri (2022). +4.2. Combining test statistics. In the non-serial case and the serial case, the limitings distributions +of the statistics n1/2rA,n are independent, so they could be combined. Littell and Folks (1971, 1973) +showed that it should be better combine the P-values of the tests statistics. However, given that we +have Gaussian limits, there is not much difference in terms of power by using instead the sum of squared +statistics +Ln,p = n +� +A⊂Nd,|A|≤p +r2 +A,n or Ln,p = n +� +A⊂Sd,|A|≤p +r2 +A,n, +where the rn,A are normalised in such a way that n1/2rA,n ⇝ N(0, 1) under the null hypothesis of +independence or randomness. This was shown numerically in Nasri (2022). In the non-serial case, one +could consider all sets A ∈ Nd, so Ln,d has approximately a chi-square distribution with 2d−d−1 degrees +of freedom, or consider only the pairs, i.e., Ln,2, which has approximately a chi-square distribution with +d(d−1) +2 +degrees of freedom. In the serial case, Ln,d has approximately a chi-square distribution with +2d−1 − 1 degrees of freedom, while Ln,2, which has approximately a chi-square distribution with d − 1 +degrees of freedom. One can also draw dependograms, i.e., graphs of n1/2rA,n plotted as a functions of + +12 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +all possible sets A or all pairs. These statistics and graphs will be implemented in the next version of +the CRAN package MixedIndTests (Nasri et al., 2022). +5. Numerical experiments +In what follows, we consider only the serial case and the following copula families: independence, Tent +map, Farlie-Gumbel-Morgenstern (FGM) (with θ = 1), and Clayton, Frank and Gaussian families with +Kendall’s tau of 0.1. Recall that the Tent map copula is the joint cdf of (U1, 2 min(U1, 1 − U1)), with +U1 ∼ U(0, 1). The generated series are all stationary and Markov, with the exception of the FGM which +is 2-Markov, as defined by (2). We consider the same set of 7 margins as in Nasri (2022), namely F1 is +Bernoulli with p = 0.8, F2 is Poisson(6), F3 is a negative binomial NB(r=1.5,p=0.2), F4 is a mixture of +0 with probability 0.1 and Poisson(10) otherwise, F5 is a mixture of 0 with probability 0.1 and N(0,1) +otherwise, F6 is a discretized Gaussian with F −1 +6 +(u) = +� +200Φ−1(u) +� +, and F7 is a discrete Pareto with +F7(k) = 1 − +1 +k+1, k ∈ N. For the tests, we considered the statistics Ln,2 and Ln,5 for Spearman, van der +Waerden, and Savage coefficients, for n ∈ {100, 250, 500}. The simulations results, based on N = 1000 +replications, appear in Table 1 for the independence and the Tent map copulas, in Table 2 for the +Farlie-Gumbel-Morgenstern copulas, and in Table 3 for the Gaussian and Frank copulas. +From the results for the independence copula in Table 1, the empirical levels of the tests are quite +satisfactory, being close to the 5% target. Next, for the Tent map copula, Savage’s test is surprisingly +good, compared to the two other coefficients, with the exception of the Bernoulli margin F1 which give +the same results for all tests. The good performance of Savage’s test might come from the fact that for +continuous margins, the theoretical coefficient is not 0, contrary to Spearman’s rho and van der Waerden +coefficients (R´emillard, 2013). Next, from Table 2, without any surprise, the tests based on Ln,2 are +not powerful for the Farlie-Gumbel-Morgenstern copula, given the calculations in Example 2, while the +best test is Ln,5 based on Spearman’s rho, as predicted. Also from the computations in Example 2, the +tests based on Savage’s coefficients are the best for the Clayton’s copula. Finally, from the results in +Table 3, as predicted, the tests based on Spearman’s rho are the best for Frank’s copula, while the tests +based on van der Waerden’s coefficient are the best for the Gaussian copula. These results all agree with +the results in Example 2, as well as the results of Genest and Verret (2005) for the bivariate case with +continuous margins. +6. Conclusion +For the non-serial and serial settings, we defined bivariate and multivariate extensions of several known +dependence measures that are usually defined for observations with continuous distributions. Even if + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +13 +we consider observations with arbitrary distributions, there is not added difficulty for computations and +from the simulation results, the results are quite good. We also deduced the locally most powerful tests +based on covariances for some known copula families, whatever the margins. These results generalise +the previous findings of Genest and Verret (2005) in the bivariate when the margins were assumed to be +continuous. +Appendix A. Auxiliary results +Here we define to important transformations: the M¨obius transform and the multilinear interpolation. +For A ∈ Nd, the M¨obius transform MA is defined by MA(f)(u) = +� +B⊂A +(−1)|A\B|f +� +uB� +� +j∈A\B +uj, +where uB ∈ [0, 1]d is such that uB +j += + + + +uj +if j ∈ B, +1 +if j ̸∈ B. +. +In particular, if f = f1 ⊗ · · · ⊗ fd, i.e., +f(u) = �d +j=1 fj(uj), and fj(1) = 1, then MA(f) = �d +j=1{fj(uj) − uj}. As a result, for any A ∈ Nd, +MA(Π) ≡ 0. Next, following Genest et al. (2017), for F = (F1, . . . , Fd), we define the interpolation +operator MF. To this end, for arbitrary u = (u1, . . . , ud) ∈ [0, 1]d and S ⊆ {1, . . ., d}, and for any +B ⊂ {1, . . ., d}, set (uF,B)j = Fj ◦ F −1 +j +(uj) if j ∈ B, and (uF,B)j = Fj +� +F −1 +j +(uj)− +� +if j /∈ B. In +particular, if Fj is continuous at F −1 +j +(uj), then (uF,B)j = uj for any B. Note that uF,S is an element +in the closure ¯RF of RF = RF1 × · · ·× RFd. Further let ℓ∞(K) be the collection of bounded real-valued +functions on K ⊆ [0, 1]d. The multilinear interpolation operator MF, is then defined for all g ∈ ℓ∞( ¯RF) +and u ∈ [0, 1]d, by MF(g)(u) = +� +B⊂{1,...,d} +g(uF,B) + + + +� +j∈B +λFj(uj) + + + + + � +j∈B∁ +{1 − λFj(uj)} + +. In particular, +if g(u) = +d +� +j=1 +gj(uj), then MF(g)(u) = +d +� +j=1 +MFj(gj)(uj). The following commutation result was proven +in Nasri (2022). +Lemma 2. For any f = f1 ⊗ · · · ⊗ fd, such that fj(1) = 1, and for any A ∈ Nd, one has +MA ◦ MF(f) = MF ◦ MA(f). +The next result is fundamental for the computations of the dependence measures. +Proposition 2. For any cdf G with mean µ and variance σ2, we have +(16) +� ∞ +−∞ +[JF {x, G(y)} − G(y)] dy = µ − GF (x). + +14 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +Proof. First, since 0 ≤ JF ≤ 1, +� +JF (x, u)dF(x) = u, and Y ∼ G is integrable, it follows that +� �� ∞ +−∞ +[JF {x, G(y)} − G(y)] dy +� +dF(x) = 0. +Next, for any c ∈ R, E [Y I{Y > c}] = +� ∞ +c +¯G(y)dy−max(0, −c) ¯G(c) and E [Y I{Y ≤ c}] = c− +� c +−∞ +G(y)dy+ +max(0, −c) ¯G(c). As a result, +(17) +µ = c + +� ∞ +c +¯G(y)dy − +� c +−∞ +G(y)dy. +Set a = F(x−) and b = F(x). Further set ¯G(y) = 1 − G(y), y0 = G−1(a) and y1 = G−1(b). Now, +suppose that ∆F (x) = b − a = 0. Then, according to (17) +−I = − +� ∞ +y1 +¯G(y)dy + +� y1 +−∞ +G(y)dy = y1 − µ = GF (x) − µ. +Suppose now that f(x) = b − a > 0. Then, +I += +� ∞ +−∞ +� +D +�G(y) − F(x−) +f(x) +� +− G(y) +� +dy = − +� y0 +−∞ +G(y)dy + +� y1 +y0 +��G(y) − a +b − a +� +− G(y) +� +dy ++ +� ∞ +y1 +¯G(y)dy += +� ∞ +y1 +¯G(y)dy − +� y1 +−∞ +G(y)dy + +� y1 +y0 +�G(y) − a +b − a +� +dy = µ − y1 + +� y1 +y0 +�G(y) − a +b − a +� +dy, +using (17). Finally, +� y1 +y0 +{G(y) − a}dy = (y1 − y0){G(y0) − a} + E +�� y1 +y0 +I{y0 < Y ≤ y}dy +� += (y1 − y0){G(y0) − a} + E [(y1 − Y )I{y0 < Y ≤ y1}] += (y1 − y0){G(y0) − a} + y1 {G(y1) − G(y0)} − +� b +a +G−1(v)dv + +� b +a +G−1(v)dv − +� G(y1) +G(y0) +G−1(v)dv += (y1 − y0){G(y0) − a} + y1 {G(y1) − G(y0)} − LG(b) + LG(a) + +� G(y0) +a +G−1(v)dv − +� G(y1) +b +G−1(v)dv += y1(b − a) − LG(b) + LG(a). +As a result, −I = LG(b) − LG(a) +b − a +− µ = GF (x) − µ. +□ + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +15 +Appendix B. Proofs +B.1. Proof of Lemma 1. Without loss of generality, drop the subscript j. Then, by Proposition 2, +s2 +n += +n−1 +n +� +i=1 +�L{Fn(Xi)} − Lj{Fn(Xi−)} +∆Fn(Xi) +− µ +�2 += +n−1 +n +� +i=1 +�� +R +�� 1 +0 +I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) +� +dx +�2 +. +Next, let M > 0 be given and choose δ ∈ (0, 1) so that K−1(1 − δ) > M and K−1(δ) < −M. To prove +the result, it suffices to show that if M is large enough, and δ is small enough, +s2 +n,1,M = n−1 +n +� +i=1 +�� M +−M +�� 1 +0 +I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) +� +dx +�2 +can be arbitrarily close to s2, while +s2 +n,2,M = n−1 +n +� +i=1 +�� ∞ +M +�� 1 +0 +I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) +� +dx +�2 +and +s2 +n,3,M = n−1 +n +� +i=1 +�� −M +−∞ +�� 1 +0 +I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) +� +dx +�2 +can be made arbitrarily small. First, as n → ∞, s2 +n,1,M converges in probability to +s2 +1,M = E +�� M +−M +�� 1 +0 +I {F(X−) + s∆F (X) ≤ K(x)} ds − K(x) +� +dx +�2 +. +Using similar arguments as in Genest and R´emillard (2004), s2 +1,M → s2 as M → ∞. Next, s2 +n,2,M = +s2 +n,2a,M + s2 +n,2b,M + s2 +n,2c,M, where +s2 +n,2a,M = n−1 +n +� +i=1 +I{Fn(Xi−) ≤ 1 − δ, Fn(Xi) ≥ δ} +× +�� ∞ +M +�� 1 +0 +I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) +� +dx +�2 +, + +16 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +s2 +n,2b,M = n−1 +n +� +i=1 +I{Fn(Xi−) > 1 − δ} +× +�� ∞ +M +�� 1 +0 +I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) +� +dx +�2 +, +s2 +n,2c,M = n−1 +n +� +i=1 +I{Fn(Xi) < δ} +× +�� ∞ +M +�� 1 +0 +I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) +� +dx +�2 +. +Now s2 +n,2a,M converges in probability to s2 +2a,M, which can be made arbitrarily small by taking M large +enough. Next, sn,2c,M = 0 since K(δ) ≤ −M. Finally, +s2 +n,2b,M ≤ +n +� +i=1 +I{Fn(Xi−) > 1 − δ} +�� ∞ +M +{1 − K(x)}dx +�2 +, +which can be made arbitrarily small since 1−K is integrable on (0, ∞). The case of sn,3,M is similar. +□ +B.2. Proof of Proposition 1. If Aj is the set of atoms of Fj, and Ij = ∪x∈Aj (Fj(x−), Fj(x)), then +for U ∼ U(0, 1), Xj = F −1 +j +(U) ∼ Fj, and +cov +� +Kj ◦ F−1 +j +(U), G−1 +j +(U) +� += +� +x∈Aj +� +Kj,Fj(x) − µj +� � Fj(x) +Fj(x−) +� +G−1 +j (u) − ˜µj +� +du + +� +{u̸∈Ij} +� +K−1 +j +(u) − µj +� � +G−1 +j (u) − ˜µj +� +du += +� +x∈Aj +∆Fj(x) +� +Kj,Fj(x) − µj +� � +Gj,Fj(x) − ˜µj +� ++ +� +{u̸∈Ij} +� +K−1 +j +(u) − µj +� � +G−1 +j (u) − ˜µj +� +du += cov +� +Kj ◦ F−1 +j +(U), Gj ◦ F−1 +j +(U) +� += cov {Kj(Xj), Gj(Xj)} . +The rest of the proof follows from using Remark 3. +□ +References +Blest, D. C. (2000). Rank correlation—an alternative measure. Aust. N. Z. J. Stat., 42(1):101–111. +Brockwell, A. E. (2007). Universal residuals: A multivariate transformation. Statist. Probab. Lett., +77(14):1473–1478. +Ferguson, T. S. (1967). +Mathematical Statistics: A Decision Theoretic Approach. Probability and +Mathematical Statistics, Vol. 1. Academic Press, New York. + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +17 +Genest, C., Neˇslehov´a, J. G., and R´emillard, B. (2014). On the empirical multilinear copula process for +count data. Bernoulli, 20(3):1344–1371. +Genest, C., Neˇslehov´a, J. G., and R´emillard, B. (2017). Asymptotic behavior of the empirical multilinear +copula process under broad conditions. J. Multivariate Anal., 159:82–110. +Genest, C., Neˇslehov´a, J. G., R´emillard, B., and Murphy, O. A. (2019). Testing for independence in +arbitrary distributions. Biometrika, 106(1):47–68. +Genest, C. and Plante, J.-F. (2003). On Blest’s measure of rank correlation. Canad. J. Statist., 31(1):35– +52. +Genest, C., Quessy, J.-F., and R´emillard, B. (2007). Asymptotic local efficiency of Cram´er-von Mises +tests for multivariate independence. Ann. Statist., 35(1):166–191. +Genest, C. and R´emillard, B. (2004). Tests of independence and randomness based on the empirical +copula process. Test, 13(2):335–370. +Genest, C. and Verret, F. (2005). Locally most powerful rank tests of independence for copula models. +J. Nonparametr. Stat., 17(5):521–539. +Ghoudi, K. and R´emillard, B. (2018). Serial independence tests for innovations of conditional mean and +variance models. TEST, 27(1):3–26. +Hoeffding, W. (1940). Maßstabinvariante korrelationstheorie f¨ur diskontinuierliche verteilungen. Arch. +Math. Wirt. Sozialforsch., 7:4–70. +Littell, R. C. and Folks, J. L. (1971). Asymptotic optimality of Fisher’s method of combining independent +tests. J. Amer. Statist. Assoc., 66:802–806. +Littell, R. C. and Folks, J. L. (1973). Asymptotic optimality of Fisher’s method of combining independent +tests. II. J. Amer. Statist. Assoc., 68:193–194. +Nasri, B. R. (2022). Tests of serial dependence for multivariate time series with arbitrary distributions. +J. Multivariate Anal., 192:Paper No. 105102. +Nasri, B. R., R´emillard, B. N., Neˇslehov´a, J. G., and Genest, C. (2022). +MixedIndTests: Tests of +Randomness and Tests of Independence. R package version 1.1.0. +Neˇslehov´a, J. (2007). On rank correlation measures for non-continuous random variables. J. Multivariate +Anal., 98(3):544–567. +R´emillard, B. (2013). Statistical Methods for Financial Engineering. CRC Press, Boca Raton, FL. +R¨uschendorf, L. (1981). Stochastically ordered distributions and monotonicity of the OC-function of +sequential probability ratio tests. Math. Operationsforsch. Statist. Ser. Statist., 12(3):327–338. + +18 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +Sklar, M. (1959). Fonctions de r´epartition `a n dimensions et leurs marges. Publ. Inst. Statist. Univ. +Paris, 8:229–231. +Table 1. Power of the proposed tests of serial independence for statistics Ln,2 and +Ln,5 of Spearman’s, van der Waerden’s, and Savage’s coefficients, for the independence +copula and the Tent map copula, based on N = 1000 replications. +Ind +Tent map +n +Margin +Spearman +van der Waerden +Savage +Spearman +van der Waerden +Savage +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +100 +F1 +3.6 +6.4 +3.6 +6.4 +3.6 +6.4 +84.8 +77.3 +84.8 +77.3 +84.8 +77.3 +F2 +5.1 +5.6 +4.5 +7.5 +3.9 +6.1 +9.1 +17.4 +6.8 +45.0 +61.1 +62.7 +F3 +6.0 +5.0 +5.3 +7.0 +5.5 +7.9 +9.2 +15.0 +5.8 +43.1 +68.5 +67.5 +F4 +5.8 +4.8 +5.4 +6.3 +4.6 +7.5 +7.7 +16.3 +4.3 +50.8 +68.0 +65.4 +F5 +5.2 +5.9 +5.0 +6.3 +3.3 +5.3 +7.9 +16.6 +5.4 +56.2 +59.6 +56.2 +F6 +4.6 +5.1 +4.7 +7.1 +3.4 +5.5 +8.5 +16.3 +6.2 +54.3 +59.7 +61.4 +F7 +5.5 +4.3 +5.7 +5.4 +5.7 +4.7 +52.0 +32.3 +57.1 +34.8 +16.8 +12.2 +250 +F1 +5.5 +6.0 +5.5 +6.0 +5.5 +6.0 +100.0 +99.7 +100.0 +99.7 +100.0 +99.7 +F2 +4.6 +4.8 +5.2 +5.7 +4.4 +6.6 +10.1 +28.2 +8.2 +68.9 +97.6 +98.4 +F3 +4.3 +5.2 +4.3 +5.9 +6.4 +7.9 +9.9 +22.8 +9.6 +60.9 +97.9 +97.4 +F4 +4.6 +5.2 +4.4 +6.4 +4.4 +6.5 +10.7 +29.1 +7.7 +64.4 +97.1 +97.2 +F5 +6.1 +5.3 +5.8 +6.6 +4.1 +9.8 +8.1 +23.7 +6.1 +72.0 +98.1 +98.2 +F6 +3.5 +4.4 +3.9 +5.3 +4.1 +8.0 +7.9 +24.3 +7.2 +71.9 +98.3 +98.2 +F7 +4.2 +5.3 +3.6 +5.3 +5.0 +5.2 +90.9 +78.7 +98.1 +92.5 +36.2 +24.7 +500 +F1 +5.4 +7.3 +5.4 +7.3 +5.4 +7.3 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +F2 +4.8 +4.2 +4.8 +5.3 +4.9 +7.7 +11.7 +33.4 +11.2 +75.3 +100.0 +100.0 +F3 +5.0 +5.0 +5.0 +5.3 +5.5 +6.8 +12.0 +29.8 +18.9 +73.3 +100.0 +100.0 +F4 +3.9 +5.8 +3.6 +5.4 +5.4 +5.6 +12.0 +37.8 +12.3 +80.4 +100.0 +99.9 +F5 +5.4 +5.1 +5.9 +5.9 +4.7 +7.6 +11.8 +27.2 +9.4 +79.5 +100.0 +100.0 +F6 +4.1 +5.1 +3.8 +4.7 +3.7 +7.1 +9.2 +29.0 +8.8 +80.5 +100.0 +100.0 +F7 +4.9 +4.9 +5.1 +5.5 +5.0 +5.4 +99.8 +99.2 +100.0 +100.0 +65.3 +49.5 +D´epartement de m´edecine sociale et pr´eventive, ´Ecole de sant´e publique, Universit´e de Montr´eal, C.P. +6128, succursale Centre-ville Montr´eal (Qu´ebec) H3C 3J7 +Email address: bouchra.nasri@umontreal.ca +GERAD and Department of Decision Sciences, HEC Montr´eal, 3000, chemin de la Cˆote-Sainte-Catherine, +Montr´eal (Qu´ebec), Canada H3T 2A7 +Email address: bruno.remillard@hec.ca + +COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA +19 +Table 2. Power of the proposed tests of serial independence for statistics Ln,2 and +Ln,5 of Spearman’s, van der Waerden’s, and Savage’s coefficients, for the Farlie-Gumbel- +Morgenstern and Clayton copula families, based on N = 1000 replications. +FGM +Clayton +n +Margin +Spearman +van der Waerden +Savage +Spearman +van der Waerden +Savage +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +100 +F1 +4.4 +10.7 +4.4 +10.7 +4.4 +10.7 +15.3 +24.6 +15.3 +24.6 +15.3 +24.6 +F2 +5.9 +15.6 +5.7 +13.4 +7.0 +8.7 +17.1 +11.8 +18.7 +17.2 +27.6 +30.8 +F3 +4.7 +14.7 +5.1 +14.2 +6.5 +9.8 +18.5 +13.8 +17.0 +15.3 +23.0 +28.6 +F4 +5.8 +14.1 +5.3 +14.1 +6.8 +10.7 +17.1 +12.4 +16.8 +15.2 +25.9 +29.4 +F5 +5.6 +14.6 +4.9 +12.4 +5.3 +8.8 +18.0 +12.4 +18.7 +16.6 +25.0 +26.7 +F6 +5.4 +14.0 +3.7 +10.0 +4.3 +7.8 +16.8 +12.2 +18.0 +16.2 +29.6 +26.8 +F7 +5.9 +12.0 +5.7 +10.1 +6.2 +12.4 +9.8 +8.9 +9.0 +8.6 +10.2 +8.9 +250 +F1 +7.2 +15.7 +7.2 +15.7 +7.2 +15.7 +36.4 +40.9 +36.4 +40.9 +36.4 +40.9 +F2 +6.8 +38.5 +6.6 +32.4 +8.1 +16.2 +39.7 +26.3 +43.5 +35.5 +61.9 +53.3 +F3 +7.5 +38.2 +7.7 +32.1 +8.7 +20.9 +39.8 +29.4 +40.8 +31.8 +59.8 +53.8 +F4 +8.0 +39.8 +8.3 +32.9 +8.4 +22.3 +40.1 +26.0 +41.5 +28.8 +57.1 +50.5 +F5 +7.8 +39.9 +8.4 +31.5 +7.9 +14.5 +46.3 +29.0 +51.1 +38.0 +68.0 +61.5 +F6 +6.0 +40.7 +6.3 +31.2 +7.4 +16.4 +45.9 +31.9 +51.0 +41.3 +66.5 +58.4 +F7 +6.9 +25.5 +5.6 +24.1 +6.2 +24.1 +18.1 +11.3 +16.8 +11.6 +19.2 +12.2 +500 +F1 +6.0 +19.1 +6.0 +19.1 +6.0 +19.1 +60.9 +61.7 +60.9 +61.7 +60.9 +61.7 +F2 +7.6 +74.8 +6.6 +64.4 +8.6 +28.0 +76.3 +58.5 +82.1 +70.4 +92.0 +85.8 +F3 +8.0 +77.7 +7.4 +70.1 +8.0 +42.9 +73.3 +55.5 +76.2 +59.4 +87.4 +81.2 +F4 +10.0 +76.3 +8.8 +69.5 +10.6 +42.2 +73.9 +55.3 +77.4 +58.2 +87.3 +78.3 +F5 +9.1 +78.6 +9.1 +67.6 +8.1 +28.6 +75.2 +54.1 +81.7 +64.8 +92.5 +82.9 +F6 +9.2 +77.5 +9.7 +67.7 +9.2 +29.2 +74.3 +53.9 +80.3 +66.8 +91.1 +83.3 +F7 +8.4 +58.0 +6.6 +52.0 +8.2 +51.2 +39.3 +23.4 +33.4 +22.0 +41.2 +23.5 + +20 +BOUCHRA R. NASRI AND BRUNO N. R´EMILLARD +Table 3. Power of the proposed tests of serial independence for statistics Ln,2 and +Ln,5 of Spearman’s, van der Waerden’s, and Savage’s coefficients, for the Gaussian and +Frank copula families, based on N = 1000 replications. +Gaussian +Frank +n +Margin +Spearman +van der Waerden +Savage +Spearman +van der Waerden +Savage +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +Ln,2 +Ln,5 +100 +F1 +8.7 +14.4 +8.7 +14.4 +8.7 +14.4 +8.4 +16.2 +8.4 +16.2 +8.4 +16.2 +F2 +16.2 +10.4 +16.4 +9.9 +11.4 +12.9 +17.3 +10.6 +15.9 +8.6 +11.5 +11.6 +F3 +16.0 +10.5 +15.8 +13.2 +11.8 +13.0 +15.2 +10.0 +13.9 +9.9 +10.9 +12.4 +F4 +14.8 +9.6 +14.5 +11.2 +13.8 +14.5 +16.6 +8.8 +14.4 +9.6 +11.4 +12.1 +F5 +15.3 +11.5 +15.8 +11.6 +12.8 +11.7 +16.3 +10.9 +14.4 +11.6 +12.1 +12.2 +F6 +13.1 +9.3 +14.1 +11.3 +10.5 +12.3 +15.8 +8.7 +13.9 +10.2 +9.4 +9.4 +F7 +12.4 +7.9 +13.1 +12.1 +11.6 +6.4 +13.1 +9.7 +13.1 +9.8 +13.4 +8.9 +250 +F1 +14.5 +19.4 +14.5 +19.4 +14.5 +19.4 +14.2 +21.5 +14.2 +21.5 +14.2 +21.5 +F2 +40.2 +21.7 +41.3 +23.6 +31.2 +24.3 +38.4 +22.6 +33.0 +22.6 +24.5 +18.5 +F3 +41.5 +23.8 +43.3 +25.9 +31.1 +25.6 +40.9 +23.2 +38.4 +22.5 +28.0 +20.8 +F4 +40.6 +22.5 +42.7 +24.5 +32.7 +25.1 +43.6 +24.6 +39.7 +23.1 +29.2 +22.2 +F5 +42.4 +26.5 +44.2 +27.9 +31.5 +27.1 +40.8 +22.4 +36.8 +21.4 +22.7 +17.2 +F6 +38.4 +22.9 +43.4 +26.6 +32.3 +25.1 +42.4 +24.4 +38.5 +20.9 +24.1 +16.0 +F7 +31.4 +20.9 +31.8 +25.2 +27.3 +17.3 +32.4 +19.8 +32.3 +19.8 +29.5 +18.1 +500 +F1 +27.2 +28.8 +27.2 +28.8 +27.2 +28.8 +23.8 +26.5 +23.8 +26.5 +23.8 +26.5 +F2 +73.3 +49.7 +78.1 +56.2 +60.1 +42.9 +76.2 +48.0 +71.0 +41.9 +47.5 +31.2 +F3 +76.0 +50.5 +77.8 +55.4 +61.1 +42.6 +74.6 +50.1 +72.4 +45.6 +54.5 +34.3 +F4 +73.8 +48.6 +77.0 +52.6 +63.3 +44.2 +75.2 +52.6 +71.7 +48.7 +54.6 +38.3 +F5 +75.9 +51.3 +79.5 +56.5 +60.9 +39.5 +73.4 +49.7 +69.3 +43.5 +47.6 +27.3 +F6 +75.5 +52.9 +80.0 +55.1 +63.2 +43.0 +74.3 +46.7 +68.3 +43.4 +46.3 +29.1 +F7 +56.8 +36.5 +59.5 +42.5 +48.7 +30.6 +61.3 +41.3 +58.8 +39.4 +58.6 +36.5 + diff --git a/-dFQT4oBgHgl3EQfKDXj/content/tmp_files/load_file.txt b/-dFQT4oBgHgl3EQfKDXj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab6e94acb9317710b288557bfa5dbd544ffb578f --- /dev/null +++ b/-dFQT4oBgHgl3EQfKDXj/content/tmp_files/load_file.txt @@ -0,0 +1,1419 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf,len=1418 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='13259v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='ME] 30 Jan 2023 COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In this article, we define extensions of copula-based dependence measures for data with arbitrary distributions, in the non-serial case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', for independent and identically distributed random vectors, as well as in serial case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', for time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' These dependence measures are covariances with respect to a multilinear copula associated with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' We also consider multivariate extensions based on M¨obius transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' We find the asymptotic distributions of the statistics under the hypothesis of independence or randomness and under contiguous alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This enables us to find out locally most powerful test statistics for some alternatives, whatever the margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Numerical experiments are performed for combinations of these statistics to assess the finite sample performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Introduction In many cases, simple measures of dependence like Kendall’s tau and Spearman’s rho, perform as well as more complex statistics like Cram´er-von Mises statistics based on empirical processes, and are gener- ally much faster to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' However, tests based of such measures are not always consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Neverthe- less, tests of independence or randomness based on copulas should always be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Here, we are in- terested in copula-based dependence measures for a sample of iid observations Xi = (Xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Xid) ∼ H, with vector of margins F = (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Fd), d ≥ 2, called the non-serial case, as well as for the serial case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', for stationary time series Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Yn with common cumulative distribution function (cdf) F, where one defines the random vectors Xt = (Yt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Yt+1−d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' hereafter, the series Y is extended in a circular way by setting Yt+n = Yt for all t ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the bivariate case, when the margins are continuous, most copula-based dependence measures are theoretically defined as the correlation ̺K(C) = cor � K−1 1 (Ui1), K−1 2 (Ui2) � , since by continuity of the margins, Ui = (Ui1, Ui2) = F(Xi) ∼ C, for a unique copula C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Here K = (K1, K2) is a given vector of cdfs, with mean µ1, µ2, and variances σ2 1, σ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The value under independence is clearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, by Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Independence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' randomness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' multilinear copula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Spearman’s rho, van der Waerden’s coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Savages’s coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Funding in partial support of this work was provided by the Fonds qu´eb´ecois de la recherche en sant´e and the Natural Sciences and Engineering Research Council of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 1 2 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD Hoeffding’s identity (Hoeffding, 1940), (1) ̺K(C) = γK(C) σ1σ2 = 1 σ1σ2 � R2 [C {K1(x1), K2(x2)} − K1(x1)K2(x2)] dx1dx2, since � K−1 1 (U1), K−1 2 (U2) � has joint cdf C ◦ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For example, suppose that (U1, U2) ∼ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Then, taking K1 = K2 = D, where D is the cdf of the uniform distribution over (0, 1), one obtains Spearman’s correlation ρS(C) = 12E(U1U2) − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The case K1 = K2 = Φ, where Φ is the cdf of the standard Gaussian distribution yields the van der Waerden coefficient ρvdw(C) = E � Φ−1(U1)Φ−1(U2) � , while if K1 = K2 is the cdf of a Bernoulli(1/2), one gets Blomqvist’s coefficient 4C(1/2, 1/2) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Note that by definition, when K1 = K2, ̺K(C+) = 1 for the complete dependence, where C+(u1, u2) = min(u1, u2) is the Fr´echet-Hoeffding upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' However, when K1 ̸= K2, the covariance γK1,K2(C) must be divided by E � K−1 1 (U1)K−1 2 (U1) � − µ1µ2 to give 1 for complete dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Blest’s coefficient (Blest, 2000, Genest and Plante, 2003) can be seen as such an example if one considers a natural modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In fact, Blest’s coefficient has been originally defined as the covariance between (1 − U1)2 and U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' An obvious modification is obtained by taking K1(u) = u1/2, u ∈ [0, 1], K2 = D, so the modified coefficient is 12E(U 2 1 U2)−2, normalised to give 1 for complete dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Genest and Plante (2003) also proposed a symmetrised Blest’s coefficient, which, in our general setting, amounts to defining γ∗ K1,K2(C) = γK1,K2(C) + γK2,K1(C) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Not all copula-based dependence measures are defined by a covariance (up to a constant), a well-known example being Kendall’s tau, defined by τ(C) = 4 � (0,1)2 C(u1, u2)dC(u1, u2) − 1 = 4E{C(U1, u2)} − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' However, Kendall’s tau and Spearman’s rho have an equivalent limiting distribution, even under a sequence of contiguous alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Estimating the dependence measures defined previously is relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the continu- ous case, the copula is replaced by the empirical copula ˆCn(u1, u2) = n−1 � i=1 I � n n + 1Fn1(Xi1) ≤ u � I � n n + 1Fn2(Xi2) ≤ u2 � , u1, u2 ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In fact, γK(C) can be estimated by γK(Cn), and according to Genest and R´emillard (2004), one has γK � ˆCn � = � R2 [Cn {K1(x1), K2(x2)} − K1(x1)K2(x2)] dx1dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Asymptotic limits and their representations are easier to work with the latter representation, being a linear functional of the empirical process �Cn(u1, u2) = n1/2 � ˆCn(u1, u2) − u1u2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' These dependence measures and their estimation work well when the margins are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' However, for applications, COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA 3 there is a need to consider arbitrary distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', when at least of the margins is not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In this case, since there are ties, one might be tempted to replace the ranks by the mid-ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' However, the asymptotic distribution might not be simple enough and it makes sense to base the dependence measures on copulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The main problem here is that the copula is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' If X ∼ H, there are infinitely many copulas satisfying Sklar’s equation H = C ◦ F (Sklar, 1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' To construct solutions for this equation, for any copula C, take V ∼ C independent of X ∼ H and set U = ψF(X, V), where Uj = ψFj(Xj, Vj) = Fj(Xj−) + Vj∆Fj(Xi), with Fj(x−) = P(Xj < x) and ∆Fj(x) = Fj(x) − Fj(x−) = P(Xj = x), j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It is known (Ferguson, 1967, R¨uschendorf, 1981, Neˇslehov´a, 2007, Brockwell, 2007) that for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}, Uj ∼ U(0, 1), and the joint cdf CC of U is a copula satisfying Sklar’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In addition, there is one interesting copula C✠ in this family, the so-called multilinear copula, obtained by taking C = Π, the independence copula, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', Π(u) = �d j=1 D(uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' One interesting property of C✠ is that if H(x) = �d j=1 Fj(xj), then CC = Π if and only if CC = C✠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a by-product, taking the empirical joint cdf Hn with the vector of margins Fn = (Fn1, Fn2), one obtains the empirical multilinear copula �C✠ n , for which an explicit expression will be given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Note that contrary to �Cn, �C✠ n is a genuine copula, so all dependence measures presented before can be computed with C✠ and its empirical counterpart �C✠ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This is the approach that we propose here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Note that the asymptotic behaviour of the associated versions of Kendall’s tau and Spearman’s rho has been studied in Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2014), and tests of independence based on �C✠ n were proposed in Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2019), while in the serial case, tests of randomness based on the serial version �C✠,s n have been studied in Nasri (2022), as well as the asymptotic behaviour of the serial versions of Kendall’s tau and Spearman’s rho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The main aim of this article is to define bivariate and multivariate extensions of the dependence measures when the margins are arbitrary, to find explicit expressions of the measures, and to study their asymptotic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' We will also look at the asymptotic distribution of the test statistics under a sequence of contiguous alternatives to be able to suggest locally powerful tests for given dependence models, in the same spirit as Genest and Verret (2005) did in the bivariate case for continuous margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' To this end, we also present a new representation of the multilinear copulas in the serial and non-serial cases that enables us to perform calculations more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Note that in both Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2019) and Nasri (2022), the main focus was on using Cram´er-von Mises statistics of related multilinear processes, which is not done here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 4 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD In Section 2, we recall the definitions and properties of multilinear copulas in a serial setting (Nasri, 2022) and non-serial setting Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2019), together with their associated M¨obius transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, in Section 3, we define the serial and non-serial versions of dependence measures, providing explicit formulas that are easy to implement, and we study their asymptotic behaviour under the null hypothesis of independence or randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Multivariate extensions similar to those defined in Genest and R´emillard (2004) and Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2014) will also be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, in Section 4, we will study the asymptotic behaviour of the proposed dependence measures under a sequence of contiguous alternatives, using the results of Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2019) and Nasri (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This will enable us to find the locally most powerful tests based of the proposed dependence measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' We will also discuss how to combine the proposed dependence measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Finally, numerical experiments will be performed in Section 5 to assess the power of the tests for finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Multilinear copulas and associated empirical processes From now on, we consider the following two settings: the non-serial case and the serial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the non-serial case, we have independent and identically distributed (iid) random vectors U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Un ∼ C, for a given copula C, and the observations are Xi = F−1(Ui), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the serial setting, we have a series of random variables U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Un ∼ U(0, 1), and the observed time series is Yt = F −1(Ut), t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , n}, where (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Un) is d-Markov process with copula C, meaning that the distribution of (Ut, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Ut+1−d) is C, C has density c, and the joint density of (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Un), evaluated at (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , un), is given by (2) n � t=d+1 c(ut, ut−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , ut+1−d) cd−1(ut−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , ut+1−d), where cd−1(ud, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , u2) = � 1 0 c(ud, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , u2, s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' We can now define the multilinear copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}, set JFj(xj, uj) = E � ψFj(Xj, uj)|Xj = xj � = P � Fj(xj−) + Vj∆Fj(xj) ≤ uj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Then, JFj(xj, uj) = \uf8f1 \uf8f2 \uf8f3 I{Fj(xj) ≤ uj}, if ∆Fj(xj) = 0, D � uj−Fj(xj−) ∆Fj (xj) � , if ∆Fj(xj) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , where D is the cdf of U ∼ U(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Note that when ∆Fj(xj) > 0, JFj(xj, uj) = 0 if uj ≤ Fj(xj−), JFj(xj, uj) = 1 if uj ≥ Fj(xj), and JFj(xj, uj) = uj − Fj(xj−) ∆Fj(xj) if Fj(xj−) ≤ uj ≤ Fj(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Using properties of conditional expectations, one obtains (3) C✠(u) = E \uf8f1 \uf8f2 \uf8f3 d � j=1 JFj(Xj, uj) \uf8fc \uf8fd \uf8fe , u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA 5 As a result, (4) �C✠ n (u) = n−1 n � i=1 d � j=1 JFnj(Xij, uj) = n−1 n � i=1 d � j=1 D �uj − Fnj(Xij−) ∆Fnj(Xij) � , u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This new expression is different from what appears in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2017, 2019), but it is easier to manipulate for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In fact, ˆC✠ n was previously defined by ˆC✠ n (u) = n−1 n � i=1 d � j=1 � λFnj(uj)I{Xij ≤ F −1 nj (uj)} + {1 − λFnj(uj)}I{Xij < F −1 nj (uj)} � , where, for any cdf G and u ∈ (0, 1), λG(u) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 u − G � G−1(u)− � ∆G {G−1(u)} , ∆G � G−1(u) � > 0, 1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, the empirical serial multilinear copula, first defined and studied in Nasri (2022), can also be written as (5) �C✠,s n (u) = n−1 n � t=1 d � j=1 D �uj − Fn(Yt+1−j−) ∆Fn(Yt+1−j) � , u ∈ [0, 1]d, where Fn(y) = n−1 n � t=1 {Yt+1−j ≤ y}, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Using the circular construction, it follows that for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}, Fn(y) = n−1 n � t=1 {Yt+1−j ≤ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Further define the empirical multilinear processes �C✠ n = n1/2 � �C✠ n − Π � and �C✠,s n = n1/2 � �C✠,s n − Π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, let Nd be the set of all subsets A of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', d} with card (A) = |A| > 1, and let Sd be the set of all elements A of Nd with A ∋ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It has been shown, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', Genest and R´emillard (2004), Ghoudi and R´emillard (2018), Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2019), Nasri (2022), that M¨obius transforms of empirical processes have nice asymptotic properties for tests of independence or tests of randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' To this end, define (6) G✠ A,n(u) = MA � �C✠ n � (u) = n−1/2 n � i=1 � j∈A � D �uj − Fnj(Xij−) ∆Fnj(Xij) � − uj � , A ∈ Nd, and (7) G✠,s A,n(u) = MA � �C✠,s n � (u) = n−1/2 n � t=1 � j∈A � D �uj − Fn(Yt+1−j−) ∆Fn(Yt+1−j) � − uj � , A ∈ Sd, where the M¨obius transform MA is defined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, for any s, t ∈ [0, 1], and any cdf G, set (8) ΓG(s, t) = s∧t−st− � x:∆G(x)>0 I{G(x−) ≤ s∧t ≤ s∨t ≤ G(x)}{(s ∧ t) − G(x−)} {G(x) − s ∨ s)} ∆G(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The main findings of Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2019) and Nasri (2022) that we need can be summarised as follows: 6 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Under the null hypothesis of independence, � G✠ A,n : A ∈ Nd � converge jointly in ℓ∞ � (0, 1)d� to independent Gaussian processes � G✠ A : A ∈ Nd � , where E � G✠ A(u)G✠ A(v) � = � j∈A ΓFj(uj, vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Fur- thermore, under the null hypothesis of randomness, � G✠,s A,n : A ∈ Sd � converge jointly in ℓ∞ � (0, 1)d� to independent Gaussian processes � G✠,s A : A ∈ Sd � , where E � G✠,s A (u)G✠,s A (v) � = � j∈A ΓF (uj, vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The formulas for the covariances in Theorem 1 follows from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='6) and (8) in Nasri (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' One can check that for any s, t ∈ [0, 1], ΓG(s, t) ≥ 0 with equality if and only if s ∧ t = 0 or s ∨ t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It is interesting to note that for sets A of size 2, G✠ A,n and G✠,s A,n are empirical multilinear copula processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In fact, for A = {j, k} ∈ Nd, j < k, G✠ A(u1, u2) = n1/2 � �C✠ A (u1, u2) − u1u2 � , where �C✠ A is the multilinear copula for the pairs (Xij, Xik), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Similarly, for any A = {1, 1 + ℓ} ∈ Sd, G✠,s A (u1, u2) = n1/2 � �C✠,s A (u1, u2) − u1u2 � , where C✠,s A is the multilinear copula for the pairs (Yt, Yt−ℓ), t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Dependence measures for arbitrary distributions From now on, let K = (K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Kd) be a vector of margins with mean µj and variance σ2 j , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}, and define Lj(u) = � u 0 K−1 j (v)dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', d}, and any cdf G, define Kj,G(x) = � 1 0 K−1 j {G(x−) + s∆G(x)} ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Then Kj,G(x) = K−1 j {G(x)}, if G is continuous at x, and Kj,G(x) = LKj{G(x)} − LKj{G(x−)} ∆G(x) , if G is not continuous at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The extension of the covariance measures are defined the following way: In the non-serial case, for any A ∈ Nd, set γK,A � �C✠ n � = n−1/2(−1)|A| � RA � G✠ A,n {K(x)} dx, while in the serial case, for any A ∈ Sd, set γK,A � �C✠,s n � = n−1/2(−1)|A| � RA � G✠,s A,n {K(x)} dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It then follows from Proposition 2 in Appendix A that for any A ∈ Nd, in the non-serial case, (9) γK,A � �C✠ n � = n−1 n � i=1 � j∈A �Lj{Fnj(Xij)} − Lj{Fnj(Xij−)} ∆Fnj(Xij) − µj � , COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA 7 while in the serial case, for any A ∈ Sd, (10) γK,A � �C✠,s n � = n−1 n � t=1 � j∈A �Lj{Fn(Yt+1−j)} − Lj{Fn(Yt+1−j−)} ∆Fn(Yt+1−j) − µj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For Spearman’s rho, Kj ≡ D, so Lj(u) = u2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For van der Waerden’s coefficient, Kj ≡ Φ, so Lj = −φ ◦ Φ−1, µj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For Savage’s coefficient, Kj(x) ≡ 1 − e−x, x ≥ 0, so Lj(u) = u − u log u, µj = 1, with the convention that 0 log 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Finally, for the modified Blest’s coefficient in the bivariate case, K−1 1 (u) = u2, u ∈ [0, 1], K2 = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a result, one gets the following formula for 12γK1,K2 � �C✠ n � : 2n−1 n � i=1 � F 2 nj(Xij−) + Fnj(Xij−)Fnj(Xij) + F 2 nj(Xij) − 1 � {Fn2(Xi2−) + Fn2(Xi2) − 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For continuous margins, ∆Fnj(Xij) = n−1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', so Lj{Fnj(Xij)} − Lj{Fnj(Xij−)} ∆Fnj(Xij) ≈ K−1 j � n n + 1Fnj(Xij) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Note that in general, n−1 �n i=1 K−1 j � n n+1Fnj(Xij) � ̸= µj, while n−1 n � i=1 Lj{Fnj(Xij)} − Lj{Fnj(Xij−)} ∆Fnj(Xij) = µj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This shows that even for continuous margins, one should use formulas (9)–(10) based on the multilinear copulas, since we do not need to work with the normalised n n+1Fnj(Xij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The following result is an immediate consequence of Theorem 1, the continuous mapping theorem, together with representations (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' When K−1 j is unbounded, one can use the same technique as in the corresponding proofs in Genest and R´emillard (2004), meaning that one integrates GA,n{K(x)} on large compact sets and show that the remainder can be made arbitrarily small, since K−1 j is square integrable by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The covariance formulas follows from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='6)-(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='7) in Nasri (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Under the hypothesis of independence, � n1/2γK,A,n � �C✠ n � : A ∈ Nd � converge jointly to independent Gaussian random variables with variance ς2 K,F,A = � j∈A ς2 Kj,Fj, where for any cdf G, (11) ς2 Kj,G = � {Kj,G{G(x)} − µ}2 dG(x) = � R2 ΓG{Kj(x), Kj(y)}dxdy, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Furthermore, under the hypothesis of randomness, � n1/2γK,A,n � �C✠,s n � : A ∈ Sd � converge jointly to independent Gaussian random variables with variance ς2 K,F,A = � j∈A ς2 Kj,F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 8 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It follows from Genest and R´emillard (2004) that σ2 j = � R2{Kj(x ∧ y) − Kj(x)Kj(y)}dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Finally, ς2 Kj,Fj = var � Kj,Fj(Xj) � , if Xj ∼ Fj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The next result is fundamental for applications since it shows how to normalised the statistics to standard Gaussian distributions in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Its proof is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the non-serial case, s2 Kj,Fnj = n−1 n � i=1 �Lj{Fnj(Xij)} − Lj{Fnj(Xij−)} ∆Fnj(Xij) − µj �2 P r −→ ς2 Kj,Fj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}, and in the serial case, s2 Kj,Fn = n−1 n � t=1 �Lj{Fn(Yt)} − Lj{Fn(Yt−)} ∆Fn(Yt) − µj �2 P r −→ ς2 Kj,F , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Asymptotic behaviour along contiguous alternatives and local power In this section, we consider contiguous alternatives of the form Cθn for the non-serial case as well as the serial case described by (2), where Cθ0 = Π, for some θ0, and θn = θ0 + n−1/2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It is assumed that the copula family Cθ is smooth enough, namely that the Conditions 1–2 in Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2019) are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' More precisely, assume that Cθ has a continuous density cθ which is square integrable, continuously differentiable in a neighbourhood of θ0, with ˙c = ∇θcθ(u)|θ=θ0, (u) ∈ (0, 1)d, ˙C(u) = � (0,u] ˙c(s)ds, and lim n→∞ � (0,1)d[n1/2[{cθn(u)}1/2 − 1] − δ ˙c(u)/2]2du = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Before stating the limiting distribution under the sequence of contiguous alternatives Cθn, for any A ∈ Nd, set qA = MA( ˙C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It follows from Lemma 2, stated in the Appendix, and proven in Nasri (2022), that in the non-serial case, MF(qA) = MA ◦MF � ˙C � = MA � ˙C✠� , while in the serial case, MF ⊗d(qA) = MA ◦ MF ⊗d � ˙C � = MA � ˙C✠,s� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Under the previous conditions, the following results were obtained by Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2019) in the non-serial case, and by Nasri (2022) in the serial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Under the sequence of contiguous alternatives Cθn, in the non-serial case, the processes G✠ A,n, A ∈ Nd, converge jointly in ℓ∞ � (0, 1)d� to G✠ A + δMA( ˙C✠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Furthermore, in the serial case, the processes G✠,s A,n, A ∈ Sd, converge jointly in ℓ∞ � (0, 1)d� to G✠,s A + δMA( ˙C✠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Nasri (2022) also considered Poisson contiguous alternatives with conditional mean λt,n = λ0 + δn−1/2Yt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In this case, for any A ∈ Sd, the processes G✠,s A,n converge jointly in ℓ∞ � (0, 1)d� to COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA 9 G✠,s A + δ λ0 I{A = {1, 2}}MF(f)(u1)MF (f)(u2), where f(u) = {LF (u) − λ0u}, and F is the cdf of the Poisson with parameter λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a corollary, we obtain the asymptotic behaviour of the proposed dependence measures under the sequence of contiguous alternatives Cθn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Under the sequence of contiguous alternatives Cθn, in the non-serial case, the random variables n1/2γK,A � �C✠ n � , A ∈ Nd, converge jointly to independent Gaussian random variables with mean δ ˙γK,A � C✠� and variance ς2 K,F,A, where (12) ˙γK,A � C✠� = � ˙cA(u) � j∈A � Kj,Fj ◦ F −1 j (uj) − µj � du, and CA is the copula restricted to components Uj with j ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Furthermore, in the serial case, the random variables n1/2γK,A � �C✠,s n � , A ∈ Sd, converge jointly to independent Gaussian random variables with mean δ ˙γK,A � C✠,s� and variance ς2 K,F,A, where (13) ˙γK,A � C✠,s� = � ˙cA(u) � j∈A � Kj,F ◦ F −1(uj) − µj � du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Note that if the margin Fj is continuous, Kj,Fj ◦ F −1 j (uj) = K−1 j (uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In particular, in the serial case, if the margin F is continuous, then ˙γK,A � C✠,s� = � ˙cA(u) � j∈A � K−1 j (uj) − µj � du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Since MA � C✠ θ � (u) = Eθ \uf8ee \uf8f0� j∈A � JFj(Xj, uj) − uj � \uf8f9 \uf8fb, Proposition 2 in Appendix A yields γK,A � C✠ θ � = (−1)|A| � RA Eθ \uf8ee \uf8f0� j∈A � JFj{Xj, Kj(xj)} − Kj(xj) � \uf8f9 \uf8fb dx = Eθ \uf8ee \uf8f0� j∈A {Kj(Xj) − µj} \uf8f9 \uf8fb , so ˙γK,F,A = ∂θ γK,A � C✠ θ ��� θ=θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a result, one obtains formulas (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In particular, if ˙cA = � j∈A Jj(uj), then in the non-serial case, for any A ∈ Nd, (14) ˙γK,A � C✠� = � j∈A � 1 0 � Kj ◦ F −1 j (uj) − µj � Jj(uj)duj = � j∈A cov � Kj ◦ F−1 j (U), Jj(U) � , where U ∼ U(0, 1), while in the serial case, for any A ∈ Sd, (15) ˙γK,A � C✠,s� = � j∈A cov � Kj ◦ F−1(U), Jj(U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 10 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Applications for local power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' First note that for many copula families satisfying the smooth- ness conditions listed at the beginning of the section, one has ˙cA(u) = � j∈A J(uj), and Jj is often a quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In this case, choosing K−1 j = Jj would make sense in order to have a non-zero mean, and hence having more local power by maximising formulas (14)–(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This is what was proposed in Genest and Verret (2005) in the bivariate case, where the margins were assumed to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In fact, the next proposition shows that this choice is also optimal for any margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The proof of the following result is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Suppose that ˙cA(u) ∝ � j∈A G−1 j (uj), u ∈ (0, 1)d, where G = (G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Gd) is a vector of margins with means (˜µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , ˜µd) and variances � ˜σ2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , ˜σ2 d � , and assume U ∼ U(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Then, in the non-serial case cov � Kj ◦ F−1 j (U), G−1 j (U) � = cov {Kj(Xj), Gj(Xj)}, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}, where Xj ∼ Fj, so ˙γK,A � C✠� = � j∈A cov {Kj(Xj), Gj(Xj)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In particular, if K = G, then ˙γK,A � C✠� = ς2 K,F,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the serial case, cov � Kj ◦ F−1(U), G−1 j (U) � = cov {Kj(X), Gj(X)}, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}, where X ∼ F, so ˙γK,A � C✠,s� = � j∈A cov {Kj(X), Gj(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In particular, if K = G, then ˙γK,A � C✠,s� = ς2 K,F,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Under the assumptions of Proposition 1, it follows from Proposition 3 in Genest and Verret (2005) that the ARE between the test based on K, and G is given by � j∈A cor2 {Kj(Xj), Gj(Xj)} in the non- serial case, and the ARE is � j∈A cor2 {Kj(X), Gj(X)} in the serial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This shows that whenever ˙cA(u) ∝ � j∈A G−1 j (uj), the ARE is maximised by taking K = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Moreover, this result is independent of the margins, although the solution might not be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This is the case for example for a Bernoulli margin in the serial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In fact, for any A ∈ Sd, nr2 A,n = Z2 A,n {pn(1−pn)}|A| , where pn = n−1 n � t=1 {Yt = 1}, and ZA,n = n−1/2 n � t=1 � j∈A [I{Yt+1−j = 1} − pn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the non-serial case, if all margins are Bernoulli, pnj = n−1 n � i=1 {Xij = 1}, and ZA,n = n−1/2 n � i=1 � j∈A [I{Xij = 1} − pnj], A ∈ Nd, then nr2 A,n = Z2 A,n � j∈A pnj(1−pnj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' We will now look at some well-known copula families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The following expressions for ˙C or ˙c come from Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' If Cθ is the equicorrelated Gaussian copula, then ˙cA(u) = � B⊂A,|B|=2 � j∈B Φ−1(uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It follows from (14)–(15) that in the non-serial case, ˙γK,A � C✠� = I{|A| = 2} � j∈A cov � Kj ◦ F−1 j (U), Φ−1(U) � , while in the serial case, ˙γK,A � C✠,s� = I{|A| = 2} � j∈A cov � Kj ◦ F−1(U), Φ−1(U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a result, van der Waerden’s coefficients should be locally the most powerful when restricted to pairs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', when |A| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA 11 For the Farlie-Gumbel-Morgensten’s copula family, ˙cA(u) = I{A = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', d}} d � j=1 (1 − 2uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' It follows that in the non-serial case, ˙γK,A � C✠� = 2d(−1)dI{A = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d}} d � j=1 cov � Kj ◦ F−1 j (U), U � , and in the serial case, ˙γK,A � C✠,s� = 2d(−1)dI{A = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', d}} d � j=1 cov � Kj ◦ F−1(U), U � , so Spearman’s rho with A = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , d} should be locally the most powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For Claytons’s copula family, ˙cA(u) = � B⊂A,|B|=2 � j∈B (1 + log uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the non-serial case, one gets ˙γK,A � C✠� = I{|A| = 2} � j∈A cov � Kj ◦ F−1 j (U), log U � , and in the serial case, ˙γK,A � C✠,s� = I{|A| = 2} � j∈A cov � Kj ◦ F−1(U), log U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a result, Savage’s coefficients for pairs should be locally the most powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Finally, for Frank’s copula family, ˙cA(u) = |A| − 1 2 + 2|A|−1 � j∈A uj − � j∈A uj, and it then follows from formulas (12)–(13) that in the non-serial case, ˙γK,A � C✠� = 2|A|−1 � j∈A cov � Kj ◦ F−1 j (U), U � , and in the serial case, ˙γK,A � C✠,s� = 2|A|−1 � j∈A cov � Kj ◦ F−1(U), U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' So even if ˙cA is not a product, ˙γK,A can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a result, Spearman’s rho for all sets should be locally the most powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The good performance of combination of Spearman’s rho for pairs was confirmed in numerical experiments in the serial case for Frank’s family;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', Nasri (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Combining test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the non-serial case and the serial case, the limitings distributions of the statistics n1/2rA,n are independent, so they could be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Littell and Folks (1971, 1973) showed that it should be better combine the P-values of the tests statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' However, given that we have Gaussian limits, there is not much difference in terms of power by using instead the sum of squared statistics Ln,p = n � A⊂Nd,|A|≤p r2 A,n or Ln,p = n � A⊂Sd,|A|≤p r2 A,n, where the rn,A are normalised in such a way that n1/2rA,n ⇝ N(0, 1) under the null hypothesis of independence or randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' This was shown numerically in Nasri (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the non-serial case, one could consider all sets A ∈ Nd, so Ln,d has approximately a chi-square distribution with 2d−d−1 degrees of freedom, or consider only the pairs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', Ln,2, which has approximately a chi-square distribution with d(d−1) 2 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In the serial case, Ln,d has approximately a chi-square distribution with 2d−1 − 1 degrees of freedom, while Ln,2, which has approximately a chi-square distribution with d − 1 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' One can also draw dependograms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', graphs of n1/2rA,n plotted as a functions of 12 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD all possible sets A or all pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' These statistics and graphs will be implemented in the next version of the CRAN package MixedIndTests (Nasri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Numerical experiments In what follows, we consider only the serial case and the following copula families: independence, Tent map, Farlie-Gumbel-Morgenstern (FGM) (with θ = 1), and Clayton, Frank and Gaussian families with Kendall’s tau of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Recall that the Tent map copula is the joint cdf of (U1, 2 min(U1, 1 − U1)), with U1 ∼ U(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The generated series are all stationary and Markov, with the exception of the FGM which is 2-Markov, as defined by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' We consider the same set of 7 margins as in Nasri (2022), namely F1 is Bernoulli with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='8, F2 is Poisson(6), F3 is a negative binomial NB(r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='5,p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='2), F4 is a mixture of 0 with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='1 and Poisson(10) otherwise, F5 is a mixture of 0 with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='1 and N(0,1) otherwise, F6 is a discretized Gaussian with F −1 6 (u) = � 200Φ−1(u) � , and F7 is a discrete Pareto with F7(k) = 1 − 1 k+1, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For the tests, we considered the statistics Ln,2 and Ln,5 for Spearman, van der Waerden, and Savage coefficients, for n ∈ {100, 250, 500}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The simulations results, based on N = 1000 replications, appear in Table 1 for the independence and the Tent map copulas, in Table 2 for the Farlie-Gumbel-Morgenstern copulas, and in Table 3 for the Gaussian and Frank copulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' From the results for the independence copula in Table 1, the empirical levels of the tests are quite satisfactory, being close to the 5% target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, for the Tent map copula, Savage’s test is surprisingly good, compared to the two other coefficients, with the exception of the Bernoulli margin F1 which give the same results for all tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The good performance of Savage’s test might come from the fact that for continuous margins, the theoretical coefficient is not 0, contrary to Spearman’s rho and van der Waerden coefficients (R´emillard, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, from Table 2, without any surprise, the tests based on Ln,2 are not powerful for the Farlie-Gumbel-Morgenstern copula, given the calculations in Example 2, while the best test is Ln,5 based on Spearman’s rho, as predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Also from the computations in Example 2, the tests based on Savage’s coefficients are the best for the Clayton’s copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Finally, from the results in Table 3, as predicted, the tests based on Spearman’s rho are the best for Frank’s copula, while the tests based on van der Waerden’s coefficient are the best for the Gaussian copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' These results all agree with the results in Example 2, as well as the results of Genest and Verret (2005) for the bivariate case with continuous margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Conclusion For the non-serial and serial settings, we defined bivariate and multivariate extensions of several known dependence measures that are usually defined for observations with continuous distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Even if COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA 13 we consider observations with arbitrary distributions, there is not added difficulty for computations and from the simulation results, the results are quite good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' We also deduced the locally most powerful tests based on covariances for some known copula families, whatever the margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' These results generalise the previous findings of Genest and Verret (2005) in the bivariate when the margins were assumed to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Auxiliary results Here we define to important transformations: the M¨obius transform and the multilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For A ∈ Nd, the M¨obius transform MA is defined by MA(f)(u) = � B⊂A (−1)|A\\B|f � uB� � j∈A\\B uj, where uB ∈ [0, 1]d is such that uB j = \uf8f1 \uf8f2 \uf8f3 uj if j ∈ B, 1 if j ̸∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In particular, if f = f1 ⊗ · · · ⊗ fd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', f(u) = �d j=1 fj(uj), and fj(1) = 1, then MA(f) = �d j=1{fj(uj) − uj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a result, for any A ∈ Nd, MA(Π) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, following Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2017), for F = (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , Fd), we define the interpolation operator MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' To this end, for arbitrary u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' , ud) ∈ [0, 1]d and S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', d}, and for any B ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', d}, set (uF,B)j = Fj ◦ F −1 j (uj) if j ∈ B, and (uF,B)j = Fj � F −1 j (uj)− � if j /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In particular, if Fj is continuous at F −1 j (uj), then (uF,B)j = uj for any B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Note that uF,S is an element in the closure ¯RF of RF = RF1 × · · ·× RFd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Further let ℓ∞(K) be the collection of bounded real-valued functions on K ⊆ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The multilinear interpolation operator MF, is then defined for all g ∈ ℓ∞( ¯RF) and u ∈ [0, 1]d, by MF(g)(u) = � B⊂{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=',d} g(uF,B) \uf8f1 \uf8f2 \uf8f3 � j∈B λFj(uj) \uf8fc \uf8fd \uf8fe \uf8ee \uf8f0 � j∈B∁ {1 − λFj(uj)} \uf8f9 \uf8fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' In particular, if g(u) = d � j=1 gj(uj), then MF(g)(u) = d � j=1 MFj(gj)(uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The following commutation result was proven in Nasri (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For any f = f1 ⊗ · · · ⊗ fd, such that fj(1) = 1, and for any A ∈ Nd, one has MA ◦ MF(f) = MF ◦ MA(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The next result is fundamental for the computations of the dependence measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' For any cdf G with mean µ and variance σ2, we have (16) � ∞ −∞ [JF {x, G(y)} − G(y)] dy = µ − GF (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 14 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' First, since 0 ≤ JF ≤ 1, � JF (x, u)dF(x) = u, and Y ∼ G is integrable, it follows that � �� ∞ −∞ [JF {x, G(y)} − G(y)] dy � dF(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, for any c ∈ R, E [Y I{Y > c}] = � ∞ c ¯G(y)dy−max(0, −c) ¯G(c) and E [Y I{Y ≤ c}] = c− � c −∞ G(y)dy+ max(0, −c) ¯G(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a result, (17) µ = c + � ∞ c ¯G(y)dy − � c −∞ G(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Set a = F(x−) and b = F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Further set ¯G(y) = 1 − G(y), y0 = G−1(a) and y1 = G−1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Now, suppose that ∆F (x) = b − a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Then, according to (17) −I = − � ∞ y1 ¯G(y)dy + � y1 −∞ G(y)dy = y1 − µ = GF (x) − µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Suppose now that f(x) = b − a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Then, I = � ∞ −∞ � D �G(y) − F(x−) f(x) � − G(y) � dy = − � y0 −∞ G(y)dy + � y1 y0 ��G(y) − a b − a � − G(y) � dy + � ∞ y1 ¯G(y)dy = � ∞ y1 ¯G(y)dy − � y1 −∞ G(y)dy + � y1 y0 �G(y) − a b − a � dy = µ − y1 + � y1 y0 �G(y) − a b − a � dy, using (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Finally, � y1 y0 {G(y) − a}dy = (y1 − y0){G(y0) − a} + E �� y1 y0 I{y0 < Y ≤ y}dy � = (y1 − y0){G(y0) − a} + E [(y1 − Y )I{y0 < Y ≤ y1}] = (y1 − y0){G(y0) − a} + y1 {G(y1) − G(y0)} − � b a G−1(v)dv + � b a G−1(v)dv − � G(y1) G(y0) G−1(v)dv = (y1 − y0){G(y0) − a} + y1 {G(y1) − G(y0)} − LG(b) + LG(a) + � G(y0) a G−1(v)dv − � G(y1) b G−1(v)dv = y1(b − a) − LG(b) + LG(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' As a result, −I = LG(b) − LG(a) b − a − µ = GF (x) − µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' □ COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA 15 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Proofs B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Without loss of generality, drop the subscript j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Then, by Proposition 2, s2 n = n−1 n � i=1 �L{Fn(Xi)} − Lj{Fn(Xi−)} ∆Fn(Xi) − µ �2 = n−1 n � i=1 �� R �� 1 0 I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) � dx �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, let M > 0 be given and choose δ ∈ (0, 1) so that K−1(1 − δ) > M and K−1(δ) < −M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' To prove the result, it suffices to show that if M is large enough, and δ is small enough, s2 n,1,M = n−1 n � i=1 �� M −M �� 1 0 I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) � dx �2 can be arbitrarily close to s2, while s2 n,2,M = n−1 n � i=1 �� ∞ M �� 1 0 I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) � dx �2 and s2 n,3,M = n−1 n � i=1 �� −M −∞ �� 1 0 I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) � dx �2 can be made arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' First, as n → ∞, s2 n,1,M converges in probability to s2 1,M = E �� M −M �� 1 0 I {F(X−) + s∆F (X) ≤ K(x)} ds − K(x) � dx �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Using similar arguments as in Genest and R´emillard (2004), s2 1,M → s2 as M → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, s2 n,2,M = s2 n,2a,M + s2 n,2b,M + s2 n,2c,M, where s2 n,2a,M = n−1 n � i=1 I{Fn(Xi−) ≤ 1 − δ, Fn(Xi) ≥ δ} × �� ∞ M �� 1 0 I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) � dx �2 , 16 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD s2 n,2b,M = n−1 n � i=1 I{Fn(Xi−) > 1 − δ} × �� ∞ M �� 1 0 I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) � dx �2 , s2 n,2c,M = n−1 n � i=1 I{Fn(Xi) < δ} × �� ∞ M �� 1 0 I {Fn(Xi−) + s∆Fn(Xi) ≤ K(x)} ds − K(x) � dx �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Now s2 n,2a,M converges in probability to s2 2a,M, which can be made arbitrarily small by taking M large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Next, sn,2c,M = 0 since K(δ) ≤ −M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Finally, s2 n,2b,M ≤ n � i=1 I{Fn(Xi−) > 1 − δ} �� ∞ M {1 − K(x)}dx �2 , which can be made arbitrarily small since 1−K is integrable on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The case of sn,3,M is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' If Aj is the set of atoms of Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' and Ij = ∪x∈Aj (Fj(x−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Fj(x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' then for U ∼ U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Xj = F −1 j (U) ∼ Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' and cov � Kj ◦ F−1 j (U),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' G−1 j (U) � = � x∈Aj � Kj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='Fj(x) − µj � � Fj(x) Fj(x−) � G−1 j (u) − ˜µj � du + � {u̸∈Ij} � K−1 j (u) − µj � � G−1 j (u) − ˜µj � du = � x∈Aj ∆Fj(x) � Kj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='Fj(x) − µj � � Gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='Fj(x) − ˜µj � + � {u̸∈Ij} � K−1 j (u) − µj � � G−1 j (u) − ˜µj � du = cov � Kj ◦ F−1 j (U),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Gj ◦ F−1 j (U) � = cov {Kj(Xj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Gj(Xj)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' The rest of the proof follows from using Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' □ References Blest, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Rank correlation—an alternative measure.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Academic Press, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' COPULA-BASED DEPENDENCE MEASURES FOR ARBITRARY DATA 17 Genest, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', Neˇslehov´a, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', and R´emillard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' On the empirical multilinear copula process for count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Bernoulli, 20(3):1344–1371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Genest, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', Neˇslehov´a, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', and R´emillard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Asymptotic behavior of the empirical multilinear copula process under broad conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Multivariate Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', 159:82–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Genest, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Stochastically ordered distributions and monotonicity of the OC-function of sequential probability ratio tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Operationsforsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=', 12(3):327–338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' 18 BOUCHRA R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' NASRI AND BRUNO N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' R´EMILLARD Sklar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Fonctions de r´epartition `a n dimensions et leurs marges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Paris, 8:229–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Power of the proposed tests of serial independence for statistics Ln,2 and Ln,5 of Spearman’s, van der Waerden’s, and Savage’s coefficients, for the independence copula and the Tent map copula, based on N = 1000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Ind Tent map n Margin Spearman van der Waerden Savage Spearman van der Waerden Savage Ln,2 Ln,5 Ln,2 Ln,5 Ln,2 Ln,5 Ln,2 Ln,5 Ln,2 Ln,5 Ln,2 Ln,5 100 F1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='3 84.' metadata={'source': 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Gaussian and Frank copula families, based on N = 1000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content=' Gaussian Frank n Margin Spearman van der Waerden Savage Spearman van der Waerden Savage Ln,2 Ln,5 Ln,2 Ln,5 Ln,2 Ln,5 Ln,2 Ln,5 Ln,2 Ln,5 Ln,2 Ln,5 100 F1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQfKDXj/content/2301.13259v1.pdf'} +page_content='4 8.' metadata={'source': 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turbulence +Spectral response between particle and fluid kinetic energy in decaying +homogeneous isotropic turbulence +M. Schiødt,1 A. Hodžić,1 F. Evrard,2, a) M. Hausmann,2 B. Van Wachem,2 and C.M. Velte1 +1)Technical University of Denmark, Kongens Lyngby, Denmark +2)Otto von Guericke University, Magdeburg, Germany +(*Electronic mail: maschi@dtu.dk) +(Dated: 1 February 2023) +In particle-laden turbulence, the Fourier Lagrangian spectrum of each phase is regularly computed, and analytically +derived response functions relate the Lagrangian spectrum of the fluid- and the particle phase. However, due to the +periodic nature of the Fourier basis, the analysis is restricted to statistically stationary flows. In the present work, +utilizing the bases of time-focalized proper orthogonal decomposition (POD), this analysis is extended to temporally +non-stationary turbulence. Studying two-way coupled particle-laden decaying homogeneous isotropic turbulence for +various Stokes numbers, it is demonstrated that the temporal POD modes extracted from the dispersed phase may be +used for the expansion of both fluid- and particle velocities. The POD Lagrangian spectrum of each phase may thus be +computed from the same set of modal building blocks, allowing the evaluation of response functions in a POD frame +of reference. Based on empirical evaluations, a model for response functions in non-stationary flows is proposed. The +related energies of the two phases is well approximated by simple analytical expressions dependent on the particle +Stokes number. It is found that the analytical expressions closely resemble those derived through Fourier analysis +of statistically stationary flows. These results suggest the existence of an inherent spectral symmetry underlying the +dynamical systems consisting of particle-laden turbulence, a symmetry which spans across stationary/non-stationary +particle-laden flow states. +I. +INTRODUCTION +Recent years have seen renewed attention directed towards +particle-laden turbulence, due to its relevance in numerous +engineering and natural settings (Brandt and Coletti (2022)). +Theoretical models and improved experimental and numer- +ical methods have led to advancements in our understand- +ing of particle dynamics, herein counting acceleration statis- +tics, preferential sampling and particle clustering to name a +few (Toschi and Bodenschatz (2009); Gustavsson and Mehlig +(2016); Maxey (2017)). +One focus of study has been the modulation of turbulence +induced by two-way coupling (Druzhinin and Elghobashi +(1999); Ferrante and Elghobashi (2003)). Here, the presence +of particles in flows under zero gravity conditions has been +shown to attenuate turbulent kinetic energy (TKE) at low +wavenumbers and augment it at higher wavenumbers, lead- +ing to an increase in dissipation (Squires and Eaton (1994)). +Inertial particles may, however, also act as sources of in- +creased turbulence energy, and the total TKE may be either +augmented or attenuated by the presence of a dispersed phase +(Ferrante and Elghobashi (2003)). A key parameter identified +in this regard is the particle Stokes number. Letournel et al. +(2020) investigated TKE totals as a function of the Stokes +number, and found an approximate threshold below which tur- +bulence was augmented, and above which it was attenuated. +Nevertheless, the same authors underlined the lack of consen- +sus on a unique criterion for turbulence modulation by parti- +cles. +a)Sibley School of Mechanical and Aerospace Engineering, Cornell Univer- +sity, Ithaca, United States +Ireland, Bragg, and Collins (2016a) investigated the large +scale single-particle velocity statistics of inertial particles in +homogeneous isotropic turbulence (HIT). Driven by the ef- +fects of inertial filtering and preferential sampling, the aver- +age particle kinetic energy normalized by the average fluid ki- +netic energy was shown to approximately follow a simple re- +lation dependent on the Stokes number. Similar studies were +conducted under gravity conditions by Good et al. (2014) and +Ireland, Bragg, and Collins (2016b). +Although the study of particle-laden turbulence has rapidly +progressed over the past decade, new theoretical tools are +still needed in order to gain further insights into the dy- +namics (Brandt and Coletti (2022)). One such tool may be +the particle proper orthogonal decomposition (PPOD) for- +mulated by Schiødt et al. (2022), where Lagrangian parti- +cle velocities are decomposed into a set of modes that rep- +resent temporal particle dynamics. +This tool is utilized in +the present study, where the extracted modes are compared +to those extracted for the fluid measured at fixed Eulerian +mesh points using the temporal formulation of POD intro- +duced by Aubry, Guyonnet, and Lima (1991). Both formu- +lations of POD are briefly outlined in section II, and the con- +straints required for direct comparisons of fluid- and particle +POD modes are listed. +Modal decomposition of fluid- and particle temporal dy- +namics allows for the evaluation of the Lagrangian spectrum +of both phases in a POD frame of reference. In the current +work, this leads to formulations of POD-based response func- +tions, that relate the energy of the two phases on a modal +level. Although response functions based on the Fourier de- +composition have previously been studied in stationary flows +(Csanady (1963); Zhang, Legendre, and Zamansky (2019); +Berk and Coletti (2021)), the advantage of the POD-based ap- +proach is that stationarity is not required, and the present study + +Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +2 +is therefore focused on the analysis of various simulations +of two-way coupled particle-laden decaying HIT. The analy- +sis culminates in analytic expressions of POD-based response +functions closely resembling those derived through Fourier +analysis of stationary flows. +Section II gives a brief outline of the formulation of POD +and the structure of the ensembles that will produce temporal +modes representing fluid- and particle dynamics. A summary +of the simulation setup is given in III, which is followed by +a presentation and discussion of results in section IV. Finally, +our conclusions are given in section V. +II. +PROPER ORTHOGONAL DECOMPOSITION +The main objective of POD is to extract a set of empirical +basis functions ϕ = {ϕα}M +α=1 that represent dominating fea- +tures of the studied dynamical system. The basis functions, +also known as modes, are extracted by solving the eigenvalue +problem +Rϕα = λαϕα, +α ∈ [1 : M], +(1) +where λ = {λα}M +α=1 are the eigenvalues connected to each +mode, and for the cases we study, these are real and sorted +such that λ1 ≥ ··· ≥ λM ≥ 0. The operator R : H → H is +defined from the ensemble of empirical data u = {u(i)}Ne +i=1, +and is dependent on the definition of the Hilbert space H for +which ϕ serves as an empirical orthonormal basis. Though +the basis is not necessarily complete in H , each ensemble +member may be decomposed into a weighted sum of modes, +thus +u(i) = +M +∑ +α=1 +c(i) +α ϕα, +i ∈ [1 : Ne], +(2) +where the weights c(i) +α are known as the projection coefficients +given by +c(i) +α = (u(i),ϕα). +(3) +Here (·,·) denotes the inner product of H . The projection +coefficients are connected to the eigenvalues λ by the relation +λα = +�� +c(i) +α c(i)∗ +α +�Ne +i=1 +� +, +α ∈ [1 : M], +(4) +where (∗) denotes both the complex conjugate transpose for +a scalar and Hermitian transpose for a vector, and ⟨{·}Ne +i=1⟩ is +the ensemble average operator. +The definition of H +and what constitutes an ensem- +ble member determines the interpretation of ϕ and λ. +In +section II A and section II B we briefly outline the discrete for- +mulations of the Eulerian- and the Lagrangian (particle) POD, +respectively, and show their dependency on the definition of +u. +A. +Eulerian POD +The most common application of POD is based on the fluid +velocity u f (x,t) ∈ RD measured at fixed mesh points in a Eu- +lerian grid at equidistant sample times. Following the classical +interpretation of POD (Lumley (1967)) an ensemble member +may in this discrete case be formed by +u(i) = +� +u(i) +f (x1,t0)∗ ··· u(i) +f (xNg,t0)∗ ··· u(i) +f (xNg,tNt−1)∗ +�∗ +. +(5) +Here u(i) +f is the i’th fluid velocity realization, xg ∈ RD, g ∈ [1 : +Ng] are the Eulerian mesh points and tn ∈ T, n ∈ [0 : Nt − 1] +are the sample times of the temporal domain T. In this case +u(i) ∈ H = RN, where N = DNgNt. H is equipped with the +standard inner product (w1,w2) = w∗ +2w1, and the operator R +in equation (1) is given by +R = +�� +u(i)u(i)∗�Ne +i=1 +� +∈ RN×N. +(6) +Solving equation (1) then results in a set of spatio-temporal +modes that are optimal with respect to energy, where λ rep- +resents the energy of each mode. However, the amount of +data needed to generate ϕ often makes this classical approach +infeasible, as several uncorrelated fluid flow realizations are +needed to generate the data. Instead, an approach popularized +by Sirovich (1987) and Aubry et al. (1988) is to extract spa- +tially orthogonal modes, with time dependent projection co- +efficients. This is what Towne, Schmidt, and Colonius (2018) +refers to as the space-only POD, and in a statistically station- +ary flow an ensemble member may be given by the fluid veloc- +ity measured at all grid points at a single sample time. From +one fluid realization several ensemble members may thus be +generated, and the ensemble average operator reduces to a +temporal average. +In the current work we will focus on what we term +the time-only POD (TPOD) and its relation to PPOD. +The TPOD is also formulated in the continuous case +(Aubry, Guyonnet, and Lima (1991); Aubry (1991)) and as an +analogy to its spatial counterpart it produces a set of tempo- +rally orthogonal modes, with spatially dependent projection +coefficients. An ensemble member is in this case given by +u(i) = +�u f (xi,t0)∗ ··· u f (xi,tNt−1)∗�∗ , +i ∈ [1 : Ne], (7) +i.e. the fluid velocity at a grid point i measured at sample +times tn. Note that Ne ≤ Ng when the ensemble members are +taken from the same fluid realization, and that the fluid flow +in that case should be homogeneous (Aubry (1991)), signify- +ing that the temporal evolution is statistically equivalent in all +grid points. The ensemble average operator then reduces to a +spatial average and the operator R is still given as in equation +(6), although here N = DNt. +The modes extracted with TPOD represent the temporal +evolution of the fluid velocity through a Eulerian mesh point, +and λ is connected to the energy +E(t) = 1 +2 +�� +u∗ +f (xi,t)u f (xi,t) +�Ne +i=1 +� +, +(8) + +Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +3 +by +Nt−1 +∑ +n=0 +E(tn) = 1 +2 +M +∑ +α=1 +λα . +(9) +B. +Particle POD +Schiødt et al. (2022) formulated PPOD as a method for +decomposing the velocity of Lagrangian particles into a +weighted sum of empirical modes. Like TPOD the method +produces a set of temporal modes, however, the modes repre- +sent the dynamics of Lagrangian particles rather than the fluid +dynamics at fixed Eulerian mesh points. The ensemble u is in +this formulation defined by the ensemble members +u(i) = +� +v(i)(t0)∗ ··· v(i)(tNt−1)∗�∗ , +i ∈ [1 : Ne], +(10) +where +v(i)(tn) = +� +v(i) +1 (tn)∗ ··· v(i) +Np(tn)∗�∗ +, +(11) +is the velocity of Np Lagrangian particles measured at sample +times tn ∈ T, n ∈ [0 : Nt −1]. Here u(i) ∈ RN with N = DNpNt, +since v(i) +p (t) ∈ RD is the velocity of a single particle. Choos- +ing Np = 1 for the remainder of the current work, we see +that PPOD and TPOD ensemble members belong to the same +Hilbert space H = RN, N = DNt. The mode-sets extracted +with respectively TPOD and PPOD are therefore in this case +directly comparable. +To generate a meaningful ensemble of Lagrangian parti- +cle velocities, the ensemble particles should belong to sim- +ilar flows or be sampled from the same flow containing +certain symmetries. +We elaborate further on this point in +section III B. +In section IV both TPOD and PPOD analysis is applied to +the Reynolds decomposed u(i) +fluct = u(i) − ⟨{u(i)}Ne +i=1⟩ rather +than u(i). Thus, E(t) in equations (8)-(9) becomes a measure +of TKE, and +u(i) = +�� +u(i)�Ne +i=1 +� ++ +M +∑ +α=1 +c(i) +α ϕα, +i ∈ [1 : Ne]. +(12) +However, ⟨{u(i)}Ne +i=1⟩ ≈ 0 for all TPOD and PPOD ensembles +considered, and we will therefore interchangeably refer to ϕ +as the mode-set spanning both the signal u(i) and u(i) +fluct. +III. +SIMULATION +In the current work we consider the simulation of one +single-phase flow and four different simulations of two-way +coupled particle-laden turbulence. All simulations are per- +formed within a periodic cube with edge length ℓ, discretized +into Ng computational cells. +A. +Dynamical equations +We apply the Euler-Lagrange point-particle approach +(Elghobashi and Truesdell (1992)) where the fluid velocity u f +is computed at each time step by numerical integration of the +incompressible Navier-Stokes equations on a Eulerian mesh, +and particle velocities are obtained by integrating the gov- +erning particle equations of motion forward in time. For the +Navier-Stokes equations a constant dynamic viscosity µf and +mass density ρ f are used, and with p denoting pressure the +equations are given by +∇·u f = 0, +(13a) +∂u f +∂t + ∇·(u f ⊗ u f) = − 1 +ρ f +∇p+ µf +ρ f +∇2u f + Fp + F . +(13b) +Here Fp is the force that the dispersed particles exert on the +carrier fluid, and F is an artificial source term applied in an +initial forcing period. In section III C the details of Fp and F +are outlined. +The particles considered are monodisperse solid spheres +with diameter dp, volume Vp and density ρp. Assuming parti- +cles are only accelerated according to drag force, the dynamic +equations for particle motion are given by +dxp +dt += vp , +(14a) +Vpρp +dvp +dt = FD = π +8 d2 +pρ fCD|u f@p − vp|(u f@p − vp), +(14b) +where xp and u f@p = u f (xp,t) are the particle position +and fluid velocity at particle position, respectively. FD de- +notes the drag force and CD is the drag coefficient given by +(Schiller and Naumann (1933)) +CD = 24 +Rep +� +1 + 0.15Re0.687 +p +� +, +(15) +and +Rep = dpρ f |u f@p − vp| +µf +, +(16) +is the particle Reynolds number. Equation (15) holds for 0 < +Rep ≤ 1000, which is the only range considered in the current +work. +A +second-order +finite-volume +solver +(Denner, Evrard, and van Wachem (2020)) is used to in- +tegrate (13) forward in time, and the Verlet scheme is used +for the forward integration of (14). +B. +Decaying homogenous isotropic turbulence +To study two-way coupling effects in an idealized test case, +we analyze a particle-laden fluid with decaying HIT. This case +is chosen over stationary HIT because the effects of the forc- +ing term F would overlap with the particle-fluid interaction + +Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +4 +energy in the latter (Abdelsamie and Lee (2012)). In addition, +the properties of decaying HIT signifies that the fluid velocity +in all Eulerian mesh points evolves in a statistically equivalent +manner. The inertial particles are thermalized to the fluid (see +section III C) and thus have a statistically equivalent evolution +throughout the temporal domain. Therefore, a meaningful en- +semble of realizations can be generated for both TPOD and +PPOD from a single simulation of particle-laden turbulence. +For TPOD, the ensemble members are formed by sampling +the fluid velocity at Ne equidistantly spaced mesh points at +sample times tn ∈ T, n ∈ [0 : Nt − 1], and for PPOD the en- +semble members are formed by randomly choosing Ne parti- +cle records to track over the same sample times. The inertial +particles are initially spaced randomly throughout the cubic +domain in order to avoid introducing bias. +C. +Forces +Each simulation can be split into two periods – a forcing +period, and a decaying period. The forcing period is the ini- +tial part of the simulation, in which HIT is obtained by apply- +ing the source term F in equation (13). This period is neces- +sary to initiate decay from a fully developed turbulent velocity +field. The forcing procedure follows the forcing scheme de- +veloped by Mallouppas, George, and van Wachem (2013) and +is the same as the one briefly outlined in Schiødt et al. (2022). +During the forcing period particles are present within the +fluid, but two-way coupling is deactivated, i.e. Fp = 0 in +equation (13). This allows for the thermalization of parti- +cles under one-way coupling conditions, which minimizes +the transitional regime when two-way coupling is activated +(Ferrante and Elghobashi (2003)). +We define the end of the forcing period as time t0 = 0s, +which also denotes the start of the decaying period. Here +F = 0, and two-way coupling is activated for the multiphase +simulations, but remains zero for the single-phase simulation. +The two-way coupling term, Fp, in equation (13) is +modelled as suggested by Crowe, Sharma, and Stock (1977) +where +Fp = − +1 +ρ fVg +Np,g +∑ +p′=1 +FD,p′ . +(17) +Here Vg is the volume of cell g in the discretized domain, and +Np,g is the number of particles present in that cell. FD,p′ is the +drag force exerted by the fluid on particle p′. +D. +Setup +1. +Fluid +We use the setup of Mallouppas, George, and van Wachem +(2017) for the fluid simulation. Here the cube edge length +is given by ℓ = 0.128m, and the domain is discretized into +Ng = 1283 computational cells. Fluid viscosity is given by +µf = 1.72×10−5Pa s, and fluid density by ρ f = 1.17kg m−3. +FIG. 1. Evolution of normalized turbulent kinetic energy. +For all of the subsequent cases studied the Taylor Reynolds +number at t0 is given by Reλ = 58.0, where the integral-, +Taylor-, and Kolmogorov length scales are respectively I = +1.129 × 10−2m, λ = 6.134 × 10−3m and η = 4.0 × 10−4m. +The Kolmogorov time scale at t0 is τη = 10−2s. The reader +is referred to Schiødt et al. (2022) for a more thorough out- +line of the temporal evolution of the fluid characteristics in +the single-phase simulation. +2. +Particles +The different multiphase simulations considered are char- +acterized by the Stokes number St(t) = τp(t)/τη(t) of the +inertial particles at t = t0. +Here τp (equation (23)) is the +particle response time. The particle diameter is set to dp = +1.0 × 10−4m, and the particle mass fraction φm ≈ 1. Since +the particle density ρp is tweaked in each case to obtain dif- +ferent Stokes numbers, this signifies that the number of par- +ticles present in the fluid varies between each case. Letting +St0 = St(t0), the Stokes numbers considered are St0 = 0.25, +St0 = 0.75, St0 = 1.5, and St0 = 3.0. +IV. +RESULTS & DISCUSSION +All subsequent results are based on fluid- and inertial parti- +cle velocities during the decaying period, which lasts for 0.4s +of physical time. The velocities are sampled every δt = 10−3 +seconds, amounting to Nt = 400 temporal samples. The tem- +poral domain is normalized with respect to the reference time +scale tref = τη(t0) = 10−2s which is shared between all sim- +ulations. +A. +Fluid statistics +Figure 1 shows the temporal evolution of the carrier phase +TKE, E(t), in the single- and multiphase simulations. The + +Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +5 +102 +103 +κ +10−13 +10−11 +10−9 +10−7 +10−5 +E(κ) +Single-phase +St0 = 0.25 +St0 = 0.75 +St0 = 1.5 +St0 = 3.0 +FIG. 2. Fourier turbulence energy spectrum E(κ) of each simulation +at final time step t/tre f = 40. +TKE is normalized by E(0), and the figure illustrates that tur- +bulence is increasingly attenuated for increasing Stokes num- +bers. However, at St0 = 0.25 there is a slight augmentation of +turbulence for t/tref > 38. Similar observations have been +reported in previous studies (Sundaram and Collins (1999); +Ferrante and Elghobashi (2003); Letournel et al. (2020)). +The Fourier turbulence energy spectrum E(κ) of the car- +rier phase at time t/tref = 40 is seen in Figure 2. +As ob- +served in previous work (Druzhinin and Elghobashi (1999); +Ferrante and Elghobashi (2003); Letournel et al. (2020)) the +presence of inertial particles modulates the spectrum, shift- +ing energy from low to high wavenumbers. The degree with +which this energy transfer occurs is dependent on the Stokes +number, where more energy is observed to be transferred at +lower Stokes numbers. +Increased energy at high wavenumbers implies more en- +ergetic small scale turbulence structures. The fluid velocity +measured over time at a fixed spatial point will therefore, on +average, contain more fluctuations for the multiphase flows +compared to the single-phase flow. +This behaviour is in- +deed observed when considering the TPOD eigenspectra of +Figure 3. +Here, the extracted modes ϕ and corresponding +eigenvalues λ are based on the 3-D fluid velocity measured in +Ne = 163 = 4096 equidistantly spaced Eulerian mesh points, +where these ensemble members are assumed to represent the +dynamics of all 1283 mesh points (see section III B). +Figure 3 shows, for all simulations, the energy λα of each +TPOD mode for α ∈ [1 : 400]. A brief glance at the eigen- +spectra depicted reveals a distinct difference of shape between +the single- and multiphase simulations. The figure also il- +lustrates that modal energy is slightly higher in the single- +phase case when the mode number is low, whereas for higher +mode numbers the modal energy is higher in the multiphase +cases. As will be shown later (Figure 6) the higher numbered +modes contain more fluctuations, and this observation there- +fore aligns well with the intuition of how TPOD modal energy +should be distributed in accordance to the spatial structures. It +is notable that the modal energy is larger for some mode num- +bers in the multiphase cases compared to the single-phase case +even though the total modal energy in the latter is larger (see +FIG. 3. TPOD eigenspectrum showing the distribution of modal en- +ergy of the carrier phase in each simulation case. +Figure 1). This further underlines the observation that a larger +fraction of energy is distributed to more rapidly fluctuating +TPOD modes when the fluid is laden with inertial particles +and two-way coupling is activated. +B. +PPOD convergence +PPOD is applied to the velocity of Ne = 4096 randomly +selected inertial particles, initially distributed throughout the +spatial domain. This is performed for all multiphase simula- +tions under the assumption that these subsets of particles rep- +resent the dynamics of all particles within each respective sim- +ulation. +Let Ea,modal(m) denote the fraction of accumulated +POD modal energy up until mode number m: +Ea,modal(m) = +m +∑ +α=1 +λα +� +M +∑ +β=1 +λβ, +m ∈ [1 : M]. +(18) +Although m ∈ [1 : M], M = min(N,Ne) = 1200, the statistic is +only shown for m ≤ 40 in Figure 4 for the sake of readability. +The figure clearly shows that almost all of the PPOD modal +FIG. 4. Convergence of PPOD accumulated modal energy. + +Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +6 +FIG. 5. (a) Convergence rates of Ea,modal are equivalent between each velocity component and (b) the extracted modes are almost completely +parallel for α ≤ 20. +energy is contained within the first ∼ 4% of modes. More- +over, it is observed that the rate of convergence towards unity +increases as the Stokes number increases. +There are several contributing factors to the observed be- +haviour of convergence. +Firstly, the particles character- +ized by higher Stokes numbers are heavier, thus requir- +ing more energy to be accelerated. +Due to inertial filter- +ing, the velocities of these particles fluctuate less around +the mean (ensemble) velocity compared to lower Stokes +number +particles +(Ayyalasomayajula, Warhaft, and Collins +(2008); Salazar and Collins (2012)). +Secondly, as seen in +Figure 1 the increasing attenuation of TKE for increasing +Stokes numbers implies a less energetic fluid surrounding the +higher Stokes number particles, and the higher Stokes num- +ber particles are thus accelerated by smaller energies than +the lower Stokes number particles. Thirdly, for lower Stokes +numbers the small scale turbulent structures of the surround- +ing fluid are more energetic (Figure 2). The particles are in +these cases accelerated by a wider range of turbulent struc- +tures resulting in more fluctuating particle velocities. Ulti- +mately, these factors imply an increase in fluctuating particle +velocities for low Stokes numbers compared to higher Stokes +numbers, and hence a wider range of PPOD modes are re- +quired to account for these particle dynamics. +The modal +energy is thus more widely distributed for the lower Stokes +number case, decreasing the convergence rate of Ea,modal. +C. +Component decomposition +In stationary flows it is commonly accepted that fluid- +and particle velocities may appropriately be decomposed +with Fourier modes spanning the temporal domain (Tchen +(1947); Csanady (1963); Hinze (1975); Glauser and George +(1992); +Delville et al. +(1999); +Citriniti and George +(2000); +Johansson, George, and Woodward +(2002); +Iqbal and Thomas (2007); Muralidhar et al. (2019)). +The +Fourier decomposition is applied such that each velocity +component is decomposed separately. In analogy to this we +now apply PPOD componentwise, i.e. with dimension D = 1 +we extract M = DNt = 400 modes and eigenvalues separately +for the particle velocities in coordinate directions x1, x2 and +x3. Figure 5a shows up until m = 40 the convergence rate of +Ea,modal for component PPOD applied to the case St0 = 0.25. +Since the particles are suspended in decaying HIT, there is +not a preferential direction, and the convergence rates are +equivalent for all velocity component. +In Figure 5b the parallelity of the extracted modes is as- +sessed by evaluating +Pi,j +α,β = |(ϕi +α,ϕj +β)|, +i, j ∈ [1 : 3], α,β ∈ [1 : M], +(19) +where ϕi +α is the α’th mode extracted for coordinate direction +xi. When Pi,j +α,β = 1 the modes are completely parallel, whereas +Pi,j +α,β = 0 indicates orthogonality. The figure shows that along +the diagonal (α = β) there is almost complete parallelity for +low mode numbers (α ≤ 20), signifying that the mode-sets ex- +tracted are basically the same. For higher mode numbers this +is not the case, however as seen in Figure 5a these modes carry +little energy, and they account for ensemble-specific variance +rather than dominating particle dynamics. The importance of +these modes is thus negligible, and it may be concluded that +PPOD analysis of velocities in coordinate direction xi in de- +caying HIT yields the same qualitative results regardless of +the value of i. +Although only shown here for St0 = 0.25, +upon closer inspection of the data it is found that this con- +clusion may be drawn for every Stokes number considered, +and similarly for fluid velocity modes extracted with compo- +nent TPOD. For the remainder of this work, we will hence +consider component PPOD and TPOD applied to velocities in +coordinate direction x1, and consider the results representative +of all coordinate directions. +D. +Modes +A sample of the modes extracted with component TPOD +(solid) for both the single- and multiphase simulations are + +Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +7 +FIG. 6. Modes (ϕα, α ∈ [1 : 12]) extracted with TPOD (solid) and PPOD (dotted) for each simulation case. The nuance indicates Stokes +number where the Stokes number increases from darker to lighter grey. The black solid lines are the TPOD modes for the single-phase case. +shown in Figure 6 alongside a corresponding sample of the +modes extracted with PPOD (dotted) in the multiphase cases. +The modes are shown as functions of t, where ϕα(t) denotes +the element of ϕα connected to sample time t. All mode-sets +resemble slightly damped harmonic oscillators, where the lo- +cal wavelength of each mode increases over time. The damp- +ing of amplitude may be attributed to the temporal decay of +TKE (Figure 1). The increase of wavelength shows that the +energy-optimalmodes for the decomposition of fluid- and par- +ticle velocities in decaying HIT are not Fourier modes, keep- +ing in mind that in the stationary HIT case the component- +wise TPOD and PPOD modes would be well approximated +by Fourier modes (see e.g. Aubry (1991) for TPOD modes). +A high correlation between all POD mode-sets is observed, +indicating that the energetically dominating fluid- and particle +dynamics do not vary considerably across Stokes numbers. +This point is further investigated in Figure 7 where the paral- +lelity between ϕ and ψ is evaluated. Here, ϕ is chosen as a +reference mode-set given by the TPOD modes extracted from +the St0 = 0.25 simulation, and ψ represents the TPOD mode- +set of the single-phase case (Figure 7a), the TPOD mode-set +of the St0 = 3.0 case (Figure 7b), and the PPOD mode-set of +the St0 = 0.25 case (Figure 7c). Though figures 7b and 7c do +not exhibit complete parallelity between ϕ and ψ, they still +illustrate a strong parallelity at lower mode numbers, and lin- +ear dependency of similarly numbered modes at higher mode +numbers. This underlines that fluid- and particle dynamics are +fairly similar across phase and Stokes number. Figure 7a also +exhibits strong parallelity at lower mode numbers, whereas +ϕ is linearly dependent on many ψ-modes for higher mode +numbers. This shows that the dominating dynamics between +the single- and multiphase flow are similar, and suggests that +the two-way coupling information which is not captured in the +single-phase modes is embedded in the higher numbered POD +modes of the multiphase mode-sets. +The high parallelity observed raises an important question: +do the extracted bases span the same vector space? As briefly +noted in section II the extracted POD-bases are not necessar- +ily complete in H , and it is therefore not guaranteed that +the velocities of one ensemble may be fully decomposed by +the modes extracted from another ensemble. However, if one +mode-set ϕ can be completely decomposed by another mode- + +Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +8 +FIG. 7. Parellelity between ϕ denoting the TPOD modes extracted +for St0 = 0.25 and ψ denoting the (a) TPOD modes of the single- +phase simulation (b) TPOD modes extracted for St0 = 3.0 and (c) +PPOD modes extracted for St0 = 0.25. +set ψ, then the ensemble members generating ϕ can also be +fully decomposed by ψ. In total nine mode-sets are extracted +in the current work (five TPOD and four PPOD) and it turns +out that each of these can fully reconstruct (down to machine +precision) the other eight mode-sets, i.e. +����� +�����ϕβ − +M +∑ +α=1 +(ϕβ,ψα)ψα +����� +����� = 0, +β ∈ [1 : M]. +(20) +All mode-sets therefore span the same vector space, and both +particle- and fluid velocities may be expanded in the same ba- +sis. Though not shown here, u f@p (equation (14)) may also be +fully decomposed with respect to the POD bases extracted in +the current study. This enables the use of a single empirically +determined basis to be used for the expansion of all data sets +in question, illustrating the versatility of the PPOD method in +the application to multiphase flows. +E. +Response - velocity +Following the procedure of Csanady (1963) it may be +shown for statistically stationary flows that the Fourier La- +grangian spectrum of the particle velocity, Ep, is connected to +the equivalent spectrum of the fluid velocity at particle posi- +tion, E f@p, by a response function H2. The relation is given +by +Ep(α) = H2(α)E f@p(α), +α ∈ [1 : M], +(21) +where Ep(α) and E f@p(α) is the ensemble averaged en- +ergy connected to the α’th Fourier mode for respectively +the dispersed- and carrier phase. An analytic expression for +H2 may be found by replacing vp and u f@p in equation +(14) with their Fourier expansions. This was also done by +Berk and Coletti (2021) where they showed that +H2(α) = +1 +1 + (ω(α)τp)2 , +α ∈ [1 : M]. +(22) +Here ω(α) is the angular frequency of the α’th Fourier mode +and +τp = ρpd2 +p +18µf +� +1 + 0.15Re0.687 +p +�−1 +. +(23) +To the authors knowledge, no analytic expressions exist for +the response function H2 in non-stationary flows at the time of +writing. However, we may study the fraction Ep(α)/E f@p(α) +and let this serve as an empirical response function. +Considering Fourier modes as a special case of POD +modes, i.e. those derived empirically for a statistically sta- +tionary flow (Glauser and George (1992)), we conjecture that +some of the properties of the Fourier basis may also apply to +the POD basis in general. Indeed, for select test functions +representing stationary dynamics, Hodži´c, Olesen, and Velte +(2022) observed a high correlation between the POD eigen- +spectrum and the analytical Fourier spectrum. Interestingly, +the correlation exceeded that of the analytical Fourier spec- +trum and the spectrum of the discrete Fourier transform +(DFT), indicating a close spectral symmetry between the ana- +lytical Fourier basis and the POD basis in locally statistically +stationary flows. +Based on these considerations, and recalling that PPOD +modes may, in our simulations, completely expand both vp +and u f@p, we hypothesize that the empirical response func- +tion based on PPOD modes follow a trend similar to H2 (equa- +tion (22)). The hypothesis is validated in Figure 8 where H2, +H2 +four and H2 +pod are shown. Here H2 +four (Figure 8b) is shown as +a reference case, representing the empirical response function +where Ep and E f@p are computed based on Fourier modes. +H2 +pod (Figure 8c) represents the empirical response function +computed based on the PPOD modes of each simulation. For +H2, τp = τp(t0) is used in each case, although the quantity +is dependent on time since the studied flow is non-stationary. +However, for the Stokes numbers and temporal domain con- +sidered it is observed that +|τp(t0)− τp(tNt−1)| +τp(t0) +≤ 8%. +(24) +It is therefore assumed that the choice of τp is reasonably rep- +resentative of the dynamics over the entire temporal domain +in each simulation case. + +0.8300 +α +1000.6 +(Pα,p) +0.4 +0.2 +0 +0300 +α +100 +(c) +300 +α +100 +100 +30Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +9 +FIG. 8. Analytic response function derived for stationary flows is +shown in (a) whereas (b) and (c) are respectively the Fourier- and +POD empirical response functions for the current non-stationary +flow. +Though equation (22) is derived based on the Fourier trans- +form, Figures 8a and 8b show little correlation between H2 +and H2 +four. The inherent periodicity of Fourier modes justifies +this result, since expansion of non-periodic signals will lead +to spectral leakage. We are studying decaying HIT, and as a +consequence, the fluid- and particle velocity signals are not +periodic and the Fourier modes do not form an appropriate +basis for the expansion of these signals (Lumley (2007)). +Conversely, Figures 8a and 8c show a high correlation +between H2 and H2 +pod, as hypothesized. +Feasibly, the re- +sult reflects some deeper spectral symmetry related to en- +ergy optimality, from which equation (22) follows, rather than +the equation strictly following from the properties of Fourier +modes. +In Figure 9 the correlation is more clearly illustrated by the +depiction of H2 +pod (markers) and H2 +fit (solid). Here H2 +fit is a +least squares fit of H2 to H2 +pod given by +H2 +fit(α) = +1 +1 + ((α − 1)ω∗τp)2 , +α ∈ [1 : M]. +(25) +The product (α − 1)ω∗ represents a "POD-frequency", and +the fitting parameter ω∗ = 9.0647 is found through minimiza- +FIG. 9. Fit of analytic response function (solid) to empirical POD +response function (markers). +tion of the objective +min +ω∗ ∑ +τp ∑ +α +����� +����� +H2 +fit(α)− H2 +pod(α) +H2 +fit(α) +����� +����� +2 +. +(26) +Summation over τp represents the fitting of ω∗ to the data of +all Stokes numbers simultaneously. The figure shows a clear +connection between the PPOD empirical response function +and a modified version of the analytical model (22). These +results show potential for the ability to approximate the modal +energy of the carrier phase sampled at the particle position in +decaying HIT, directly from the Stokes number and particle +velocities. +F. +Response - relative velocity +Csanady (1963) derived a relation between the mean square +relative velocity and E f@p. Inspired by this, and extending the +considerations of the previous section, we conjecture that +Erel(α) = H2 +rel(α)E f@p(α), +α ∈ [1 : M], +(27) +where Erel(α) is the ensemble averaged energy of the relative +velocity, urel = u f@p −vp, connected to the α’th POD mode +(PPOD), and +H2 +rel(α) = ((α − 1)ω∗τp)2H2 +fit(α), +α ∈ [1 : M]. +(28) +Figure 10 +depicts +the +empirically +evaluated +fraction +Erel(α)/E f@p(α) (markers) and the fit H2 +rel(α) (solid). +The quantities are plotted against the log-scaled α to high- +light the fit at the energetically dominant modes. +Visual +inspection shows a good model fit for α ≤ 30 and St0 ≤ 1.5, +whereas the fit and the empirical results diverge at higher +mode numbers. +Noting that the modes ϕα, α ∈ [1 : 30], +account for more than 99% of the total modal energy in +each case, it can be argued that the fit is appropriate at the + +Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence +10 +FIG. 10. H2 +rel (solid) plotted against the fraction Erel/E f@p (markers) +shows a fairly good fit for Stokes numbers St0 ≤ 1.5 and α ≤ 30. +energetically dominant modes for St0 ≤ 1.5. For St0 = 3.0 the +model fit at the energetically dominant modes is not on point. +This suggests a range of Stokes numbers at which the model +is appropriate. However, in all cases, the trend of Erel/E f@p +is similar to that of H2 +rel, highlighting the similarities between +the models derived through Fourier analysis of stationary +flows and POD analysis of the current non-stationary flow. +Equations (21), (25), (27) and (28) provide a method for +approximating E f@p and Erel in particle-laden decaying HIT +based on particle velocity measurements. +The method re- +quires knowledge of τp, and if this quantity changes signifi- +cantly over the considered temporal domain, or if it has rapid +fluctuations, the current accurateness of the model may deteri- +orate since it was assumed that τp = τp(t0) for the generation +of the fit. +V. +CONCLUSIONS +A study of the temporal dynamics of two-way coupled +particle-laden decaying HIT for various Stokes numbers was +conducted. +Using time-focalized formulations of POD – +TPOD and PPOD for respectively the decomposition of fluid- +and particle velocities – sets of energy-optimal modes were +extracted representing the temporal dynamics of the two +phases. For both phases it was observed that the extracted +modes resembled damped harmonic oscillators, where the lo- +cal wavelength of each mode increased over time. Moreover, +the modes exhibited a high correlation in the dominating dy- +namics between the carrier- and dispersed phase. +The TPOD eigenspectrum of each simulation was inspected +and compared to the eigenspectrum of a corresponding single- +phase simulation. A distinct difference of shape between the +single- and multiphase spectra was observed. In addition, the +TPOD spectra were compared to the Fourier turbulence en- +ergy spectrum generated at the final time step of each simula- +tion. Here an increase of energy at high wavenumbers of the +turbulence spectrum was observed to correlate with a relative +increase in the TPOD eigenspectrum at high mode numbers. +It was demonstrated that the POD mode-set extracted from +the velocity of one phase could span the velocity of both +phases. Therefore, the Lagrangian spectrum based on PPOD +modes could be computed for both the carrier- and dispersed +phase. A relation between these spectra was evaluated empir- +ically giving rise to analytical expressions of response func- +tions in a PPOD frame of reference. The response functions +related the modal energy of the inertial particles to that of the +surrounding fluid through simple expressions dependent on +the Stokes number. Notably, the expressions, fitting the data +of the current non-stationary flow, resembled those derived +through Fourier analysis of stationary flows. This suggested a +deeper symmetry between POD and Fourier spectra. +The current PPOD analysis was applied to an ideal test case +of a non-stationary flow. The results outlined, and the theoret- +ical applicablity of PPOD to any non-stationary flow, indicate +that PPOD analysis may provide insightful dynamical infor- +mation for the Lagrangian dynamics of alternative flows in +future studies of particle-laden turbulence. +ACKNOWLEDGMENTS +AH and CMV acknowledge financial support from the Eu- +ropean Research council: This project has received funding +from the European Research Council (ERC) under the Euro- +pean Unions Horizon 2020 research and innovation program +(grant agreement No 803419). +MS acknowledges financial support from the Poul Due +Jensen Foundation: Financial support from the Poul Due +Jensen Foundation (Grundfos Foundation) for this research is +gratefully acknowledged. +DECLARATION OF INTEREST +The authors report no conflicts of interest. +DATA AVAILABILITY STATEMENT +The data that forms the basis of this study is available from +the corresponding author upon reasonable request. +REFERENCES +Abdelsamie, A. 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T., and Colonius, T., “Spectral proper orthogonal +decomposition and its relationship to dynamic mode decomposition and +resolvent analysis,” Journal of Fluid Mechanics 847, 821–867 (2018). +Zhang, Z., Legendre, D., and Zamansky, R., “Model for the dynamics of +micro-bubbles in high-reynolds-number flows,” Journal of Fluid Mechan- +ics 879, 554–578 (2019). + +arXiv:2301.13621v1 [physics.flu-dyn] 31 Jan 2023 +1 +(Dated: 1 February 2023) +1 + +I. +A. +1. +2 + diff --git a/-dFRT4oBgHgl3EQfrjff/content/tmp_files/load_file.txt b/-dFRT4oBgHgl3EQfrjff/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5765338930139705e39b321d050a34155707bba7 --- /dev/null +++ b/-dFRT4oBgHgl3EQfrjff/content/tmp_files/load_file.txt @@ -0,0 +1,565 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf,len=564 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='13621v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='flu-dyn] 31 Jan 2023 Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Schiødt,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Hodžić,1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Evrard,2, a) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Hausmann,2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Van Wachem,2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Velte1 1)Technical University of Denmark, Kongens Lyngby, Denmark 2)Otto von Guericke University, Magdeburg, Germany (*Electronic mail: maschi@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='dk) (Dated: 1 February 2023) In particle-laden turbulence, the Fourier Lagrangian spectrum of each phase is regularly computed, and analytically derived response functions relate the Lagrangian spectrum of the fluid- and the particle phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' However, due to the periodic nature of the Fourier basis, the analysis is restricted to statistically stationary flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In the present work, utilizing the bases of time-focalized proper orthogonal decomposition (POD), this analysis is extended to temporally non-stationary turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Studying two-way coupled particle-laden decaying homogeneous isotropic turbulence for various Stokes numbers, it is demonstrated that the temporal POD modes extracted from the dispersed phase may be used for the expansion of both fluid- and particle velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The POD Lagrangian spectrum of each phase may thus be computed from the same set of modal building blocks, allowing the evaluation of response functions in a POD frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Based on empirical evaluations, a model for response functions in non-stationary flows is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The related energies of the two phases is well approximated by simple analytical expressions dependent on the particle Stokes number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' It is found that the analytical expressions closely resemble those derived through Fourier analysis of statistically stationary flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' These results suggest the existence of an inherent spectral symmetry underlying the dynamical systems consisting of particle-laden turbulence, a symmetry which spans across stationary/non-stationary particle-laden flow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' INTRODUCTION Recent years have seen renewed attention directed towards particle-laden turbulence, due to its relevance in numerous engineering and natural settings (Brandt and Coletti (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Theoretical models and improved experimental and numer- ical methods have led to advancements in our understand- ing of particle dynamics, herein counting acceleration statis- tics, preferential sampling and particle clustering to name a few (Toschi and Bodenschatz (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Gustavsson and Mehlig (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Maxey (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' One focus of study has been the modulation of turbulence induced by two-way coupling (Druzhinin and Elghobashi (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Ferrante and Elghobashi (2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here, the presence of particles in flows under zero gravity conditions has been shown to attenuate turbulent kinetic energy (TKE) at low wavenumbers and augment it at higher wavenumbers, lead- ing to an increase in dissipation (Squires and Eaton (1994)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Inertial particles may, however, also act as sources of in- creased turbulence energy, and the total TKE may be either augmented or attenuated by the presence of a dispersed phase (Ferrante and Elghobashi (2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A key parameter identified in this regard is the particle Stokes number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Letournel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2020) investigated TKE totals as a function of the Stokes number, and found an approximate threshold below which tur- bulence was augmented, and above which it was attenuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Nevertheless, the same authors underlined the lack of consen- sus on a unique criterion for turbulence modulation by parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' a)Sibley School of Mechanical and Aerospace Engineering, Cornell Univer- sity, Ithaca, United States Ireland, Bragg, and Collins (2016a) investigated the large scale single-particle velocity statistics of inertial particles in homogeneous isotropic turbulence (HIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Driven by the ef- fects of inertial filtering and preferential sampling, the aver- age particle kinetic energy normalized by the average fluid ki- netic energy was shown to approximately follow a simple re- lation dependent on the Stokes number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Similar studies were conducted under gravity conditions by Good et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2014) and Ireland, Bragg, and Collins (2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Although the study of particle-laden turbulence has rapidly progressed over the past decade, new theoretical tools are still needed in order to gain further insights into the dy- namics (Brandt and Coletti (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' One such tool may be the particle proper orthogonal decomposition (PPOD) for- mulated by Schiødt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2022), where Lagrangian parti- cle velocities are decomposed into a set of modes that rep- resent temporal particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This tool is utilized in the present study, where the extracted modes are compared to those extracted for the fluid measured at fixed Eulerian mesh points using the temporal formulation of POD intro- duced by Aubry, Guyonnet, and Lima (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Both formu- lations of POD are briefly outlined in section II, and the con- straints required for direct comparisons of fluid- and particle POD modes are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Modal decomposition of fluid- and particle temporal dy- namics allows for the evaluation of the Lagrangian spectrum of both phases in a POD frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In the current work, this leads to formulations of POD-based response func- tions, that relate the energy of the two phases on a modal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Although response functions based on the Fourier de- composition have previously been studied in stationary flows (Csanady (1963);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Zhang, Legendre, and Zamansky (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Berk and Coletti (2021)), the advantage of the POD-based ap- proach is that stationarity is not required, and the present study Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 2 is therefore focused on the analysis of various simulations of two-way coupled particle-laden decaying HIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The analy- sis culminates in analytic expressions of POD-based response functions closely resembling those derived through Fourier analysis of stationary flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Section II gives a brief outline of the formulation of POD and the structure of the ensembles that will produce temporal modes representing fluid- and particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A summary of the simulation setup is given in III, which is followed by a presentation and discussion of results in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Finally, our conclusions are given in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' PROPER ORTHOGONAL DECOMPOSITION The main objective of POD is to extract a set of empirical basis functions ϕ = {ϕα}M α=1 that represent dominating fea- tures of the studied dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The basis functions, also known as modes, are extracted by solving the eigenvalue problem Rϕα = λαϕα, α ∈ [1 : M], (1) where λ = {λα}M α=1 are the eigenvalues connected to each mode, and for the cases we study, these are real and sorted such that λ1 ≥ ··· ≥ λM ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The operator R : H → H is defined from the ensemble of empirical data u = {u(i)}Ne i=1, and is dependent on the definition of the Hilbert space H for which ϕ serves as an empirical orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Though the basis is not necessarily complete in H , each ensemble member may be decomposed into a weighted sum of modes, thus u(i) = M ∑ α=1 c(i) α ϕα, i ∈ [1 : Ne], (2) where the weights c(i) α are known as the projection coefficients given by c(i) α = (u(i),ϕα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (3) Here (·,·) denotes the inner product of H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The projection coefficients are connected to the eigenvalues λ by the relation λα = �� c(i) α c(i)∗ α �Ne i=1 � , α ∈ [1 : M], (4) where (∗) denotes both the complex conjugate transpose for a scalar and Hermitian transpose for a vector, and ⟨{·}Ne i=1⟩ is the ensemble average operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The definition of H and what constitutes an ensem- ble member determines the interpretation of ϕ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In section II A and section II B we briefly outline the discrete for- mulations of the Eulerian- and the Lagrangian (particle) POD, respectively, and show their dependency on the definition of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Eulerian POD The most common application of POD is based on the fluid velocity u f (x,t) ∈ RD measured at fixed mesh points in a Eu- lerian grid at equidistant sample times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Following the classical interpretation of POD (Lumley (1967)) an ensemble member may in this discrete case be formed by u(i) = � u(i) f (x1,t0)∗ ··· u(i) f (xNg,t0)∗ ··· u(i) f (xNg,tNt−1)∗ �∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (5) Here u(i) f is the i’th fluid velocity realization, xg ∈ RD, g ∈ [1 : Ng] are the Eulerian mesh points and tn ∈ T, n ∈ [0 : Nt − 1] are the sample times of the temporal domain T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In this case u(i) ∈ H = RN, where N = DNgNt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' H is equipped with the standard inner product (w1,w2) = w∗ 2w1, and the operator R in equation (1) is given by R = �� u(i)u(i)∗�Ne i=1 � ∈ RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (6) Solving equation (1) then results in a set of spatio-temporal modes that are optimal with respect to energy, where λ rep- resents the energy of each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' However, the amount of data needed to generate ϕ often makes this classical approach infeasible, as several uncorrelated fluid flow realizations are needed to generate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Instead, an approach popularized by Sirovich (1987) and Aubry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (1988) is to extract spa- tially orthogonal modes, with time dependent projection co- efficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This is what Towne, Schmidt, and Colonius (2018) refers to as the space-only POD, and in a statistically station- ary flow an ensemble member may be given by the fluid veloc- ity measured at all grid points at a single sample time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' From one fluid realization several ensemble members may thus be generated, and the ensemble average operator reduces to a temporal average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In the current work we will focus on what we term the time-only POD (TPOD) and its relation to PPOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The TPOD is also formulated in the continuous case (Aubry, Guyonnet, and Lima (1991);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Aubry (1991)) and as an analogy to its spatial counterpart it produces a set of tempo- rally orthogonal modes, with spatially dependent projection coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' An ensemble member is in this case given by u(i) = �u f (xi,t0)∗ ··· u f (xi,tNt−1)∗�∗ , i ∈ [1 : Ne], (7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' the fluid velocity at a grid point i measured at sample times tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Note that Ne ≤ Ng when the ensemble members are taken from the same fluid realization, and that the fluid flow in that case should be homogeneous (Aubry (1991)), signify- ing that the temporal evolution is statistically equivalent in all grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The ensemble average operator then reduces to a spatial average and the operator R is still given as in equation (6), although here N = DNt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The modes extracted with TPOD represent the temporal evolution of the fluid velocity through a Eulerian mesh point, and λ is connected to the energy E(t) = 1 2 �� u∗ f (xi,t)u f (xi,t) �Ne i=1 � , (8) Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 3 by Nt−1 ∑ n=0 E(tn) = 1 2 M ∑ α=1 λα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (9) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Particle POD Schiødt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2022) formulated PPOD as a method for decomposing the velocity of Lagrangian particles into a weighted sum of empirical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Like TPOD the method produces a set of temporal modes, however, the modes repre- sent the dynamics of Lagrangian particles rather than the fluid dynamics at fixed Eulerian mesh points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The ensemble u is in this formulation defined by the ensemble members u(i) = � v(i)(t0)∗ ··· v(i)(tNt−1)∗�∗ , i ∈ [1 : Ne], (10) where v(i)(tn) = � v(i) 1 (tn)∗ ··· v(i) Np(tn)∗�∗ , (11) is the velocity of Np Lagrangian particles measured at sample times tn ∈ T, n ∈ [0 : Nt −1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here u(i) ∈ RN with N = DNpNt, since v(i) p (t) ∈ RD is the velocity of a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Choos- ing Np = 1 for the remainder of the current work, we see that PPOD and TPOD ensemble members belong to the same Hilbert space H = RN, N = DNt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The mode-sets extracted with respectively TPOD and PPOD are therefore in this case directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' To generate a meaningful ensemble of Lagrangian parti- cle velocities, the ensemble particles should belong to sim- ilar flows or be sampled from the same flow containing certain symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' We elaborate further on this point in section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In section IV both TPOD and PPOD analysis is applied to the Reynolds decomposed u(i) fluct = u(i) − ⟨{u(i)}Ne i=1⟩ rather than u(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Thus, E(t) in equations (8)-(9) becomes a measure of TKE, and u(i) = �� u(i)�Ne i=1 � + M ∑ α=1 c(i) α ϕα, i ∈ [1 : Ne].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (12) However, ⟨{u(i)}Ne i=1⟩ ≈ 0 for all TPOD and PPOD ensembles considered, and we will therefore interchangeably refer to ϕ as the mode-set spanning both the signal u(i) and u(i) fluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' SIMULATION In the current work we consider the simulation of one single-phase flow and four different simulations of two-way coupled particle-laden turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' All simulations are per- formed within a periodic cube with edge length ℓ, discretized into Ng computational cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Dynamical equations We apply the Euler-Lagrange point-particle approach (Elghobashi and Truesdell (1992)) where the fluid velocity u f is computed at each time step by numerical integration of the incompressible Navier-Stokes equations on a Eulerian mesh, and particle velocities are obtained by integrating the gov- erning particle equations of motion forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' For the Navier-Stokes equations a constant dynamic viscosity µf and mass density ρ f are used, and with p denoting pressure the equations are given by ∇·u f = 0, (13a) ∂u f ∂t + ∇·(u f ⊗ u f) = − 1 ρ f ∇p+ µf ρ f ∇2u f + Fp + F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (13b) Here Fp is the force that the dispersed particles exert on the carrier fluid, and F is an artificial source term applied in an initial forcing period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In section III C the details of Fp and F are outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The particles considered are monodisperse solid spheres with diameter dp, volume Vp and density ρp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Assuming parti- cles are only accelerated according to drag force, the dynamic equations for particle motion are given by dxp dt = vp , (14a) Vpρp dvp dt = FD = π 8 d2 pρ fCD|u f@p − vp|(u f@p − vp), (14b) where xp and u f@p = u f (xp,t) are the particle position and fluid velocity at particle position, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' FD de- notes the drag force and CD is the drag coefficient given by (Schiller and Naumann (1933)) CD = 24 Rep � 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='15Re0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='687 p � , (15) and Rep = dpρ f |u f@p − vp| µf , (16) is the particle Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Equation (15) holds for 0 < Rep ≤ 1000, which is the only range considered in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A second-order finite-volume solver (Denner, Evrard, and van Wachem (2020)) is used to in- tegrate (13) forward in time, and the Verlet scheme is used for the forward integration of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Decaying homogenous isotropic turbulence To study two-way coupling effects in an idealized test case, we analyze a particle-laden fluid with decaying HIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This case is chosen over stationary HIT because the effects of the forc- ing term F would overlap with the particle-fluid interaction Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 4 energy in the latter (Abdelsamie and Lee (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In addition, the properties of decaying HIT signifies that the fluid velocity in all Eulerian mesh points evolves in a statistically equivalent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The inertial particles are thermalized to the fluid (see section III C) and thus have a statistically equivalent evolution throughout the temporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Therefore, a meaningful en- semble of realizations can be generated for both TPOD and PPOD from a single simulation of particle-laden turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' For TPOD, the ensemble members are formed by sampling the fluid velocity at Ne equidistantly spaced mesh points at sample times tn ∈ T, n ∈ [0 : Nt − 1], and for PPOD the en- semble members are formed by randomly choosing Ne parti- cle records to track over the same sample times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The inertial particles are initially spaced randomly throughout the cubic domain in order to avoid introducing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Forces Each simulation can be split into two periods – a forcing period, and a decaying period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The forcing period is the ini- tial part of the simulation, in which HIT is obtained by apply- ing the source term F in equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This period is neces- sary to initiate decay from a fully developed turbulent velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The forcing procedure follows the forcing scheme de- veloped by Mallouppas, George, and van Wachem (2013) and is the same as the one briefly outlined in Schiødt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' During the forcing period particles are present within the fluid, but two-way coupling is deactivated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Fp = 0 in equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This allows for the thermalization of parti- cles under one-way coupling conditions, which minimizes the transitional regime when two-way coupling is activated (Ferrante and Elghobashi (2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' We define the end of the forcing period as time t0 = 0s, which also denotes the start of the decaying period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here F = 0, and two-way coupling is activated for the multiphase simulations, but remains zero for the single-phase simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The two-way coupling term, Fp, in equation (13) is modelled as suggested by Crowe, Sharma, and Stock (1977) where Fp = − 1 ρ fVg Np,g ∑ p′=1 FD,p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (17) Here Vg is the volume of cell g in the discretized domain, and Np,g is the number of particles present in that cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' FD,p′ is the drag force exerted by the fluid on particle p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Setup 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Fluid We use the setup of Mallouppas, George, and van Wachem (2017) for the fluid simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here the cube edge length is given by ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='128m, and the domain is discretized into Ng = 1283 computational cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Fluid viscosity is given by µf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='72×10−5Pa s, and fluid density by ρ f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='17kg m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Evolution of normalized turbulent kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' For all of the subsequent cases studied the Taylor Reynolds number at t0 is given by Reλ = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0, where the integral-, Taylor-, and Kolmogorov length scales are respectively I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='129 × 10−2m, λ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='134 × 10−3m and η = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0 × 10−4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The Kolmogorov time scale at t0 is τη = 10−2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The reader is referred to Schiødt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2022) for a more thorough out- line of the temporal evolution of the fluid characteristics in the single-phase simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Particles The different multiphase simulations considered are char- acterized by the Stokes number St(t) = τp(t)/τη(t) of the inertial particles at t = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here τp (equation (23)) is the particle response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The particle diameter is set to dp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0 × 10−4m, and the particle mass fraction φm ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Since the particle density ρp is tweaked in each case to obtain dif- ferent Stokes numbers, this signifies that the number of par- ticles present in the fluid varies between each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Letting St0 = St(t0), the Stokes numbers considered are St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25, St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='75, St0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='5, and St0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' RESULTS & DISCUSSION All subsequent results are based on fluid- and inertial parti- cle velocities during the decaying period, which lasts for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='4s of physical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The velocities are sampled every δt = 10−3 seconds, amounting to Nt = 400 temporal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The tem- poral domain is normalized with respect to the reference time scale tref = τη(t0) = 10−2s which is shared between all sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Fluid statistics Figure 1 shows the temporal evolution of the carrier phase TKE, E(t), in the single- and multiphase simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 5 102 103 κ 10−13 10−11 10−9 10−7 10−5 E(κ) Single-phase St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25 St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='75 St0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='5 St0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Fourier turbulence energy spectrum E(κ) of each simulation at final time step t/tre f = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' TKE is normalized by E(0), and the figure illustrates that tur- bulence is increasingly attenuated for increasing Stokes num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' However, at St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25 there is a slight augmentation of turbulence for t/tref > 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Similar observations have been reported in previous studies (Sundaram and Collins (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Ferrante and Elghobashi (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Letournel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The Fourier turbulence energy spectrum E(κ) of the car- rier phase at time t/tref = 40 is seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' As ob- served in previous work (Druzhinin and Elghobashi (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Ferrante and Elghobashi (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Letournel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2020)) the presence of inertial particles modulates the spectrum, shift- ing energy from low to high wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The degree with which this energy transfer occurs is dependent on the Stokes number, where more energy is observed to be transferred at lower Stokes numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Increased energy at high wavenumbers implies more en- ergetic small scale turbulence structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The fluid velocity measured over time at a fixed spatial point will therefore, on average, contain more fluctuations for the multiphase flows compared to the single-phase flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This behaviour is in- deed observed when considering the TPOD eigenspectra of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here, the extracted modes ϕ and corresponding eigenvalues λ are based on the 3-D fluid velocity measured in Ne = 163 = 4096 equidistantly spaced Eulerian mesh points, where these ensemble members are assumed to represent the dynamics of all 1283 mesh points (see section III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Figure 3 shows, for all simulations, the energy λα of each TPOD mode for α ∈ [1 : 400].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A brief glance at the eigen- spectra depicted reveals a distinct difference of shape between the single- and multiphase simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The figure also il- lustrates that modal energy is slightly higher in the single- phase case when the mode number is low, whereas for higher mode numbers the modal energy is higher in the multiphase cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' As will be shown later (Figure 6) the higher numbered modes contain more fluctuations, and this observation there- fore aligns well with the intuition of how TPOD modal energy should be distributed in accordance to the spatial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' It is notable that the modal energy is larger for some mode num- bers in the multiphase cases compared to the single-phase case even though the total modal energy in the latter is larger (see FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' TPOD eigenspectrum showing the distribution of modal en- ergy of the carrier phase in each simulation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This further underlines the observation that a larger fraction of energy is distributed to more rapidly fluctuating TPOD modes when the fluid is laden with inertial particles and two-way coupling is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' PPOD convergence PPOD is applied to the velocity of Ne = 4096 randomly selected inertial particles, initially distributed throughout the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This is performed for all multiphase simula- tions under the assumption that these subsets of particles rep- resent the dynamics of all particles within each respective sim- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Let Ea,modal(m) denote the fraction of accumulated POD modal energy up until mode number m: Ea,modal(m) = m ∑ α=1 λα � M ∑ β=1 λβ, m ∈ [1 : M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (18) Although m ∈ [1 : M], M = min(N,Ne) = 1200, the statistic is only shown for m ≤ 40 in Figure 4 for the sake of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The figure clearly shows that almost all of the PPOD modal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Convergence of PPOD accumulated modal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (a) Convergence rates of Ea,modal are equivalent between each velocity component and (b) the extracted modes are almost completely parallel for α ≤ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' energy is contained within the first ∼ 4% of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' More- over, it is observed that the rate of convergence towards unity increases as the Stokes number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' There are several contributing factors to the observed be- haviour of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Firstly, the particles character- ized by higher Stokes numbers are heavier, thus requir- ing more energy to be accelerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Due to inertial filter- ing, the velocities of these particles fluctuate less around the mean (ensemble) velocity compared to lower Stokes number particles (Ayyalasomayajula, Warhaft, and Collins (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Salazar and Collins (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Secondly, as seen in Figure 1 the increasing attenuation of TKE for increasing Stokes numbers implies a less energetic fluid surrounding the higher Stokes number particles, and the higher Stokes num- ber particles are thus accelerated by smaller energies than the lower Stokes number particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Thirdly, for lower Stokes numbers the small scale turbulent structures of the surround- ing fluid are more energetic (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The particles are in these cases accelerated by a wider range of turbulent struc- tures resulting in more fluctuating particle velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Ulti- mately, these factors imply an increase in fluctuating particle velocities for low Stokes numbers compared to higher Stokes numbers, and hence a wider range of PPOD modes are re- quired to account for these particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The modal energy is thus more widely distributed for the lower Stokes number case, decreasing the convergence rate of Ea,modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Component decomposition In stationary flows it is commonly accepted that fluid- and particle velocities may appropriately be decomposed with Fourier modes spanning the temporal domain (Tchen (1947);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Csanady (1963);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Hinze (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Glauser and George (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Delville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Citriniti and George (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Johansson, George, and Woodward (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Iqbal and Thomas (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Muralidhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The Fourier decomposition is applied such that each velocity component is decomposed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In analogy to this we now apply PPOD componentwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' with dimension D = 1 we extract M = DNt = 400 modes and eigenvalues separately for the particle velocities in coordinate directions x1, x2 and x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Figure 5a shows up until m = 40 the convergence rate of Ea,modal for component PPOD applied to the case St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Since the particles are suspended in decaying HIT, there is not a preferential direction, and the convergence rates are equivalent for all velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In Figure 5b the parallelity of the extracted modes is as- sessed by evaluating Pi,j α,β = |(ϕi α,ϕj β)|, i, j ∈ [1 : 3], α,β ∈ [1 : M], (19) where ϕi α is the α’th mode extracted for coordinate direction xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' When Pi,j α,β = 1 the modes are completely parallel, whereas Pi,j α,β = 0 indicates orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The figure shows that along the diagonal (α = β) there is almost complete parallelity for low mode numbers (α ≤ 20), signifying that the mode-sets ex- tracted are basically the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' For higher mode numbers this is not the case, however as seen in Figure 5a these modes carry little energy, and they account for ensemble-specific variance rather than dominating particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The importance of these modes is thus negligible, and it may be concluded that PPOD analysis of velocities in coordinate direction xi in de- caying HIT yields the same qualitative results regardless of the value of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Although only shown here for St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25, upon closer inspection of the data it is found that this con- clusion may be drawn for every Stokes number considered, and similarly for fluid velocity modes extracted with compo- nent TPOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' For the remainder of this work, we will hence consider component PPOD and TPOD applied to velocities in coordinate direction x1, and consider the results representative of all coordinate directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Modes A sample of the modes extracted with component TPOD (solid) for both the single- and multiphase simulations are Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Modes (ϕα, α ∈ [1 : 12]) extracted with TPOD (solid) and PPOD (dotted) for each simulation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The nuance indicates Stokes number where the Stokes number increases from darker to lighter grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The black solid lines are the TPOD modes for the single-phase case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' shown in Figure 6 alongside a corresponding sample of the modes extracted with PPOD (dotted) in the multiphase cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The modes are shown as functions of t, where ϕα(t) denotes the element of ϕα connected to sample time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' All mode-sets resemble slightly damped harmonic oscillators, where the lo- cal wavelength of each mode increases over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The damp- ing of amplitude may be attributed to the temporal decay of TKE (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The increase of wavelength shows that the energy-optimalmodes for the decomposition of fluid- and par- ticle velocities in decaying HIT are not Fourier modes, keep- ing in mind that in the stationary HIT case the component- wise TPOD and PPOD modes would be well approximated by Fourier modes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Aubry (1991) for TPOD modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A high correlation between all POD mode-sets is observed, indicating that the energetically dominating fluid- and particle dynamics do not vary considerably across Stokes numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This point is further investigated in Figure 7 where the paral- lelity between ϕ and ψ is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here, ϕ is chosen as a reference mode-set given by the TPOD modes extracted from the St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25 simulation, and ψ represents the TPOD mode- set of the single-phase case (Figure 7a), the TPOD mode-set of the St0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0 case (Figure 7b), and the PPOD mode-set of the St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25 case (Figure 7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Though figures 7b and 7c do not exhibit complete parallelity between ϕ and ψ, they still illustrate a strong parallelity at lower mode numbers, and lin- ear dependency of similarly numbered modes at higher mode numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This underlines that fluid- and particle dynamics are fairly similar across phase and Stokes number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Figure 7a also exhibits strong parallelity at lower mode numbers, whereas ϕ is linearly dependent on many ψ-modes for higher mode numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This shows that the dominating dynamics between the single- and multiphase flow are similar, and suggests that the two-way coupling information which is not captured in the single-phase modes is embedded in the higher numbered POD modes of the multiphase mode-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The high parallelity observed raises an important question: do the extracted bases span the same vector space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' As briefly noted in section II the extracted POD-bases are not necessar- ily complete in H , and it is therefore not guaranteed that the velocities of one ensemble may be fully decomposed by the modes extracted from another ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' However, if one mode-set ϕ can be completely decomposed by another mode- Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Parellelity between ϕ denoting the TPOD modes extracted for St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25 and ψ denoting the (a) TPOD modes of the single- phase simulation (b) TPOD modes extracted for St0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0 and (c) PPOD modes extracted for St0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' set ψ, then the ensemble members generating ϕ can also be fully decomposed by ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In total nine mode-sets are extracted in the current work (five TPOD and four PPOD) and it turns out that each of these can fully reconstruct (down to machine precision) the other eight mode-sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' ����� �����ϕβ − M ∑ α=1 (ϕβ,ψα)ψα ����� ����� = 0, β ∈ [1 : M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (20) All mode-sets therefore span the same vector space, and both particle- and fluid velocities may be expanded in the same ba- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Though not shown here, u f@p (equation (14)) may also be fully decomposed with respect to the POD bases extracted in the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This enables the use of a single empirically determined basis to be used for the expansion of all data sets in question, illustrating the versatility of the PPOD method in the application to multiphase flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Response - velocity Following the procedure of Csanady (1963) it may be shown for statistically stationary flows that the Fourier La- grangian spectrum of the particle velocity, Ep, is connected to the equivalent spectrum of the fluid velocity at particle posi- tion, E f@p, by a response function H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The relation is given by Ep(α) = H2(α)E f@p(α), α ∈ [1 : M], (21) where Ep(α) and E f@p(α) is the ensemble averaged en- ergy connected to the α’th Fourier mode for respectively the dispersed- and carrier phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' An analytic expression for H2 may be found by replacing vp and u f@p in equation (14) with their Fourier expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This was also done by Berk and Coletti (2021) where they showed that H2(α) = 1 1 + (ω(α)τp)2 , α ∈ [1 : M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (22) Here ω(α) is the angular frequency of the α’th Fourier mode and τp = ρpd2 p 18µf � 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='15Re0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='687 p �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (23) To the authors knowledge, no analytic expressions exist for the response function H2 in non-stationary flows at the time of writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' However, we may study the fraction Ep(α)/E f@p(α) and let this serve as an empirical response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Considering Fourier modes as a special case of POD modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' those derived empirically for a statistically sta- tionary flow (Glauser and George (1992)), we conjecture that some of the properties of the Fourier basis may also apply to the POD basis in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Indeed, for select test functions representing stationary dynamics, Hodži´c, Olesen, and Velte (2022) observed a high correlation between the POD eigen- spectrum and the analytical Fourier spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Interestingly, the correlation exceeded that of the analytical Fourier spec- trum and the spectrum of the discrete Fourier transform (DFT), indicating a close spectral symmetry between the ana- lytical Fourier basis and the POD basis in locally statistically stationary flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Based on these considerations, and recalling that PPOD modes may, in our simulations, completely expand both vp and u f@p, we hypothesize that the empirical response func- tion based on PPOD modes follow a trend similar to H2 (equa- tion (22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The hypothesis is validated in Figure 8 where H2, H2 four and H2 pod are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here H2 four (Figure 8b) is shown as a reference case, representing the empirical response function where Ep and E f@p are computed based on Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' H2 pod (Figure 8c) represents the empirical response function computed based on the PPOD modes of each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' For H2, τp = τp(t0) is used in each case, although the quantity is dependent on time since the studied flow is non-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' However, for the Stokes numbers and temporal domain con- sidered it is observed that |τp(t0)− τp(tNt−1)| τp(t0) ≤ 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (24) It is therefore assumed that the choice of τp is reasonably rep- resentative of the dynamics over the entire temporal domain in each simulation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='8300 α 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='6 (Pα,p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='2 0 0300 α 100 (c) 300 α 100 100 30Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Analytic response function derived for stationary flows is shown in (a) whereas (b) and (c) are respectively the Fourier- and POD empirical response functions for the current non-stationary flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Though equation (22) is derived based on the Fourier trans- form, Figures 8a and 8b show little correlation between H2 and H2 four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The inherent periodicity of Fourier modes justifies this result, since expansion of non-periodic signals will lead to spectral leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' We are studying decaying HIT, and as a consequence, the fluid- and particle velocity signals are not periodic and the Fourier modes do not form an appropriate basis for the expansion of these signals (Lumley (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Conversely, Figures 8a and 8c show a high correlation between H2 and H2 pod, as hypothesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Feasibly, the re- sult reflects some deeper spectral symmetry related to en- ergy optimality, from which equation (22) follows, rather than the equation strictly following from the properties of Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In Figure 9 the correlation is more clearly illustrated by the depiction of H2 pod (markers) and H2 fit (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here H2 fit is a least squares fit of H2 to H2 pod given by H2 fit(α) = 1 1 + ((α − 1)ω∗τp)2 , α ∈ [1 : M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (25) The product (α − 1)ω∗ represents a "POD-frequency", and the fitting parameter ω∗ = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0647 is found through minimiza- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Fit of analytic response function (solid) to empirical POD response function (markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' tion of the objective min ω∗ ∑ τp ∑ α ����� ����� H2 fit(α)− H2 pod(α) H2 fit(α) ����� ����� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (26) Summation over τp represents the fitting of ω∗ to the data of all Stokes numbers simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The figure shows a clear connection between the PPOD empirical response function and a modified version of the analytical model (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' These results show potential for the ability to approximate the modal energy of the carrier phase sampled at the particle position in decaying HIT, directly from the Stokes number and particle velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Response - relative velocity Csanady (1963) derived a relation between the mean square relative velocity and E f@p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Inspired by this, and extending the considerations of the previous section, we conjecture that Erel(α) = H2 rel(α)E f@p(α), α ∈ [1 : M], (27) where Erel(α) is the ensemble averaged energy of the relative velocity, urel = u f@p −vp, connected to the α’th POD mode (PPOD), and H2 rel(α) = ((α − 1)ω∗τp)2H2 fit(α), α ∈ [1 : M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' (28) Figure 10 depicts the empirically evaluated fraction Erel(α)/E f@p(α) (markers) and the fit H2 rel(α) (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The quantities are plotted against the log-scaled α to high- light the fit at the energetically dominant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Visual inspection shows a good model fit for α ≤ 30 and St0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='5, whereas the fit and the empirical results diverge at higher mode numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Noting that the modes ϕα, α ∈ [1 : 30], account for more than 99% of the total modal energy in each case, it can be argued that the fit is appropriate at the Spectral response between particle and fluid kinetic energy in decaying homogeneous isotropic turbulence 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' H2 rel (solid) plotted against the fraction Erel/E f@p (markers) shows a fairly good fit for Stokes numbers St0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='5 and α ≤ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' energetically dominant modes for St0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' For St0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content='0 the model fit at the energetically dominant modes is not on point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This suggests a range of Stokes numbers at which the model is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' However, in all cases, the trend of Erel/E f@p is similar to that of H2 rel, highlighting the similarities between the models derived through Fourier analysis of stationary flows and POD analysis of the current non-stationary flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Equations (21), (25), (27) and (28) provide a method for approximating E f@p and Erel in particle-laden decaying HIT based on particle velocity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The method re- quires knowledge of τp, and if this quantity changes signifi- cantly over the considered temporal domain, or if it has rapid fluctuations, the current accurateness of the model may deteri- orate since it was assumed that τp = τp(t0) for the generation of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' CONCLUSIONS A study of the temporal dynamics of two-way coupled particle-laden decaying HIT for various Stokes numbers was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Using time-focalized formulations of POD – TPOD and PPOD for respectively the decomposition of fluid- and particle velocities – sets of energy-optimal modes were extracted representing the temporal dynamics of the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' For both phases it was observed that the extracted modes resembled damped harmonic oscillators, where the lo- cal wavelength of each mode increased over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Moreover, the modes exhibited a high correlation in the dominating dy- namics between the carrier- and dispersed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The TPOD eigenspectrum of each simulation was inspected and compared to the eigenspectrum of a corresponding single- phase simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A distinct difference of shape between the single- and multiphase spectra was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' In addition, the TPOD spectra were compared to the Fourier turbulence en- ergy spectrum generated at the final time step of each simula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Here an increase of energy at high wavenumbers of the turbulence spectrum was observed to correlate with a relative increase in the TPOD eigenspectrum at high mode numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' It was demonstrated that the POD mode-set extracted from the velocity of one phase could span the velocity of both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Therefore, the Lagrangian spectrum based on PPOD modes could be computed for both the carrier- and dispersed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' A relation between these spectra was evaluated empir- ically giving rise to analytical expressions of response func- tions in a PPOD frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The response functions related the modal energy of the inertial particles to that of the surrounding fluid through simple expressions dependent on the Stokes number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' Notably, the expressions, fitting the data of the current non-stationary flow, resembled those derived through Fourier analysis of stationary flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' This suggested a deeper symmetry between POD and Fourier spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The current PPOD analysis was applied to an ideal test case of a non-stationary flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' The results outlined, and the theoret- ical applicablity of PPOD to any non-stationary flow, indicate that PPOD analysis may provide insightful dynamical infor- mation for the Lagrangian dynamics of alternative flows in future studies of particle-laden turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' ACKNOWLEDGMENTS AH and CMV acknowledge financial support from the Eu- ropean Research council: This project has received funding from the European Research Council (ERC) under the Euro- pean Unions Horizon 2020 research and innovation program (grant agreement No 803419).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' MS acknowledges financial support from the Poul Due Jensen Foundation: Financial support from the Poul Due Jensen Foundation (Grundfos Foundation) for this research is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' DECLARATION OF INTEREST The authors report no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT The data that forms the basis of this study is available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' REFERENCES Abdelsamie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFRT4oBgHgl3EQfrjff/content/2301.13621v1.pdf'} +page_content=' and Lee, C.' metadata={'source': 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--git a/59FIT4oBgHgl3EQf8Ss2/content/tmp_files/2301.11401v1.pdf.txt b/59FIT4oBgHgl3EQf8Ss2/content/tmp_files/2301.11401v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b38c8d5fc865c4e2fad8a4c3194572a06df37c4 --- /dev/null +++ b/59FIT4oBgHgl3EQf8Ss2/content/tmp_files/2301.11401v1.pdf.txt @@ -0,0 +1,2104 @@ +Causal Bandits without Graph Learning +Mikhail Konobeev 1 Jalal Etesami 2 3 Negar Kiyavash 1 2 +Abstract +We study the causal bandit problem when the +causal graph is unknown and develop an efficient +algorithm for finding the parent node of the re- +ward node using atomic interventions. We derive +the exact equation for the expected number of in- +terventions performed by the algorithm and show +that under certain graphical conditions it could +perform either logarithmically fast or, under more +general assumptions, slower but still sublinearly +in the number of variables. We formally show +that our algorithm is optimal as it meets the uni- +versal lower bound we establish for any algorithm +that performs atomic interventions. Finally, we +extend our algorithm to the case when the reward +node has multiple parents. Using this algorithm +together with a standard algorithm from bandit +literature leads to improved regret bounds. +1. Introduction +Multi-armed bandit (MAB) settings provide a rich theoret- +ical context for formalizing and analyzing sequential ex- +perimental design procedures. Each arm in a MAB setting +represents an experiment/action and the consequence of +pulling an arm is represented by a stochastic reward signal. +The objective of a learner in a MAB problem is to select a +sequence of arms over a time horizon in order to either find +an arm that results in the maximum reward or to maximize +the cumulative reward during this time horizon. Bandit prob- +lems have a growing list of applications in various domains +such as marketing (Huo & Fu, 2017; Sawant et al., 2018), +recommendation systems (Heckel et al., 2019; Silva et al., +2022), clinical trials (Liu et al., 2020), etc. An important +assumption in classical MAB is that the rewards for the +arms are independent. However, this assumption is often +violated in practice because of interdependencies among the +1School of Computer and Communication Sciences, EPFL, +Lausanne, Switzerland 2College of Management of Technology, +EPFL, Lausanne, Switzerland 3Department of Computer Science, +TUM, Munich, Germany. Correspondence to: Mikhail Konobeev +. +rewards of various arms. To capture such interdependencies, +different structural bandit settings have been proposed such +as linear bandits (Abbasi-Yadkori et al., 2011), contextual +bandits (Agrawal & Goyal, 2013; Lattimore & Szepesv´ari, +2020), and causal bandits (Lattimore et al., 2016; Lee & +Bareinboim, 2018) with the latter being the main focus of +this paper. +In causal bandit setting, the dependencies between the re- +wards of different actions is captured by a causal graph and +actions are modeled as interventions on variables of the +causal graph (Lattimore et al., 2016). Causal bandits can ef- +fectively model complex real-world problems. For instance, +marketing strategists can adaptively adjust their strategy +which can be modeled as interventions made in their adver- +tisement network to maximize revenue (Nair et al., 2021; +Zhang et al., 2022). +A major drawback of most existing work in causal bandit lit- +erature is the limiting assumption that the underlying causal +graph is given upfront (Lattimore et al., 2016), which is +frequently violated in most real-world applications. Similar +to Lu et al. (2021), we also study the causal bandit problem +when the underlying causal graph is unknown. However, +unlike Lu et al. (2021) our work does not assume the knowl- +edge of the essential graph of the causal graph. Our main +contributions are summarized as follows. +• We propose (Section 4) and analyze (Section 5) a RAn- +domized Parent Search algorithm (RAPS) which does +not assume the knowledge of the causal graph (or the +essential graph of the causal graph). In our analysis we +derive the exact equation for the expected number of +interventions performed by RAPS on any graph as well +as graphical conditions under which RAPS works in a +fast or slow, but still sublinear in the number of nodes, +regime. +• Based on RAPS we propose a method that improves +upon standard bandit algorithm using causal structure +of the arms and derive upper bounds for the regret of +this method (Section 6). +• We prove a universal lower bound on any algorithm +that attempts to discover the parent node of the reward +node with atomic interventions and show that RAPS +matches this bound exactly (Section 7). +arXiv:2301.11401v1 [stat.ML] 26 Jan 2023 + +Causal Bandits without Graph Learning +2. Related Work +In recent years, several work on Causal Bandit problem +(Lattimore et al., 2016; Sen et al., 2017; Lee & Bareinboim, +2018; Nair et al., 2021; De Kroon et al., 2022) have shown +that incorporation of causal structure improves upon the +performance of standard bandit MAB algorithms. However, +the aforementioned work relay on a limiting assumption +that the underlying causal graph is given. In this work, we +remove this assumption. +When the causal graph is unknown, a natural approach is to +first learn it through observations and interventions. Prob- +lem of learning a causal graph from a mix of observations +and interventions has been extensively studied in causal +structure learning literature (Hauser & B¨uhlmann, 2014; +Hu et al., 2014; Shanmugam et al., 2015). Yet learning the +entire underlying causal graph might not be necessary for +a learner in order to maximize its reward. Further, merely +learning the essential graph requires more than linear (in +terms of variables/nodes in the graph) number of conditional +independence tests (Mokhtarian et al., 2022). Instead, we +propose an algorithm that discovers the parents of the reward +node in sublinear number of atomic interventions. Discover- +ing the parents suffices as when the underlying causal graph +does not contain unobserved variables, it is known that the +best intervention is always over the set of parent nodes of +the reward node R (Lee & Bareinboim, 2018). +De Kroon et al. (2022) propose a causal bandit algorithm +which does not require any prior knowledge of the causal +structure and uses separating sets estimated in an online +fashion. Their theoretical result holds only when a true +separating set is known. The authors do not provide a final +bound on the regret. The closest work to our paper is that of +Lu et al. (2021), in which the authors derive regret bounds +for an algorithm based on central node interventions. How- +ever, they assume the essential graph is known to the learner +while our algorithm makes no such assumption. +3. Preliminaries +A Probabilistic Causal Model (PCM) (Pearl, 2009) is a +Directed Acyclic Graph (DAG) G = (V, E) over a set of +random variables V with edges E and a distribution P over +the variables in V that factorizes with respect to G in the +sense that the distribution over V could be written as a prod- +uct of conditional distributions of each variable given its +parents. We denote the number of vertices in V by n and +assume that each variable X ∈ V takes value from a finite +set [K] := {1, . . . , K}. The set of ancestors and descen- +dants of a node X in G are denoted by AG(X) and DG(X), +respectively. In both cases, we might omit writing G when +it is clear from the context. In our definition a node is its +own ancestor and descendant and we will use horizontal bar +to exclude it, for example, for ancestors we will write ¯ +A(X) +for A(X) \ {X}. For a given subset S ⊆ V, we define +AG(S) := ∪X∈SAG(X) and DG(S) := ∪X∈SDG(X). +The vertex-induced subgraph over nodes in S is denoted +by GS. To simplify the notation, we use AS(X) (sim- +ilarly, DS(X)) for the set of ancestors (respectively, de- +scendants) of X in the induced subgraph GS. In addition, +we will use superscript c to denote the non-ancestors/non- +descendants, for example, Ac +S(X) = S\AS(X). A collider +on a path X1, . . . , Xℓ between two nodes X1, Xℓ ∈ V is +a node Xj with 1 < j < ℓ such that Xj is a children of +both Xj−1 and Xj+1, i.e., Xj−1 → Xj ← Xj+1. For +two sets A, B, we denote their symmetric difference by +A△B := (A ∪ B) \ (A ∩ B) and assume that all binary set +operations have the same precedence. +3.1. Problem Setting +In a causal bandit (Lattimore et al., 2016), a learner L per- +forms a set of interventions, i.e. actions, at each round t ∈ +[T] by setting a subset of variables Xt = (X1, . . . , Xℓ) ⊆ +V to some values xt ∈ [K]ℓ, denoted by do(Xt = xt). +Playing the empty arm denoted by do() corresponds to ob- +serving a sample from the distribution P underlying the +PCM. The goal of the learner is to maximize a designated +reward variable R. When there is only one parent node of +the reward node in the graph G the causal bandit corresponds +to standard stochastic K-armed bandit. In what follows, we +assume that the reward node lies outside of the set of vari- +ables V, and thus we implicitly work with a subgraph over +the nodes V \ {R}. We start by assuming that P ∈ V is +the only parent of R and generalize our results to multiple +parent nodes in V in Section 8. We also allow for the reward +node to have no parents in V which we denote by writing +P = ∅. The case when P = ∅ corresponds to having an +empty set of variables and thus we have A(∅) = D(∅) = ∅. +The learner does not know the underlying DAG over the +variables in V and cannot intervene directly on the reward +variable R. +Performance of a learner L can be measured in terms of +cumulative regret which takes into account the rewards re- +ceived from all the interactions performed, +RT +L(G, P) := T max +X⊆V +max +x∈[K]|X| E[R|do(X = x)] +− +T +� +t=1 +E[R|do(Xt = xt)], +or simple regret which only focuses on the reward of the +final intervention, predicted to be the best by the learner +after T interactions, +rT +L(G, P) := max +X⊆V +max +X∈[K]|X| E[R|do(X = x)] +− E[R|do(XT +1 = xT +1)], + +Causal Bandits without Graph Learning +where, do(XT +1 = xT +1) is the intervention estimated to +be the best by the learner L after performing T interactions +and |X| denotes the number of variables in X. +Remark. +Note that in both definitions of regret, the +learner is compared against an oracle that always selects the +best intervention. When the underlying DAG G does not +contain any unobserved variables, it is known that the best +intervention is always over the set of parent nodes of the re- +ward node R (Lee & Bareinboim, 2018). Thus, in this work, +we focus on a learner L that performs interventions to detect +the set of parent nodes of the reward node and then finds +the best assignment to P in order to minimize regret. In +Section 4 we present our algorithm for the case when there +is at most one parent node P in the graph G. This algorithm +uses only atomic interventions, i.e. interventions of size one, +and is analyzed in Section 5. In Section 8, we generalize this +algorithm to the case of multiple parent nodes. Our results +in Sections 5, 7 and 8 can bound both simple and cumulative +regret, for conciseness, we present only a cumulative regret +bound in the main text in section Section 6 and extend it to +a simple regret bound in Appendix E.1. +4. Randomized Parent Search Algorithm +In this section we present our learner, i.e., RAndomized +Parent Search algorithm (RAPS) shown in Algorithm 1 that +finds the parent node of the reward node or reports that +the reward node does not have a parent node in V. This +algorithm defines a recursive function REC with single ar- +gument denoted by C — the so called candidate set of nodes +in G which might contain P — and this function is called +initially with all the nodes in the graph as its argument. +After the parent node is discovered, one could use standard +algorithms from bandit literature (see, for example, Latti- +more & Szepesv´ari, 2020) to find the best intervention over +it to minimize the simple or cumulative regret. As an ex- +ample we present a bound on cumulative regret using the +combination of RAPS and a standard bandit algorithm such +as UCB (Capp´e et al., 2013) in Section 6. +We will explain RAPS first with an example. Consider the +graph with four nodes in Figure 1 where P is the parent of +reward node R. The algorithm starts by calling the recursive +function with C = V = {X1, X2, X3, P}. Assume that dur- +ing this call the recursive function samples X3. Changing +the value of this node should allow the learner to determine +the descendants which are in this case {X2, X3} and do not +include P. The learner realizes this because R does not +change unless P changes. Thus, there will be another call of +the recursive function with C = V \ {X2, X3} = {X1, P} +on Line 12. After that, if in the recursion the node X1 is sam- +pled, the same function is called on Line 8 with C = {P}. +This is because P is the only descendant of X1 not including +X1 in the graph over the nodes in {X1, P}. Lastly, the algo- +Algorithm 1 RAndomized Parent Search algorithm (RAPS) +Require: Set of nodes V of G given as input +Output: The parent node P ∈ V of the reward node or ∅ +if there is no parent node in V +1: RAPS works by calling REC(C = V) defined as follows +2: function REC(C) +3: +if C = ∅ then +4: +return ∅ +5: +X ∼ Unif(C) +6: +Intervene on X to determine if P ∈ DC(X) +7: +if P ∈ DC(X) then +8: +ˆP ← REC(DC(X) \ {X}) +9: +if ˆP = ∅ then +10: +return X +11: +return ˆP +12: +return REC(C \ DC(X)) +X1 +P +X2 +X3 +Figure 1: An example of DAG with a single parent node P. +rithm will have to sample P and return it as the discovered +parent node. +In general, RAPS intervenes on a randomly selected node +X ∈ C on Line 6. Several interventions on X should be +sufficient to determine the descendants of X and whether +P ∈ DC(X). This is because changing the value of X +should change the values of the descendants of X and we +can determine if P ∈ DC(X) by checking if the value of +the reward variable R changes. Notice that if Y ̸∈ C, then +none of the descendants of Y are in C which means that +it is possible to find DC(X) simply by taking DG(X) ∩ +C. For simplicity, the analysis in Section 5 assumes that +the descendants of an arbitrary node X and whether P is +among them could be inferred from asymptotically constant +number of interventions on the node X. At the same time, +in Section 6.1, we discuss how this information could be +obtained with high probability from observing samples of all +the variables in the graph with and without an intervention +on X. In order to bound the regret, we need to analyze +the number of interventions performed by a learner L on +a given graph G to find the parent node P. We denote +this quantity by NL(G, P) and unless stated otherwise, we +assume that the learner uses the RAPS algorithm to find P. +In what follows, we first present the exact expression to +compute the expected number of interventions performed +by RAPS in general setting. Next, we introduce classes of +DAGs for which this expected value is either asymptotically +logarithmic or sublinear in the number of nodes n. + +Causal Bandits without Graph Learning +5. Analysis of RAPS +We start by stating the exact expression for the expected +number of interventions performed by Algorithm 1. The +proof is in Appendix A. +5.1. Expected Number of Interventions +Theorem 5.1. The expected number of interventions per- +formed by a learner that uses Algorithm 1 to determine the +parent node P is given by +E[N(G, P)] = +� +X∈V +1 +|A(P)△A(X) \ {X}| + 1. +(1) +Next, we present two conditions under which RAPS per- +forms sublinearly. +In the “fast” regime, it requires +O(log(n)) expected number of interventions, while in the +“slow” regime, it requires O +� +n +logd(n) +� +expected number of +interventions with d being the maximum degree in the skele- +ton of G. It is noteworthy that our algorithm even in the +slow regime outperforms the na¨ıve exploration method that +requires Ω(n) interventions. In Section 7, we introduce a +universal lower bound on the expected number of interven- +tions required by any learner to find the parent node and +show that Equation (1) matches this lower bound. +5.2. Fast Regime +In order to introduce the condition under which RAPS per- +forms fast, we first characterize the candidate sets C ⊆ V, +that is the sets of nodes that the recursive function Algo- +rithm 1 could be called with as an argument. To this end, +we define the following family of subsets of V. +Definition 5.2. A candidate family of a graph G with a +parent node P is a family of subsets given by +CG(P) := +� +Dc(W) +�� W ⊆ Ac(P) +� +∪ +� +Dc +D(X)(W) \ {X} +��X ∈ A(P), W ⊆ Ac +D(X)(P) +� +. +Let W be an arbitrary set of non-ancestors of P. All de- +scendants of these non-ancestors could be removed from +the starting candidate set C = V. This corresponds to the +first line in the definition of CG(P). At the same time, the +algorithm might also reduce the set of candidate nodes if it +discovers an ancestor of the parent node P. This happens +when the recursive function is called on Line 8. Let X be an +intervened on ancestor of P, then the candidate set reduces +to the subset of descendants of X. This set might again +exclude arbitrary non-ancestors of P previously denoted by +W, but this time in the subgraph over D(X). We provide +an example of the candidate family for the line graph in +Figure 2 below. Next lemma shows that when the recursive +P +X1 +X2 +. . . +Xn−1 +Figure 2: An example of a line graph, such graphs satisfy +the condition of Theorem 5.4. +function in Algorithm 1 is called, its argument belongs to +the candidate family CG(P). The proof is in Appendix B. +Lemma 5.3. All possible arguments C with which the recur- +sive function in Algorithm 1 is called are contained within +the candidate family CG(P). +At a high level, Algorithm 1 performs O(log n) interven- +tions if for each C ∈ CG(P) of large size, the number of +ancestors of P in GC is large, or the number of non-ancestors +of P each of which has large number of descendants is large. +The latter condition could be interpreted as the condition +that the non-descendants of P asymptotically form a line +graph. This is captured formally by the following result, +proved in Appendix B. +Theorem 5.4. For a constant 0 < α < 1 and C ∈ C, let +the set of “heavy” non-ancestors to be +HC(α) := +� +X ∈ Ac +C(P) +�� |DC(X)| ≥ α |C| +� +. +(2) +Assume that G is such that for any C ∈ CG(P) at least one +of the following holds +• |HC(α)| ≥ β |C|, +• |AC(P)| ≥ γ |C|, +• |C| ≤ c logk(n), +for fixed 0 < α, β, γ < 1, c ∈ R>0, and k ≥ 1. Then, +E[N(G, P)] = O(logk n). +The assumption of Theorem 5.4 states that for all candidate +sets considered by the recursive function in Algorithm 1 +either the cardinality of C is upper bounded by c logk(n) or +in the subgraphs GC one of the following holds: +1. there is a β-fraction of nodes that are among the non- +ancestors of P and have at least an α-fraction of nodes +as their descendants, +2. there is a γ-fraction of nodes that are ancestors of P. +Notice that in the condition of Theorem 5.4 there is no re- +striction on the structure of the ancestors of P. Thus, if the +number of ancestors of P is sufficient we have that RAPS +enjoys logarithmic bound on the expected number of in- +terventions. At the same time, when there are more than +constant number of non-ancestors, Theorem 5.4 requires +them to have a certain structure. Consider, for example, the +line graph in Figure 2. In this case the candidate family con- +sists of the sets {X1, . . . , Xi−1} and {P, X1, . . . , Xi−1} +for i ∈ [n − 1]. The condition of Theorem 5.4 still holds +since for every candidate set C ∈ CG(P) it holds that there + +Causal Bandits without Graph Learning +X1 +X2 +. . . +Xn/2 +P +Xn/2+1 +Xn/2+2 +. . . +Xn−1 +Figure 3: An example of a graph where P has n children, but +only half of them form a line graph (X1 → · · · → Xn/2). +O(logk n) expected number of interventions still suffices +because the other half of the nodes are all children of Xn/2. +is a β-fraction of “heavy” non-ancestors of P in GC. Such +non-ancestors contain many other non-ancestors of P as +their descendants. To see this, let C = {X1, . . . , Xi−1} for +some i ∈ [n − 1] (the other case is similar) and consider the +set +� +X1, . . . , X⌊i/2⌋ +� +. Each node in this set has at least i/2 +descendants and there are at least i/2 − 1 such nodes. Thus, +the condition holds for α, β close to 1 +2. Note also that even +if P is not the first node in a topological ordering of a graph +but its’ non-descendants still satisfy the condition of The- +orem 5.4, then the RAPS succeeds in O(logk n) expected +number of interventions. +At the same time, the condition of Theorem 5.4 for non- +descendants in subgraphs over candidate sets of large size +is more general than just requiring all non-descendants to +form a line graph. First, notice that if the parent node has +Ω(n) children, each with only one parent, then the algorithm +requires Ω(n) interventions no matter how the remaining +nodes are arranged. This case is similar to the case of d-ary +trees to be discussed in Section 7 where it is shown that such +trees require at least Ω +� +n +logd n +� +interventions. However, if, +for example, half of the nodes form a line and the other +half are all children of the last node on the line as shown in +Figure 3, then the condition of Theorem 5.4 is still satisfied +and RAPS remains in the fast regime in terms of the number +of interventions. +5.2.1. ERD ˝OS-R´ENYI RANDOM GRAPHS +In addition to providing instances for which the condition +of Theorem 5.4 holds, we show that Erd˝os-R´enyi random +DAGs with large enough edge probability p also satisfy this +condition. While originally, Erd˝os-R´enyi model was pro- +posed for undirected graphs, it naturally extends to DAGs +as well (Hu et al., 2014). For this, label the nodes in V from +1 to n. An Erd˝os-R´enyi random DAG Gn,p is generated as +follows. First, select a permutation π over [n] uniformly at +random. Next, for two nodes i, j ∈ [n] such that i < j draw +an edge {i, j} with probability p. Finally, if an edge {i, j} +is picked, orient it as i → j if π(i) < π(j) and i ← j oth- +erwise. For such randomly generated DAG, the following +result, proved in Appendix C, holds. +Corollary 5.5. The family of Erd˝os-R´enyi random DAGs +satisfies the condition of Theorem 5.4 in expectation if +p ≥ 1 − +� +1−c +logk n−1 +�1/(logk n−1) +, for any constantc ∈ [0, 1]. +Therefore, for such graphs, RAPS requires E[N(G, P)] = +O(logk n) expected number of interventions. +As shown in Remark C.2 in Appendix C, the result of Corol- +lary 5.5 holds for p ≥ log(logk(n)−1)−log(1−c) +logk(n)−1 +and asymp- +totically the lower bound for p in Corollary 5.5 behaves as +Θ +� +k log log(n) +logk(n) +� +. +5.3. Slow Regime +In this section, we provide a bound on the expected number +of interventions of RAPS under a more relaxed assumption +than the condition in Theorem 5.4. The following theorem +states that if there are at most O +� +n +logd(n) +� +nodes X ∈ +V \ {P} such that all paths between X and P are inactive, +then the expected number of interventions required by RAPS +is bounded by O +� +n +logd(n) +� +, where d is the maximum degree +in the skeleton of G. Under the faithfulness assumption +(Pearl, 2009), the aforementioned condition means that there +are at most O +� +n +logd(n) +� +nodes in V which are independent +with the parent node P. +Theorem 5.6. Let G be an arbitrary DAG in which there +are at most O +� +n +logd(n) +� +nodes X ∈ V \ {P} such that +either P is disconnected with X, or all paths between P +and X are blocked by colliders. Then, +E[N(G, P)] = O +� +n +logd n +� +, +where d is the maximum degree in the skeleton of G. +The proof of Theorem 5.6 is in Appendix D. +6. Regret Analysis +6.1. Determining if X is an Ancestor of P and Finding +the Descendants of X +On Line 6 of Algorithm 1 RAPS needs to determine whether +an arbitrary node X is an ancestor of P in GC or not. To +do so, it must intervene on X setting it to all K possible +values that it can take. Let B be the number of times that +the same intervention setting do(X = x) for some x ∈ +[K] is performed. The average value of reward ¯Rdo(X=x) +under this intervention is compared to the average value ¯R +obtained from the samples of the observational distribution. +We use a similar argument to Lu et al. (2021) to obtain +bounds on the required number of samples B from the + +Causal Bandits without Graph Learning +interventional and observational distributions to determine +whether X ∈ AC(P) and estimate all Y ∈ DC(X) with +high probability. +Assume that the reward variable minus its expectation, both +under observational distribution or any intervention, is 1- +subgaussian. We need the following additional assumptions. +Assumption +6.1 +(Ancestoral +Effect +Identifiabil- +ity). There exists an ε +> +0, +such that for any +two variables X, Y +∈ +V with X +∈ +A(Y ) in G, +|P {Y = y|do(X = x)} − P {Y = y}| +> +ε for some +x, y ∈ [K]. +Assumption 6.2 (Reward Identifiability). We assume that +for all X +∈ A(P), there exists x ∈ [K] such that +|E[R] − E[R|do(X = x)]| > ∆, for some universal con- +stant ∆ > 0. +The second assumption is the same as in Lu et al. (2021) +and allows us to determine if an arbitrary node X is an +ancestor of P. More specifically, this is done by comparing +�� ¯R − ¯Rdo(X=x)�� for all x ∈ [K] with ∆/2 and conclud- +ing that X is an ancestor of P in G (and therefore in GC +where C is an argument passed to the recursive function in +Algorithm 1) if for some x ∈ [K] the absolute difference +exceeds the threshold. The first assumption allows to de- +termine the descendants of X after an intervention on it. +More specifically, we consider as the descendants the set of +nodes Y ∈ C such that for some x, y ∈ [K] the absolute dif- +ference +��� ˆP(Y = y) − ˆP(Y = y|do(X = x) +��� exceeds ε/2, +where ˆP(·), ˆP(·|do(X = x)) are the empirical distributions +over Y without any intervention and under intervention +do(X = x). The following lemma proven in Appendix E +provides the condition for the number of samples B such +the criteria described above will determine for every node if +it is an ancestor of P and correctly obtain all its descendants +with high probability. +Lemma 6.3. Let +AX = +� +∃x ∈ [K] : +��� ¯R − ¯Rdo(X=x)��� > ∆/2 +� +, +DX,Y = +� +∃x, y ∈ [K] : +��� ˆP(Y = y) − ˆP(Y = y|do(X = x)) +��� > ε/2 +� +for any X, Y ∈ V and Ac +X, Dc +X,Y be their compliments. +Define E as the event that for every node we correctly de- +termine its descendants and whether it is an ancestor of P +using the criteria described above, i.e., +E = +� +X∈A(P ) +AX ∩ +� +X∈Ac(P ) +Ac +X +∩ +� +X∈V +� +� +Y ∈ ¯ +D(X) +DX,Y ∩ +� +Y ∈Dc(X) +Dc +X,Y +� +Then it holds that P {E} +≥ +1 − δ +if B += +max +� +32 +∆2 log +� 8nK +δ +� +, 8 +ε2 log +� +8n2K2 +δ +�� +. +6.2. Regret Bounds +Our regret bound is given for conditional regret defined as +follows: +RT +L(G, P | E) := T max +X⊆V +max +x⊆[K]|X| E[R|do(X = x)] +− +T +� +t=1 +E[R|do(Xt = xt), E], +where E is the event defined in Lemma 6.3. A straightfor- +ward corollary of Theorem 5.1, Theorem 5.4, Theorem 5.6 +and Lemma 6.3 is the following result proved in Appendix E. +Corollary 6.4. Assume that P ̸= ∅, i.e., the reward vari- +able has a parent in V. A learner that uses Algorithm 1 +and then runs a standard bandit algorithm such as UCB +achieves the following bound for the conditional regret: +RT +L(G, P | E) = += O +� +K max +� 1 +∆2 , 1 +ε2 +� +log +�nK +δ +� +E[N(G, P)] ++ +� +KT log T +� +. +The bound above improves on the Ω +�√ +nKT +� +lower +bound that applies to any multi-armed bandit algorithm that +does not use the causal structure of the arms. For example, +in the fast regime, setting δ = min +� +1, +� +K log(T)/T +� +and ignoring the quantities not relating to n, K, T, our +algorithm achieves RT +L(G, P) ≤ RT +L(G, P|E) + O(δT) = +O +� +K log(n) log +� +nK ∨ n +� +KT/ log T +� ++ √KT log T +� +1 +since with probability at most δ the event E from Lemma 6.3 +does not hold and in this case we can only bound the regret +asymptotically by T. We provide similar bounds on simple +regret in Appendix E.1. +7. Universal Lower Bound +In this section we show that the result of Section 5.1 is tight +in the sense that any algorithm that finds the parent node P +(or determines it does not exist in the graph G) requires at +least the number of interventions performed by RAPS. +Theorem 7.1. Fix a causal graph G and a parent node P. +Any learner L that correctly identifies the parent node P, +1a ∨ b stands for max {a, b}. + +Causal Bandits without Graph Learning +P +X1 +X2 +X3 +. . . +Xn−1 +Figure 4: An example of a graph with a skeleton that is a +line graph and n/2 − 1 colliders (n is assumed to be even). +Our lower bound in Theorem 7.1 implies that any learner +requires Ω(n) atomic interventions to discover P in this +graph. +for any graph obtained from G by relabeling of the nodes +and having P take one of n vertices or P = ∅, satisfies +E[NL(G, P)] ≥ +� +X∈V +1 +|AG(P)△AG(X) \ {X}| + 1. +The proof is in Appendix F. Consider, for example, a null +graph G = (V, ∅). The number of ancestors of every vertex +is equal to one and the expression in Theorem 7.1 becomes +Ω(n). At the same time, even if the graph is connected and +its skeleton is a line graph it is possible to have a lower +bound of Ω(n) by having all vertices separated from P by +colliders as in Figure 4. In this figure, every node Xi where +1 ≤ i < n − 1 is odd is a collider on the path between +Xi−1 and Xi+1 and assume that X0 ≡ P. The number +of ancestors of every node Xj, where 1 < j < n − 1 is +even, equals one leading to Ω(n) lower bound. If a graph +is connected and has no colliders, Theorem 5.6 results in +O +� +n +logd(n) +� +upper bound on the number of interventions. +The upper bound in Theorem 5.6 is tight for perfect d-ary +trees. In such trees the number of non-common ancestors +between P (possibly P = ∅) and any node X is lower +bounded by the distance from X to the root assuming that +X comes from one of the subtrees, other than the subtree +containing P. Thus, considering only the last term in the +summation results in +E[NL(G, P)] ≥ +logd(n+1) +� +h=1 +(d − 1)dh−1 +h + 1 +≥ +(n + 1)(d − 1) +d(logd(n + 1) + 1), +which matches the asymptotic upper bound of Theorem 5.6. +By adding an extra knowledge about the essential graph, the +algorithm in Greenewald et al. (2019) can detect the parent +node with at most O(log n) number of atomic interventions. +8. Generalization to Multiple Parent Nodes +Let P be the set of all parent nodes of the reward node. We +generalize Algorithm 1 to an algorithm that finds all the +parent nodes of the reward node by repeatedly discovering +each of the parent nodes in Algorithm 2. +Algorithm 2 Generalization of Algorithm 1 to Graphs with +Multiple Parent Nodes of the Reward Node +1: ˆP ← ∅, S ← V +2: while True do +3: +ˆP ← the result of running Algorithm 1 providing it +with S and ˆP +▷ See remark +4: +if ˆP = ∅ then +5: +break +6: +ˆP ← ˆP ∪ +� +ˆP +� +7: +S ← S \ D( ˆP) +return ˆP +P1 +P2 +R +Figure 5: An example DAG with multiple parent nodes. +Remark. +On Line 3 Algorithm 2 calls Algorithm 1 to find +a next parent node P with the starting candidate set being +equal to S. While previously Algorithm 1 could use only +atomic interventions to determine if an arbitrary X ∈ V is +an ancestor of P, in this case this algorithm will intervene on +ˆP ∪ {X} to find if there exists a realization that changes R +by changing the value of X while keeping the other values +in the intervention set constant. This would imply that X +is an ancestor of some parent node in P. Algorithm 2 uses +the observation that if P, P ′ ∈ P and P ∈ ¯ +A(P ′), then P ′ +will be discovered by Algorithm 1, but not P. This happens +because even if P is intervened on, the algorithm would +have to exclude the descendants of P before returning P +as the parent of the reward node. Afterwards, by recalling +Algorithm 1 and providing it with ˆP that contains P ′, it will +be able to discover P as another parent node. +As an example, consider the graph in Figure 5. During the +first call to Algorithm 1 the node P2 will be discovered. In +the second call to Algorithm 1 with ˆP = {P2}, there needs +to be an intervention on P2 in order to cut the causal link +from P1 to P2 to determine whether P1 is a parent of the +reward node R. +The following theorem proved in Appendix G generalizes +the conditions of Theorems 5.4 and 5.6 such that the Al- +gorithm 2 discovers each parent node with the number of +interventions as in Theorems 5.4 and 5.6, respectively. This +is done by considering all graphs from which the descen- +dants of some subsequence of parent nodes were removed. +The nodes in the subsequence are selected as the last nodes +in some topological ordering of the nodes in P. +Theorem 8.1. Let τ(P) be the set of all topological order- +ings of the parent nodes P. Assume that the condition of + +Causal Bandits without Graph Learning +10 +5 +10 +1 +p +101 +102 +103 +No. interventions +Eq. 1 +experiment +(a) +0 +500 +1000 +n +101 +103 +No. interventions +linear +n/log2(n) +log2(n) +experiment +(b) +0 +500 +1000 +n +101 +103 +No. interventions +linear +n/log2(n) +log2(n) +experiment +(c) +0 +500 +1000 +n +100 +200 +No. interventions +(d) +RAPS, | | = 20 +RAPS, | | = 10 +RAPS, | | = 5 +21log2(n) +11log2(n) +6log2(n) +Figure 6: (a) Comparison between Equation (1) of from Theorem 5.1 and the experimental number of interventions of +Algorithm 1 on Erd˝os-R´enyi random DAGs. (b-c) The results of running RAPS on Erd˝os-R´enyi random DAGs with large +and small p. (d) Results of running Algorithm 2 to discover multiple parent nodes. +Theorem 5.4 holds for all graph-parent-node pairs in +� +(GV\D(P), ∅) +� +∪ +� +(GV\S(τ,i), τi)|τ ∈ τ(P), i ∈ [|P|], +S(τ, i) = +� +P ∈τ[i+1:] +D(P), |V \ S(τ, i)| > c logk(n) +� +, +where τ[i + 1 :] consists of the last |P| − i elements of +τ, τi is the i-th element of τ and c > 0 is some constant. +Then the expected number of interventions required by Algo- +rithm 2 to find all parent nodes is O +� +|P| logk n +� +. Similarly, +assume that all graphs of size at least +cn +logd(n) and graph- +parent-node pairs in the set above satisfy the condition of +Theorem 5.6. Then the expected number of interventions +required by Algorithm 2 is O +� +|P|n +logd n +� +. +Theorem 8.1 gives general conditions for the upper bounds +on the number of interventions. Below, we combine our +result for Erd˝os-R´enyi graphs from Section 5.2.1 with the +result of Theorem 8.1 to arrive at a condition on the prob- +ability p such that Algorithm 2 discovers all parents of the +reward node. The proof is in Appendix G. +Corollary 8.2. Let Gn,p be an Erd˝os-R´enyi graph with +p ≥ 1 − +� +1−c0 +logk(c1 logk(n))−1 +�1/(logk(c1 logk(n))−1) +for some +constants c0 ∈ [0, 1] and c1 ∈ R>0, then to discover P, Al- +gorithm 2 needs E[N(G, P)] = O +� +|P| logk(n) +� +expected +number of interventions. +9. Experiments +In this section we discuss our experimental results aimed at +testing our theoretical findings. In all figures we obtain the +average experimentally required number of interventions to +discover the parent node and the standard deviation over 20 +independent runs of RAPS. +Firstly, we confirm that the Equation (1) could be used to +compute the expected number of interventions required to +discover the parent node in Figure 6a. In this experiment +we generate Erd˝os-R´enyi random DAG as described in Sec- +tion 5.2.1 with different values of p and for each such DAG +compute the expected number of interventions as predicted +in Equation (1), as well as perform 20 independent runs on +the same graph of Algorithm 1. The average over those 20 +runs and the standard deviation are shown as the line and the +shaded area, respectively, similar to the other figures. For +this experiment we set the number of nodes in all graphs +n = 1000. Notice also that Equation (1) matches the lower +bound in Theorem 7.1, thus in Figure 6a we show that the +performance of our algorithm matches the performance of +the best possible algorithm. +Secondly, in Figure 6b we confirm the result of Corollary 5.5 +by showing that when p = 1 − +� +0.5 +log2(n)−1 +�1/(log2(n)−1) +in +Erd˝os-R´enyi random DAG obtained as discussed in Sec- +tion 5.2.1, then the number of interventions required scales +as O(log n). At the same time, in Figure 6c we show that +when p = ln n +n , then the number of interventions scales as +n +log n. +Lastly, we verify the result of Corollary 8.2 in Figure 6d. +On this figure we see that in Erd˝os-R´enyi graphs with p as +in the lower bound of Corollary 8.2 (with c0 = 0.5, c1 = 1 +and k = 1) the number of interventions required to discover +|P| parents grows as (|P| + 1) log(n). +10. Conclusion +We proposed a causal bandit algorithm that does not require +the knowledge of the graph structure of the causal graph in- +cluding the knowledge of the essential graph. Our algorithm +matches the universal lower bound that holds for all algo- +rithms attempting to discover the parent node using atomic +interventions and is generalized to multiple parent nodes of +the reward node. + +Causal Bandits without Graph Learning +References +Abbasi-Yadkori, Y., P´al, D., and Szepesv´ari, C. Improved +algorithms for linear stochastic bandits. 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In 18th Annual Symposium on +Foundations of Computer Science (sfcs 1977), pp. 222– +227. IEEE Computer Society, 1977. +Zhang, J., Chen, Y., and Singh, A. Causal bandits: Online +decision-making in endogenous settings. arXiv preprint +arXiv:2211.08649, 2022. + +Causal Bandits without Graph Learning +Algorithm 3 An algorithm equivalent to Algorithm 1. +Require: Set of nodes V of G given as input +Output: The parent node P ∈ V of the reward node or ∅ if there is no parent node in V +1: Sample a random permutation τ of nodes in V +2: ˆP ← ∅, i ← 0 +3: for X ∈ τ do +4: +i ← i + 1 +5: +if A(P)△A(X) ∩ (τ1, . . . , τi−1) ̸= ∅ then +6: +continue +7: +Intervene on X to determine if P ∈ D(X) +8: +if P ∈ D(X) then +9: +ˆP ← X +10: return ˆP +A. Exact Number of Interventions +Theorem 5.1. The expected number of interventions performed by a learner that uses Algorithm 1 to determine the parent +node P is given by +E[N(G, P)] = +� +X∈V +1 +|A(P)△A(X) \ {X}| + 1. +(1) +Proof. First we show that Algorithm 1 is equivalent to Algorithm 3 in the sense that the same sequences of nodes are +intervened on by both algorithms with the same probability. Although the Algorithm 3 is not practical because it uses the +graph structure on Line 5, it allows us to present a proof for this theorem. This algorithm first samples a permutation τ +of nodes in V and then intervenes on a node X ∈ τ only if none of the nodes in A(P)△A(X) appeared before X in the +permutation. As an example, suppose that Algorithm 3 selects permutation (X3, X2, X1, P) in Figure 1. Then the nodes +intervened on by Algorithm 3 are the same as the nodes intervened on by Algorithm 1 in the example in Section 4. This is +because X2 is a descendant of X3 (and X3 is not an ancestor of P) and therefore it will not be intervened on. Moreover, if +Algorithm 3 selects permutations (X3, X1, X2, P) and (X3, X1, P, X2), the resulting intervention sequences would be the +same run of Algorithm 3. +By induction on the intervened on nodes, the base case is clear since in both Algorithm 1 and Algorithm 3 the first node +is sampled with probability 1/n and always intervened on. Let W = (W1, . . . , Wl) be a uniformly random permutation +of any l elements of V and W′ be a subsequence of W with an element of W ∈ W included in W′ if no element +of A(P)△A(W) \ {W} was included before it. For the example in Figure 1 and sequence W = (X3, X2) we have +W′ = (X3) since, as mentioned before, X3 ∈ A(P)△A(X2) \ {X2}. Note that in Algorithm 1 a node could be intervened +on only when it was not intervened on before and none of the non-common ancestors of that node and the parent node were +intervened on. Thus, let S = +� +X ∈ V : A(P)△A(X) ∪ {X} ∩ W′ = ∅ +� +be the set of nodes that could be intervened on +by Algorithm 1 in the next round. The probability that a node X ∈ S is sampled by Algorithm 1 given the sequence W′ is +1/s, where s = |S|. On the other hand, the probability that a node X ∈ S is intervened on next by Algorithm 3 is +P {X is sampled before A(P)△A(X) \ {X}|W} +(3) += +n−l−s +� +k=0 +P {Wl+1:l+k ∩ (A(P)△A(X) \ {X}) = ∅ and Wl+1+k = X|W} +(4) += +n−l−s +� +k=0 +�n−l−s +k +� +k!(n − l − k − 1)! +(n − l)! +(5) += +n−l−s +� +k=0 +(n − l − s)!(n − l − k − 1)! +(n − l − s − k)!(n − l)! +(6) += 1 +s +n−l−s +� +k=0 +�n−l−k−1 +s−1 +� +�n−l +s +� += 1 +s, +(7) + +Causal Bandits without Graph Learning +where Wl+1:l+k consists of k elements sampled uniformly at random after sampling W, we used the fact that there are +n − l − s “good“ elements from which we need to sample k elements before sampling Wl+1+k = X while the remaining +elements could come in any order, and to get the last equality we used the hockey-stick identity (Jones, 1994). The +hockey-stick identity states that for any n, r ∈ N such that n ≥ r it holds that �n +i=r +�i +r +� += +�n+1 +r+1 +� +. By the chain rule of +probability, the intermediate result holds. +Next, with W = (W1, . . . , Wn) being a uniformly random permutation corresponding to a run of Algorithm 3, and defining +AX = {Algorithm 3 intervenes on X}, W0, and k ≥ 1. Then, E[N(G, P)] = O(logk n). +Proof. Based on the definition of Algorithm 1, it is straightforward to see that the following recursion holds +T(C) = 1 +|C| +� +X∈AC(P ) +T( ¯DC(X)) + 1 +|C| +� +X∈Ac +C(P ) +T(Dc +C(X)) + 1, +(21) +where T(C) denotes the number of interventions performed by the recursive function in Algorithm 1 given candidate +set C. We will show that T(C) ≤ c′ log |C| + c logk(n) for some c′ > 0 by considering two cases: |C| ≤ c log(n) and +|C| > c log(n). The case where |C| ≤ c log(n) is straightforward since each node in C is intervened on at most once and +|C| ≤ c logk(n). +Next, consider the case |C| > c logk(n). We provide a proof for the case when |HC(α)| ≥ β |C| then comment on why +the result holds when |AC(P)| ≥ γ |C|. By Lemma 5.3 we have that for all X ∈ AC(P) it holds that ¯DC(X) ∈ C and for +all X ∈ Ac +C(P) it holds that Dc +C(X) ∈ C, thus the condition of the theorem holds for the recursive calls and we can use +induction hypothesis after which it is left to check that +c′ +|C| +� +X:X∈AC(P )∧|DC(X)|>1 +log(|DC(X)| − 1) + c′ +|C| +� +X∈Ac +C(P ) +log(|C| − |DC(X)|) + 1 +(22) ++ c +|C| +� +X∈AC(P ) +logk(n) + c +|C| +� +X∈Ac +C(P ) +logk(n) +(23) +is bounded above by c′ log |C| + c logk(n). First, note that the Equation (23) is bounded by c logk(n) since AC(X) ∪ +Ac +C(X) = C. Next, consider the Equation (22). Note that for X ̸∈ HC(α) we can upper bound |DC(X)| − 1 and +|C| − |DC(X)| by |C|. At the same time, for X ∈ HC(α) we have |C| − |DC(X)| ≤ (1 − α) |C|. Then our goal is to show +c′(|C| − |HC(α)|) +|C| +log |C| + c′ |HC(α)| +|C| +log((1 − α) |C|) + 1 +(24) +≤ c′ log |C| = c′(|C| − |HC(α)|) +|C| +log |C| + c′ |HC(α)| +|C| +log |C| , +(25) +rearranging we get +|C| +c′ |HC(α)| ≤ log +1 +1 − α = log +� +α +1 − α + 1 +� +. +(26) +Notice that since for any x > −1 it holds that +x +1+x ≤ log(x + 1), it suffices to show +|C| +c′ |HC(α)| ≤ α ⇐⇒ |C| ≤ c′ |HC(α)| α. +(27) +From our assumption on |HC(α)| it suffices to pick c′ ≥ +1 +αβ . For the base case consider C = {X, Y } then for X ̸= P we +have that either there is a single edge P → X or P is disconnected with X for the condition of the theorem to hold. Then +the recursion in Equation (21) could be rewritten as +T({X, Y }) = 1 +2T({Y }) + 1 +2T({X}) + 1, +(28) +from which it follows that +T({X, Y }) = 2 ≤ c log(2). +(29) +For the case when |AC(P)| ≥ γ |C| consider the set H′ of size at least γ +2 |C| of the ancestors of P which are closest to P in +the topologically sorted order in the graph GC. Each node X ∈ H′ has at most |C| − γ +2 |C| descendants. The rest of the +proof follows similar steps as for the case when |HC(α)| ≥ β |C|. + +Causal Bandits without Graph Learning +C. Fast Parent Discovery in Erd˝os-R´enyi Graphs +In this subsection we show that Erd˝os-R´enyi graphs satisfy the condition for fast discovery of the parent node for sufficiently +large values of p. We first state the following theorem. +Theorem C.1. Let Gn,p be Erd˝os-R´enyi random DAG with probability of each edge between n nodes being equal to p. +Assume p ≥ 1 − +� +1−c +n−1 +�1/(n−1) +for some constant c ∈ [0, 1], and denote by X, Y the first and last nodes in the topological +order, respectively. It holds that +E |D(X)| ≥ cn, and +(30) +E |A(Y )| ≥ cn. +(31) +Proof. In the proof we show by induction that the expected number of descendants of the root node (the first node in +topological order) is lower bounded by cn. The expected number of the ancestors could be lower bounded using the same +reasoning. Denote by pn,i the probability that there are exactly i descendants of the root node in the graph Gn,p and note +that +E |D(X)| = +n +� +i=1 +ipn,i. +(32) +Furthermore, pn,i satisfies the following recursion: +pn,i = (1 − (1 − p)i−1)pn−1,i−1 + (1 − p)ipn−1,i +(33) += pn−1,i−1 + (1 − p)i−1((1 − p)pn−1,i − pn−1,i−1), +(34) +with p1,1 = 1 and pn,i = 0 if i > n or i = 0. Thus, we can write +E |D(X)| = +n +� +i=1 +ipn−1,i−1 + +n +� +i=1 +i(1 − p)i−1((1 − p)pn−1,i − pn−1,i−1) +(35) += +n +� +i=1 +ipn−1,i−1 − +n +� +i=1 +(1 − p)i−1pn−1,i−1 +(36) ++ +n +� +i=1 +� +i(1 − p)ipn−1,i − (i − 1)(1 − p)i−1pn−1,i−1 +� +. +(37) +Note that +n +� +i=1 +� +i(1 − p)ipn−1,i − (i − 1)(1 − p)i−1pn−1,i−1 +� += 0 +(38) +as a telescoping sum and by induction hypothesis we have that +n +� +i=1 +ipn−1,i−1 = +n +� +i=1 +(i − 1)pn−1,i−1 + +n +� +i=1 +pn−1,i−1 ≥ c(n − 1) + 1, +(39) +therefore it is left to prove +n +� +i=1 +(1 − p)i−1pn−1,i−1 ≤ 1 − c. +(40) +To prove this we first show that pn,i ≤ (1 − p)n−i, again by induction. This holds for n = 1 and all i or i = 0 and all n > 1. +Furthermore by induction hypothesis we have +pn,i ≤ (1 − (1 − p)i−1)(1 − p)n−i + (1 − p)i(1 − p)n−1−i +(41) += (1 − p)n−i + (1 − p)n−1 + (1 − p)n−1 = (1 − p)n−i. +(42) + +Causal Bandits without Graph Learning +Using this result together with the fact that pn−1,0 = 0 we have +n +� +i=1 +(1 − p)i−1pn−1,i−1 ≤ (n − 1)(1 − p)n−1 ≤ 1 − c, +(43) +where the last inequality follows from the assumption of the theorem. To finish the proof, note that for n = 1 we have +E |D(X)| = 1 ≥ c. +Corollary 5.5. The family of Erd˝os-R´enyi random DAGs satisfies the condition of Theorem 5.4 in expectation if p ≥ +1 − +� +1−c +logk n−1 +�1/(logk n−1) +, for any constantc ∈ [0, 1]. Therefore, for such graphs, RAPS requires E[N(G, P)] = O(logk n) +expected number of interventions. +Proof. From our lower bound on p and Theorem C.1 it follows that for all subgraphs of size at least logk n the first and last +nodes in the topological order have at least cn descendants and ancestors respectively. Let C be an arbitrary candidate set of +size larger than 4 logk n considered by Algorithm 1 when run on the graph Gn,p. Let j ∈ [m] with m = |C| be the index +of P in the topologically sorted order in the subgraph of Gm,p over nodes in C. We will comment bellow on the situation +when P ̸∈ C. We consider two cases. First, if j ≤ m/2, then consider m/4 subgraphs each consisting of m − m/2 − i last +nodes for i ∈ [m/4] of the original graph Gn. The size of each of these subgraph is at least logk n and by Theorem C.1 we +have that each node at index m/2 + i − 1 in the topological order of the original graph Gm has at least cm/4 descendants. +Since there are m/4 such nodes, the first condition of Theorem 5.4 is satisfied. The same happens for the cases when +P ̸∈ C because in that case all the nodes in C are non-ancestors of P since for P ̸∈ C it must be the case that the algorithm +intervened on P at some point before considering C. Second, if j > m/2, then by Theorem C.1 we have that the number of +ancestors of P is at least cn/2 which means that the second condition of Theorem 5.4 is satisfied. +Remark C.2. Note that since (1−1/n)n ≤ e−1 we have log(n/c) log (1 − 1/n) ≤ − log(n/c) +n +and thus (1 − 1/n)log(n/c) ≤ +� c +n +�1/n. Using this together with the Bernoulli inequality (1 + x)r ≥ 1 + rx for x ≥ −1 and r ≥ 1 we get that assuming +logk n ≥ 1 + max(1, (1 − c)e) the condition of Corollary 5.5 is satisfied for p ≥ +log(logk(n)−1)−log(1−c) +logk(n)−1 +. Additionally, +by using L’H´opital’s rule and the fact that limn→∞ (1/n)1/n = 1, we get that the two lower bounds for p presented in +Corollary 5.5 and here are asymptotically equivalent. +D. Sublinear Upper Bound +Theorem 5.6. Let G be an arbitrary DAG in which there are at most O +� +n +logd(n) +� +nodes X ∈ V \ {P} such that either P +is disconnected with X, or all paths between P and X are blocked by colliders. Then, +E[N(G, P)] = O +� +n +logd n +� +, +where d is the maximum degree in the skeleton of G. +Proof. Let AX = {the algorithm intervenes on the node X}, then +E[N(G, P)] = E[ +� +X∈V +I {AX}] = +� +X∈V +P(AX). +(44) +We split the sum above into three parts. First, consider the nodes X that are at distance at most m from the parent node for +some m to be specified later. There are at most dm+1 such nodes and for each of them we bound the probability P(AX) by +1. Similarly, for all nodes X such that there is no collider-free path between P and X we also bound the probability P(AX) +by 1. Each of the leftover nodes has a collider-free path of length at least m to P. We will show that this means that the +probability P(AX) ≤ 2/m. Define +BX = {the algorithm intervenes on the node P before intervening on the node X} , +(45) + +Causal Bandits without Graph Learning +then using the law of total probability we can write +P(AX) = P(AX|BX)P(BX) + P(AX|Bc +X)P(Bc +X), +(46) +where Bc +X stands for the complement of the event BX, i.e. +Bc +X = {the algorithm intervenes on node P after intervening on the node X} . +(47) +Consider the probability P(AX|BX). For this probability not to be zero it must be the case that the node X is a descendant +of the parent node P. Therefore, there must be a directed path of length at least m from P to X. Note that any node on +this path except for the node P cannot be intervened on before the node X is intervened on. Therefore, by the time the +algorithm intervenes on the node X there are at least m nodes in the set of candidate nodes from which it has to sample the +node X and hence +P(AX|BX) ≤ 1 +m. +(48) +Next, consider the probability P(Bc +X). If there is a directed path from P to X, then no node on this path could have been +intervened on before the round at which the algorithm intervenes on X since otherwise X would have been removed from +the candidate set. Similarly, if there is a directed path from X to P, then intervening on any node on the path means +excluding X from the candidate set. Thus, in these two cases P(Bc +X) ≤ 1/(m + 1) as there are at least m + 1 nodes on +the path of length at least m. Lastly, consider the case when there are no directed paths between P and X. Since for this +X there exists a collider-free path, there must be a path containing exactly one ancestor of both X and P. Intervening on +any node on this path other than the node which is an ancestor of both X and P means excluding X from the candidate +set because the intervened on node would be an ancestor of X or P but not both. Thus, there are at least m nodes in the +candidate set by the time the algorithm intervenes on X and therefore +P(Bc +X) ≤ 1 +m. +(49) +Bounding the other probabilities by one gives P(AX) ≤ 2/m. Bounding the number of nodes in the third group by n and +combining all of the above we get +E[N(G, P)] = O +� +dm+1 + n +m + +n +logd(n) +� +. +(50) +Finally, setting m = logd +� +n +logd(n) +� +− 1 finishes the proof. +E. Regret Bounds +Lemma 6.3. Let +AX = +� +∃x ∈ [K] : +��� ¯R − ¯Rdo(X=x)��� > ∆/2 +� +, +DX,Y = +� +∃x, y ∈ [K] : +��� ˆP(Y = y) − ˆP(Y = y|do(X = x)) +��� > ε/2 +� +for any X, Y ∈ V and Ac +X, Dc +X,Y be their compliments. Define E as the event that for every node we correctly determine +its descendants and whether it is an ancestor of P using the criteria described above, i.e., +E = +� +X∈A(P ) +AX ∩ +� +X∈Ac(P ) +Ac +X +∩ +� +X∈V +� +� +Y ∈ ¯ +D(X) +DX,Y ∩ +� +Y ∈Dc(X) +Dc +X,Y +� +Then it holds that P {E} ≥ 1 − δ if B = max +� +32 +∆2 log +� 8nK +δ +� +, 8 +ε2 log +� +8n2K2 +δ +�� +. + +Causal Bandits without Graph Learning +Proof. By Hoeffding’s inequality for bounded random variables for fixed X, Y ∈ V with X ∈ A(Y ) and x, y ∈ [K] it +holds that +��� ˆP(Y = y|do(X = x)) − P {Y = y|do(X = x)} +��� ≥ +� +1 +2B log +�8n2K2 +δ +� +(51) +with probability at most +δ +4n2K2 and +��� ˆP(Y = y) − P {Y = y} +��� ≥ +� +1 +2B log +�8 +δ +� +(52) +with probability at most δ +4. Additionally, by Hoeffding’s inequality for 1-subgaussian random variables we have that for +fixed X ∈ V and x ∈ [K] it holds that +���E[R|do(X = x)] − ¯Rdo(X=x)��� ≥ +� +2 +B log +�8nK +δ +� +(53) +with probability at most +δ +4nK . Moreover, +�� ¯R − E[R] +�� ≥ +� +2 +B log +�8 +δ +� +(54) +with probability at most δ +4. Consider the event which is the union of the above bad events. By union bound we have that the +probability of this bad event is at most δ. Note under the complement of this bad event for X ̸∈ A(P) and all x ∈ [K] by +Assumption 6.2 and the choice of B as in the statement of Lemma 6.3 we have +��� ¯R − ¯Rdo(X=x)��� ≤ +�� ¯R − E[R] +�� + +��� ¯Rdo(X=x) − E[R|do(X = x)] +��� ≤ ∆/2, +(55) +and for some x ∈ [K] +��� ¯R − ¯Rdo(X=x)��� ≥ |E[R] − E[R|do(X = x)]| − +�� ¯R − E[R] +�� − +���E[R|do(X = x) − ¯Rdo(X=x)] +��� > ∆/2, +(56) +Similarly, under the complement of the same event we get that for Y ̸∈ D(X) and all x, y ∈ [K] it holds that +��� ˆP(Y = y) − ˆP(Y = y|do(X = x) +��� ≤ +(57) +≤ +��� ˆP(Y = y) − P {Y = y} +��� + +��� ˆP(Y = y|do(X = x)) − P {Y = y|do(X = x)} +��� +(58) +≤ ε/2, +(59) +and if Y ∈ D(X), then for some x, y ∈ [K] +��� ˆP(Y = y) − ˆP(Y = y|do(X = x) +��� ≥ |P {Y = y|do(X = x)} − P {Y = y}| +(60) +− +���P {Y = y|do(X = x)} − ˆP(Y = y|do(X = x)) +��� +(61) +− +��� ˆP(Y = y) − P {Y = y} +��� +(62) +> ε/2. +(63) +Corollary 6.4. Assume that P ̸= ∅, i.e., the reward variable has a parent in V. A learner that uses Algorithm 1 and then +runs a standard bandit algorithm such as UCB achieves the following bound for the conditional regret: +RT +L(G, P | E) = += O +� +K max +� 1 +∆2 , 1 +ε2 +� +log +�nK +δ +� +E[N(G, P)] ++ +� +KT log T +� +. + +Causal Bandits without Graph Learning +Proof. By lemma Lemma 6.3 it holds that for every node we can correctly identify whether that node is an ancestor +of P and all the descendants of that node with probability at least 1 − δ using the criteria described in Section 6.1. +That means that Algorithm 1 will correctly discover the parent node under the same good event in BKE[N(G, P)] +interventions. Subsequently running a standard bandit algorithm such as UCB to find an optimal intervention on P will lead +to +� +KT log(T) regret bound (Lattimore & Szepesv´ari, 2020). The bounds of Theorems 5.4 and 5.6 lead to bounds on the +conditional regret. +E.1. Simple Regret +In this subsection we provide an upper bound on simple regret. First, similar to how it is done in the main text, we define +conditional simple regret: +rT +L(G, P|E) = max +X⊆V +max +x∈[K]|X| E[R|do(X = x)] − E[R|do(XT +1 = xT +1), E], +(64) +where E is the event that all descendants of any node in V are correctly identified together with whether any node is +an ancestor of P, defined in Lemma 6.3. Suppose that the learner runs a standard bandit algorithm that is designed to +minimize cumulative regret, for example, UCB (Capp´e et al., 2013), from round N(G, P) + 1 to round T. After that the +final intervention do(X = x) for arbitrary x ∈ [K] and X ∈ V is sampled with probability +1 +T +T +� +t=1 +I {It = do(X = x)} , +(65) +where It is the intervention performed in round t. Standard conversion of cumulative regret bound to simple regret bound +(see e.g., Lattimore & Szepesv´ari, 2020, Proposition 33.2) leads to Corollary E.1 stated below. +Corollary E.1. For the learner described above we can bound conditional simple regret as follows: +rT +L(G, P|E) = O +�K +T max +� 1 +∆2 , 1 +ε2 +� +log +�nK +δ +� +E[N(G, P)] + +� +K log(T)/T +� +. +(66) +The bound on conditional simple regret implies bounds on simple regret. For example, in the fast regime of our algorithm, +ignoring the quantities not related to n, K, T and setting δ = min +� +1, +� +K log(T)/T +� +we get +rT +L(G, P) ≤ rT +L(G, P|E) + O(δ) = O +�K +T log(n) log +� +nK ∨ n +� +KT/ log(T) +� ++ +� +K log(T)/T +� +, +(67) +since with probability at most δ the event E does not hold and in this case we bound the simple regret by a constant. +F. Universal Lower Bound +Theorem 7.1. Fix a causal graph G and a parent node P. Any learner L that correctly identifies the parent node P, for +any graph obtained from G by relabeling of the nodes and having P take one of n vertices or P = ∅, satisfies +E[NL(G, P)] ≥ +� +X∈V +1 +|AG(P)△AG(X) \ {X}| + 1. +Proof. The proof uses Yao’s principle (Yao, 1977) from which it follows that we need to show that the best deterministic +algorithm performs at least the number of interventions in the lower bound of the theorem against some distribution over +graphs G. Thus, in what follows, let P be the probability measure over the random graphs obtained by randomly and +uniformly relabeling the nodes of the graph G, such a random graph will be denoted by H. Assume that the learner L does +not intervene on the same node twice. Moreover, assume that if it intervenes on a non-ancestor of P it will not subsequently +intervene on any of its’ descendants and that if it intervenes on an ancestor of P it will not subsequently intervene on any of +the non-descendants of that ancestor. We can make this assumption as any learner which does not satisfy it would intervene + +Causal Bandits without Graph Learning +on the same nodes with the same results as a new learner which avoids making these redundant interventions. We denote by +D the set of all learners that satisfy this assumption. Our goal is to lower bound +inf +L∈D EH[N(H, L)] = inf +L∈D E +� � +X∈V +I {AX} +� += inf +L∈D +� +X∈V +P {AX} , +(68) +where AX = {the learner L intervenes on the node X}. Let Z be a node selected uniformly at random and independently +from the sampling of the graph and the parent node, then +� +X∈V +P {AX} = n +� +X∈V +P {Z = X} P {AX} = nP {A} , +(69) +where A = {the learner L intervenes on a randomly selected node Z}. Note that for learner L there are only two ways not +to intervene on any node Z. The first is to intervene either on an ancestor of Z which is not an ancestor of P and the second +is to intervene on an ancestor of P which is not an ancestor of Z. Using this we get +P {A} = P {L intervenes on Z before any node in AH(P)△AH(Z) \ {Z}} . +(70) +We note that any deterministic learner L could be represented by a sequence of nodes W1, . . . , Wn with Wi ̸= Wj for +i ̸= j, and for a random graph H the learner intervenes on the node Wi if there is no element of AH(P)△AH(Wi) in the +sequence W c logk(n) +� +, +where τ[i + 1:] consists of the last |P| − i elements of τ, τi is the i-th element of τ and c > 0 is some constant. Then the +expected number of interventions required by Algorithm 2 to find all parent nodes is O +� +|P| logk n +� +. Similarly, assume that +all graphs of size at least +cn +logd(n) and graph-parent-node pairs in the set above satisfy the condition of Theorem 5.6. Then +the expected number of interventions required by Algorithm 2 is O +� +|P|n +logd n +� +. + +Causal Bandits without Graph Learning +Proof. As noted in the main text, Algorithm 2 discovers the parent nodes in a reverse topological order, let τ ∈ τ(P) be +such an order and i = +��� ˆP +��� ≤ |P| be a number of the iteration of the while loop in Algorithm 2. First, assume i < |P|. +We argue that the expected number of interventions during a call of Algorithm 1 is the same as the expected number of +interventions done by Algorithm 1 when there is only one parent node P which is equal to the (|P| − i)-th element of τ +and ˆP = ∅ on the graph GV\S(τ,|P|−i). If i = |P|, then we need to show that the call to Algorithm 1 with ˆP = P on the +graph GV\D(P) is the same as running Algorithm 1 on the graph-parent-node pair (GV\D(P), ∅) with ˆP = ∅. Proving these +results leads to the proof of the result of the theorem since by the assumption of the theorem we have that (GV\S(τ,|P|−i), P) +and (GV\D(P), ∅) satisfy the assumptions of either Theorem 5.4 or Theorem 5.6. Let X be an arbitrary node, intervened on +during the call of Algorithm 1 by Algorithm 2. If X is an ancestor of P in GC for some C, then X is also an ancestor of P +in GV\S(τ,|P|−i)∩C since no ancestor of P is contained in S(τ, |P| − i) because of its definition. Then in there will be a +recursive call for the candidate set ¯DC(X). At the same time, if X is not an ancestor of any node in ˆP in GC then X is not +an ancestor of any such node in GV\S(|P|−i)∩C since it is a subgraph of the graph GC, and therefore there will be a recursive +call for the candidate set Dc +C(X). Finally, if X is not an ancestor of P in GC but there exist some P ′ ∈ P such that P ′ ̸= P +and X is an ancestor of P ′ in GC, then the call to Algorithm 1 by Algorithm 2 will return P ′ which is a contradiction. Thus, +by induction on the elements of P we have that the sequences of candidate sets C with which the recursive function of +Algorithm 1 is called when this algorithm is called by Algorithm 2 and the sequences of of candidate sets C with which +the recursive function of Algorithm 1 is called when this algorithm is executed on GV\S(τ,|P|−i) are equally likely and we +conclude that the expected number of interventions in these two cases is the same. +Corollary 8.2. Let Gn,p be an Erd˝os-R´enyi graph with p ≥ 1− +� +1−c0 +logk(c1 logk(n))−1 +�1/(logk(c1 logk(n))−1) +for some constants +c0 ∈ [0, 1] and c1 ∈ R>0, then to discover P, Algorithm 2 needs E[N(G, P)] = O +� +|P| logk(n) +� +expected number of +interventions. +Proof. The minimum p in the condition of Corollary 5.5 grows with decreasing n. Therefore, for the condition of +Theorem 8.1 it suffices that for the smallest subgraph GV\D(P) considered by Algorithm 2 the condition of Corollary 5.5 +holds. However, this graph needs to be of size at least c1 logk(n) since otherwise the bound is trivial. Plugging n′ = +c1 logk(n) as n in the condition of Corollary 5.5 leads to the desired result. + diff --git a/59FIT4oBgHgl3EQf8Ss2/content/tmp_files/load_file.txt b/59FIT4oBgHgl3EQf8Ss2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..02047163b814e50c09c0dfa8c6f971f72370302a --- /dev/null +++ b/59FIT4oBgHgl3EQf8Ss2/content/tmp_files/load_file.txt @@ -0,0 +1,966 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf,len=965 +page_content='Causal Bandits without Graph Learning Mikhail Konobeev 1 Jalal Etesami 2 3 Negar Kiyavash 1 2 Abstract We study the causal bandit problem when the causal graph is unknown and develop an efficient algorithm for finding the parent node of the re- ward node using atomic interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' We derive the exact equation for the expected number of in- terventions performed by the algorithm and show that under certain graphical conditions it could perform either logarithmically fast or, under more general assumptions, slower but still sublinearly in the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' We formally show that our algorithm is optimal as it meets the uni- versal lower bound we establish for any algorithm that performs atomic interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' Finally, we extend our algorithm to the case when the reward node has multiple parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' Using this algorithm together with a standard algorithm from bandit literature leads to improved regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' Introduction Multi-armed bandit (MAB) settings provide a rich theoret- ical context for formalizing and analyzing sequential ex- perimental design procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' Each arm in a MAB setting represents an experiment/action and the consequence of pulling an arm is represented by a stochastic reward signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' The objective of a learner in a MAB problem is to select a sequence of arms over a time horizon in order to either find an arm that results in the maximum reward or to maximize the cumulative reward during this time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' Bandit prob- lems have a growing list of applications in various domains such as marketing (Huo & Fu, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' Sawant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=', 2018), recommendation systems (Heckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=', 2022), clinical trials (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=', 2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' An important assumption in classical MAB is that the rewards for the arms are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' However, this assumption is often violated in practice because of interdependencies among the 1School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland 2College of Management of Technology, EPFL, Lausanne, Switzerland 3Department of Computer Science, TUM, Munich, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FIT4oBgHgl3EQf8Ss2/content/2301.11401v1.pdf'} +page_content=' Correspondence to: Mikhail Konobeev 0, parametrized by the arc-length s. Following Bishop [6], +we start recalling the notion of relatively parallel fields along x. Let t = x′ be the unit tangent vector to x and let +d in the plane perpendicular to t. We say that d is relatively parallel field along x if d′ = ct for some constant c (see +Figure 1). +Figure 1: The field d is relatively parallel along the curve x: d′ = ct where c is a constant. +Remark 2.1. By means of a standard parallel transport argument, see [6, Thm. 1] for details, we can ensure that there exists +at least one relatively parallel field along x. +A relatively parallel adapted frame along x, briefly RPAF along x, is an orthonormal frame {t, d1, d2} along the +curve such that both d1, d2 are relatively parallel fields along x. As a consequence, there are smooth coefficients +2 + +ui, i = 1, 2, such that +� +� +� +� +� +t′ = u2 d1 − u1d2, +d′ +1 = −u2 t, +d′ +2 = u1 t. +(2.1) +Clearly, any orthonormal frame satisfying (2.1) is a RPAF along any curve with t as tangent vector. +We notice that the system (2.1) is similar to the classical Serret-Frenet system which can be defined for a C3- +curve. Precisely, if x ∈ C3((0, L); R3) is parametrized by the arc-length, then t = x′. Assume |t′| ̸= 0 everywhere +and let κ = |t′| the (positive) curvature of x. We set +n = t′ +|t′|, +b = t × n, +usually referred as the principal normal and the binormal respectively. Directly, we have +b′ = τn, +for some coefficient τ which is called the torsion of the curve x. Hence, the Serret-Frenet orthonormal frame +{t, n, b} satisfies the system +� +� +� +� +� +t′ = κ n, +n′ = −κ t − τb, +b′ = τn. +(2.2) +We notice that system (2.2) is well-defined since |t′| ̸= 0. The moving orthonormal frame {t, n, b} is not an RPAF +along x since both n and b are not relatively parallel along x. Nevertheless, the Serret-Frenet system contains +important information about x. Indeed, in (2.2) the coefficients κ and τ are geometric invariants of x: the curvature +and the torsion of x do not depend on the parametrization of x. +Both RPAF and Serret-Frenet frames are special cases of the general frame along the curve, which looks like +� +� +� +� +� +t′ = u2 d1 − u1d2, +d′ +1 = −u2 t + u3d2, +d′ +2 = u1 t − u3d1, +(2.3) +where u1, u2 and u3 are smooth coefficients. +To deal with non-smooth curves, the idea is to prescribe the coefficients ui ∈ L2(0, L), (u1, u2 are called flexural +densities, while u3 is the twist density), and look for a solution of (2.3). More precisely, let us fix the following initial +conditions +t(0) = t0, +d1(0) = d0 +1, +d2(0) = d0 +2, +(2.4) +such that {t0, d1 +0, d20} is an orthonormal basis in R3. By classical results [12], there exists a unique orthonormal +frame {t, d1, d2} ∈ (W1,2((0, L); R3))3 satisfying (2.3) and (2.4). By integration we can therefore reconstruct the +curve x as +x(s) = t0 + +� s +0 t(r) dr. +In particular, we get x ∈ W2,2((0, L); R3). This approach has been introduced and developed by Gonzalez et +al. (see [11] and [17]) and it has been defined the framed curve approach. In the case where u3 = 0, we call again +{t, d1, d2} a RPAF along the curve x. +From now on, we will focus only on systems (2.1) and (2.2) and we compare them. A natural question arises: +are κ and τ related with the coefficients u1, u2 of an RPAF along x? In the next Lemma we show the relation +between ui’s and κ, τ. +3 + +Lemma 2.2. Let ui : [0, L] → R, i = 1, 2, be such that the unique solution x of (2.1) and (2.4) is of class C3((0, L); R3). +Assume that |t′| = |x′′| ̸= 0 everywhere. Let κ, τ be the curvature and the torsion of x. There exists ϑ ∈ C1([0, L]) such +that +� +� +� +� +� +ϑ′ = τ +u1 = κ sin ϑ +u2 = κ cos ϑ. +(2.5) +In particular, +� +u2 +1 + u2 +2 = κ. +Proof. Since the curve is regular, both systems +� +� +� +� +� +t′ = u2 d1 − u1d2 +d′ +1 = −u2 t +d′ +2 = u1 t +(2.6) +and +� +� +� +� +� +t′ = κ n +n′ = −κ t − τb +b′ = τn. +(2.7) +hold. +Comparing (2.7)1 and (2.6)1, it follows +κn = t′ = u2 d1 − u1d2, +(2.8) +whence, remembering that d1 and d2 are orthonormal, +κ2 = u2 +1 + u2 +2. +(2.9) +This suggests to set +u1 = κ sin ϑ, +u2 = κ cos ϑ, +(2.10) +for a suitable function ϑ ∈ C1((0, L)). +To relate ϑ to the torsion τ, we notice that since κ ̸= 0 and x ∈ C3((0, L); R3), we get κ ∈ C1((0, L)). Hence, +differentiating (2.10) and using (2.8), we obtain +u′ +1 = κ′ sin ϑ + κϑ′ cos ϑ = κ′ +κ u1 + u2ϑ′ +(2.11) +u′ +2 = κ′ cos ϑ − κϑ′ sin ϑ = κ′ +κ u2 − u1ϑ′. +(2.12) +On the other hand, differentiating (2.8), it follows +κ′n + κn′ = u′ +2d1 + u2d1′ − u′ +1d2 − u1d2′. +Substituting the expressions of n′, d1′, d2′ from (2.6) and (2.7) respectively, and using (2.8), (2.9), (2.11) and (2.12), +one easily gets +κ′n − κτb = −ϑ′(u1d1 + u2d2) + κ′ +κ (u2d1 − u1d2) = −ϑ′(u1d1 + u2d2) + κ′ +κ κn. +This simplifies into +κτb = ϑ′(u1d1 + u2d2). +Squaring this relation and using (2.9), it follows immediately τ = ±ϑ′, which is the thesis. +4 + +Remark 2.3. Equation (2.8), together with (2.10), implies for κ ̸= 0 +n = cos ϑd1 − sin ϑd2 +and since b is perpendicular to n, it is immediate to check that +b = sin ϑd1 + cos ϑd2. +Therefore, ϑ(s) is, for every s, the angle of a rotation R(s) in the plane perpendicular to t(s) at x(s), which can be thought +as a space rotation around t(s). i.e. leaving the tangent vector fixed: R(s)t(s) = t(s). +Remark 2.4. Lemma 2.2 shows that, assuming smoothness of the curve, the curvature and the torsion are related to the +flexural densities u1 and u2 through the twist ϑ of the moving frame. Nevertheless, from the system (2.5), it is clear that u1 +and u2 are not geometric invariants of the curve. +In order to extract geometric invariants of a curve of class W2,2 from a RPAF along it the we need to understand +better the “degrees of freedom” of the RPAF. The next proposition is only stated in [6]. +Figure 2: Two RPAFs along x: they differe by a constant angle of rotation ϑ. +Proposition 2.5. If {t, d1, d2} is a RPAF, then the totality of RPAFs consists of frames of the form +R(t, d1, d2) +where R is the rotation introduced in Remark 2.3 and it is independent of s, see Figure 2. +Proof. Obviously, if (t, d1, d2) is an RPAF and R does not depend on s, then R(t, d1, d2) is an RPAF. +Let {t, d1, d2} and {t, ˜d1, ˜d2} be two RPAFs defined on the curve x. Then, necessarily in the plane perpen- +dicular to t, we have (d1, d2) are related to ( ˜d1, ˜d2) through a rotation which depends on the applied point s, +namely +� ˜d1 +˜d2 +� += R +� +d1 +d2 +� += +� +cos ϑ +− sin ϑ +sin ϑ +cos ϑ +� � +d1 +d2 +� +, +(2.13) +5 + +where R is an rotation matrix which in general depends on s. We want to prove that actually R does not depend +on s. Using the expression of ( ˜d1, ˜d2) in Eq. (2.13), we get +˜d′ +1 = − sin ϑϑ′(s)d1 + cos ϑd′ +1 − ϑ′ cos ϑd2(s) − sin ϑd′ +2, +˜d′ +2 = cos ϑϑ′d1 + sin ϑd′ +1 − ϑ′ sin ϑd2(s) + cos ϑd′ +2. +Since the couple (d1, d2) satisfies Eq. (2.1), the above expressions simplify into +˜d′ +1 = ϑ′ [− sin ϑd1 − cos ϑd2] +� +�� +� +A1 ++ cos ϑu2t − sin ϑu1t, +˜d′ +2 = ϑ′ [cos ϑd1 − sin ϑd2] +� +�� +� +A2 +− sin ϑu2t + cos ϑu1t. +Since both {t, d1, d2} and {t, ˜d1, ˜d2} were chosen to be RPAFs, i.e. the components of the derivatives of ˜d′ +1 and ˜d′ +2 +are allowed only along the tangential direction t, this implies that both A1 and A2 have to be zero. Hence, ϑ′ = 0 +which gives the thesis. +Figure 3: Along a straight-line segment the twist of a RPAF is not arbitrary. +Remark 2.6. The definition of a RPAF and the Proposition 2.5 can be easily adapted to a curve in Rn for any n > 3. +An immediate consequence of Proposition 2.5 is the following remark. +Remark 2.7. If x has a straight-line piece then a RPAF along such a piece of the curve must be constant (see Figure 3). +3 +From the unit tangent field to geometric invariants +In this section we prescribe t ∈ W1,2((0, L); R3) with |t| = 1 everywhere, so that, by integration, we obtain a +curve x, parametrized by the arc-length. We want to define directly geometric invariants of x. +We begin with the following proposition. +6 + +Proposition 3.1. Let d0 +1, d0 +2 ∈ R3 be such that {t(0), d0 +1, d0 +2} is an orthonormal basis in R3. Then the integral equations +u1(s) = −d0 +2 · t′(s) − +� s +0 u1(r)t(r) · t′(s) dr, +(3.1) +u2(s) = d0 +1 · t′(s) − +� s +0 u2(r)t(r) · t′(s) dr. +(3.2) +have a unique solution u1, u2 ∈ L2(0, L). Moreover, if we set +d1(s) = d0 +1 − +� s +0 u2(r)t(r) dr, +d2(s) = d0 +2 + +� s +0 u1(r)t(r) dr +(3.3) +then {t, d1, d2} is an orthonormal basis in R3 and it is the unique solution of (2.1)-(2.4). +Proof. The integral equations (3.1) and (3.2) are Volterra integral equations of the second kind with kernel in L2. +Applying [18, Sec. 1.5] we get existence and uniqueness of solutions u1, u2 ∈ L2(0, L). Then, if we define d1, d2 +by means of (3.3) we get d′ +1 = −u2t and d′ +2 = u1t. At this point, from (3.1) - (3.2), we obtain +u1 = −d2 · t′, +u2 = d1 · t′. +As a consequence +(t · d1)′ = t′ · d1 + t · d′ +1 = u2 − u2 = 0, +(t · d2)′ = t′ · d2 + t · d′ +2 = −u1 + u1 = 0. +This means that t · di = t(0) · d0 +i = 0 for i = 1, 2. Furthermore, we also easily obtain +(d1 · d1)′ = (d2 · d2)′ = (d1 · d2)′ = 0. +Hence, {t, d1, d2} is an orthonormal basis in R3. Finally, t′ = u2d1 − u1d2 and this yields the conclusion. +To define geometric invariants of x, following Giusteri et al. [9], we introduce +u := u2 + iu1 +(3.4) +where u1 and u2 are the solutions of (3.1) and (3.2) respectively, having fixed the initial data d0 +1 and d0 +2. In order +to simplify the arguments, we denote by ϑ(u) the unique argument of u in [−π, π). In particular, we notice that +u = |u|eiϑ(u). +Theorem 3.2. Let {d0 +1, d0 +2} and { �d0 +1, �d0 +2} be such that {t(0), d0 +1, d0 +2} and {t(0), �d0 +1, �d0 +2} are two orthonormal basis in R3. +Let (u1, u2) and (�u1, �u2) be the respective solutions of (3.1) and (3.2). Then, the following hold true +|u| = |�u| +(3.5) +and +ϑ(u) − ϑ( ˜u) = α, +(3.6) +where α is a constant independent of s. +Proof. First of all, to show the first relation (3.5), we notice, by (2.1)1, that +|u|2 = u2 +1 + u2 +2 = |t′|2 = �u2 +1 + �u2 +2 = |�u|2 . +Next, to verify (3.6), we observe directly by Proposition 3.1 that +u1 = −t′ · d2, +u2 = t′ · d1, +7 + +from which we get +u = t′(s) · (d1 − i d2) . +Moreover, from Proposition 2.5, we can write the rotated frame ( �d1, �d2) in the plane perpendicular to t as +� �d1 = Ad1 + Bd2 +�d2 = Γd1 + ∆d2 +where A, B, Γ, ∆ are constants. Then, it can be easily seen that +˜u = ˜u2 + i ˜u1 = t′ · (Ad1 + Bd2(s) − iΓd1 − i∆d2) = (Au2 − Bu1) + i (∆u1 − Γu2) . +Since the matrix +� +A +B +Γ +∆ +� +is a constant rotation, it follows that ˜u = ueiα where the angle α depends only on A, B, Γ, ∆, and this yields the +conclusion. +4 +Conclusions and remarks +In this work, we compared the Serret-Frenet approach with the RPAF one. +In the first, starting from a moving frame {t, n, b} ∈ (W1,2((0, L); R3))3 such that t′ · b = 0, we can define the +curvature and the torsion of x as follows +κ = t′ · n, +τ = n′ · b. +We stress the fact that κ and τ are always defined in a weak sense and they are L2 functions. However, in +general, it is not true that starting from t ∈ W1,2((0, L); R3) with |t| = 1, there exists a moving frame {t, n, b} ∈ +(W1,2((0, L); R3))3 satisfying +� +� +� +� +� +� +� +� +� +� +� +x′ = t, +t′ = κn, +n′ = −κt + τb, +b′ = −τn. +(4.1) +Indeed, to have n ∈ W1,2((0, L); R3), we must require, whenever |t′| ̸= 0 +t′ +|t′| ∈ W1,2((0, L); R3). +(4.2) +The condition (4.2) could not be true without further assumptions on t: observe that in general we cannot say +more than t′ ∈ L2([0, L]; R3). +For a RPAF system, we immediately notice that the moving frame {t, n, b} is not a RPAF since n′ and b′ are not +parallel to t. By means of Remark 2.1, on a curve a RPAF generated by t ∈ W1,2((0, L); R3) always exists. In this +sense, the RPAF approach is more general since we can deal with any curve x ∈ W2,2((0, L); R3). Nevertheless, +by Theorem 3.2, the curvature and the torsion are defined as (see (3.5) and (3.6)) +κ = +� +u2 +1 + u2 +2 +and +τ = ϑ′, +where u2 + iu1 = κeiϑ and u1, u2 are the solutions of (3.1) and (3.2). The main drawback of this approach stems +in τ which is defined only in the sense of distributions: choosing discontinuous coefficients u1 and u2, we indeed +get the angle ϑ to be a discontinuous function. We remark that this fact cannot happen for a frame of type (4.1), +where τ ∈ L2((0, L)). +8 + +To study variational problems for elastic curves related to functionals of type +F[x] = +� +x f (κ, τ) dℓ, +one needs to introduce a weak notion of curvature and torsion. For instance, in [5, 1], we used the framed curve +approach. Precisely, we considered functionals of the following type +F[t|n|b] = +� L +0 +f (t′ · n, n′ · b) ds, +where the independent variable is the moving frame {t, n, b}. On the other hand, the formulation of the func- +tional F in terms of RPAF is harder since the torsion is defined only in the sense of distributions. +As a conclusion, it seems that the RPAF’s approach is not suitable for performing a variational analysis of a +functional depending on curvature and torsion. Nevertheless, it could be a useful approach to study geometric +properties of curves since curvature and torsion turn out to be well-defined in a weaker framework. +Acknowledgements +The authors thank Marco Degiovanni and Giulio Giusteri and for helpful suggestions and fruitful discussions. +GB is supported by the European Research Council (ERC), under the European Union’s Horizon 2020 re- +search and innovation programme, through the project ERC VAREG - Variational approach to the regularity of the +free boundaries (grant agreement No. 853404). GB and LL are supported by italian Gruppo Nazionale per l’Analisi +Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale per l’Alta Matematica (IN- +dAM). AM is supported by italian Gruppo Nazionale per la Fisica Matematica (GNFM) of Istituto Nazionale per +l’Alta Matematica (INdAM). +References +[1] F. Ballarin, G. Bevilacqua, L. Lussardi, and A. Marzocchi. Elastic membranes spanning deformable bound- +aries. arXiv preprint arXiv:2207.13614, 2022. +[2] G. Bevilacqua, L. Lussardi, and A. Marzocchi. Soap film spanning electrically repulsive elastic protein links. +Atti della Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche, Matematiche e Naturali, 96(S3):1, 2018. +[3] G. Bevilacqua, L. Lussardi, and A. Marzocchi. +Soap film spanning an elastic link. +Quarterly of Applied +Mathematics, 77(3):507–523, 2019. +[4] G. Bevilacqua, L. Lussardi, and A. Marzocchi. Dimensional reduction of the Kirchhoff-Plateau problem. +Journal of Elasticity, 140(1):135–148, 2020. +[5] G. Bevilacqua, L. Lussardi, and A. Marzocchi. Variational analysis of inextensible elastic curves. Proceedings +of the Royal Society A, 478(2260):20210741, 2022. +[6] R. L. Bishop. There is more than one way to frame a curve. The American Mathematical Monthly, 82(3):246–251, +1975. +[7] L. Freddi, P. Hornung, M. G. Mora, and R. Paroni. A corrected Sadowsky functional for inextensible elastic +ribbons. Journal of Elasticity, 123(2):125–136, 2016. +[8] L. Freddi, P. Hornung, M. G. Mora, and R. Paroni. A variational model for anisotropic and naturally twisted +ribbons. SIAM Journal on Mathematical Analysis, 48(6):3883–3906, 2016. +[9] G. G. Giusteri and E. Fried. Importance and effectiveness of representing the shapes of Cosserat rods and +framed curves as paths in the special Euclidean algebra. Journal of Elasticity, 132(1):43–65, 2018. +9 + +[10] G. G. Giusteri, L. Lussardi, and E. Fried. Solution of the Kirchhoff–Plateau problem. Journal of Nonlinear +Science, 27(3):1043–1063, 2017. +[11] O. Gonzalez, J. H. Maddocks, F. Schuricht, and H. Von Der Mosel. Global curvature and self-contact of +nonlinearly elastic curves and rods. Calculus of Variations and Partial Differential Equations, 14(1):29–68, 2002. +[12] P. Hartman. Ordinary Differential Equations, volume 38. SIAM, 1982. +[13] J. Langer and D. A. Singer. Knotted elastic curves in R3. Journal of the London Mathematical Society, 2(3):512– +520, 1984. +[14] J. Langer and D. A. Singer. The total squared curvature of closed curves. Journal of Differential Geometry, +20(1):1–22, 1984. +[15] J. Langer and D. A. Singer. Curve straightening and a minimax argument for closed elastic curves. Topology, +24(1):75–88, 1985. +[16] C. Mantegazza, A. Pluda, and M. Pozzetta. A survey of the elastic flow of curves and networks. Milan +Journal of Mathematics, 89(1):59–121, 2021. +[17] F. Schuricht. Global injectivity and topological constraints for spatial nonlinearly elastic rods. Journal of +Nonlinear Science, 12(5), 2002. +[18] F. G. Tricomi. Integral equations. Dover Publications, Inc., New York, 1985. Reprint of the 1957 original. +10 + diff --git a/5tE1T4oBgHgl3EQf6wUC/content/tmp_files/load_file.txt b/5tE1T4oBgHgl3EQf6wUC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1944f0e6c5d2288aca1d5a39b5122146daf71ec2 --- /dev/null +++ b/5tE1T4oBgHgl3EQf6wUC/content/tmp_files/load_file.txt @@ -0,0 +1,399 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf,len=398 +page_content='Geometric invariants of non-smooth framed curves GIULIA BEVILACQUA 1* − LUCA LUSSARDI 2† − ALFREDO MARZOCCHI 3‡ 1 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I–56127 Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 2 DISMA “G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Lagrange”, Politecnico di Torino, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='so Duca degli Abruzzi 24, I-10129 Torino, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 3 Dipartimento di Matematica e Fisica “N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Tartaglia", Università Cattolica del Sacro Cuore, via della Garzetta 48, I-25133 Brescia, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' January 10, 2023 Abstract We compare the Serret-Frenet frame with a relatively parallel adapted frame (RPAF) introduced by Bishop [6] to parametrize W2,2-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Next, we derive the geometric invariants, curvature and torsion, with the RPAF asso- ciated to the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Finally, we discuss applications of the two approaches in variational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Mathematics Subject Classification (2020): 53A04, 74B20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Keywords: curvature, torsion, relatively parallel adapted frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 1 Introduction The study of curvature and torsion of spatial curves is contained in differential geometry where high regularity on the curve is customary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In such a setting, the curvature κ and torsion τ generate the Serret-Frenet frame, which is an orthonormal basis along the curve, where τ is defined only in points with κ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' The Serret-Frenet system takes the form � � � � � t′ = κ n, n′ = −κ t − τb, b′ = τn, where t is tangent to the curve parametrized by the arc-length, n is the principal normal and b is the binormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Variational analysis of elastic curves has been widely developed by Langer and Singer [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We also mention a recent interesting research on elastic networks by Mantegazza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' [16], where the authors consider essentially the Euler elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Nevertheless, to extend the study to more general energy functionals, one may need to introduce a suitable weak notion of curvature and torsion defined everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' More than 40 years ago, Bishop [6] introduced another way to frame a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Precisely, he defined the relatively parallel adapted frame (RPAF) as an orthonormal basis along the curve such that normal vectors to the curve have tangential derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' This requires less regularity than in the classical case, for instance it can be set for a W2,2-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Precisely, a RPAF is given by � � � � � t′ = u2 d1 − u1d2, d′ 1 = −u2 t, d′ 2 = u1 t, giulia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='bevilacqua@dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='it †luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='lussardi@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='it ‡alfredo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='marzocchi@unicatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='it 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='03525v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='AP] 9 Jan 2023 where t is tangent to the curve and (d1, d2) are orthogonal to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We notice that the coefficients u1, u2 are not geometric invariants of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Another way to set the problem is the framed curve approach, which has been reconsidered by Schuricht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' [11, 17] to study elastic rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We also mention [9], where the RPAF has been obtained within the theory of Cosserat rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Framed curves have been recently used to study variational problems related to elastic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We refer to [7, 8], where they derived a corrected version of the Sadowsky functional within the theory of elastic ribbons and to [10, 2, 3, 4], where the authors investigated equilibrium shapes of a system in which a closed flexible filament is spanned by a liquid film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Other results can be found in [5, 1], where the framed curve approach has been employed to derive first-order necessary conditions for minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In this paper, in Section 2, we recall the Bishop framework introducing the relatively parallel adapted frames (RPAF) and we compare them with the Serret-Frenet frames in the smooth case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Next, in Section 3, we derive the geometric invariants κ and τ as functions of the coefficients u1, u2 of the RPAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Finally, in Section 4, we conclude that the framed curve approach seems to be useful to study variational problems involving energy functionals, while the RPAF approach is less suitable for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' However, the framed curve approach requires more than a W2,2-curve, whereas in the RPAF one, curvature and torsion are well-defined for any W2,2-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 2 Relatively adapted frames along a curve Let us consider a curve x ∈ C2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3), where L > 0, parametrized by the arc-length s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Following Bishop [6], we start recalling the notion of relatively parallel fields along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Let t = x′ be the unit tangent vector to x and let d in the plane perpendicular to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We say that d is relatively parallel field along x if d′ = ct for some constant c (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Figure 1: The field d is relatively parallel along the curve x: d′ = ct where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' By means of a standard parallel transport argument, see [6, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 1] for details, we can ensure that there exists at least one relatively parallel field along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' A relatively parallel adapted frame along x, briefly RPAF along x, is an orthonormal frame {t, d1, d2} along the curve such that both d1, d2 are relatively parallel fields along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' As a consequence, there are smooth coefficients 2 ui, i = 1, 2, such that � � � � � t′ = u2 d1 − u1d2, d′ 1 = −u2 t, d′ 2 = u1 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) Clearly, any orthonormal frame satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) is a RPAF along any curve with t as tangent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We notice that the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) is similar to the classical Serret-Frenet system which can be defined for a C3- curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Precisely, if x ∈ C3((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3) is parametrized by the arc-length, then t = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Assume |t′| ̸= 0 everywhere and let κ = |t′| the (positive) curvature of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We set n = t′ |t′|, b = t × n, usually referred as the principal normal and the binormal respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Directly, we have b′ = τn, for some coefficient τ which is called the torsion of the curve x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Hence, the Serret-Frenet orthonormal frame {t, n, b} satisfies the system � � � � � t′ = κ n, n′ = −κ t − τb, b′ = τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) We notice that system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) is well-defined since |t′| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' The moving orthonormal frame {t, n, b} is not an RPAF along x since both n and b are not relatively parallel along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Nevertheless, the Serret-Frenet system contains important information about x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Indeed, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) the coefficients κ and τ are geometric invariants of x: the curvature and the torsion of x do not depend on the parametrization of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Both RPAF and Serret-Frenet frames are special cases of the general frame along the curve, which looks like � � � � � t′ = u2 d1 − u1d2, d′ 1 = −u2 t + u3d2, d′ 2 = u1 t − u3d1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='3) where u1, u2 and u3 are smooth coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' To deal with non-smooth curves, the idea is to prescribe the coefficients ui ∈ L2(0, L), (u1, u2 are called flexural densities, while u3 is the twist density), and look for a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' More precisely, let us fix the following initial conditions t(0) = t0, d1(0) = d0 1, d2(0) = d0 2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='4) such that {t0, d1 0, d20} is an orthonormal basis in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' By classical results [12], there exists a unique orthonormal frame {t, d1, d2} ∈ (W1,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3))3 satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' By integration we can therefore reconstruct the curve x as x(s) = t0 + � s 0 t(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In particular, we get x ∈ W2,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' This approach has been introduced and developed by Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (see [11] and [17]) and it has been defined the framed curve approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In the case where u3 = 0, we call again {t, d1, d2} a RPAF along the curve x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' From now on, we will focus only on systems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) and we compare them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' A natural question arises: are κ and τ related with the coefficients u1, u2 of an RPAF along x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In the next Lemma we show the relation between ui’s and κ, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 3 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Let ui : [0, L] → R, i = 1, 2, be such that the unique solution x of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='4) is of class C3((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Assume that |t′| = |x′′| ̸= 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Let κ, τ be the curvature and the torsion of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' There exists ϑ ∈ C1([0, L]) such that � � � � � ϑ′ = τ u1 = κ sin ϑ u2 = κ cos ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5) In particular, � u2 1 + u2 2 = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Since the curve is regular, both systems � � � � � t′ = u2 d1 − u1d2 d′ 1 = −u2 t d′ 2 = u1 t (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='6) and � � � � � t′ = κ n n′ = −κ t − τb b′ = τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='7) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Comparing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='7)1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='6)1, it follows κn = t′ = u2 d1 − u1d2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='8) whence, remembering that d1 and d2 are orthonormal, κ2 = u2 1 + u2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='9) This suggests to set u1 = κ sin ϑ, u2 = κ cos ϑ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='10) for a suitable function ϑ ∈ C1((0, L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' To relate ϑ to the torsion τ, we notice that since κ ̸= 0 and x ∈ C3((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3), we get κ ∈ C1((0, L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Hence, differentiating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='10) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='8), we obtain u′ 1 = κ′ sin ϑ + κϑ′ cos ϑ = κ′ κ u1 + u2ϑ′ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='11) u′ 2 = κ′ cos ϑ − κϑ′ sin ϑ = κ′ κ u2 − u1ϑ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='12) On the other hand, differentiating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='8), it follows κ′n + κn′ = u′ 2d1 + u2d1′ − u′ 1d2 − u1d2′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Substituting the expressions of n′, d1′, d2′ from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='7) respectively, and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='8), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='12), one easily gets κ′n − κτb = −ϑ′(u1d1 + u2d2) + κ′ κ (u2d1 − u1d2) = −ϑ′(u1d1 + u2d2) + κ′ κ κn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' This simplifies into κτb = ϑ′(u1d1 + u2d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Squaring this relation and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='9), it follows immediately τ = ±ϑ′, which is the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 4 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='8), together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='10), implies for κ ̸= 0 n = cos ϑd1 − sin ϑd2 and since b is perpendicular to n, it is immediate to check that b = sin ϑd1 + cos ϑd2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Therefore, ϑ(s) is, for every s, the angle of a rotation R(s) in the plane perpendicular to t(s) at x(s), which can be thought as a space rotation around t(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' leaving the tangent vector fixed: R(s)t(s) = t(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2 shows that, assuming smoothness of the curve, the curvature and the torsion are related to the flexural densities u1 and u2 through the twist ϑ of the moving frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Nevertheless, from the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5), it is clear that u1 and u2 are not geometric invariants of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In order to extract geometric invariants of a curve of class W2,2 from a RPAF along it the we need to understand better the “degrees of freedom” of the RPAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' The next proposition is only stated in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Figure 2: Two RPAFs along x: they differe by a constant angle of rotation ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' If {t, d1, d2} is a RPAF, then the totality of RPAFs consists of frames of the form R(t, d1, d2) where R is the rotation introduced in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='3 and it is independent of s, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Obviously, if (t, d1, d2) is an RPAF and R does not depend on s, then R(t, d1, d2) is an RPAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Let {t, d1, d2} and {t, ˜d1, ˜d2} be two RPAFs defined on the curve x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Then, necessarily in the plane perpen- dicular to t, we have (d1, d2) are related to ( ˜d1, ˜d2) through a rotation which depends on the applied point s, namely � ˜d1 ˜d2 � = R � d1 d2 � = � cos ϑ − sin ϑ sin ϑ cos ϑ � � d1 d2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='13) 5 where R is an rotation matrix which in general depends on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We want to prove that actually R does not depend on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Using the expression of ( ˜d1, ˜d2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='13), we get ˜d′ 1 = − sin ϑϑ′(s)d1 + cos ϑd′ 1 − ϑ′ cos ϑd2(s) − sin ϑd′ 2, ˜d′ 2 = cos ϑϑ′d1 + sin ϑd′ 1 − ϑ′ sin ϑd2(s) + cos ϑd′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Since the couple (d1, d2) satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1), the above expressions simplify into ˜d′ 1 = ϑ′ [− sin ϑd1 − cos ϑd2] � �� � A1 + cos ϑu2t − sin ϑu1t, ˜d′ 2 = ϑ′ [cos ϑd1 − sin ϑd2] � �� � A2 − sin ϑu2t + cos ϑu1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Since both {t, d1, d2} and {t, ˜d1, ˜d2} were chosen to be RPAFs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' the components of the derivatives of ˜d′ 1 and ˜d′ 2 are allowed only along the tangential direction t, this implies that both A1 and A2 have to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Hence, ϑ′ = 0 which gives the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Figure 3: Along a straight-line segment the twist of a RPAF is not arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' The definition of a RPAF and the Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5 can be easily adapted to a curve in Rn for any n > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' An immediate consequence of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5 is the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' If x has a straight-line piece then a RPAF along such a piece of the curve must be constant (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 3 From the unit tangent field to geometric invariants In this section we prescribe t ∈ W1,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3) with |t| = 1 everywhere, so that, by integration, we obtain a curve x, parametrized by the arc-length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We want to define directly geometric invariants of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We begin with the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 6 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Let d0 1, d0 2 ∈ R3 be such that {t(0), d0 1, d0 2} is an orthonormal basis in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Then the integral equations u1(s) = −d0 2 · t′(s) − � s 0 u1(r)t(r) · t′(s) dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) u2(s) = d0 1 · t′(s) − � s 0 u2(r)t(r) · t′(s) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) have a unique solution u1, u2 ∈ L2(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Moreover, if we set d1(s) = d0 1 − � s 0 u2(r)t(r) dr, d2(s) = d0 2 + � s 0 u1(r)t(r) dr (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='3) then {t, d1, d2} is an orthonormal basis in R3 and it is the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' The integral equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) are Volterra integral equations of the second kind with kernel in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Applying [18, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5] we get existence and uniqueness of solutions u1, u2 ∈ L2(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Then, if we define d1, d2 by means of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='3) we get d′ 1 = −u2t and d′ 2 = u1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' At this point, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2), we obtain u1 = −d2 · t′, u2 = d1 · t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' As a consequence (t · d1)′ = t′ · d1 + t · d′ 1 = u2 − u2 = 0, (t · d2)′ = t′ · d2 + t · d′ 2 = −u1 + u1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' This means that t · di = t(0) · d0 i = 0 for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Furthermore, we also easily obtain (d1 · d1)′ = (d2 · d2)′ = (d1 · d2)′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Hence, {t, d1, d2} is an orthonormal basis in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Finally, t′ = u2d1 − u1d2 and this yields the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' To define geometric invariants of x, following Giusteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' [9], we introduce u := u2 + iu1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='4) where u1 and u2 are the solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) respectively, having fixed the initial data d0 1 and d0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In order to simplify the arguments, we denote by ϑ(u) the unique argument of u in [−π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In particular, we notice that u = |u|eiϑ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Let {d0 1, d0 2} and { �d0 1, �d0 2} be such that {t(0), d0 1, d0 2} and {t(0), �d0 1, �d0 2} are two orthonormal basis in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Let (u1, u2) and (�u1, �u2) be the respective solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Then, the following hold true |u| = |�u| (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5) and ϑ(u) − ϑ( ˜u) = α, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='6) where α is a constant independent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' First of all, to show the first relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5), we notice, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1)1, that |u|2 = u2 1 + u2 2 = |t′|2 = �u2 1 + �u2 2 = |�u|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Next, to verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='6), we observe directly by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1 that u1 = −t′ · d2, u2 = t′ · d1, 7 from which we get u = t′(s) · (d1 − i d2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Moreover, from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5, we can write the rotated frame ( �d1, �d2) in the plane perpendicular to t as � �d1 = Ad1 + Bd2 �d2 = Γd1 + ∆d2 where A, B, Γ, ∆ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Then, it can be easily seen that ˜u = ˜u2 + i ˜u1 = t′ · (Ad1 + Bd2(s) − iΓd1 − i∆d2) = (Au2 − Bu1) + i (∆u1 − Γu2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Since the matrix � A B Γ ∆ � is a constant rotation, it follows that ˜u = ueiα where the angle α depends only on A, B, Γ, ∆, and this yields the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 4 Conclusions and remarks In this work, we compared the Serret-Frenet approach with the RPAF one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In the first, starting from a moving frame {t, n, b} ∈ (W1,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3))3 such that t′ · b = 0, we can define the curvature and the torsion of x as follows κ = t′ · n, τ = n′ · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We stress the fact that κ and τ are always defined in a weak sense and they are L2 functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' However, in general, it is not true that starting from t ∈ W1,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3) with |t| = 1, there exists a moving frame {t, n, b} ∈ (W1,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3))3 satisfying � � � � � � � � � � � x′ = t, t′ = κn, n′ = −κt + τb, b′ = −τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) Indeed, to have n ∈ W1,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3), we must require, whenever |t′| ̸= 0 t′ |t′| ∈ W1,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) The condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2) could not be true without further assumptions on t: observe that in general we cannot say more than t′ ∈ L2([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' For a RPAF system, we immediately notice that the moving frame {t, n, b} is not a RPAF since n′ and b′ are not parallel to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' By means of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1, on a curve a RPAF generated by t ∈ W1,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3) always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' In this sense, the RPAF approach is more general since we can deal with any curve x ∈ W2,2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Nevertheless, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2, the curvature and the torsion are defined as (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='6)) κ = � u2 1 + u2 2 and τ = ϑ′, where u2 + iu1 = κeiϑ and u1, u2 are the solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' The main drawback of this approach stems in τ which is defined only in the sense of distributions: choosing discontinuous coefficients u1 and u2, we indeed get the angle ϑ to be a discontinuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' We remark that this fact cannot happen for a frame of type (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content='1), where τ ∈ L2((0, L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 8 To study variational problems for elastic curves related to functionals of type F[x] = � x f (κ, τ) dℓ, one needs to introduce a weak notion of curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' For instance, in [5, 1], we used the framed curve approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Precisely, we considered functionals of the following type F[t|n|b] = � L 0 f (t′ · n, n′ · b) ds, where the independent variable is the moving frame {t, n, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' On the other hand, the formulation of the func- tional F in terms of RPAF is harder since the torsion is defined only in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' As a conclusion, it seems that the RPAF’s approach is not suitable for performing a variational analysis of a functional depending on curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Nevertheless, it could be a useful approach to study geometric properties of curves since curvature and torsion turn out to be well-defined in a weaker framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Acknowledgements The authors thank Marco Degiovanni and Giulio Giusteri and for helpful suggestions and fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' GB is supported by the European Research Council (ERC), under the European Union’s Horizon 2020 re- search and innovation programme, through the project ERC VAREG - Variational approach to the regularity of the free boundaries (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 853404).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' GB and LL are supported by italian Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale per l’Alta Matematica (IN- dAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' AM is supported by italian Gruppo Nazionale per la Fisica Matematica (GNFM) of Istituto Nazionale per l’Alta Matematica (INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Journal of Nonlinear Science, 12(5), 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' [18] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Tricomi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Dover Publications, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=', New York, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' Reprint of the 1957 original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQf6wUC/content/2301.03525v1.pdf'} diff --git a/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf b/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..715f70cf6bac5c5384ad0ed38513ab98a1acaf9a --- /dev/null +++ b/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6cce0265e8a5a8f1e1301cf1bef80c82926035cb2272cb263899fce0b9b507c0 +size 1837118 diff --git 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Biomedical image segmentation plays a significant role in +computer-aided diagnosis. However, existing CNN based methods rely +heavily on massive manual annotations, which are very expensive and +require huge human resources. In this work, we adopt a coarse-to-fine +strategy and propose a self-supervised correction learning paradigm for +semi-supervised biomedical image segmentation. Specifically, we design +a dual-task network, including a shared encoder and two independent +decoders for segmentation and lesion region inpainting, respectively. In +the first phase, only the segmentation branch is used to obtain a rela- +tively rough segmentation result. In the second step, we mask the de- +tected lesion regions on the original image based on the initial segmenta- +tion map, and send it together with the original image into the network +again to simultaneously perform inpainting and segmentation separately. +For labeled data, this process is supervised by the segmentation anno- +tations, and for unlabeled data, it is guided by the inpainting loss of +masked lesion regions. Since the two tasks rely on similar feature in- +formation, the unlabeled data effectively enhances the representation of +the network to the lesion regions and further improves the segmenta- +tion performance. Moreover, a gated feature fusion (GFF) module is +designed to incorporate the complementary features from the two tasks. +Experiments on three medical image segmentation datasets for differ- +ent tasks including polyp, skin lesion and fundus optic disc segmenta- +tion well demonstrate the outstanding performance of our method com- +pared with other semi-supervised approaches. The code is available at +https://github.com/ReaFly/SemiMedSeg. +1 +Introduction +Medical image segmentation is an essential step in computer-aided diagnosis. +In practice, clinicians use various types of images to locate lesions and analyze +diseases. An automated and accurate medical image segmentation technique is +bound to greatly reduce the workload of clinicians. +⋆ Corresponding author is Guanbin Li (liguanbin@mail.sysu.edu.cn). +arXiv:2301.04866v1 [cs.CV] 12 Jan 2023 + +2 +Zhang et al. +With the vigorous development of deep learning, the FCN [15], UNet [19] +and their variants [12,23] have achieved superior segmentation performance for +both natural images and medical images. However, these methods rely heavily on +labeled data, which is time-consuming to acquire especially for medical images. +Therefore, many studies adopt semi-supervised learning to alleviate this issue, +including GAN-based methods [9,24], consistency training [17,20], pseudo label- +ing [11] and so on. For instance, Mean Teacher (MT) [20] and its variants [13,22] +employ the consistency training for labeled data and unlabeled data by updat- +ing teacher weights via an exponential moving average of consecutive student +models. Recently, some works [1, 14] integrate self-supervised learning such as +jigsaw puzzles [16] or contrastive learning [4] to semi-supervised segmentation +and achieve competitive results. However, few of them try to dig deeply into +the context and structural information of unlabeled images to supplement the +semantic segmentation. +In this work, we also consider introducing self-supervised learning to semi- +supervised segmentation. In contrast to +[1, 14], we make full use of massive +unlabeled data to exploit image internal structure and boundary characteris- +tics by utilizing pixel-level inpainting as an auxiliary self-supervised task, which +is combined with semantic segmentation to construct a dual-task network. As +the inpainting of normal non-lesion image content will only introduce additional +noise for lesion segmentation, we design a coarse-to-fine pipeline and then en- +hance the network’s representations with the help of massive unlabeled data in +the correction stage by only masking the lesion area for inpainting based on the +initial segmentation result. Specifically, in the first phase, only the segmentation +branch is used to acquire a coarse segmentation result, while in the second step, +the masked and original images are sent into the network again to simultane- +ously perform lesion region inpainting and segmentation separately. Since the +two tasks rely on similar feature information, we also design a gated feature +fusion (GFF) module to incorporate complementary features for improving each +other. Compared with the most related work [2] which introduces a reconstruc- +tion task for unlabeled data, their two tasks lack deep interaction and feature +reuse, thus cannot collaborate and facilitate each other. Besides, our network not +only makes full use of massive unlabeled data, but also explores more complete +lesion regions for limited labeled data through the correction phase, which can +be seen as “image-level erase [21]” or “reverse attention [3]”. +Our contribution is summarized as follows. (1) We propose a novel self- +supervised semi-supervised learning paradigm for general lesion region segmenta- +tion of medical imaging, and verify that the pretext self-supervised learning task +of inpainting the lesion region at the pixel level can effectively enhance the fea- +ture learning and greatly reduce the algorithm’s dependence on large-scale dense +annotation. (2) We propose a dual-task framework for semi-supervised medical +image segmentation. Through introducing the inpainting task, we create supervi- +sion signals for unlabeled data to enhance the network’s representation learning +of lesion regions and also exploit additional lesion features for labeled data, thus +effectively correct the initial segmentation results. (3) We evaluate our method + +Self-Supervised Correction Learning +3 +on three tasks, including polyp, skin lesion and fundus optic disc segmentation, +under a semi-supervision setting. The experimental results demonstrate that +our method achieves superior performance compared with other state-of-the-art +semi-supervised methods. +2 +Methodology +2.1 +Overview +In this work, we adopt a coarse-to-fine strategy and propose a self-supervised +correction learning paradigm for semi-supervised biomedical image segmenta- +tion. Specifically, we introduce inpainting as the pretext task of self-supervised +learning to take advantage of massive unlabeled data and thus construct a dual- +task network, as shown in Fig. 1. Our proposed framework is composed of a +shared encoder, two decoders and five GFF modules placed on each layer of +both decoders. We utilize ResNet34 [8] pretrained on the ImageNet [5] as our +encoder, which consists of five blocks in total. Accordingly, the decoder branch +also has five blocks. Each decoder block is composed of two Conv-BN-ReLU +combinations. For the convenience of expression, we use Eseg, Dseg to represent +the encoder and decoder of the segmentation branch, and Einp and Dinp for +those of the inpainting branch. +Fig. 1. The overview of our network. Both encoders share weights. G1-G5 represent five +GFF modules. The red and blue arrows denote the input and output of our network +in the first and second stage respectively. +In the first step, given the image x ∈ RH×W ×C, in which H,W,C are the +height, width and channels of the image respectively, we use the segmenta- +tion branch Eseg, Dseg with skip-connections, the traditional U-shape encoder- +decoder structure, to obtain a coarse segmentation map ˆycoarse and then mask +the original input based on its binary result ycoarse by the following formulas: +ˆycoarse = Dseg(Eseg(x)) +(1) + +G5G4G3G2G1 +pare +welgh +G1G2 G3 / G4 | G54 +Zhang et al. +xmask = x × (1 − ycoarse) +(2) +In the second phase, the original image x and the masked image xmask are +sent into Eseg and Einp simultaneously to extract features eseg and einp. Obvi- +ously, eseg is essential for the inpainting task, and since the initial segmentation +is usually inaccurate and incomplete, einp may also contain important residual +lesion features for the correction of the initial segmentation. In order to adap- +tively select the useful features of einp and achieve complementary fusion of eseg +and einp, we design the GFF modules (G1-G5) and place them on each decoder +layer of both branches. Specifically, for the ith layer, the features ei +seg and ei +inp +are delivered into Gi through skip-connections to obtain the fusion ei = Gi(ei +seg, +ei +inp), and then sent to the corresponding decoder layer. Thus, both Gi of the +two branches shown in Fig. 1 actually share parameters, taking the same input +and generating the identical output. To enhance the learning of the GFF mod- +ules, we adopt a deep supervision strategy and each layer of the two decoder +branches generate a segmentation result and an inpainting result respectively by +the following formulas: +ˆyi +fine = +� +Di +seg([ei, di+1 +seg ]), +i = 1, 2, 3, 4 +Di +seg(ei), +i = 5 +(3) +ˆxi = +� +Di +inp([ei, di+1 +inp ]), +i = 1, 2, 3, 4 +Di +inp(ei), +i = 5 +(4) +Where [· , ·] denotes the concatenation process, and di+1 +seg , di+1 +inp represent the +features from previous decoder layers. The deep supervision strategy can also +avoid Dinp directly copying the features of the low-level eseg to complete the +inpainting task without in-depth lesion feature mining. The output of the last +layer ˆy1 +fine is the final segmentation result of our network. +Fig. 2. Gated Feature Fusion Module + +inp +c +conv +conv +conv +1-Self-Supervised Correction Learning +5 +2.2 +Gated Feature Fusion (GFF) +To better incorporate complementary features and filter out the redundant in- +formation, we design the GFF modules placed on each decoder layer to integrate +the features delivered from the corresponding encoder layer of two branches. The +details are shown in Fig. 2. Our GFF module consists of a reset gate and a select +gate. Specifically, for the Gi placed on the ith decoder layer, the value of two +gates is calculated as follows: +ri = σ(Wr +� +ei +seg, ei +inp] +� +(5) +si = σ(Ws +� +ei +seg, ei +inp] +� +(6) +Where Wr, Ws denote the convolution process, taking the concatenation of ei +seg +and ei +inp as input. σ represents the Sigmoid function. ri and si represent the +value of the reset gate and the select gate, respectively. Since the input of the +inpainting branch is the masked image, the reset gate is necessary to suppress +massive invalid background information. And then the select gate achieves adap- +tive and complementary feature fusion between the reintegrated features ˜ei and +the original segmentation feature ei +seg by the following operations: +˜ei = W +� +ri × ei +inp, ei +seg +� +) +(7) +ei = si × ˜ei + (1 − si) × ei +seg +(8) +where W also represents the convolution process to make the reintegrated fea- +tures ˜ei have the same dimension with ei +seg. +2.3 +Loss Function +We only calculate loss in the second stage. The labeled dataset and unlabeled +dataset are denoted as Dl and Du. For the labeled data xl ∈ Dl, yl is the Ground +Truth. Since we adopt the deep supervision strategy, the overall loss is the sum +of the combination of Binary CrossEntropy (BCE) loss and Dice loss between +each output and the Ground Truth: +Lseg(xl) = +5 +� +i=1 +Li +BCE(ˆyi +l, yi +l) + Li +Dice(ˆyi +l, yi +l) +(9) +where ˆyi +l, yi +l denote the segmentation map ˆyi +fine of the ith decoder layer and the +corresponding down-sampling Ground Truth yl. +For unlabeled data xu ∈ Du, the inpainting loss is the sum of L1 loss between +each inpainting image and the original image in the masked region: +Linp(xu) = +5 +� +i=1 +yi +u × +��ˆxi +u − xi +u +�� +(10) + +6 +Zhang et al. +where ˆxi +u, xi +u and yi +u represent the inpainting image, down-sampling original +image and binary segmentation result of the ith decoder layer, respectively. In +the end, the total loss function is formulated as follows: +L = λ1 +� +xl∈Dl +Lseg(xl) + λ2 +� +xu∈Du +Linp(xu) +(11) +where λ1, λ2 are weights balancing the segmentation loss and the inpainting loss. +And we set λ1 = 2 and λ2 = 1 in our experiments. +3 +Experimental Results +3.1 +Dataset and Evaluation Metric +We conduct experiments on a variety of medical image segmentation tasks to +verify the effectiveness and robustness of our approach, including polyp, skin +lesion and fundus optic disc segmentation, respectively. +Polyp Segmentation We use the publicly available kvasir-SEG [10] dataset +containing 1000 images, and randomly select 600 images as the training set, 200 +images as the validation set, and the remaining as the test set. +Skin Lesion Segmentation We utilize the ISBI 2016 skin lesion dataset [7] to +evaluate our method performance. This dataset consists of 1279 images, among +which 900 are used for training and the others for testing. +Optic Disc Segmentation The Rim-one r1 dataset [6] is utilized in our ex- +periments, which has 169 images in total. We randomly split the dataset into a +training set and a test set with the ratio of 8:2. +Evaluation Metric Referring to common semi-supervised segmentation set- +tings [13, 22], for all datasets, we randomly use 20% of the training set as +the labeled data, 80% as the unlabeled data and adopt five metrics to quan- +titively evaluate the performance of our approach and other methods, including +“Dice Similarity Coefficient (Dice)”, “Intersection over Union (IoU)”, “Accuracy +(Acc)”, “Recall (Rec)” and “Specificity (Spe)”. +3.2 +Implementation Details +Data pre-processing In our experiments, since the image resolution of all +datasets varies greatly, we uniformly resize all images into a fixed size of 320×320 +for training and testing. And in the training stage, we use data augmentation, +including random horizontal and vertical flips, rotation, zoom, and finally all the +images are randomly cropped to 256 × 256 as input. +Training details Our method is implemented using PyTorch [18] framework. +We set batch size of the training process to 4, and use SGD optimizer with +a momentum of 0.9 and a weight decay of 0.00001 to optimize the model. A +poly learning rate policy is adopted to adjust the initial learning rate, which is +lr = init lr ×(1− +iter +max iter)power, where init lr = 0.001, power = 0.9). The total +number of epochs is set to 80. + +Self-Supervised Correction Learning +7 +Table 1. Comparison with other state-of-the-art methods and ablation study on the +Kvasir-SEG dataset +Methods +Data +Dice +IoU +Acc +Rec +Spe +Supervised +600L (All) +89.48 +83.69 +97.34 +91.06 +98.58 +Supervised +120L +84.40 +76.18 +96.09 +85.35 +98.55 +DAN [24] +120L + 480U +85.77 +78.12 +96.37 +86.86 +98.53 +MT [20] +120L + 480U +85.99 +78.84 +96.21 +86.81 +98.79 +UA-MT [22] +120L + 480U +85.70 +78.34 +96.38 +88.51 +98.40 +TCSM V2 [13] +120L + 480U +86.17 +79.15 +96.38 +87.14 +98.76 +MASSL [2] +120L + 480U +86.45 +79.61 +96.34 +89.18 +98.32 +Ours +120L + 480U +87.14 +80.49 +96.42 +90.78 +97.89 +Ours (add) +120L + 480U +85.59 +78.56 +96.12 +87.98 +98.26 +Ours (concat) +120L + 480U +86.09 +78.98 +96.21 +90.54 +97.63 +3.3 +Comparisons with the State-of-the-Art +In our experiments, ResNet34 [8] based UNet [19] is utilized as our baseline, +which is trained using all training set and our selected 20% labeled data sep- +arately in a fully-supervised manner. Besides, we compare our method with +other state-of-the-art approaches, including DAN [24], MASSL [2], MT [20] +and its variants (UA-MT [22], TCSM V2 [13]). All comparison methods adopt +ResNet34UNet as the backbone and use the same experimental settings for a fair +comparison. On the Kvasir-SEG dataset, Table. 1 shows that our method ob- +tains the outstanding performance compared with other semi-supervised meth- +ods, with Dice of 87.14%, which is 2.74% improvement over the baseline only +using the 120 labeled data, outperforming the second best method by 0.69%. On +the ISBI 2016 skin lesion dataset, we obtain a 90.95% Dice score, which is +superior to other semi-supervised methods and very close to the score of 91.38% +achieved by the baseline using all training set images. On the Rim-one r1 +dataset, we can conclude that our method achieves the best performance over +five metrics, further demonstrating the effectiveness of our method. Note that +detailed results on the latter two datasets are listed in the supplementary mate- +rial due to the space limitation. Some visual segmentation results are shown in +Fig. 3 (col.1-8). +3.4 +Ablation study +Effectiveness of our approach with different ratio of labeled data We +draw the curves of Dice score under three settings in Fig. 4. To verify that our +proposed framework can mine residual lesion features and enhance the lesion +representation by GFF modules in the second stage, we conduct experiments +and draw the blue line. The blue line denotes that our method uses the same +labeled data with the baseline (the red line) to perform the two-stage process, +without utilizing any unlabeled data. Note that we only calculate the segmenta- +tion loss for the labeled data. The performance gains compared with the baseline + +8 +Zhang et al. +Fig. 3. Visual comparison of various lesion segmentation from state-of-the-art methods. +Our proposed method consistently produces segmentation results closest to the ground +truth. The inpainting result is shown in the rightmost column. +show that our network mines useful lesion information in the second stage. The +green line means that our method introduces the remaining as unlabeled data +for the inpainting task, further enhancing the feature representation learning of +the lesion regions and improving the segmentation performance, especially when +only a small amount of labeled data is used. When using 100% labeled data, +the green line is equivalent to the blue line since no additional unlabeled data is +utilized to do the inpainting task, thus maintaining the same results. +Effectiveness of the GFF modules To verify the effectiveness of the GFF +modules, we also design two variants, which merge features by directly addition +and concatenation, denoting as Ours (add) and Ours (concat) respectively. In +Table. 1, we can observe performance degradation by both approaches compared +with our method, proving that the GFF module plays a significant role in filter- +ing redundant information and improving the model performance. +Fig. 4. The performance of our method with different ratio of labeled data on the +Kvasir-SEG dataset. + +90 +88 +Dice Score(%) +86 +84 +82 +ResNetUnet (fully-supervised) +Ours (fully-supervised) +Ours (semi-supervised) +80 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Labeled Data Percentage(%)Self-Supervised Correction Learning +9 +4 +Conclusions +In this paper, we believe that massive unlabeled data contains rich context and +structural information, which is significant for lesion segmentation. Based on +this, we introduce the self-supervised inpainting branch for unlabeled data, co- +operating with the main segmentation task for labeled data, to further enhance +the representation for lesion regions, thus refine the segmentation results. We +also design the GFF module for better feature selection and aggregation from +the two tasks. Experiments on various medical datasets have demonstrated the +superior performance of our method. +Acknowledgement +This work is supported in part by the Key-Area Research and Development Pro- +gram of Guangdong Province (No. 2020B0101350001), in part by the Guangdong +Basic and Applied Basic Research Foundation (No. 2020B1515020048), in part +by the National Natural Science Foundation of China (No. 61976250) and in +part by the Guangzhou Science and technology project (No. 202102020633). +References +1. Chaitanya, K., Erdil, E., Karani, N., Konukoglu, E.: Contrastive learning of global +and local features for medical image segmentation with limited annotations. arXiv +preprint arXiv:2006.10511 (2020) +2. Chen, S., Bortsova, G., Ju´arez, A.G.U., van Tulder, G., de Bruijne, M.: Multi- +task attention-based semi-supervised learning for medical image segmentation. In: +Shen, D., et al. (eds.) MICCAI 2019. LNCS, vol. 11766, pp. 457–465. 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In: Descoteaux, M., Maier-Hein, L., Franz, A., Jannin, P., Collins, D.L., +Duchesne, S. (eds.) MICCAI 2017. LNCS, vol. 10435, pp. 408–416. Springer, Cham +(2017). https://doi.org/10.1007/978-3-319-66179-7 47 + diff --git a/9NE4T4oBgHgl3EQfDAuV/content/tmp_files/load_file.txt b/9NE4T4oBgHgl3EQfDAuV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f20769de6c5a04d307ac91cf44ac1d765a2fbd91 --- /dev/null +++ b/9NE4T4oBgHgl3EQfDAuV/content/tmp_files/load_file.txt @@ -0,0 +1,527 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf,len=526 +page_content='Self-Supervised Correction Learning for Semi-Supervised Biomedical Image Segmentation Ruifei Zhang1, Sishuo Liu2, Yizhou Yu2,3, and Guanbin Li1,4⋆ 1Sun Yat-sen University, Guangzhou, China 2The University of Hong Kong, Pokfulam, Hong Kong 3Deepwise AI Lab, Beijing, China 4Shenzhen Research Institute of Big Data, Shenzhen, China Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Biomedical image segmentation plays a significant role in computer-aided diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' However, existing CNN based methods rely heavily on massive manual annotations, which are very expensive and require huge human resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In this work, we adopt a coarse-to-fine strategy and propose a self-supervised correction learning paradigm for semi-supervised biomedical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Specifically, we design a dual-task network, including a shared encoder and two independent decoders for segmentation and lesion region inpainting, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In the first phase, only the segmentation branch is used to obtain a rela- tively rough segmentation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In the second step, we mask the de- tected lesion regions on the original image based on the initial segmenta- tion map, and send it together with the original image into the network again to simultaneously perform inpainting and segmentation separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' For labeled data, this process is supervised by the segmentation anno- tations, and for unlabeled data, it is guided by the inpainting loss of masked lesion regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Since the two tasks rely on similar feature in- formation, the unlabeled data effectively enhances the representation of the network to the lesion regions and further improves the segmenta- tion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Moreover, a gated feature fusion (GFF) module is designed to incorporate the complementary features from the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Experiments on three medical image segmentation datasets for differ- ent tasks including polyp, skin lesion and fundus optic disc segmenta- tion well demonstrate the outstanding performance of our method com- pared with other semi-supervised approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='com/ReaFly/SemiMedSeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 1 Introduction Medical image segmentation is an essential step in computer-aided diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In practice, clinicians use various types of images to locate lesions and analyze diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' An automated and accurate medical image segmentation technique is bound to greatly reduce the workload of clinicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' ⋆ Corresponding author is Guanbin Li (liguanbin@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='04866v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='CV] 12 Jan 2023 2 Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' With the vigorous development of deep learning, the FCN [15], UNet [19] and their variants [12,23] have achieved superior segmentation performance for both natural images and medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' However, these methods rely heavily on labeled data, which is time-consuming to acquire especially for medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Therefore, many studies adopt semi-supervised learning to alleviate this issue, including GAN-based methods [9,24], consistency training [17,20], pseudo label- ing [11] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' For instance, Mean Teacher (MT) [20] and its variants [13,22] employ the consistency training for labeled data and unlabeled data by updat- ing teacher weights via an exponential moving average of consecutive student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Recently, some works [1, 14] integrate self-supervised learning such as jigsaw puzzles [16] or contrastive learning [4] to semi-supervised segmentation and achieve competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' However, few of them try to dig deeply into the context and structural information of unlabeled images to supplement the semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In this work, we also consider introducing self-supervised learning to semi- supervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In contrast to [1, 14], we make full use of massive unlabeled data to exploit image internal structure and boundary characteris- tics by utilizing pixel-level inpainting as an auxiliary self-supervised task, which is combined with semantic segmentation to construct a dual-task network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' As the inpainting of normal non-lesion image content will only introduce additional noise for lesion segmentation, we design a coarse-to-fine pipeline and then en- hance the network’s representations with the help of massive unlabeled data in the correction stage by only masking the lesion area for inpainting based on the initial segmentation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Specifically, in the first phase, only the segmentation branch is used to acquire a coarse segmentation result, while in the second step, the masked and original images are sent into the network again to simultane- ously perform lesion region inpainting and segmentation separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Since the two tasks rely on similar feature information, we also design a gated feature fusion (GFF) module to incorporate complementary features for improving each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Compared with the most related work [2] which introduces a reconstruc- tion task for unlabeled data, their two tasks lack deep interaction and feature reuse, thus cannot collaborate and facilitate each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Besides, our network not only makes full use of massive unlabeled data, but also explores more complete lesion regions for limited labeled data through the correction phase, which can be seen as “image-level erase [21]” or “reverse attention [3]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Our contribution is summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' (1) We propose a novel self- supervised semi-supervised learning paradigm for general lesion region segmenta- tion of medical imaging, and verify that the pretext self-supervised learning task of inpainting the lesion region at the pixel level can effectively enhance the fea- ture learning and greatly reduce the algorithm’s dependence on large-scale dense annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' (2) We propose a dual-task framework for semi-supervised medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Through introducing the inpainting task, we create supervi- sion signals for unlabeled data to enhance the network’s representation learning of lesion regions and also exploit additional lesion features for labeled data, thus effectively correct the initial segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' (3) We evaluate our method Self-Supervised Correction Learning 3 on three tasks, including polyp, skin lesion and fundus optic disc segmentation, under a semi-supervision setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The experimental results demonstrate that our method achieves superior performance compared with other state-of-the-art semi-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='1 Overview In this work, we adopt a coarse-to-fine strategy and propose a self-supervised correction learning paradigm for semi-supervised biomedical image segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Specifically, we introduce inpainting as the pretext task of self-supervised learning to take advantage of massive unlabeled data and thus construct a dual- task network, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Our proposed framework is composed of a shared encoder, two decoders and five GFF modules placed on each layer of both decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' We utilize ResNet34 [8] pretrained on the ImageNet [5] as our encoder, which consists of five blocks in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Accordingly, the decoder branch also has five blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Each decoder block is composed of two Conv-BN-ReLU combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' For the convenience of expression, we use Eseg, Dseg to represent the encoder and decoder of the segmentation branch, and Einp and Dinp for those of the inpainting branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The overview of our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Both encoders share weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' G1-G5 represent five GFF modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The red and blue arrows denote the input and output of our network in the first and second stage respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In the first step, given the image x ∈ RH×W ×C, in which H,W,C are the height, width and channels of the image respectively, we use the segmenta- tion branch Eseg, Dseg with skip-connections, the traditional U-shape encoder- decoder structure, to obtain a coarse segmentation map ˆycoarse and then mask the original input based on its binary result ycoarse by the following formulas: ˆycoarse = Dseg(Eseg(x)) (1) G5G4G3G2G1 pare welgh G1G2 G3 / G4 | G54 Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' xmask = x × (1 − ycoarse) (2) In the second phase, the original image x and the masked image xmask are sent into Eseg and Einp simultaneously to extract features eseg and einp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Obvi- ously, eseg is essential for the inpainting task, and since the initial segmentation is usually inaccurate and incomplete, einp may also contain important residual lesion features for the correction of the initial segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In order to adap- tively select the useful features of einp and achieve complementary fusion of eseg and einp, we design the GFF modules (G1-G5) and place them on each decoder layer of both branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Specifically, for the ith layer, the features ei seg and ei inp are delivered into Gi through skip-connections to obtain the fusion ei = Gi(ei seg, ei inp), and then sent to the corresponding decoder layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Thus, both Gi of the two branches shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 1 actually share parameters, taking the same input and generating the identical output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' To enhance the learning of the GFF mod- ules,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' we adopt a deep supervision strategy and each layer of the two decoder branches generate a segmentation result and an inpainting result respectively by the following formulas: ˆyi fine = � Di seg([ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' di+1 seg ]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 4 Di seg(ei),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' i = 5 (3) ˆxi = � Di inp([ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' di+1 inp ]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 4 Di inp(ei),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' i = 5 (4) Where [· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' ·] denotes the concatenation process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' and di+1 seg ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' di+1 inp represent the features from previous decoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The deep supervision strategy can also avoid Dinp directly copying the features of the low-level eseg to complete the inpainting task without in-depth lesion feature mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The output of the last layer ˆy1 fine is the final segmentation result of our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Gated Feature Fusion Module inp c conv conv conv 1-Self-Supervised Correction Learning 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='2 Gated Feature Fusion (GFF) To better incorporate complementary features and filter out the redundant in- formation, we design the GFF modules placed on each decoder layer to integrate the features delivered from the corresponding encoder layer of two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The details are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Our GFF module consists of a reset gate and a select gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Specifically, for the Gi placed on the ith decoder layer, the value of two gates is calculated as follows: ri = σ(Wr � ei seg, ei inp] � (5) si = σ(Ws � ei seg, ei inp] � (6) Where Wr, Ws denote the convolution process, taking the concatenation of ei seg and ei inp as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' σ represents the Sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' ri and si represent the value of the reset gate and the select gate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Since the input of the inpainting branch is the masked image, the reset gate is necessary to suppress massive invalid background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' And then the select gate achieves adap- tive and complementary feature fusion between the reintegrated features ˜ei and the original segmentation feature ei seg by the following operations: ˜ei = W � ri × ei inp, ei seg � ) (7) ei = si × ˜ei + (1 − si) × ei seg (8) where W also represents the convolution process to make the reintegrated fea- tures ˜ei have the same dimension with ei seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='3 Loss Function We only calculate loss in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The labeled dataset and unlabeled dataset are denoted as Dl and Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' For the labeled data xl ∈ Dl, yl is the Ground Truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Since we adopt the deep supervision strategy, the overall loss is the sum of the combination of Binary CrossEntropy (BCE) loss and Dice loss between each output and the Ground Truth: Lseg(xl) = 5 � i=1 Li BCE(ˆyi l, yi l) + Li Dice(ˆyi l, yi l) (9) where ˆyi l, yi l denote the segmentation map ˆyi fine of the ith decoder layer and the corresponding down-sampling Ground Truth yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' For unlabeled data xu ∈ Du, the inpainting loss is the sum of L1 loss between each inpainting image and the original image in the masked region: Linp(xu) = 5 � i=1 yi u × ��ˆxi u − xi u �� (10) 6 Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' where ˆxi u, xi u and yi u represent the inpainting image, down-sampling original image and binary segmentation result of the ith decoder layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In the end, the total loss function is formulated as follows: L = λ1 � xl∈Dl Lseg(xl) + λ2 � xu∈Du Linp(xu) (11) where λ1, λ2 are weights balancing the segmentation loss and the inpainting loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' And we set λ1 = 2 and λ2 = 1 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 3 Experimental Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='1 Dataset and Evaluation Metric We conduct experiments on a variety of medical image segmentation tasks to verify the effectiveness and robustness of our approach, including polyp, skin lesion and fundus optic disc segmentation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Polyp Segmentation We use the publicly available kvasir-SEG [10] dataset containing 1000 images, and randomly select 600 images as the training set, 200 images as the validation set, and the remaining as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Skin Lesion Segmentation We utilize the ISBI 2016 skin lesion dataset [7] to evaluate our method performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' This dataset consists of 1279 images, among which 900 are used for training and the others for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Optic Disc Segmentation The Rim-one r1 dataset [6] is utilized in our ex- periments, which has 169 images in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' We randomly split the dataset into a training set and a test set with the ratio of 8:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Evaluation Metric Referring to common semi-supervised segmentation set- tings [13, 22], for all datasets, we randomly use 20% of the training set as the labeled data, 80% as the unlabeled data and adopt five metrics to quan- titively evaluate the performance of our approach and other methods, including “Dice Similarity Coefficient (Dice)”, “Intersection over Union (IoU)”, “Accuracy (Acc)”, “Recall (Rec)” and “Specificity (Spe)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='2 Implementation Details Data pre-processing In our experiments, since the image resolution of all datasets varies greatly, we uniformly resize all images into a fixed size of 320×320 for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' And in the training stage, we use data augmentation, including random horizontal and vertical flips, rotation, zoom, and finally all the images are randomly cropped to 256 × 256 as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Training details Our method is implemented using PyTorch [18] framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' We set batch size of the training process to 4, and use SGD optimizer with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='9 and a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='00001 to optimize the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' A poly learning rate policy is adopted to adjust the initial learning rate, which is lr = init lr ×(1− iter max iter)power, where init lr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='001, power = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The total number of epochs is set to 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Self-Supervised Correction Learning 7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Comparison with other state-of-the-art methods and ablation study on the Kvasir-SEG dataset Methods Data Dice IoU Acc Rec Spe Supervised 600L (All) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='48 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='69 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='34 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='06 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='58 Supervised 120L 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='40 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='18 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='09 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='35 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='55 DAN [24] 120L + 480U 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='77 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='12 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='37 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='86 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='53 MT [20] 120L + 480U 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='99 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='84 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='21 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='81 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='79 UA-MT [22] 120L + 480U 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='70 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='34 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='38 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='51 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='40 TCSM V2 [13] 120L + 480U 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='17 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='15 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='38 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='14 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='76 MASSL [2] 120L + 480U 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='45 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='61 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='34 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='18 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='32 Ours 120L + 480U 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='14 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='49 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='42 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='78 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='89 Ours (add) 120L + 480U 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='59 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='56 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='12 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='98 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='26 Ours (concat) 120L + 480U 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='09 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='98 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='21 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='54 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='3 Comparisons with the State-of-the-Art In our experiments, ResNet34 [8] based UNet [19] is utilized as our baseline, which is trained using all training set and our selected 20% labeled data sep- arately in a fully-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Besides, we compare our method with other state-of-the-art approaches, including DAN [24], MASSL [2], MT [20] and its variants (UA-MT [22], TCSM V2 [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' All comparison methods adopt ResNet34UNet as the backbone and use the same experimental settings for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' On the Kvasir-SEG dataset, Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 1 shows that our method ob- tains the outstanding performance compared with other semi-supervised meth- ods, with Dice of 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='14%, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='74% improvement over the baseline only using the 120 labeled data, outperforming the second best method by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='69%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' On the ISBI 2016 skin lesion dataset, we obtain a 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='95% Dice score, which is superior to other semi-supervised methods and very close to the score of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='38% achieved by the baseline using all training set images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' On the Rim-one r1 dataset, we can conclude that our method achieves the best performance over five metrics, further demonstrating the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Note that detailed results on the latter two datasets are listed in the supplementary mate- rial due to the space limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Some visual segmentation results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 3 (col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='1-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content='4 Ablation study Effectiveness of our approach with different ratio of labeled data We draw the curves of Dice score under three settings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' To verify that our proposed framework can mine residual lesion features and enhance the lesion representation by GFF modules in the second stage, we conduct experiments and draw the blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The blue line denotes that our method uses the same labeled data with the baseline (the red line) to perform the two-stage process, without utilizing any unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Note that we only calculate the segmenta- tion loss for the labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The performance gains compared with the baseline 8 Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Visual comparison of various lesion segmentation from state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Our proposed method consistently produces segmentation results closest to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The inpainting result is shown in the rightmost column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' show that our network mines useful lesion information in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The green line means that our method introduces the remaining as unlabeled data for the inpainting task, further enhancing the feature representation learning of the lesion regions and improving the segmentation performance, especially when only a small amount of labeled data is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' When using 100% labeled data, the green line is equivalent to the blue line since no additional unlabeled data is utilized to do the inpainting task, thus maintaining the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Effectiveness of the GFF modules To verify the effectiveness of the GFF modules, we also design two variants, which merge features by directly addition and concatenation, denoting as Ours (add) and Ours (concat) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' In Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 1, we can observe performance degradation by both approaches compared with our method, proving that the GFF module plays a significant role in filter- ing redundant information and improving the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' The performance of our method with different ratio of labeled data on the Kvasir-SEG dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 90 88 Dice Score(%) 86 84 82 ResNetUnet (fully-supervised) Ours (fully-supervised) Ours (semi-supervised) 80 10 20 30 40 50 60 70 80 90 100 Labeled Data Percentage(%)Self-Supervised Correction Learning 9 4 Conclusions In this paper, we believe that massive unlabeled data contains rich context and structural information, which is significant for lesion segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Based on this, we introduce the self-supervised inpainting branch for unlabeled data, co- operating with the main segmentation task for labeled data, to further enhance the representation for lesion regions, thus refine the segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' We also design the GFF module for better feature selection and aggregation from the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Experiments on various medical datasets have demonstrated the superior performance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Acknowledgement This work is supported in part by the Key-Area Research and Development Pro- gram of Guangdong Province (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 2020B0101350001), in part by the Guangdong Basic and Applied Basic Research Foundation (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 2020B1515020048), in part by the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 61976250) and in part by the Guangzhou Science and technology project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' 202102020633).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=' Chaitanya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQfDAuV/content/2301.04866v1.pdf'} +page_content=', Erdil, E.' 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Hu1,2, Lei Hsin Kuo1, and Jia Liu1 +1Department of Mathematics and Statistics, University of West Florida +2College of Information Engineering, Hubei University of Chinese Medicine +Abstract +Complex network analysis has brought significant advances in uncovering network mesoscopic +properties. Community detection is one of the significant features of understanding real-world +complex systems. In this paper, we propose a High-order node proximity Spectral Clustering on +Ratios-of-Eigenvectors (SCOREH+) algorithm for finding communities in complex networks. This +algorithm preserves high-order transitivity information of the network affinity matrix. First, we +construct the high-order proximity matrix from the original affinity matrix using the Radial Basis +Functions (RBFs) and Katz index. Furthermore, we obtain the normalized Laplacian matrix and +the normalized leading eigenvectors. The ratios of the leading eigenvectors aid in mitigating the +effect of degree heterogeneity. Moreover, we implement a procedure that joins an additional eigen- +vector (the (K + 1)th leading eigenvector) to the spectrum domain for clustering if the network is +considered to be a “weak signal” graph. Finally, we apply the K-means algorithm to the spectrum +domain for acquiring the node labels. We compare our SCOREH+ algorithm with spectral cluster- +ing (SC), Spectral Clustering on Ratios-of-Eigenvectors (SCORE), and SCORE+. To demonstrate +the high effectiveness of our algorithm, we conducted comparison experiments on 11 real-world +networks and several synthetic networks with noise. The experimental results demonstrate that +our SCOREH+ outperforms SC, SCORE, and SCORE+ on most of these networks. In addition, +we find that by tuning the RBFs and their shaping parameters, we can obtain state-of-the-art +community structures on all real-world networks and even on noisy synthetic networks. +1 +Introduction +Complex networks model a large number of entities and their relations as nodes and edges in real- +world scenarios, which are ubiquitous and can be applied to any data as long as pair-wise interactions +exist among the objects. +Non-trivial topological features preserved in the network structure have +attracted researchers from various fields, for example, biology [1], climate [2], sociology, epidemiology +[3,4], etc. In complex networks, community detection is one of the kinds that discovers the clusters in +the network model to mine latent information among the objects. The nodes are densely connected by +edges within clusters while they are sparsely connected between clusters. Researchers have developed +various algorithms to discover community structures, for example, Walktrap [5], Infomap [6], Louvain +algorithm [7], deep learning-based algorithms [8–10], diffusion-based algorithms [11,12], etc. +Spectral clustering, rooted in graph theory, is one of the state-of-the-art algorithms for detecting +communities in complex networks. A modern spectral clustering algorithm consists of three procedures: +(1) regularization of a suitable adjacency or Laplacian matrix, (2) a form of spectral truncation, and +(3) a K-means algorithm on the reduced spectral-domain [13]. For the first step, the formation and +selection of the graph proximity method and Laplacian are significant. A similarity matrix models +the local neighborhood relationships of pair-wise data points. Researchers typically use the network +affinity matrix as the node similarity representation. However, some researchers use similarity measures +to compute a new similarity matrix. +Radial Basis Functions (RBFs) are commonly used kernels +in the construction similarity matrices [14, 15]. +Furthermore, the Laplacian matrix is obtained by +subtracting the similarity matrix from the degree matrix. In the third step, the number of clusters +is a prerequisite. Nonetheless, by decomposing the Laplacian matrix, the first large gap between two +1 +arXiv:2301.02885v1 [cs.SI] 7 Jan 2023 + +eigenvalues generally indicates the number of clusters. That is to say, the number of eigenvalues before +this gap is the number of clusters [16]. However, this approach lacks a theoretical justification [17]. +The biggest challenges of spectral clustering and community detection are presented as follows: +• Despite the effectiveness of spectral clustering in real networks, using the network affinity matrix +as the pair-wise nodes’ similarity information is insufficient to capture local information. +• Given the number of clusters K, researchers generally preserve the exact top K eigenvectors +for posting K-means clustering. However, for some networks, other eigenvectors may also carry +information for clustering [18]. +• RBFs have many genres, and they all have a shaping parameter to tune to achieve good results +for application scenarios. A bad choice for an RBF can cause serious bad results. +To address these challenges, we propose an effective community detection algorithm: High-order +node proximity Spectral Clustering on Ratios-of-Eigenvectors (SCOREH+). It utilizes RBFs to ap- +proximate the similarity matrix from the original affinity matrix and considers its high-order proximity. +We also analyze the Eigen distribution and then determine if one more eigenvector is necessary. Fur- +thermore, we give a range of optimal RBF shaping parameters. To test our algorithm and demonstrate +its superiority, we collected 11 real-world networks that span various areas, ranging from social sciences +to political sciences. In addition, we also generate benchmark networks using a criterion first presented +by Lancichinetti, Fortunato, and Radicchi [19], in short, the LFR benchmark. Taking advantage of +these datasets, we analyzed the results from multi-views concerning the number of clusters and mixing +parameters. +Overall, our paper makes the following contributions: +• Our algorithm casts the nodes as the basis of an approximation space. Then, we use the Katz +index to compute the high-order proximity of the nodes. +Similarity only measures the close +relationship between a node and its neighbors. In this approach, we can preserve more node- +local information. +• By calculating the gap between the Kth and the (K + 1)th eigenvalue, we can easily determine +if an additional eigenvector is worth maintaining for clustering. The (K + 1)th eigenvector as an +additional feature can facilitate the clustering performance for some networks. +• We run our algorithm on various real-world networks and synthetic networks generated using +various parameters. Numerical results show that each RBF has a shaping parameter domain +where we can achieve optimal results. This finding can help tune RBF parameters in future wide +applications. +The remainder of this paper is organized as follows. +Section 2 demonstrates the related work +of spectral clustering, SCORE/SCORE+ algorithms, high-order proximity applications, and RBF +applications. Section 3 illustrates the algorithm, including its design, steps, and pseudocode. Section +4 shows the experimental design, data sets, evaluation metrics, experimental results, and their analyses. +Finally, we give the conclusion and future work in Section 5. +2 +Related Work +In this section, we present the related work concerned with the community detection algorithms, +SCORE, and SCORE+ algorithms, Higher-order proximities, and RBF applications, respectively. +2.1 +Spectral Clustering +Community detection algorithm is a basic tool that enables us to discover the organizational principles +in the network, it plays a huge role in solving various complex problems in real life, and it is also a +research hotspot in various disciplines [20], [21]. +Zhuang et al. [22] proposed a novel modular-based dynamic community detection algorithm, Dy- +naMo, which aims to detect dynamic network communities in a more efficient manner, which is as +effective as repeated application of static algorithms. Experimental results show that DynaMo outper- +forms comparison algorithms in terms of effectiveness. Shang et al. [23] proposed a local community +2 + +detection algorithm based on high-order structure and edge information (HSEI). Use edge information +to mine the membership strength between nodes and communities,so as to obtain more complete local +community members. +Spectral clustering is one of the common community detection algorithms, it uses information from +the eigenvalues of the Laplacian of an affinity matrix and maps the nodes to a low-dimensional space +where data is more separable, enabling us to perform Eigen decomposition and form clusters [24]. +Zhu et al. [25] proposed a Low-rank Sparse Subspace (LSS) clustering method to learn an affinity +matrix from the original low-dimension features. +The affinity matrix is learned dynamically at a +fast speed so that the optimal clustering results are promised. Hu et al. [26] utilized a probabilistic +Dirichlet process to infer the number of clusters from the affinity matrix and then applied the gaussian +mixture model to the feature matrix to acquire communities. As a result, the detection accuracy is +improved. Yang et al. [8] developed a robust deep learning framework for discriminative embedding and +spectral clustering. The method is efficient and of high precision on benchmark datasets. Sharma et +al. [27] constructed the self-adaptive mixture similarity measure (SAM) and combined it with spectral +clustering for uncertain datasets. Their method was verified to be more effective than other state-of- +the-art methods using experimental comparisons and null hypothesis significance tests. +2.2 +SCORE and SCORE+ +Jin et al. [28] first proposed a Spectral Clustering On Ratios-of-Eigenvectors (SCORE) algorithm, +which uses the entry-wise ratios between eigenvectors for clustering to improve the effectiveness of +spectral clustering. SCORE effectively removes the effect of degree heterogeneity by taking entry-wise +ratios between the first leading eigenvector and each of the other eigenvectors. The SCORE can be +extended in various directions. First, SCORE applies to a large class of methods that utilize scaling- +invariant mapping. Second, the Degree Corrected Block Model (DCBM) [29] can be generalized to +more realistic models, where the spectral methods could continue to work well. +Jin et al. proposed Mixed-SCORE [30] in 2017 and derived the convergence rate of this algorithm +using delicate spectral analysis. In particular, this algorithm enjoys tight row-wise deviation bounds for +the rational number region. It solves the mixed membership estimation problem and has been applied +to four network data sets with encouraging results. Inspired by SCORE, Ke et al. [31] proposed a +pre-SVD normalization, which adopted the SCORE method to normalize singular vectors. This novel +algorithm discovered a low-dimensional post-SVD simplex. The authors provided theoretical properties +and carefully studied the rate-optimality. As a result, pre-SVD normalization has a faster convergence +rate than existing methods in a wide variety of real applications. +Furthermore, SCORE provided novel ideas about computing communities in networks. Later in +2018, Jin et al. [18] applied two normalizations and Eigen selections to improve the performance +of SCORE. The new algorithm SCORE+ demonstrated the rationality of Laplacian regularization +as a pre-PCA normalization and retained an additional eigenvector as a post-PCA normalization. +SCORE+ has two tuning parameters, but each is easy to set and not sensitive. It is guided by common +sense in this paper. Therefore, SCORE is fast, and SCORE+ is slightly slower. The experimental +results showed that the clustering error rate was reduced dramatically compared to SCORE on testing +networks. Researchers [32, 33] have borrowed ideas from SCORE and SCORE+ for the community +detection field. +2.3 +Higher-Order Proximities +In networks, to measure the similarity of every pair of nodes, the adjacency matrix, and Laplacian ma- +trix represent the first-order proximity, which simulates the local pair-wise proximity between vertices. +Cosine similarity, Euclidean similarity, Jaccard similarity, et al., are also popularly used. However, +these similar methods can only preserve local information by using its connectivity to its neighbors. +They are not sufficient to fully simulate the pair-wise proximity between nodes. Therefore, how to +preserve high-order proximity has become a hot topic recently. People have also explored higher-order +similarities to simulate the strength between two nodes [34,35]. +Three commonly used high-order proximities are Common Neighbors and Propagation [36], Katz +Proximity [37], and Eigenvector Centrality [38]. The Katz index was proposed by Katz [37] to compute +the similarity of two nodes in a heterogeneous network by computing the walks between two nodes. +3 + +It has been used in graph embedding [39], complex networks [40], and relationship prediction in +networks [41,42]. +Ou et al. [39] proposed applying multiple high-order proximity measurements, e.g., Katz index +[37] on the graph embedding task. +This work has attracted much attention. +Plenty of proximity +measurements have emerged in the last century. Cosine similarity is a simple index used to determine +the number of common neighbors between two nodes. The Katz index is a widely used measurement +that considers the total number of walks between two nodes rather than the shortest one. +Furthermore, the Katz similarity has been used to predict new potential lncRNA and environmental +factors (EF) associations [43]. The author computed the high-order similarity value of every pair of +nodes with the Katz measure in a heterogeneous network through the number of walks. Zhang et +al. [44] proposed using the linear combination of the polynomial function of the affinity matrix to +preserve the high-order proximity. Their method preserves arbitrary-order proximities in the network +embedding tasks. +2.4 +RBF applications +The Gaussian similarity function exp(−r2/(2c2)) is the most common Radial Basis Function (RBF) or +similarity function in the neural networks, where r is the distance between the two nodes and c is the +shaping parameter. This function is equivalent to the Gaussian RBF in Table 1. The Multiquadric +(MQ) RBF is effective in geographical data sets, and the density of the local dataset determines the +shaping parameter c. The selection of RBFs used for the interpolation matrix is strongly problem- +dependent. On the other hand, the interpolation matrix, which is the same weight matrix we use in the +complex network, is highly ill-conditioned. Therefore, the selection of RBFs and the parameters are +significant. Zhang et al. [45] proposed a framework that integrates the attention mechanism and auto- +kernel learning. The hyperparameter tuning for kernels largely facilitated improving the performance +of graph convolutional networks. +In the network sciences, researchers consider undirected graphs where the weighted adjacency +matrix W = WT is symmetric. The structures of the networks could be studied by exploring the +structures of the matrix W [46]. The pattern of the vertices and edges of the adjacency graph or the +corresponding adjacency matrix may reveal information about the network’s divisions, clusters, and +communities. The first step is to transform the given data set into a graph called a “similarity graph”. +The goal of constructing the similarity graph is to model the local neighborhood relationships from +the network data. +This paper considers the construction based on the Radial Basis Function (RBF) interpolations +and forms a high-order proximity matrix. +3 +The High-order Proximity Preserved Spectral Clustering +We present a community detection algorithm: High-order node proximity Spectral Clustering on +Ratios-of-Eigenvectors (SCOREH+). Firstly, we introduce an efficient spectral clustering algorithm +that uses the ratios of eigenvectors and the Eigen selection strategy. Then, we derive how to preserve +higher-order proximities in the networks using the RBF and Katz index to capture more node-local +information. Before introducing the detailed derivation, we clarify the symbols and definitions that +will be used. +3.1 +Notations +We define an undirected graph G = {V, E} with n nodes (vertices) and m edges (links), and V, E +are the node (vertex) set and edge (link) set of the network, respectively. Let An×n be the affinity +matrix of a network with n nodes and m edges, and Aij ̸= 0 if an edge exists between nodes vi and +vj otherwise 0. A node v ∈ V is represented by {vi}n +i=1 = {Avi}n +i=1. The degree matrix is denoted +by Dn×n, where the diagonal value Dii is the degree of the corresponding node i and the off-diagonal +elements are 0. The Laplacian matrix L is obtained by the equation L = D − A. We assume that the +number of clusters K is given for each network. +4 + +3.2 +The Algorithm +The algorithm constructs the high-order proximity matrix while preserving the high-order transitivity +information from the original affinity matrix using the RBF technique and Katz index. From the +high-order proximity matrix, we obtain the normalized graph Laplacian. Next, we normalize the K +leading eigenvectors of the proximity matrix by dividing the leading eigenvectors into an additional +(K + 1)th eigenvector, which will be used to cluster if the network is considered to be a “weak signal”. +3.2.1 +Radial Basis Functions +For a given node vector {vi}n +i=1 ∈ A, which contains n distinct elements in the computational domain, +Ω ⊆ Rd. The approximation that utilizes RBF is an unknown function f, which can be expressed as +linear combinations of data norms. +f (v) ≈ ˜f (v) = +n +� +i=1 +αiΦ (∥v − vi∥) , +v ∈ Ω +(1) +where ∥·∥ represents the Euclidean norm on Rd, αi is the coefficients, and Φ : Rd → R is called a +Radial Basis Function (RBF) [47] if, +Φ (v) = Φ (u) , +whenever +∥v∥ = ∥u∥, +v, u ∈ Rd +(2) +In Table 1, we list the most common RBFs that have been widely used in neural networks and the +numerical approximations. The symbol c is a shaping parameter, and the symbol r in the table denotes +the Euclidean distance of v ∈ Rd from the original point, r = ∥v∥2 = +��d +i=1 x2 +i +Table 1: Some common choices of RBFs +Choice of RBF +Definition +Multiquadric (MQ) +Φ(r, c) = +√ +c2 + r2 +Inverse Multiquadric (IMQ) +Φ(r, c) = 1/ +√ +c2 + r2 +Gaussian +Φ(r, c) = exp +� +r2/c2� +Follows Equation (1) with the collocation scheme, +˜f (vi) = f (vi) , +i = 1, 2, · · · , n +(3) +leads to a system of linear equations, +Wα = f +(4) +where α = [α1, · · · , αn]T , f = [f (v1) , · · · , f (vn)]T , and a matrix Wij = Φ (rij) ∈ Rn×n. The distance +matrix, rij, contained within W can be expressed as follows, +rij = +� +�� +∥u1−v1∥2 +· · · +∥u1−vn∥2 +... +... +... +∥un−v1∥2 +· · · +∥un−vn∥2 +� +�� , +i, j =1,· · ·, n +(5) +After applying RBF to the data sets, we obtain the similarity graph and the weighted matrix Wij, +with entries Wij = Φ(∥vi − vj∥2), i = 1, · · · , n, also the interpolation matrix. Wij consists of the +functions serving as the basis of the approximation space. For distinct data points in the data sets +and a constant shape parameter c, Wij is a nonsingular matrix. Both the choice of the RBF and its +corresponding shaping parameter plays an important role in calculating the final interpolations and +partitions in the resulting graph. +5 + +3.2.2 +Higher-Order Proximities +The Katz index [37, 39] computes the relative influence of a node within a network. We call nodes +that are directly connected to a node as immediate neighbors. Therefore, the Katz index measures +the number of immediate neighbors and the immediate neighbors of its immediate neighbors. It is a +weighted summation of the path node-set between two nodes. The weight of a path is an exponential +function of its length (the number of nodes on this path). We formularize the Katz index as: +K = (I − β · W)−1 · β · W +(6) +where β is a decay parameter, it determines the weight of a path decay speed as the length of the +path grows. β should be properly set to preserve the series convergence. In practice, β must be smaller +than the spectral radius of the weighted matrix W. Conventionally, in this paper, we set β to 0.0025. +The pseudocode for computing the high-order proximity of an affinity matrix using Gaussian RBF is +shown in Algorithm 1. +This algorithm first generates a list of N shaping parameters c (Line 2). Then iteratively find +an optimal shaping parameter (Line 3-4) where GaussianRBF(·,·) computes the distance using the +Gaussian RBF. We can compute MQ and iMQ RBFs using the procedures analogous to Algorithm 1 +by substituting Line 4 with the respective RBF distances. +Algorithm 1: High-order Proximity (HOP) +Input +: Affinity matrix: A ∈ Rn×n +Output: High-order matrix: K +1 x ← linspace(0, 1, n); +2 c ← linspace(0, 1, 100); +3 for i ← 1 to n do +4 +B ← GaussianRBF(ci, DMatrix(xT , xT )); +5 C ← optimal B; +6 ˆC ← C · A; +7 K ← Katz( ˆC); +3.2.3 +Normalized Eigens +The spectrum (eigenvalues) of the network similarity matrix is used in spectral clustering. +This +procedure works similarly to dimensionality reduction. +Using RBF and Katz index, we obtained the high-order similarity matrix K. The network’s diagonal +matrix D = diag(K) is obtained. The normalized Laplacian matrix Lσ can then be formed. The graph +laplacian matrix with ridge regularization σ is: +Lσ = (D + σ · dmax · I)− 1 +2 K(D + σ · dmax · I)− 1 +2 +(7) +where dmax is the maximum node degree of the network. The empirical setting of σ is 0.1. +Next, we compute K +1 largest eigenvalues ˆλ and their corresponding eigenvectors ˆΞ, a.k.a. K +1 +leading eigenvectors, and sort them in non-descending order by ˆλ. Consequently, the feature matrix’s +dimension is reduced from n × n to n × (K + 1). The reduced feature matrix, Θ, is expressed as: +ˆΘ = ˆΞ · Diag(ˆλ) +(8) +where Diag(ˆλ) forms a diagonal matrix Mn×n where Mij = ˆλi for all i = j such that 0 ≤ i, j < n; +otherwise 0. +3.2.4 +Eigen-selection and Clustering +We support that the weighted matrix K contains “signal” and “noise” network information. A network +with “strong signal” has a large gap between the Kth and the (K + 1)th eigenvectors of its Laplacian +6 + +matrix. We have a threshold t > 0 to determine whether a network has a weak signal profile by, +ˆλK+1 +ˆλK +≥ 1 − t +(9) +If the above equation (9) is satisfied, then we say that the weighted matrix is of “weak signal” profile. +Therefore, the (K + 1)th eigenvector contains useful information as the Kth for community detection. +Consequently, we consider it as one more feature that contributes to label clustering. Finally, we apply +the K-means algorithm to the new feature matrix with K +1 dimensions to compute the communities. +The pseudocode for community detection is below in Algorithm 2 and the implementation of the +algorithms in this paper is publicly accessible on https://github.com/yz24/RBF-SCORE. In Algo- +rithm 2, Line 1 computes the high-order proximity of a network from the original affinity matrix using +Algorithm 1 and returns a matrix with the same dimension as its input. Then, collect eigenpairs +(eigenvalue, eigenvector) of K and arrange them in decreasing order by eigenvalues (Lines 2 to 6). +Lines 8 to 9 determine if the network has a strong or weak signal profile and assign k′ accordingly. +The algorithm returns a list of clustered node labels. +Algorithm 2: High-Order Node Proximity Spectral Clustering on Ratios-of-Eigenvectors +(SCOREH+) +Input +: A ∈ Rn×n, σ > 0, t ∈ (0, 1), k ∈ N≥1 +Output: Node labels: ˆy +1 K ← HOP(A); +2 D ← diag(K); +3 Lσ ← (D + σ · dmax · I)− 1 +2 K(D + σ · dmax · I)− 1 +2 ; +4 k′ ← k + 1; +5 ˆλ, ˆΞ ← eigsh(Lσ, k′); +6 ˆΛ ← Diag(ˆλ) ; +7 Θ ← ˆΞ · ˆΛ; +8 if +ˆλr +ˆλk < 1 − t then +9 +k′ ← k; +10 ˆE ← { +ˆΘh +ˆΘ1 , · · · , +ˆΘk′ +ˆΘ1 }; +11 ˆy ← K-means(ˆE, k); +4 +Experiments +We conduct experiments on several real-world and synthetic datasets. We first introduce the experi- +ment design and then show the results and analyses on these two types of datasets. Furthermore, we +also analyze the RBF parameter selections and the domains for optimal results. For synthetic networks, +we demonstrate the noise resistance ability of our algorithm compared to the baseline algorithms. +4.1 +Datasets +4.1.1 +Real-world Networks +We tested our algorithm and other algorithms (SC, SCORE, and SCORE+) on 11 public real-world +network datasets from various areas, including social sciences, political sciences, and so forth. These +networks are Les Mis´erable [48], Karate [49], Football [50], Dolphins [51], Blog [52], Simmons [53], +Caltech [53], UKfaculty [54], Github, Facebook [55], and Polbooks 1. The descriptions for each of +them as shown below, and the basic statistics are in Table 2. +1 This network was not published, but it is public on this website: http://www.orgnet.com/divided.html +7 + +Table 2: Statistics of real-world datasets. For 11 real-world datasets, the number of nodes is n, the +number of edges is m, the number of clusters is k, the minimum and maximum of the node degrees +and the average shortest path length are min(d), max(d) of each network. +No. +Dataset +Source +n +m +k +min(d) +max(d) +1 +Les Mis´erable +[48] +77 +254 +11 +1 +36 +2 +Caltech +[53] +590 +12, 822 +172 +1 +179 +3 +Dolphins +[51] +62 +159 +2 +1 +12 +4 +Football +[50] +110 +568 +11 +7 +12 +5 +Karate +[49] +34 +78 +2 +1 +17 +6 +Polbooks +[56] +92 +374 +2 +1 +24 +7 +Blog +[52] +1, 222 +16, 714 +2 +1 +351 +8 +Simmons +[57] +1, 137 +24, 257 +4 +1 +293 +9 +UKfaculty +[54] +79 +552 +3 +2 +39 +10 +Github +[55] +37, 700 +289, 003 +2 +1 +9, 458 +11 +Facebook +[55] +22, 470 +171, 002 +4 +1 +709 +4.1.2 +Synthetic Networks +We generate a series of synthetic networks using the LFR criteria, first proposed by Lancichinetti, +Fortunato, and Radicchi [19]. A network can be generated in terms of the given parameters: the +power-law exponent for the degree distribution is τ1, the power-law exponent for the community size +distribution is τ2, the number of nodes is N, the average degree is ⟨k⟩, the minimum of communities +is c, and the mixing parameter is µ. Most importantly, µ, one of the key parameters, controls the +fraction of edges between communities. Thus, it reflects the amount of noise in the network. When +µ = 0, all links are within community links; when µ = 1, all links are between nodes from different +communities. +We generate networks by setting the number of nodes ranging from 150 to 10, 000, and the key +parameter µ (mixing parameter) from 0.15 to 0.85. +4.2 +Evaluation Metrics +We use two evaluation metrics: modularity and normalized mutual information (NMI), to verify the +performance of the proposed SCOREH+. The former does not require the ground truth of the network +while the latter does. For both of them, the higher the metrics are, the better the communities are. +4.2.1 +Modularity +Newman and Girvan [58] proposed modularity Q to assess the quality of the detected community +structure. It represents the difference between the actual number of connections and the expected +number of connections in random graphs. The equation is as follows: +Q = +k +� +s=1 +� +ls +l − +�ds +2l +�2� +(10) +Where k is the number of communities in the network, l is the sum of all edges in the network, ls is +the sum of all edges in the community s, and ds is the sum of the degree of all nodes in s. +4.2.2 +Normalized Mutual Information (NMI) +Let ˆy be the list of community labels obtained from an algorithm, and y be the list of ground-truth +labels. Denote H(·) as the entropy function for a graph partitioning. Then, the mutual information +between the ground truth and the detected labels is: +MI(ˆy, y) = +|ˆy| +� +i=1 +|y| +� +j=1 +|ˆyi ∩ yj| +n +log +�n|ˆyi ∩ yj| +|ˆyi||yj| +� +(11) +Then the normalized mutual information is: +NMI(ˆy, y) = +MI(ˆy, y) +mean(H(ˆy), H(y)) +(12) +Because a higher metric value of modularity Q and NMI represent better community discovery, we +bold the highest values and underline the second highest values in the comparison tables. +8 + +(a) +(b) +Figure 1: The comparison plots of Modularity and NMI on real-world datasets. +4.3 +Experiment Design +Our experiments were carried out on a 16.0 GB RAM, 1.90GHz Intel(R) Core(TM) i7-8650U CPU. +we select spectral clustering (SC) [24], Spectral Clustering on Ratios-of-Eigenvectors (SCORE) [28], +and SCORE+ [18] as baseline algorithms for comparisons. We test the proposed algorithm and the +baselines on the above-mentioned two types of data sets. For each single network, we run each model +ten times on it, evaluate two metrics, and then report the mean and variance of the results in the form +of mean (variance). +4.4 +Simulation Experiments and Result Analyses: Real-World Networks +In this subsection, we compare our algorithm SCOREH+ with SC, SCORE, and SCORE+ on real- +world networks. We report the experimental results by scoring the quality of the discovered commu- +nities with Modularity and NMI, see Table 3 and Table 4, respectively. Moreover, we visualize the +community structures computed from our algorithm and other baselines. We compare and analyze the +topological structures of four small networks: Karate, Dolphin, Polbooks, and UKfalculty. +Furthermore, we analyze the RBF shaping parameters and their influence on the final community +discovery. We find that the choice of shaping parameters can make a difference in the results. However, +the shaping parameters have a relatively small domain to achieve an optimal result. +Table 3: Numerical results on real-world networks (Modularity) +No. +Dataset +SC +SCORE +SCORE+ +SCOREH+ +1 +Les Mis´erable +−0.019(0.021) +0.386(0.057) +0.239(0.038) +0.486(0.002) +2 +Caltech +0.39(0.0) +0.372(0.002) +0.373(0.001) +0.368(0.003) +3 +Dolphins +0.379(0.0) +0.276(0.0) +0.353(0.0) +0.379(0.0) +4 +Football +0.624(0.0) +0.622(0.021) +0.624(0.0) +0.622(0.01) +5 +Karate +0.36(0.0) +0.371(0.0) +0.36(0.0) +0.371(0.0) +6 +Polbooks +0.479(0.0) +0.473(0.0) +0.479(0.0) +0.479(0.0) +7 +Blogs +0.0(0.0) +0.423(0.0) +0.424(0.0) +0.415(0.001) +8 +Simmons +0.482(0.0) +0.462(0.0) +0.46(0.0) +0.447(0.004) +9 +UKfaculty +0.442(0.0) +0.262(0.131) +0.44(0.0) +0.442(0.0) +10 +Github +0.11(0.0) +0.185(0.01) +0.271(0.0) +0.281(0.0) +11 +Facebook +0.154(0.0) +0.145(0.0) +0.097(0.0) +0.175(0.0) +Table 4: Numerical results on real-world networks (NMI) +No. +Dataset +SC +SCORE +SCORE+ +SCOREH+ +1 +Les Mis´erable +0.377(0.038) +0.686(0.032) +0.616(0.022) +0.752(0.008) +2 +Caltech +0.63(0.001) +0.58(0.003) +0.574(0.003) +0.637(0.006) +3 +Dolphins +1.0(0.0) +0.588(0.0) +0.811(0.0) +1.0(0.0) +4 +Football +0.934(0.0) +0.946(0.022) +0.934(0.0) +0.958(0.012) +5 +Karate +0.836(0.0) +1.0(0.0) +0.836(0.0) +1.0(0.0) +6 +Polbooks +0.87(0.0) +0.924(0.0) +0.87(0.0) +0.87(0.004) +7 +Blogs +0.006(0.0) +0.725(0.0) +0.751(0.0) +0.646(0.001) +8 +Simmons +0.702(0.001) +0.621(0.0) +0.615(0.001) +0.658(0.005) +9 +UKfaculty +0.95(0.0) +0.658(0.187) +0.917(0.0) +0.95(0.0) +10 +Github +0.12(0.0) +0.212(0.01) +0.241(0.0) +0.307(0.0) +11 +Facebook +0.08(0.0) +0.11(0.0) +0.132(0.0) +0.146(0.0) +The modularity and NMI values of each algorithm are illustrated in Figure 1(a) and 1(b). It is +9 + +0'02 +0'02 +vtinsluboM +2.0 +0'S2 +0'32 +0'42 +0'22 +0'e2 +. . . .[.0 +S.0 +E.0 +0'4 +M +2.0 +a.0 +1.0 +8.0 +e.0 +一 +:2C :2C0BE +:2COBE+ ++ 2COBEH+Bjoa2 +nktscf) +C9]fGc +KSL9TG +c!funpFigure 2: Optimal shaping parameters with RBF choices on real-world networks +clearly shown that our algorithm achieves the best and second-best on most of the networks. +4.4.1 +RBF selection and tuning of shaping parameters on real-world networks +In section 3.2, we discussed the three common RBFs and their definitions. Each RBF has a shaping +parameter c, and it can affect the final graph interpolation matrix, which, as a consequence, can cause +a difference in the final result. Therefore, we fine-tune the shaping parameter c and find the optimal +shaping parameter where NMI is the criterion. +Table 3 and 4 show that our algorithm has already achieved the best results on Les Miserable, Cal- +tech, Dolphins, Football, Karate, UKfaculty, Facebook, and Github without tuning the RBF shaping +parameter. +We would like to see the domain of optimal shaping parameters for each RBF. Therefore, we +experiment on the general range of each RBF and report the results in Figure 2. From Figure 2, iMQ +RBF is more stable with a constantly small shaping parameter, which means that using iMQ RBF +is more likely to achieve optimal results after a small number of iterations. Moreover, although the +shaping parameter ranges from [0, 1] for iMQ RBF, the empirical experiments from these 11 real-world +networks show that the optimal parameter falls into [0, 0.1]. This aids in fine-tuning the parameters +in the future. +Table 5: Optimal RBF and corresponding shaping parameter +No. +Datasets +optimal RBF +shaping parameter +NMI +Modularity +1 +Les Mis´erable +Gaussian +0.64 +0.752 +0.48 +2 +Calttech +Gaussian +0.57 +0.646 +0.4 +3 +Dolphins +MQ +0 +1 +0.379 +4 +Football +MQ +0.303 +0.958 +0.62 +5 +Karate +iMQ +0.0245 +1 +0.371 +6 +Polbooks +iMQ +0.0318 +0.924 +0.473 +7 +Blog +iMQ +0.0664 +0.789 +0.415 +8 +Simmons +MQ +1.1616 +0.703 +0.481 +9 +UKfaculty +Gaussian +0.53 +0.95 +0.442 +10 +Github +Gaussian +0.21 +0.307 +0.281 +11 +Facebook +Gaussian +0.17 +0.146 +0.175 +In addition, our algorithm can perform better if we fine-tune the shaping parameters. Table 5 lists +the optimal RBF for each network and the corresponding optimal shaping parameter. We can also +interpret that the algorithm has no preferences for any RBFs, and all the common RBFs work well +on some networks. The results demonstrate that overall, with the optimal RBF, our algorithm made +improvements on Caltech, Blogs, and Simmons. We can obtain state-of-the-art community structures +on all 11 networks by tuning the RBF and shaping parameters. +4.4.2 +Analysis of Karate Network +We draw the network structure using the node size to differentiate the node degree and the node color +to distinguish the community label. +Zachary’s karate club network is a social network that has been widely used to test community +detection algorithms. This network contains 34 nodes and 78 edges, and it was divided into two com- +munities because of the contradictions between the “president” and the “instructor” of the karate club. +10 + +-MQ →.iMQ gaussian +Optimal Shaping Parameter(c) +4 +3.5 +w +2.5 +2 +1.5 +0.5 +Blog +Karate +Caltech +Github +Dolphins +Football +UKfaculty +Polbooks +Les Miserable +Real-world Networks(a) +(b) +(c) +(d) +Figure 3: The topological displays for the Karate network from SCORE, SCORE+, and our SCOREH+ +algorithms (3(a) is from the ground-truth of the network). +(a) +(b) +(c) +(d) +Figure 4: +The topological displays for the Dolphins network from SCORE, SCORE+, and our +SCOREH+ algorithms (4(a) is from the ground-truth of the network). +The real topological structure of this network is present as Figure 3(a) and the community detected +by SCORE, SCORE+, and our SCOREH+ are present as Figure 3(b), 3(c), and 3(d), respectively. +Note that the numbering of the subgraphs for the other three networks is the same as in Karate. +For this network, SCORE+ misclassified the number 3 node while SCORE and our SCOREH+ +achieved state-of-the-art results. +4.4.3 +Analysis of Dolphins Network +The Dolphins network contained an undirected social network in a community living off Doubtful +Sound, New Zealand. This network is constructed from frequent associations between 62 nodes (dol- +phins) [51]. It has two communities. +For the Dolphins network, the results demonstrate that SCORE misclassified the numbers 2, 8, 20, +27, 28, and 40 nodes, and SCORE+ made mistakes on the numbers 8 and 40 nodes. Refer to Figure +4, and our SCOREH+ perfectly acquired the community structure. +4.4.4 +Analysis of Polbooks Network +The Polbooks network contained books on American politics, published around the 2004 presidential +election and sold by Amazon.com. +This data set was not published, but we can access it on V. +Krebs’ website 2. In this network, the edges between books represent that they were sold together. +Typically, this network has two communities. However, the network structure is difficult to discover +since “neutral” or “moderate” people can buy books promoting both parties. The experiments by +detecting communities from the constructed network also demonstrate this viewpoint. +As shown in Figure 5, both of the SCORE+ and SCOREH+ algorithms made mistakes on the 50 +and 67 nodes, whereas the SCORE misclassified only one node: the 66 node. However, it is this small +one-node difference that improved the NMI of the SCORE algorithm on this network, from 0.87 to +0.924. +4.4.5 +Analysis of UKfaculty Network +The UKfaculty network [54] is a personal friendship network of UK university faculty, and the school +affiliation of each individual is stored as a node label. It consists of 81 vertices (individuals) and 817 +weighted edges and two communities. +2 http://www.orgnet.com/divided.html +11 + +6 +19 +46 +4 +30 +32 +15 +9 +51 +34 +1 +41 +53 +8 +44 +38 +21 +8 +17 +50 +48 +39 +31 +55 +35 +11 +40 +58 +62 +43 +20 +10 +1452 +66 +22 +9 +46 +60 +4 +30 +15 +6 +51 +34 +41 +53 +X +44 +38 +2 +21 +8 +17 +50 +48 +39 +31 +55 +1) +40 +58 +62 +43 +10 +20 +632 +28 +24 +29 +20 +34 +2 +3 +1 +9 +30 +4 +33 +31 +14 +2 +10 +8 +3 +932 +28 +24 +29 +20 +34 +2 +3 +1 +9 +30 +4 +33 +31 +14 +2 +10 +8 +3 +925 +32 +28 +1 +29 +20 +11 +34 +2 +6 +16 +3 +1 +9 +30 +4 +33 +31 +14 +27 +21 +10 +8 +15 +23 +1932 +28 +24 +29 +20 +34 +2 +3 +1 +9 +30 +4 +33 +31 +14 +2 +10 +8 +3 +952 +66 +22 +9 +46 +60 +4 +30 +15 +6 +51 +34 +41 +53 +X +44 +38 +2 +21 +8 +17 +50 +48 +39 +31 +55 +1) +40 +58 +62 +43 +10 +20 +652 +66 +22 +19 +46 +30 +15 +51 +1 +34 +41 +53 +44 +38 +21 +8 +50 +48 +39 +31 +35 +(11 +40 +58 +62 +43 +10 +20 +14 +5 +6(a) +(b) +(c) +(d) +Figure 5: +The topological displays for the Polbooks network from SCORE, SCORE+, and our +SCOREH+ algorithms (5(a) is from the ground-truth of the network). The Polbooks network, like +Dolphins and Karate, has two communities. +(a) +(b) +(c) +(d) +Figure 6: The topological displays for the UKfaculty network from SCORE, SCORE+, and our +SCOREH+ algorithms (6(a) is from the ground-truth of the network). +For this network, the experimental comparisons in Figure 6 indicate that SCORE misclassified +the numbers 9, 38, 59, 61, 79 nodes, and SCORE+ made mistakes on the numbers 59 and 61 nodes. +However, our SCOREH+ only has one misclassified node at node number 61. +4.4.6 +Community structures of other Networks +As the number of communities grows, it becomes harder to compare the structural quality detected by +algorithms to demonstrate the superiority of our algorithm. In that case, for Caltech, Football, Blog, +and Simmons, we present the topological plots in Figure 7 detected by our SCOREH+ algorithm. +The Football network [50] contained the relationships of American football games between Division +IA colleges during the regular season in Fall 2000. The Blog is a directed network of hyperlinks between +weblogs on US politics, recorded in 2005 by Adamic and Glance [52]. In this paper, we use it as an +undirected network. For Simmons and Caltech [53], they are part of the Facebook friendship networks, +recorded in 2005. +Our algorithm can detect communities from the above four networks. Moreover, the communities +show clear cluster properties where the nodes of the same color are close to each other, while the nodes +are sparsely connected between different communities. For example, the Caltech social network has +eight large clusters and hundreds of small clusters. Different colors represent their labels. The Blog +(a) +(b) +(c) +(d) +Figure 7: The topological displays for Caltech, Football, Blog, and Simmons, respectively, are from +the results of our SCOREH+ algorithms. +12 + +24 +38 +10 +39 +48 +22 +36 +31 +89 +61 +44 +45 +35 +30 +51 +75 +16 +76 +54 +58 +41 +25 +20 +56 +50 +6 +53 +34 +83 +46 +74 +69 +5 +63 +18 +43 +40 +91) +8 +32 +65 +60 +90 +81 +27 +92 +(11 +3 +84 +42 +9 +33 +26 +(14) +64 +86 +72 +66 +7 +(15) +67 +47 +19 +85 +28 +62 +23 +13 +71 +68 +87 +17 +82 +70 +21 +248 +10 +22 +36 +89 +31 +61 +44 +45 +35 +30 +51 +16 +54) +76 +325 +41 +20 +56 +50 +6 +34 +7469 +46 +83 +5 +91 +63 +18) +(43 +8 +￥81（65 +60 +90 +27 +92 +(11 +42) +3 +84 +9 +33 +26 +64 +86 +66 +67 +47 +71 +7024 +48 +10 +22 +36 +31 +61 +44 +45 +35 +30 +51 +16) +54) +76 +325 +41 +20 +6 +34 +7469 +46 +83) +5 +91 +63 +18 +(43) +40 +32 +8 +65 +60 +90 +27 +92 +11 +3 +84 +42 +9 +33 +26 +64 +4 +86 +66 +67 +47 +19 +62 +71 +68 +82 +7024 +48 +10 +22 +36 +31 +61 +44 +45 +35 +30 +51 +16) +54) +76 +325 +41 +20 +6 +34 +7469 +46 +83) +5 +91 +63 +18 +(43) +40 +32 +8 +65 +60 +90 +27 +92 +11 +3 +84 +42 +9 +33 +26 +64 +4 +86 +66 +67 +47 +19 +62 +71 +68 +82 +7056 +20 +77 +36 +2 +31 +9 +15 +44 +46 +61 +19 +29 +X +34 +39 +37 +66 +18 +64 +27 +68 +9 +42 +75 +10 +7414 +26 +50 +56 +20 +77 +3 +53 +2 +31 +9 +15 +44 +48 +43 +51 +46 +61 +19 +29 +34 +38 +39 +37 +66 +18 +57 +64 +27 +68 +13 +6 +42 +75 +7 +10 +22 +16 +74 +678 +26 +56 +20 +77 +36 +2 +31 +9 +15 +52 +44 +48 +51 +61 +6 +29 +39 +37 +66 +18 +64 +27 +68 +6 +75 +10 +16 +7420 +77 +36 +2 +31 +9 +15 +44 +43 +51 +46 +61 +19 +29 +34 +39 +37 +66 +18 +64 +27 +68 +13 +6 +2 +10 +74 +6(a) +(b) +(c) +(d) +Figure 8: The comparison plots of Modularity on LFR datasets with different N. +network has two communities in reality, but some “blogs” are neutral (green on nodes), and the pros +are neither of the two parties. It is also reasonable and natural to split this network into three clusters. +4.5 +Simulation Experiments and Result Analyses: Synthetic Networks +In this subsection, we use the networks generated from the LFR criteria. The parameters τ1 and τ2 are +fixed to be 1.0 and 1.5, respectively, for all networks. The number of nodes ranges from 150 to 10, 000, +and the key parameter µ (mixing parameter) from 0.15 to 0.85. Since µ determines the network noise, +we only compare the results with 0.35 ≤ µ ≤ 0.65. When µ is too small (µ ≤ 0.25), the network will +be too easy to discover for all the algorithms, while it will be too hard when µ is too large (µ > 0.65). +4.5.1 +Comparisons for the number of nodes +We fix the mixing parameter µ and compare the performance acquired from SC, SCORE, SCORE+, +and our SCOREH+. As we can observe from the modularity comparisons (Figure 8) and NMI compar- +isons (Figure 9), when µ is relatively small, SCORE+ and SCOREH+ can obtain excellent community +structure, especially for small networks. However, as µ grows, the superiority of SCOREH+ is obvious. +The reason is that our SCOREH+ can preserve more local information, and this property makes a +difference when the network is noisy. +4.5.2 +Comparisons with respect to mixing parameter µ +We have compared the performance of each algorithm when the mixing parameter µ is fixed. Next, we +show how mixing parameters affect the results on the same scale as networks. The modularity (Figure +10) and NMI (Figure 11) comparisons show that µ greatly affects the performance of algorithms. This +is reasonable since it determines the difficulty of a network. When µ is very small, every algorithm can +detect nearly perfect communities, while a large µ can result in a modularity metric with approximately +0, representing a nearly random community discovery. +We attach the detailed modularity and NMI comparison tables in Appendix A.1 and Appendix +A.2, respectively. +13 + +0 +0'02 +7.0 +vtiisluboM +21.0 +2COBEH+ +S.0 +2COBE+ +0'S2 +2COBE +E.0 +2C +0'32 +0'4 +0'42Wnwp6l ot woq62(w) +021 +300 +200 +008 +0001 +5000 +3000 +2000 +0008 +-0'02 10 +0'02 +vtinsluboM +2COBEH+ +7.0 +2COBE + +21.0 +2COBE +i2C +S.0 +0'S2 +E.0 +2E.0Wnwp6l ot woq62(w) +J20 +300 +200 +008 +0001 +5000 +3000 +2000 +0008 +-0'02S0.0- +vtiisluboM +:2COKEHL+ +0'03 +2COBE+ +80.0 +-2CObE +D2 +0'J3 +81.0 +0'S3Wnwp6l ot woq62(w) +021 +300 +200 +008 +0001 +5000 +3000 +2000 +0008 +10.0-S0.0- +vtinsluboM +0'03 +80.0 +ET.0 +:2COBEH+ +2COBE+ +81.0 +2COBE +2C +0'S3Wnwp6l ot woq62(w) +021 +300 +200 +008 +0001 +5000 +3000 +2000 +0008 +-001(a) +(b) +(c) +(d) +Figure 9: The comparison plots of NMI on LFR datasets with different N. +(a) +(b) +(c) +(d) +Figure 10: The comparison plots of Modularity on LFR datasets with different µ. +14 + +7.0 +S.0 +.0 +0'4 +0'2 +2COBEH+ +a.0 +2COBE+ +1.0 +ZCOBE +8.0 +e.0Mnwp6l ot oq62(m) +021 +300 +200 +008 +000F +5000 +3000 +2000 +0000F0008 +0 :NMIComparisonsonsyntheticnetworkwithμ=0.45 +0.9 +0.8 +0.7 ++.SCORE +0.6 +NMI +.SCORE+ +0.5 +SCOREH+ +0.4 +0.3 +0.2 +0.1 +0 +150 +300 +500 +800 +1000 +2000 +3000 +5000 +800010000 +Number of Nodes(N)7.0 +o'S +E.0 +0'4 +0'2 +a.0 +2COBEH+ +1.0 +2COBEi+ +8.0 +2COBE +e.0 +Wwl cowbgl120u2 ou 2ufy6f!c u6fmolk m!f h=0'22?Mnwp6l ot oq62(m) +021 +300 +200 +008 +000F +5000 +3000 +2000 +0000F0008 +0 :7.0 +S.0 +E.0 +0'4 +IMV +0'2 +2COBEH+ +a.0 +2COBE +1.0 +2COBE +8.0 +e.0Mnwp6l ot oq62(m) +021 +300 +200 +008 +000F +5000 +3000 +2000 +0000F0008 +0 :0'02 +27.0 +vinsluboM +0'S2 +0'32 +2COKEHL+ +0'42 +2COBE+ +0'22 +2COBE +0'2 +92 +21.0Wixiua bgigwefei(h) +27.0 +0'S2 +0'32 +0'42 +0'22 +0'e2 +0'12 +28.0 +-0'02 0'02 +27.0 +vinsluboM +0'S2 +0'32 +2COKEHL+ +0'42 +2COBE+ +0'22 +2COBE +0'2 +92 +0'12W!xiua bgigwefei(h) +27.0 +0'S2 +0'32 +0'42 +0'22 +0'e2 +0'12 +28.0 +-0'02 0'02 +27.0 +vinsluboM +0'S2 +0'32 +2COKEHL+ +0'42 +2COBE+ +0'22 +2COBE +0'e2 +92 +27.0W!xiua bgigwefei(h) +27.0 +0'S2 +0'32 +0'42 +0'22 +2a.0 +0'12 +28.0 +-0'02 0'02 +27.0 +vinsluboM +0'S2 +0'32 +2COKEHL+ +0'42 +2COBE+ +0'22 +2COBE +0'e2 +92 +27.0W!xiua bgigwefei(h) +27.0 +0'S2 +0'32 +0'42 +0'22 +2a.0 +0'12 +28.0 +-0'02 (a) +(b) +(c) +(d) +Figure 11: The comparison plots of NMI on LFR datasets with different µ. +5 +Conclusion +We studied spectral clustering and proposed a novel algorithm to detect communities in complex +networks. The algorithm harnessed RBF to cast the node vector into an approximation domain and, +at the same time, preserve high-order information. This technique assures a higher performance in a +noisy network than other baseline algorithms. Furthermore, joining an additional eigenvector when +a network has a weak signal can preserve more information for clustering. The choice of RBFs and +shaping parameters are key to a good result. Our experiments demonstrate that the optimal parameter +generally falls into a small range. For example, the best parameter for iMQ RBF is rather small. This +provides an experience when we fine-tune parameters on a new network. Most importantly, the optimal +parameter settings give the best community structure quality and outperform any other algorithm. +In future work, we would reduce the time complexity and apply our algorithm to large-scale net- +works. 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Girvan, “Finding and evaluating community structure in networks,” +Physical review E, vol. 69, no. 2, p. 026113, 2004. +18 + +A +Additional Results for Subsection 4.5 +A.1 +Modularity Tables +Table 6: Numerical results on synthetic networks with N=2,000 and N=5,000 (Modularity) +No. +µ +N=2,000 +N=5,000 +SC +SCORE +SCORE+ +SCOREH+ +SC +SCORE +SCORE+ +SCOREH+ +1 +0.15 +0.049(0.006) +0.001(0.0) +0.698(0.0) +0.698(0.0) +0.056(0.001) +0.0(0.0) +0.712(0.0) +0.712(0.0) +2 +0.25 +0.013(0.002) +0.0(0.0) +0.55(0.0) +0.55(0.0) +0.025(0.001) +0.0(0.0) +0.551(0.0) +0.551(0.0) +3 +0.35 +−0.009(0.003) +0.0(0.0) +0.411(0.0) +0.411(0.0) +0.006(0.001) +0.0(0.0) +0.422(0.0) +0.422(0.0) +4 +0.45 +−0.019(0.001) +0.0(0.0) +0.269(0.0) +0.279(0.0) +−0.009(0.001) +0.0(0.0) +0.284(0.0) +0.287(0.002) +5 +0.55 +−0.026(0.001) +0.0(0.0) +0.126(0.008) +0.156(0.001) +−0.011(0.001) +0.0(0.0) +0.113(0.008) +0.17(0.001) +6 +0.65 +−0.027(0.0) +−0.0(0.0) +0.044(0.003) +0.095(0.001) +−0.018(0.0) +0.0(0.0) +0.055(0.005) +0.085(0.002) +7 +0.75 +−0.03(0.0) +−0.0(0.0) +0.016(0.003) +0.069(0.001) +−0.018(0.0) +0.0(0.0) +0.023(0.001) +0.054(0.0) +8 +0.85 +−0.026(0.0) +0.0(0.0) +−0.004(0.002) +0.059(0.001) +−0.02(0.0) +−0.0(0.0) +−0.002(0.001) +0.041(0.0) +Table 7: Numerical results on synthetic networks with N=8,000 and N=10,000 (Modularity) +No. +µ +N=8,000 +N=10,000 +SC +SCORE +SCORE+ +SCOREH+ +SC +SCORE +SCORE+ +SCOREH+ +1 +0.15 +0.046(0.002) +0.0(0.0) +0.71(0.0) +0.71(0.0) +0.05(0.001) +0.0(0.0) +0.718(0.0) +0.718(0.0) +2 +0.25 +0.031(0.0) +0.0(0.0) +0.495(0.0) +0.563(0.0) +0.03(0.001) +0.0(0.0) +0.571(0.0) +0.571(0.0) +3 +0.35 +0.016(0.001) +0.0(0.0) +0.388(0.0) +0.428(0.0) +0.017(0.001) +0.0(0.0) +0.329(0.0) +0.425(0.0) +4 +0.45 +0.004(0.0) +0.0(0.0) +0.249(0.001) +0.302(0.003) +0.003(0.001) +0.0(0.0) +0.228(0.0) +0.297(0.0) +5 +0.55 +−0.009(0.0) +−0.0(0.0) +0.135(0.0) +0.185(0.002) +−0.007(0.0) +0.0(0.0) +0.105(0.003) +0.185(0.003) +6 +0.65 +−0.014(0.0) +0.0(0.0) +0.047(0.004) +0.082(0.003) +−0.011(0.0) +0.0(0.0) +0.053(0.003) +0.09(0.001) +7 +0.75 +−0.02(0.0) +0.0(0.0) +0.018(0.001) +0.045(0.0) +−0.019(0.0) +0.0(0.0) +0.02(0.0) +0.044(0.0) +8 +0.85 +−0.016(0.0) +0.0(0.0) +0.006(0.0) +0.036(0.0) +−0.016(0.0) +−0.0(0.0) +0.002(0.0) +0.029(0.0) +A.2 +NMI Tables +Table 8: Numerical results on synthetic networks with N=2,000 and N=5,000 (NMI) +No. +µ +N=2,000 +N=5,000 +SC +SCORE +SCORE+ +SCOREH+ +SC +SCORE +SCORE+ +SCOREH+ +1 +0.15 +0.407(0.013) +0.029(0.001) +1.0(0.0) +1.0(0.0) +0.423(0.005) +0.019(0.002) +1.0(0.0) +1.0(0.0) +2 +0.25 +0.296(0.005) +0.024(0.001) +1.0(0.0) +1.0(0.0) +0.424(0.007) +0.018(0.001) +1.0(0.0) +1.0(0.0) +3 +0.35 +0.207(0.005) +0.023(0.001) +0.996(0.0) +0.993(0.0) +0.43(0.006) +0.015(0.001) +0.998(0.0) +0.997(0.0) +4 +0.45 +0.169(0.006) +0.028(0.001) +0.952(0.001) +0.953(0.0) +0.314(0.008) +0.012(0.001) +0.969(0.0) +0.966(0.003) +5 +0.55 +0.067(0.004) +0.025(0.001) +0.65(0.002) +0.67(0.004) +0.234(0.006) +0.018(0.001) +0.848(0.003) +0.842(0.006) +6 +0.65 +0.056(0.002) +0.025(0.0) +0.294(0.009) +0.36(0.003) +0.057(0.002) +0.015(0.001) +0.482(0.002) +0.502(0.007) +7 +0.75 +0.044(0.002) +0.026(0.002) +0.104(0.006) +0.127(0.005) +0.046(0.001) +0.015(0.0) +0.21(0.004) +0.243(0.002) +8 +0.85 +0.07(0.004) +0.032(0.001) +0.072(0.002) +0.064(0.003) +0.044(0.002) +0.014(0.001) +0.045(0.002) +0.041(0.003) +Table 9: Numerical results on synthetic networks with N=8,000 and N=10,000 (NMI) +No. +µ +N=8,000 +N=10,000 +SC +SCORE +SCORE+ +SCOREH+ +SC +SCORE +SCORE+ +SCOREH+ +1 +0.15 +0.462(0.007) +0.011(0.001) +1.0(0.0) +1.0(0.0) +0.468(0.002) +0.014(0.001) +1.0(0.0) +1.0(0.0) +2 +0.25 +0.439(0.008) +0.011(0.0) +0.979(0.0) +0.999(0.0) +0.477(0.005) +0.009(0.0) +1.0(0.0) +0.999(0.0) +3 +0.35 +0.469(0.002) +0.014(0.001) +0.98(0.0) +0.998(0.0) +0.448(0.003) +0.013(0.001) +0.965(0.0) +0.995(0.0) +4 +0.45 +0.477(0.002) +0.013(0.001) +0.959(0.001) +0.978(0.002) +0.479(0.001) +0.01(0.0) +0.947(0.001) +0.975(0.001) +5 +0.55 +0.375(0.002) +0.012(0.001) +0.911(0.0) +0.925(0.001) +0.402(0.001) +0.012(0.0) +0.874(0.001) +0.905(0.001) +6 +0.65 +0.125(0.004) +0.013(0.001) +0.613(0.001) +0.605(0.006) +0.208(0.011) +0.012(0.0) +0.699(0.001) +0.657(0.005) +7 +0.75 +0.026(0.001) +0.01(0.0) +0.186(0.002) +0.22(0.001) +0.029(0.001) +0.007(0.0) +0.254(0.001) +0.282(0.002) +8 +0.85 +0.047(0.001) +0.012(0.001) +0.122(0.002) +0.129(0.002) +0.036(0.001) +0.01(0.001) +0.053(0.001) +0.074(0.002) +19 + diff --git a/DNE1T4oBgHgl3EQfEAOl/content/tmp_files/load_file.txt b/DNE1T4oBgHgl3EQfEAOl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb0d300c3383a7b958449dd046ba30efe5620b8d --- /dev/null +++ b/DNE1T4oBgHgl3EQfEAOl/content/tmp_files/load_file.txt @@ -0,0 +1,2340 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf,len=2339 +page_content='SCOREH+: A High-Order Node Proximity Spectral Clustering on Ratios-of-Eigenvectors Algorithm for Community Detection Yanhui Zhu1, Fang Hu1,2, Lei Hsin Kuo1, and Jia Liu1 1Department of Mathematics and Statistics, University of West Florida 2College of Information Engineering, Hubei University of Chinese Medicine Abstract Complex network analysis has brought significant advances in uncovering network mesoscopic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Community detection is one of the significant features of understanding real-world complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In this paper, we propose a High-order node proximity Spectral Clustering on Ratios-of-Eigenvectors (SCOREH+) algorithm for finding communities in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This algorithm preserves high-order transitivity information of the network affinity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' First, we construct the high-order proximity matrix from the original affinity matrix using the Radial Basis Functions (RBFs) and Katz index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Furthermore, we obtain the normalized Laplacian matrix and the normalized leading eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The ratios of the leading eigenvectors aid in mitigating the effect of degree heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Moreover, we implement a procedure that joins an additional eigen- vector (the (K + 1)th leading eigenvector) to the spectrum domain for clustering if the network is considered to be a “weak signal” graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Finally, we apply the K-means algorithm to the spectrum domain for acquiring the node labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We compare our SCOREH+ algorithm with spectral cluster- ing (SC), Spectral Clustering on Ratios-of-Eigenvectors (SCORE), and SCORE+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' To demonstrate the high effectiveness of our algorithm, we conducted comparison experiments on 11 real-world networks and several synthetic networks with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The experimental results demonstrate that our SCOREH+ outperforms SC, SCORE, and SCORE+ on most of these networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In addition, we find that by tuning the RBFs and their shaping parameters, we can obtain state-of-the-art community structures on all real-world networks and even on noisy synthetic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 1 Introduction Complex networks model a large number of entities and their relations as nodes and edges in real- world scenarios, which are ubiquitous and can be applied to any data as long as pair-wise interactions exist among the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Non-trivial topological features preserved in the network structure have attracted researchers from various fields, for example, biology [1], climate [2], sociology, epidemiology [3,4], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In complex networks, community detection is one of the kinds that discovers the clusters in the network model to mine latent information among the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The nodes are densely connected by edges within clusters while they are sparsely connected between clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Researchers have developed various algorithms to discover community structures, for example, Walktrap [5], Infomap [6], Louvain algorithm [7], deep learning-based algorithms [8–10], diffusion-based algorithms [11,12], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Spectral clustering, rooted in graph theory, is one of the state-of-the-art algorithms for detecting communities in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' A modern spectral clustering algorithm consists of three procedures: (1) regularization of a suitable adjacency or Laplacian matrix, (2) a form of spectral truncation, and (3) a K-means algorithm on the reduced spectral-domain [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For the first step, the formation and selection of the graph proximity method and Laplacian are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' A similarity matrix models the local neighborhood relationships of pair-wise data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Researchers typically use the network affinity matrix as the node similarity representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, some researchers use similarity measures to compute a new similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Radial Basis Functions (RBFs) are commonly used kernels in the construction similarity matrices [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Furthermore, the Laplacian matrix is obtained by subtracting the similarity matrix from the degree matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In the third step, the number of clusters is a prerequisite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Nonetheless, by decomposing the Laplacian matrix, the first large gap between two 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='02885v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='SI] 7 Jan 2023 eigenvalues generally indicates the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' That is to say, the number of eigenvalues before this gap is the number of clusters [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, this approach lacks a theoretical justification [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The biggest challenges of spectral clustering and community detection are presented as follows: Despite the effectiveness of spectral clustering in real networks, using the network affinity matrix as the pair-wise nodes’ similarity information is insufficient to capture local information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Given the number of clusters K, researchers generally preserve the exact top K eigenvectors for posting K-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, for some networks, other eigenvectors may also carry information for clustering [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' RBFs have many genres, and they all have a shaping parameter to tune to achieve good results for application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' A bad choice for an RBF can cause serious bad results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' To address these challenges, we propose an effective community detection algorithm: High-order node proximity Spectral Clustering on Ratios-of-Eigenvectors (SCOREH+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' It utilizes RBFs to ap- proximate the similarity matrix from the original affinity matrix and considers its high-order proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We also analyze the Eigen distribution and then determine if one more eigenvector is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Fur- thermore, we give a range of optimal RBF shaping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' To test our algorithm and demonstrate its superiority, we collected 11 real-world networks that span various areas, ranging from social sciences to political sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In addition, we also generate benchmark networks using a criterion first presented by Lancichinetti, Fortunato, and Radicchi [19], in short, the LFR benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Taking advantage of these datasets, we analyzed the results from multi-views concerning the number of clusters and mixing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Overall, our paper makes the following contributions: Our algorithm casts the nodes as the basis of an approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Then, we use the Katz index to compute the high-order proximity of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Similarity only measures the close relationship between a node and its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In this approach, we can preserve more node- local information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' By calculating the gap between the Kth and the (K + 1)th eigenvalue, we can easily determine if an additional eigenvector is worth maintaining for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The (K + 1)th eigenvector as an additional feature can facilitate the clustering performance for some networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We run our algorithm on various real-world networks and synthetic networks generated using various parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Numerical results show that each RBF has a shaping parameter domain where we can achieve optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This finding can help tune RBF parameters in future wide applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Section 2 demonstrates the related work of spectral clustering, SCORE/SCORE+ algorithms, high-order proximity applications, and RBF applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Section 3 illustrates the algorithm, including its design, steps, and pseudocode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Section 4 shows the experimental design, data sets, evaluation metrics, experimental results, and their analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Finally, we give the conclusion and future work in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2 Related Work In this section, we present the related work concerned with the community detection algorithms, SCORE, and SCORE+ algorithms, Higher-order proximities, and RBF applications, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 Spectral Clustering Community detection algorithm is a basic tool that enables us to discover the organizational principles in the network, it plays a huge role in solving various complex problems in real life, and it is also a research hotspot in various disciplines [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [22] proposed a novel modular-based dynamic community detection algorithm, Dy- naMo, which aims to detect dynamic network communities in a more efficient manner, which is as effective as repeated application of static algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Experimental results show that DynaMo outper- forms comparison algorithms in terms of effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [23] proposed a local community 2 detection algorithm based on high-order structure and edge information (HSEI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Use edge information to mine the membership strength between nodes and communities,so as to obtain more complete local community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Spectral clustering is one of the common community detection algorithms, it uses information from the eigenvalues of the Laplacian of an affinity matrix and maps the nodes to a low-dimensional space where data is more separable, enabling us to perform Eigen decomposition and form clusters [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [25] proposed a Low-rank Sparse Subspace (LSS) clustering method to learn an affinity matrix from the original low-dimension features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The affinity matrix is learned dynamically at a fast speed so that the optimal clustering results are promised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [26] utilized a probabilistic Dirichlet process to infer the number of clusters from the affinity matrix and then applied the gaussian mixture model to the feature matrix to acquire communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' As a result, the detection accuracy is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [8] developed a robust deep learning framework for discriminative embedding and spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The method is efficient and of high precision on benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [27] constructed the self-adaptive mixture similarity measure (SAM) and combined it with spectral clustering for uncertain datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Their method was verified to be more effective than other state-of- the-art methods using experimental comparisons and null hypothesis significance tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 SCORE and SCORE+ Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [28] first proposed a Spectral Clustering On Ratios-of-Eigenvectors (SCORE) algorithm, which uses the entry-wise ratios between eigenvectors for clustering to improve the effectiveness of spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' SCORE effectively removes the effect of degree heterogeneity by taking entry-wise ratios between the first leading eigenvector and each of the other eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The SCORE can be extended in various directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' First, SCORE applies to a large class of methods that utilize scaling- invariant mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Second, the Degree Corrected Block Model (DCBM) [29] can be generalized to more realistic models, where the spectral methods could continue to work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' proposed Mixed-SCORE [30] in 2017 and derived the convergence rate of this algorithm using delicate spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In particular, this algorithm enjoys tight row-wise deviation bounds for the rational number region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' It solves the mixed membership estimation problem and has been applied to four network data sets with encouraging results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Inspired by SCORE, Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [31] proposed a pre-SVD normalization, which adopted the SCORE method to normalize singular vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This novel algorithm discovered a low-dimensional post-SVD simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The authors provided theoretical properties and carefully studied the rate-optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' As a result, pre-SVD normalization has a faster convergence rate than existing methods in a wide variety of real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Furthermore, SCORE provided novel ideas about computing communities in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Later in 2018, Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [18] applied two normalizations and Eigen selections to improve the performance of SCORE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The new algorithm SCORE+ demonstrated the rationality of Laplacian regularization as a pre-PCA normalization and retained an additional eigenvector as a post-PCA normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' SCORE+ has two tuning parameters, but each is easy to set and not sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' It is guided by common sense in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Therefore, SCORE is fast, and SCORE+ is slightly slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The experimental results showed that the clustering error rate was reduced dramatically compared to SCORE on testing networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Researchers [32, 33] have borrowed ideas from SCORE and SCORE+ for the community detection field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='3 Higher-Order Proximities In networks, to measure the similarity of every pair of nodes, the adjacency matrix, and Laplacian ma- trix represent the first-order proximity, which simulates the local pair-wise proximity between vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Cosine similarity, Euclidean similarity, Jaccard similarity, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=', are also popularly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, these similar methods can only preserve local information by using its connectivity to its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' They are not sufficient to fully simulate the pair-wise proximity between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Therefore, how to preserve high-order proximity has become a hot topic recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' People have also explored higher-order similarities to simulate the strength between two nodes [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Three commonly used high-order proximities are Common Neighbors and Propagation [36], Katz Proximity [37], and Eigenvector Centrality [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The Katz index was proposed by Katz [37] to compute the similarity of two nodes in a heterogeneous network by computing the walks between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 3 It has been used in graph embedding [39], complex networks [40], and relationship prediction in networks [41,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Ou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [39] proposed applying multiple high-order proximity measurements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=', Katz index [37] on the graph embedding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This work has attracted much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Plenty of proximity measurements have emerged in the last century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Cosine similarity is a simple index used to determine the number of common neighbors between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The Katz index is a widely used measurement that considers the total number of walks between two nodes rather than the shortest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Furthermore, the Katz similarity has been used to predict new potential lncRNA and environmental factors (EF) associations [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The author computed the high-order similarity value of every pair of nodes with the Katz measure in a heterogeneous network through the number of walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [44] proposed using the linear combination of the polynomial function of the affinity matrix to preserve the high-order proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Their method preserves arbitrary-order proximities in the network embedding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4 RBF applications The Gaussian similarity function exp(−r2/(2c2)) is the most common Radial Basis Function (RBF) or similarity function in the neural networks, where r is the distance between the two nodes and c is the shaping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This function is equivalent to the Gaussian RBF in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The Multiquadric (MQ) RBF is effective in geographical data sets, and the density of the local dataset determines the shaping parameter c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The selection of RBFs used for the interpolation matrix is strongly problem- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' On the other hand, the interpolation matrix, which is the same weight matrix we use in the complex network, is highly ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Therefore, the selection of RBFs and the parameters are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [45] proposed a framework that integrates the attention mechanism and auto- kernel learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The hyperparameter tuning for kernels largely facilitated improving the performance of graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In the network sciences, researchers consider undirected graphs where the weighted adjacency matrix W = WT is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The structures of the networks could be studied by exploring the structures of the matrix W [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The pattern of the vertices and edges of the adjacency graph or the corresponding adjacency matrix may reveal information about the network’s divisions, clusters, and communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The first step is to transform the given data set into a graph called a “similarity graph”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The goal of constructing the similarity graph is to model the local neighborhood relationships from the network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This paper considers the construction based on the Radial Basis Function (RBF) interpolations and forms a high-order proximity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 3 The High-order Proximity Preserved Spectral Clustering We present a community detection algorithm: High-order node proximity Spectral Clustering on Ratios-of-Eigenvectors (SCOREH+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Firstly, we introduce an efficient spectral clustering algorithm that uses the ratios of eigenvectors and the Eigen selection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Then, we derive how to preserve higher-order proximities in the networks using the RBF and Katz index to capture more node-local information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Before introducing the detailed derivation, we clarify the symbols and definitions that will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 Notations We define an undirected graph G = {V, E} with n nodes (vertices) and m edges (links), and V, E are the node (vertex) set and edge (link) set of the network, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Let An×n be the affinity matrix of a network with n nodes and m edges, and Aij ̸= 0 if an edge exists between nodes vi and vj otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' A node v ∈ V is represented by {vi}n i=1 = {Avi}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The degree matrix is denoted by Dn×n, where the diagonal value Dii is the degree of the corresponding node i and the off-diagonal elements are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The Laplacian matrix L is obtained by the equation L = D − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We assume that the number of clusters K is given for each network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 The Algorithm The algorithm constructs the high-order proximity matrix while preserving the high-order transitivity information from the original affinity matrix using the RBF technique and Katz index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' From the high-order proximity matrix, we obtain the normalized graph Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Next, we normalize the K leading eigenvectors of the proximity matrix by dividing the leading eigenvectors into an additional (K + 1)th eigenvector, which will be used to cluster if the network is considered to be a “weak signal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 Radial Basis Functions For a given node vector {vi}n i=1 ∈ A, which contains n distinct elements in the computational domain, Ω ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The approximation that utilizes RBF is an unknown function f, which can be expressed as linear combinations of data norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' f (v) ≈ ˜f (v) = n � i=1 αiΦ (∥v − vi∥) , v ∈ Ω (1) where ∥·∥ represents the Euclidean norm on Rd, αi is the coefficients, and Φ : Rd → R is called a Radial Basis Function (RBF) [47] if, Φ (v) = Φ (u) , whenever ∥v∥ = ∥u∥, v, u ∈ Rd (2) In Table 1, we list the most common RBFs that have been widely used in neural networks and the numerical approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The symbol c is a shaping parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' and the symbol r in the table denotes the Euclidean distance of v ∈ Rd from the original point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' r = ∥v∥2 = ��d i=1 x2 i Table 1: Some common choices of RBFs Choice of RBF Definition Multiquadric (MQ) Φ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' c) = √ c2 + r2 Inverse Multiquadric (IMQ) Φ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' c) = 1/ √ c2 + r2 Gaussian Φ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' c) = exp � r2/c2� Follows Equation (1) with the collocation scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' ˜f (vi) = f (vi) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' n (3) leads to a system of linear equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Wα = f (4) where α = [α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' αn]T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' f = [f (v1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' f (vn)]T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' and a matrix Wij = Φ (rij) ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The distance matrix, rij, contained within W can be expressed as follows, rij = � �� ∥u1−v1∥2 · · ∥u1−vn∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' ∥un−v1∥2 · · ∥un−vn∥2 � �� , i, j =1,· · ·, n (5) After applying RBF to the data sets, we obtain the similarity graph and the weighted matrix Wij, with entries Wij = Φ(∥vi − vj∥2), i = 1, · · · , n, also the interpolation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Wij consists of the functions serving as the basis of the approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For distinct data points in the data sets and a constant shape parameter c, Wij is a nonsingular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Both the choice of the RBF and its corresponding shaping parameter plays an important role in calculating the final interpolations and partitions in the resulting graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 Higher-Order Proximities The Katz index [37, 39] computes the relative influence of a node within a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We call nodes that are directly connected to a node as immediate neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Therefore, the Katz index measures the number of immediate neighbors and the immediate neighbors of its immediate neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' It is a weighted summation of the path node-set between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The weight of a path is an exponential function of its length (the number of nodes on this path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We formularize the Katz index as: K = (I − β · W)−1 · β · W (6) where β is a decay parameter, it determines the weight of a path decay speed as the length of the path grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' β should be properly set to preserve the series convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In practice, β must be smaller than the spectral radius of the weighted matrix W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Conventionally, in this paper, we set β to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The pseudocode for computing the high-order proximity of an affinity matrix using Gaussian RBF is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This algorithm first generates a list of N shaping parameters c (Line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Then iteratively find an optimal shaping parameter (Line 3-4) where GaussianRBF(·,·) computes the distance using the Gaussian RBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We can compute MQ and iMQ RBFs using the procedures analogous to Algorithm 1 by substituting Line 4 with the respective RBF distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Algorithm 1: High-order Proximity (HOP) Input : Affinity matrix: A ∈ Rn×n Output: High-order matrix: K 1 x ← linspace(0, 1, n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2 c ← linspace(0, 1, 100);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 3 for i ← 1 to n do 4 B ← GaussianRBF(ci, DMatrix(xT , xT ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 5 C ← optimal B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 6 ˆC ← C · A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 7 K ← Katz( ˆC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='3 Normalized Eigens The spectrum (eigenvalues) of the network similarity matrix is used in spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This procedure works similarly to dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Using RBF and Katz index, we obtained the high-order similarity matrix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The network’s diagonal matrix D = diag(K) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The normalized Laplacian matrix Lσ can then be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The graph laplacian matrix with ridge regularization σ is: Lσ = (D + σ · dmax · I)− 1 2 K(D + σ · dmax · I)− 1 2 (7) where dmax is the maximum node degree of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The empirical setting of σ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Next, we compute K +1 largest eigenvalues ˆλ and their corresponding eigenvectors ˆΞ, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' K +1 leading eigenvectors, and sort them in non-descending order by ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Consequently, the feature matrix’s dimension is reduced from n × n to n × (K + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The reduced feature matrix, Θ, is expressed as: ˆΘ = ˆΞ · Diag(ˆλ) (8) where Diag(ˆλ) forms a diagonal matrix Mn×n where Mij = ˆλi for all i = j such that 0 ≤ i, j < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4 Eigen-selection and Clustering We support that the weighted matrix K contains “signal” and “noise” network information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' A network with “strong signal” has a large gap between the Kth and the (K + 1)th eigenvectors of its Laplacian 6 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We have a threshold t > 0 to determine whether a network has a weak signal profile by, ˆλK+1 ˆλK ≥ 1 − t (9) If the above equation (9) is satisfied, then we say that the weighted matrix is of “weak signal” profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Therefore, the (K + 1)th eigenvector contains useful information as the Kth for community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Consequently, we consider it as one more feature that contributes to label clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Finally, we apply the K-means algorithm to the new feature matrix with K +1 dimensions to compute the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The pseudocode for community detection is below in Algorithm 2 and the implementation of the algorithms in this paper is publicly accessible on https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='com/yz24/RBF-SCORE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In Algo- rithm 2, Line 1 computes the high-order proximity of a network from the original affinity matrix using Algorithm 1 and returns a matrix with the same dimension as its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Then, collect eigenpairs (eigenvalue, eigenvector) of K and arrange them in decreasing order by eigenvalues (Lines 2 to 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Lines 8 to 9 determine if the network has a strong or weak signal profile and assign k′ accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The algorithm returns a list of clustered node labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Algorithm 2: High-Order Node Proximity Spectral Clustering on Ratios-of-Eigenvectors (SCOREH+) Input : A ∈ Rn×n, σ > 0, t ∈ (0, 1), k ∈ N≥1 Output: Node labels: ˆy 1 K ← HOP(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2 D ← diag(K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 3 Lσ ← (D + σ · dmax · I)− 1 2 K(D + σ · dmax · I)− 1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4 k′ ← k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 5 ˆλ, ˆΞ ← eigsh(Lσ, k′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 6 ˆΛ ← Diag(ˆλ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 7 Θ ← ˆΞ · ˆΛ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 8 if ˆλr ˆλk < 1 − t then 9 k′ ← k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 10 ˆE ← { ˆΘh ˆΘ1 , · · · , ˆΘk′ ˆΘ1 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 11 ˆy ← K-means(ˆE, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4 Experiments We conduct experiments on several real-world and synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We first introduce the experi- ment design and then show the results and analyses on these two types of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Furthermore, we also analyze the RBF parameter selections and the domains for optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For synthetic networks, we demonstrate the noise resistance ability of our algorithm compared to the baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 Datasets 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 Real-world Networks We tested our algorithm and other algorithms (SC, SCORE, and SCORE+) on 11 public real-world network datasets from various areas, including social sciences, political sciences, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' These networks are Les Mis´erable [48], Karate [49], Football [50], Dolphins [51], Blog [52], Simmons [53], Caltech [53], UKfaculty [54], Github, Facebook [55], and Polbooks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The descriptions for each of them as shown below, and the basic statistics are in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 1 This network was not published, but it is public on this website: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='orgnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='com/divided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='html 7 Table 2: Statistics of real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For 11 real-world datasets, the number of nodes is n, the number of edges is m, the number of clusters is k, the minimum and maximum of the node degrees and the average shortest path length are min(d), max(d) of each network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Dataset Source n m k min(d) max(d) 1 Les Mis´erable [48] 77 254 11 1 36 2 Caltech [53] 590 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 822 172 1 179 3 Dolphins [51] 62 159 2 1 12 4 Football [50] 110 568 11 7 12 5 Karate [49] 34 78 2 1 17 6 Polbooks [56] 92 374 2 1 24 7 Blog [52] 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 222 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 714 2 1 351 8 Simmons [57] 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 137 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 257 4 1 293 9 UKfaculty [54] 79 552 3 2 39 10 Github [55] 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 700 289,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 003 2 1 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 458 11 Facebook [55] 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 470 171,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 002 4 1 709 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 Synthetic Networks We generate a series of synthetic networks using the LFR criteria, first proposed by Lancichinetti, Fortunato, and Radicchi [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' A network can be generated in terms of the given parameters: the power-law exponent for the degree distribution is τ1, the power-law exponent for the community size distribution is τ2, the number of nodes is N, the average degree is ⟨k⟩, the minimum of communities is c, and the mixing parameter is µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Most importantly, µ, one of the key parameters, controls the fraction of edges between communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Thus, it reflects the amount of noise in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' When µ = 0, all links are within community links;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' when µ = 1, all links are between nodes from different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We generate networks by setting the number of nodes ranging from 150 to 10, 000, and the key parameter µ (mixing parameter) from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='15 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 Evaluation Metrics We use two evaluation metrics: modularity and normalized mutual information (NMI), to verify the performance of the proposed SCOREH+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The former does not require the ground truth of the network while the latter does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For both of them, the higher the metrics are, the better the communities are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 Modularity Newman and Girvan [58] proposed modularity Q to assess the quality of the detected community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' It represents the difference between the actual number of connections and the expected number of connections in random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The equation is as follows: Q = k � s=1 � ls l − �ds 2l �2� (10) Where k is the number of communities in the network, l is the sum of all edges in the network, ls is the sum of all edges in the community s, and ds is the sum of the degree of all nodes in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 Normalized Mutual Information (NMI) Let ˆy be the list of community labels obtained from an algorithm, and y be the list of ground-truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Denote H(·) as the entropy function for a graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Then, the mutual information between the ground truth and the detected labels is: MI(ˆy, y) = |ˆy| � i=1 |y| � j=1 |ˆyi ∩ yj| n log �n|ˆyi ∩ yj| |ˆyi||yj| � (11) Then the normalized mutual information is: NMI(ˆy, y) = MI(ˆy, y) mean(H(ˆy), H(y)) (12) Because a higher metric value of modularity Q and NMI represent better community discovery, we bold the highest values and underline the second highest values in the comparison tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 8 (a) (b) Figure 1: The comparison plots of Modularity and NMI on real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='3 Experiment Design Our experiments were carried out on a 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 GB RAM, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='90GHz Intel(R) Core(TM) i7-8650U CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' we select spectral clustering (SC) [24], Spectral Clustering on Ratios-of-Eigenvectors (SCORE) [28], and SCORE+ [18] as baseline algorithms for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We test the proposed algorithm and the baselines on the above-mentioned two types of data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For each single network, we run each model ten times on it, evaluate two metrics, and then report the mean and variance of the results in the form of mean (variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4 Simulation Experiments and Result Analyses: Real-World Networks In this subsection, we compare our algorithm SCOREH+ with SC, SCORE, and SCORE+ on real- world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We report the experimental results by scoring the quality of the discovered commu- nities with Modularity and NMI, see Table 3 and Table 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Moreover, we visualize the community structures computed from our algorithm and other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We compare and analyze the topological structures of four small networks: Karate, Dolphin, Polbooks, and UKfalculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Furthermore, we analyze the RBF shaping parameters and their influence on the final community discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We find that the choice of shaping parameters can make a difference in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, the shaping parameters have a relatively small domain to achieve an optimal result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Table 3: Numerical results on real-world networks (Modularity) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Dataset SC SCORE SCORE+ SCOREH+ 1 Les Mis´erable −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='019(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='386(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='057) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='239(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='038) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='486(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='002) 2 Caltech 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='39(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='372(0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='132(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='146(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0) The modularity and NMI values of each algorithm are illustrated in Figure 1(a) and 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=" It is 9 0'02 0'02 vtinsluboM 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0 0'S2 0'32 0'42 0'22 0'e2 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0 0'4 M 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 一 :2C :2C0BE :2COBE+ + 2COBEH+Bjoa2 nktscf) C9]fGc KSL9TG c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='funpFigure 2: Optimal shaping parameters with RBF choices on real-world networks clearly shown that our algorithm achieves the best and second-best on most of the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 RBF selection and tuning of shaping parameters on real-world networks In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2, we discussed the three common RBFs and their definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Each RBF has a shaping parameter c, and it can affect the final graph interpolation matrix, which, as a consequence, can cause a difference in the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Therefore, we fine-tune the shaping parameter c and find the optimal shaping parameter where NMI is the criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Table 3 and 4 show that our algorithm has already achieved the best results on Les Miserable, Cal- tech, Dolphins, Football, Karate, UKfaculty, Facebook, and Github without tuning the RBF shaping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We would like to see the domain of optimal shaping parameters for each RBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Therefore, we experiment on the general range of each RBF and report the results in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' From Figure 2, iMQ RBF is more stable with a constantly small shaping parameter, which means that using iMQ RBF is more likely to achieve optimal results after a small number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Moreover, although the shaping parameter ranges from [0, 1] for iMQ RBF, the empirical experiments from these 11 real-world networks show that the optimal parameter falls into [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This aids in fine-tuning the parameters in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Table 5: Optimal RBF and corresponding shaping parameter No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Datasets optimal RBF shaping parameter NMI Modularity 1 Les Mis´erable Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='752 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='48 2 Calttech Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4 3 Dolphins MQ 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='379 4 Football MQ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='62 5 Karate iMQ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0245 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='371 6 Polbooks iMQ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='473 7 Blog iMQ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='415 8 Simmons MQ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='481 9 UKfaculty Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='442 10 Github Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='281 11 Facebook Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='175 In addition, our algorithm can perform better if we fine-tune the shaping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Table 5 lists the optimal RBF for each network and the corresponding optimal shaping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We can also interpret that the algorithm has no preferences for any RBFs, and all the common RBFs work well on some networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The results demonstrate that overall, with the optimal RBF, our algorithm made improvements on Caltech, Blogs, and Simmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We can obtain state-of-the-art community structures on all 11 networks by tuning the RBF and shaping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 Analysis of Karate Network We draw the network structure using the node size to differentiate the node degree and the node color to distinguish the community label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Zachary’s karate club network is a social network that has been widely used to test community detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This network contains 34 nodes and 78 edges, and it was divided into two com- munities because of the contradictions between the “president” and the “instructor” of the karate club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 10 MQ →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='iMQ gaussian Optimal Shaping Parameter(c) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5 w 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5 Blog Karate Caltech Github Dolphins Football UKfaculty Polbooks Les Miserable Real-world Networks(a) (b) (c) (d) Figure 3: The topological displays for the Karate network from SCORE, SCORE+, and our SCOREH+ algorithms (3(a) is from the ground-truth of the network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 4: The topological displays for the Dolphins network from SCORE, SCORE+, and our SCOREH+ algorithms (4(a) is from the ground-truth of the network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The real topological structure of this network is present as Figure 3(a) and the community detected by SCORE, SCORE+, and our SCOREH+ are present as Figure 3(b), 3(c), and 3(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Note that the numbering of the subgraphs for the other three networks is the same as in Karate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For this network, SCORE+ misclassified the number 3 node while SCORE and our SCOREH+ achieved state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='3 Analysis of Dolphins Network The Dolphins network contained an undirected social network in a community living off Doubtful Sound, New Zealand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This network is constructed from frequent associations between 62 nodes (dol- phins) [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' It has two communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For the Dolphins network, the results demonstrate that SCORE misclassified the numbers 2, 8, 20, 27, 28, and 40 nodes, and SCORE+ made mistakes on the numbers 8 and 40 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Refer to Figure 4, and our SCOREH+ perfectly acquired the community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4 Analysis of Polbooks Network The Polbooks network contained books on American politics, published around the 2004 presidential election and sold by Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This data set was not published, but we can access it on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Krebs’ website 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In this network, the edges between books represent that they were sold together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Typically, this network has two communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, the network structure is difficult to discover since “neutral” or “moderate” people can buy books promoting both parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The experiments by detecting communities from the constructed network also demonstrate this viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' As shown in Figure 5, both of the SCORE+ and SCOREH+ algorithms made mistakes on the 50 and 67 nodes, whereas the SCORE misclassified only one node: the 66 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, it is this small one-node difference that improved the NMI of the SCORE algorithm on this network, from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='87 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5 Analysis of UKfaculty Network The UKfaculty network [54] is a personal friendship network of UK university faculty, and the school affiliation of each individual is stored as a node label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' It consists of 81 vertices (individuals) and 817 weighted edges and two communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='orgnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='com/divided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='html ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='11 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='6(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='Figure 5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='The topological displays for the Polbooks network from SCORE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' SCORE+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' and our SCOREH+ algorithms (5(a) is from the ground-truth of the network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The Polbooks network, like Dolphins and Karate, has two communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 6: The topological displays for the UKfaculty network from SCORE, SCORE+, and our SCOREH+ algorithms (6(a) is from the ground-truth of the network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For this network, the experimental comparisons in Figure 6 indicate that SCORE misclassified the numbers 9, 38, 59, 61, 79 nodes, and SCORE+ made mistakes on the numbers 59 and 61 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, our SCOREH+ only has one misclassified node at node number 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='6 Community structures of other Networks As the number of communities grows, it becomes harder to compare the structural quality detected by algorithms to demonstrate the superiority of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In that case, for Caltech, Football, Blog, and Simmons, we present the topological plots in Figure 7 detected by our SCOREH+ algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The Football network [50] contained the relationships of American football games between Division IA colleges during the regular season in Fall 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The Blog is a directed network of hyperlinks between weblogs on US politics, recorded in 2005 by Adamic and Glance [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In this paper, we use it as an undirected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For Simmons and Caltech [53], they are part of the Facebook friendship networks, recorded in 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Our algorithm can detect communities from the above four networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Moreover, the communities show clear cluster properties where the nodes of the same color are close to each other, while the nodes are sparsely connected between different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For example, the Caltech social network has eight large clusters and hundreds of small clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Different colors represent their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The Blog (a) (b) (c) (d) Figure 7: The topological displays for Caltech, Football, Blog, and Simmons, respectively, are from the results of our SCOREH+ algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='38 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='74 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='6(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='Figure 8: The comparison plots of Modularity on LFR datasets with different N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' network has two communities in reality, but some “blogs” are neutral (green on nodes), and the pros are neither of the two parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' It is also reasonable and natural to split this network into three clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5 Simulation Experiments and Result Analyses: Synthetic Networks In this subsection, we use the networks generated from the LFR criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The parameters τ1 and τ2 are fixed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5, respectively, for all networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The number of nodes ranges from 150 to 10, 000, and the key parameter µ (mixing parameter) from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='15 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Since µ determines the network noise, we only compare the results with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='35 ≤ µ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' When µ is too small (µ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='25), the network will be too easy to discover for all the algorithms, while it will be too hard when µ is too large (µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 Comparisons for the number of nodes We fix the mixing parameter µ and compare the performance acquired from SC, SCORE, SCORE+, and our SCOREH+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' As we can observe from the modularity comparisons (Figure 8) and NMI compar- isons (Figure 9), when µ is relatively small, SCORE+ and SCOREH+ can obtain excellent community structure, especially for small networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' However, as µ grows, the superiority of SCOREH+ is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The reason is that our SCOREH+ can preserve more local information, and this property makes a difference when the network is noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2 Comparisons with respect to mixing parameter µ We have compared the performance of each algorithm when the mixing parameter µ is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Next, we show how mixing parameters affect the results on the same scale as networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The modularity (Figure 10) and NMI (Figure 11) comparisons show that µ greatly affects the performance of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This is reasonable since it determines the difficulty of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' When µ is very small, every algorithm can detect nearly perfect communities, while a large µ can result in a modularity metric with approximately 0, representing a nearly random community discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' We attach the detailed modularity and NMI comparison tables in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='1 and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=" 13 0 0'02 7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 vtiisluboM 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 2COBEH+ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0 2COBE+ 0'S2 2COBE E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0 2C 0'32 0'4 0'42Wnwp6l ot woq62(w) 021 300 200 008 0001 5000 3000 2000 0008 0'02 10 0'02 vtinsluboM 2COBEH+ 7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 2COBE + 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 2COBE i2C S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0 0'S2 E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content='0 2E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0Wnwp6l ot woq62(w) J20 300 200 008 0001 5000 3000 2000 0008 0'02S0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0- vtiisluboM :2COKEHL+ 0'03 2COBE+ 80." metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0 0'S2 0'32 0'42 0'22 2a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0 0'12 28." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content="0 0'02 (a) (b) (c) (d) Figure 11: The comparison plots of NMI on LFR datasets with different µ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' 5 Conclusion We studied spectral clustering and proposed a novel algorithm to detect communities in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The algorithm harnessed RBF to cast the node vector into an approximation domain and, at the same time, preserve high-order information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This technique assures a higher performance in a noisy network than other baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Furthermore, joining an additional eigenvector when a network has a weak signal can preserve more information for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' The choice of RBFs and shaping parameters are key to a good result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Our experiments demonstrate that the optimal parameter generally falls into a small range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' For example, the best parameter for iMQ RBF is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' This provides an experience when we fine-tune parameters on a new network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Most importantly, the optimal parameter settings give the best community structure quality and outperform any other algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' In future work, we would reduce the time complexity and apply our algorithm to large-scale net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Moreover, finding a correlation between some metrics and the optimal RBF shaping parameter may facilitate optimizing the algorithm iteratively without computing the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Guerrero, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Montoya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfEAOl/content/2301.02885v1.pdf'} +page_content=' Ba˜nos, A.' metadata={'source': 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Thorpe1 * +AJTHOR@UNM.EDU +Cyrus Neary2 * +CNEARY@UTEXAS.EDU +Franck Djeumou2 * +FDJEUMOU@UTEXAS.EDU +Meeko M. K. Oishi1 +OISHI@UNM.EDU +Ufuk Topcu2 +UTOPCU@UTEXAS.EDU +1University of New Mexico +2University of Texas at Austin +Abstract +Data-driven control algorithms use observations of system dynamics to construct an implicit +model for the purpose of control. However, in practice, data-driven techniques often require exces- +sive sample sizes, which may be infeasible in real-world scenarios where only limited observations +of the system are available. Furthermore, purely data-driven methods often neglect useful a priori +knowledge, such as approximate models of the system dynamics. We present a method to incor- +porate such prior knowledge into data-driven control algorithms using kernel embeddings, a non- +parametric machine learning technique based in the theory of reproducing kernel Hilbert spaces. +Our proposed approach incorporates prior knowledge of the system dynamics as a bias term in +the kernel learning problem. We formulate the biased learning problem as a least-squares problem +with a regularization term that is informed by the dynamics, that has an efficiently computable, +closed-form solution. Through numerical experiments, we empirically demonstrate the improved +sample efficiency and out-of-sample generalization of our approach over a purely data-driven base- +line. We demonstrate an application of our method to control through a target tracking problem +with nonholonomic dynamics, and on spring-mass-damper and F-16 aircraft state prediction tasks. +1. Introduction +The practical deployment of autonomous systems demands algorithms that can account for stochas- +ticity and unexpected events due to humans in the loop or dramatic changes in the environment. +Model-based approaches to stochastic optimal control (Bertsekas and Shreve, 1978; Bertsekas, +2012) offer an analytic representation that is highly generalizable, but often rely upon strict model +assumptions, and can become inaccurate when deployed in new environments. They are particularly +susceptible to model misspecifications, which can lead to inaccurate predictions that may lead to +unpredictable or unsafe behaviors. Data-driven control can account for poorly-characterized distur- +bances, but typically neglect prior system knowledge. Additionally, these methods (Mauroy et al., +2020; Ansari and Murphey, 2016; Rudy et al., 2017) often exhibit poor data efficiency, meaning they +require excessive sample sizes in order to adequately characterize the dynamical system behavior. +We present a method to incorporate (potentially) imperfect knowledge of the system dynamics in +kernel embeddings in order to numerically estimate expectations in stochastic optimal control and +* These authors contributed equally to this work. +© 2023 A.J. Thorpe, C. Neary, F. Djeumou, M.M.K. Oishi & U. Topcu. +arXiv:2301.03565v1 [eess.SY] 9 Jan 2023 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +Empirical +Distribution ++ +Bias +Bias +Prior System +Knowledge +Data-Driven +Kernel Embedding +Physics-Informed +Kernel Embedding +Prior Knowledge +Bias Term +Reproducing Kernel +Hilbert Space (RKHS) +True Embedding +Figure 1: Physics-informed kernel embeddings combine data and prior system knowledge to more +accurately estimate the expectation operator in an RKHS. +state prediction problems. Specifically, we propose physics-informed kernel embeddings, a non- +parametric statistical learning technique based in reproducing kernel Hilbert spaces (RKHS) that +incorporates prior knowledge of the dynamics as inductive bias. As shown in Thorpe and Oishi +(2021); Thorpe et al. (2022b,a), data-driven reformulations of stochastic optimal control problems +using kernel embeddings can efficiently be solved as a linear program by exploiting the mathemati- +cal properties of the RKHS. However, despite the applicability to control, these techniques have thus +far not seen widespread popularity, and presently do not take prior system knowledge into account. +We modify the regularized least-squares problem used to learn kernel embeddings with an addi- +tional bias term that encodes prior knowledge of the dynamics (Figure 1). We present a representer +theorem, which provides a closed-form solution to the learning problem. Finally, we describe how +the proposed physics-informed kernel embeddings may be applied to solve approximate stochas- +tic optimal control problems. We experimentally demonstrate our approach on state prediction and +control tasks, including a spring-mass-damper system with a limited sample of system observations, +a highly nonlinear F-16 aircraft, and a target tracking problem with nonholonomic dynamics. +2. Related Work +Many approaches in data-driven control construct implicit black-box representations using sparse +regression over a library of nonlinear functions (Kaiser et al., 2018), spectral properties of the col- +lected data (Proctor et al., 2016), Koopman theory (Abraham et al., 2017; Korda and Mezi´c, 2018), +or Gaussian processes (Krause and Ong, 2011; Gahlawat et al., 2020). However, these approaches +often suffer from high computational costs, expensive hyperparameter tuning, or nonconvexity of +the surrogate functions, and are not readily amenable to incorporating prior dynamics. +Methods to incorporate a priori knowledge into learned models of physical systems have been +studied extensively over the past several years (e.g. Djeumou and Topcu, 2022; Djeumou et al., +2022b; Ahmadi and Khadir, 2020). In particular, a number of recent works use neural networks +to parametrize the unknown or unmodeled terms in differential equations (Djeumou et al., 2022a; +Chen et al., 2018; Rackauckas et al., 2020). This approach allows for the inclusion of general forms +physics knowledge into data-driven models , such as for so-called Lagrangian and Hamiltonian neu- +ral networks (Cranmer et al., 2020; Lutter et al., 2019; Zhong and Leonard, 2020; Allen-Blanchette +et al., 2020; Greydanus et al., 2019; Matsubara et al., 2020; Toth et al., 2020; Finzi et al., 2020), +and it also enables learning control-oriented dynamics models (Zhong et al., 2020a,b; Roehrl et al., +2 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +2020; Duong and Atanasov, 2021; Gupta et al., 2020; Menda et al., 2019; Zhong et al., 2021; Shi +et al., 2019). However, although these methods take advantage of physics-based knowledge, they +often require extensive training data and training time. +Our proposed approach is based in the theory of kernel embeddings of distributions (Song et al., +2009; Smola et al., 2007), which have been applied to Markov models (Gr¨unew¨alder et al., 2012b; +Nishiyama et al., 2012; Song et al., 2010a), statistical inference (Song et al., 2009, 2010b), policy +synthesis (Lever and Stafford, 2015) and recently used to solve stochastic optimal control problems +(Thorpe and Oishi, 2021; Thorpe et al., 2022b,a). However, existing approaches to kernel-based +control typically neglect prior knowledge of the system dynamics, or seek to encode structure di- +rectly into the kernel (Cheng et al., 2016) or learning prior (Geist and Trimpe, 2020), which yields +a highly specialized solution that does not generalize well to all systems or problem domains. +3. Problem Formulation +Let (X, BX ) be a Borel space called the state space and (U, BU) be a compact Borel space called +the control or input space. We consider discrete-time stochastic systems of the form +xt+1 = f(xt, ut, θ, wt), +t = 0, 1, . . . , N, +(1) +where xt ∈ X is the state of the system at time t, ut ∈ U is the control action, θ ∈ Θ are model +parameters, and wt are independent random variables representing the stochastic disturbance. The +system evolves from an initial condition x0 ∈ X (which may be taken from an initial distribution +P0 on X). For notational convenience, we can represent the dynamics in (1) via a stochastic kernel +Q : BX × X × U → [0, 1] that assigns a probability measure Q(· | x, u) to every (x, u) ∈ X × U +on the measurable space (X, BX ), as shown in Bertsekas and Shreve (1978). +We presume the dynamics in (1) are unknown, meaning we do not have direct knowledge of the +system dynamics or the uncertainty. Instead, we presume that a sample S, consisting of observations +taken independently and identically distributed (i.i.d.) from the system evolution is available, e.g. +observations of the system transitions S = {(x1, u1, y1), . . . , (xM, uM, yM)}. where xi and ui +are taken from X and U, respectively, and yi ∼ Q(· | xi, ui). Such a sample may be collected +from high-fidelity simulation or via observations of the system evolution from system trajectories. +In addition, we presume that we have prior (potentially imperfect) knowledge of the dynamics, +˜f : X × U → X. Such prior knowledge may be available, for instance, if we only have access +to a first-order approximation of the dynamics, if the deterministic dynamics are available but the +stochastic uncertainty is unknown, or if the model parameters θ are poorly estimated. +We solve, under the conditions above, two problems: 1) state prediction, where we seek to +estimate the expected future state of the system after taking an action u in a given state x, +Ey∼Q(·|x,u)[y], +(2) +and 2) unconstrained stochastic optimal control, which can generally be written as +min +u∈U +Ey∼Q(·|x,u)[c(y)], +(3) +where c : X → R is a (well-posed) arbitrary cost function that could capture, e.g. LQR, MPC, or +other typical control objectives. We focus on (2) and (3) because they are representative of common +problems in controls. As shown in Thorpe and Oishi (2021), by embedding the integral operator of +3 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +the stochastic kernel Q as an element in a high-dimensional space of functions known as a reproduc- +ing kernel Hilbert space (RKHS), we can approximate the expected value as a linear operation in the +RKHS, and the approximate kernel-based reformulation of (3) can be solved as a linear program. +This is important because it provides a data-driven approach that is potentially amenable to run-time +implementations. The main challenges are twofold: kernel embeddings neglect important informa- +tion about the dynamics and are therefore more susceptible to errors, and they are susceptible to +common sampling issues such as limited sample information. +The key contribution of this paper is a method to incorporate potentially imperfect knowledge +of the system dynamics in the kernel embedding to numerically estimate (2) and (3). We propose +physics-informed kernel embeddings, that incorporates prior dynamics knowledge in kernel distribu- +tion embeddings, and apply our proposed technique to the problem of state prediction and control. +4. Physics-Informed Kernel Embeddings of Distributions +4.1. Kernel Embeddings of Distributions +Define the kernel k : X × X → R, which is a positive definite function (Steinwart and Christmann, +2008, Definition 4.15). According to the Moore-Aronszajn theorem (Aronszajn, 1950), given a +positive definite kernel k, there exists a corresponding RKHS H of functions from X to R which +satisfies the following properties: (i) For all x ∈ X, k(x, ·) ∈ H , and (ii) For all f ∈ H and x ∈ X, +f(x) = ⟨f, k(x, ·)⟩H , which is known as the reproducing property. Similarly, let l : U × U → R +be a reproducing kernel over U and let U be its associated RKHS. +Note that expectations Ey∼Q(·|x,u)[c(y)] are linear in the function argument c. As shown in +Gr¨unew¨alder et al. (2012b), assuming the kernel k is measurable and bounded, there exists an ele- +ment m(x, u) ∈ H called the kernel distribution embedding, such that by the reproducing property, +⟨c, m(x, u)⟩H = Ey∼Q(·|x,u)[c(y)]. We can compute an empirical estimate ˆm(x, u) of the embed- +ding m(x, u) using data S. As shown in Gr¨unew¨alder et al. (2012a), the estimate ˆm(x, u) can be +computed as the solution to a regularized least-squares (RLS) problem, +ˆm = arg min +f∈V +1 +2λ +M +� +i=1 +∥k(yi, ·) − f(xi, ui)∥2 +H + 1 +2∥f∥2 +V, +(4) +where V is a vector-valued RKHS of functions from X × U to H (see Gr¨unew¨alder et al., 2012a +and Micchelli and Pontil, 2005) and λ > 0 is the regularization parameter. The solution to (4) is +given by a well-known class of theorems known as representer theorems Sch¨olkopf et al. (2001). +4.2. Incorporating Prior Knowledge of the Dynamics in the Kernel Embedding +Following Sch¨olkopf et al. (2001), we propose to learn a physics-informed kernel embedding esti- +mate via the following biased RLS problem, +ˆm0 = arg min +f∈V +1 +2λ +M +� +i=1 +∥k(yi, ·) − f(xi, ui)∥2 +H + 1 +2∥f∥2 +V − ⟨f, f0⟩V, +(5) +which differs from (4) in that it includes an additional penalty term ⟨f, f0⟩V, where f0 ∈ V is a +user-specified bias term. As discussed in Sch¨olkopf et al. (2001), this is a way to introduce bias into +4 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +the regularization, and penalizes the difference between the learned function and the bias f0, instead +of only the RKHS norm ∥f∥2 +V. The solution to (5) can be characterized via a representer theorem, +which we present as Theorem 1. +Theorem 1 If ˆf ∈ V minimizes the risk functional in (5), it is unique and has the form +ˆf = +M +� +i=1 +βik(xi, ·)l(ui, ·) + f0, +(6) +where the coefficients βi ∈ H , i = 1, . . . , M, are the unique solution of the set of linear equations, +M +� +j=1 +(k(xi, xj)l(ui, uj) + λδij)βj = k(yi, ·) − f0(xi, ui), +i = 1, . . . , M. +(7) +Proof The proof is similar to (Micchelli and Pontil, 2005, Theorem 4.1). Let f be any element of V +such that f(xi, ui) = k(yi, ·), which minimizes the least-squared error of the data. Let g = f − ˆf, +and note that 1 +2∥f∥2 +V can be expanded as 1 +2∥f∥2 +V = 1 +2∥g + ˆf∥2 +V = 1 +2∥g∥2 +V + ⟨g, ˆf⟩V + 1 +2∥ ˆf∥2 +V. Let +E(f) be the risk functional, +E(f) = 1 +2λ +� +i +∥k(yi, ·) − f(xi, ui)∥2 +H + 1 +2∥f∥2 +V − ⟨f, f0⟩V. +(8) +Taking the difference between the risk E(f) from (8) and the risk E( ˆf) using (6) , and using the +above expansion, we obtain +E(f) − E( ˆf) = 1 +2λ +� +i +∥g(xi, ui)∥2 +H − 2 +� +i +⟨k(yi, ·) − ˆf(xi, ui), g(xi, ui)⟩H ++ ⟨g, ˆf⟩V + 1 +2∥g∥2 +V − ⟨g, f0⟩V, +(9) +where the final term uses the fact that ⟨ ˆf, f0⟩V − ⟨f, f0⟩V = ⟨ ˆf − f, f0⟩V = −⟨g, f0⟩V. Using the +fact that for any g ∈ V and f ∈ H , ⟨f, g(x, u)⟩H = ⟨g, k(xi, ·)l(ui, ·)f⟩V, which comes from +well-known properties of the vector-valued RKHS V (Micchelli and Pontil, 2005, Proposition 2.1), +and equations (6) and (7), we have that +⟨g, ˆf⟩V − 2 +� +i +⟨k(yi, ·) − ˆf(xi, ui), g(xi, ui)⟩H = 0. +(10) +Then, using (10) in (9), we have that +E(f) = E( ˆf) + 1 +2λ +� +i +∥g(xi, ui)∥2 +H + 1 +2∥g∥2 +V − ⟨g, f0⟩V ≥ E( ˆf), +(11) +from which we conclude that ˆf is the unique minimizer of E (uniqueness follows from the convexity +of V), which concludes the proof. +In practical terms, Theorem 1 shows that the solution ˆm0 to (5) can be represented as a combi- +nation of two elements in the RKHS: a bias term f0, and a linear combination of kernel functions +�M +i=1 βik(xi, ·)l(ui, ·) that represents the data-driven part. +5 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +Now it remains to choose a bias f0. A natural choice for f0 is given by +f0(x, u) = k( ˜f(x, u), ·), +(12) +such that for any c ∈ H , ⟨c, f0(x, u)⟩H = ⟨c, k( ˜f(x, u), ·)⟩H = c( ˜f(x, u)) by the reproducing +property. Then, using the solution ˆm0 to the RLS problem in (5) given by Theorem 1 and the bias +term f0 in (12), we have that for any function c ∈ H , +Ey∼Q(·|x,u)[c(y)] ≈ ⟨c, ˆm0(x, u)⟩H = c⊤WK(x, u) − ˜c⊤WK(x, u) + c( ˜f(x, u)), +(13) +where c ∈ RM and ˜c ∈ RM are vectors with elements ci = c(yi) and ˜ci = c( ˜f(xi, ui)), re- +spectively, W = (G + λI)−1, where G ∈ RM×M, is a positive semi-definite matrix with elements +Gij = k(xi, xj)l(ui, uj), and K(x, u) ∈ RM is a vector that depends on x and u that has elements +[K(x, u)]i = k(xi, x)l(ui, u). +The estimate in (13) has a simple interpretation via addition and subtraction of the cost over the +approximate dynamics from the expected cost, Ey∼Q(·|x,u)[c(y) − c( ˜f(x, u))] + c( ˜f(x, u)). Specif- +ically, the first term c⊤WK(x, u) on the right-hand side of (13) corresponds to the purely data- +driven kernel distribution embedding estimate; the second term ˜c⊤WK(x, u) represents a kernel +distribution embedding with the training data {(xi, ui, ˜f(xi, ui))}M +i=1, where we substitute the ap- +proximate dynamics over the data points ˜f(xi, ui) for the observations yi in the dataset S; and the +third term g( ˜f(x, u)) is a correction that shifts the estimate such that it is centered around ˜f. +4.3. Control Using Physics-Informed Kernel Embeddings +In this section, we demonstrate how physics-informed kernel embeddings can be used to solve the +kernel-based control problem in (3). A stochastic policy π : BU × X → [0, 1] for the system in (1) +is a stochastic kernel that assigns a probability measure π(· | x) to every x ∈ X on (U, BU). As +shown in Thorpe and Oishi (2021); Thorpe et al. (2022b), we can represent the stochastic policy π +as a kernel embedding p(x) in the RKHS U —a linear combination of kernels over a user-specified +control set A = {˜uj}P +j=1, given by p(x) = �P +j=1 γj(x)l(˜uj, ·), where γ(x) ∈ RP are real coeffi- +cients that depend on the value of x. +We use the physics-informed kernel embedding ˆm0 (in place of the embedding ˆm) as in (13) to +estimate the expected cost with respect to Q. Using ˆm0 in (3), the policy embedding p(x) can be +found as the solution to the following problem, +min +γ(x)∈RP +c⊤WR(x)γ(x) − ˜c⊤WR(x)γ(x) + C(x)⊤γ(x) +(14a) +s.t. +1⊤γ(x) = 1, +0 ⪯ γ(x) +(14b) +where c ∈ RM and ˜c ∈ RM are as in (13), W ∈ RM×M is a real matrix as in (13), R(x) ∈ RM×P +is a real matrix that depends on x, with elements [R(x)]ij = k(xi, x)l(ui, ˜uj), and C(x) ∈ RP is +a vector with elements [C(x)]j = c( ˜f(x, ˜uj)). Notably, the problem in (14) is a linear program, +and can be solved efficiently. According to Boyd et al. (2004), since we seek to minimize a linear +combination by choosing non-negative weights, it is immediately clear that we should allocate as +much weight as possible to the smallest terms. Thus, the solution is a vector γ∗(x) ∈ RP of all +zeros except at the index corresponding to the control action in A that gives the lowest expected +cost, where it is one. See Thorpe and Oishi (2021) for more details. +6 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +(Ours) Physics-informed kernel embedding +True dynamics +Sampled transition data +Kernel embedding w/o physics information +Approximate dynamics +−0.1 +0 +0.1 +−0.1 +0 +0.1 +−0.1 +0 +0.1 +˙q +−0.1 +0 +0.1 +q +−0.1 +0 +0.1 +−0.1 +0 +0.1 +Figure 2: Phase-space trajectories predicted by the physics-informed kernel embedding (green) us- +ing imperfect dynamics (gray dashed), and by a purely data-driven embedding (blue). +Top row: our approach accurately predicts the system behavior, despite having imperfect +system knowledge and data from a limited region of the state space. Bottom row: our +approach demonstrates better performance with smaller sample sizes. +5. Numerical Results +In all experiments, we use a Gaussian kernel function k(x, x′) = exp(−∥x − x′∥2/2σ2), σ > 0, +and the hyperparameters σ and λ are chosen via cross-validation. See Sch¨olkopf et al. (2002); Song +et al. (2009) and Li et al. (2022) for a detailed discussion of parameter selection. Code to reproduce +all experiments is available at https://github.com/ajthor/socks. +5.1. Spring-Mass-Damper System +For the purpose of analysis, we first consider the prediction problem in (2) with an (uncontrolled) +spring-mass-damper system. The equations of motion are given by m¨q = −b ˙q − kq. We pre- +sume that we have access to imperfect system dynamics ˜f(x) = −(k/m)q, corresponding to an +undamped spring-mass system. We generate a synthetic dataset S = {(xi, yi)}M +i=1 with varying +sample sizes M = 10, 50, 100, 500, where the states xi are taken randomly from a bounded re- +gion of X, and yi = f(xi) are corresponding next states at the subsequent timestep. We consider +two cases for the sample: 1) the states xi are taken within the region [0, 0.15] × [0, 0.15], meaning +we only have information within a limited operating regime, and 2) the states are taken within the +region [−0.15, 0.15] × [−0.15, 0.15], which fully encompasses the operating region. +Using the sample S, we then compute the physics-informed kernel embedding ˆm0 using (5) +with σ = 0.2, and use ˆm0 to predict the system evolution via (2) over N = 100 time steps from a +fixed initial condition x0 = [0.1, 0.1]⊤. To provide a baseline for comparison, we also use the purely +data-driven embedding ˆm, computed using S via (4) (see Thorpe and Oishi, 2021) to compute (2). +7 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +The top row of Figure 2 shows the performance of our approach for sample sizes M = 10, 50, +100, 500 when data is collected from a limited region of the state space. Our approach demonstrates +good empirical performance, and accurately predicts the evolution of the system despite having +imperfect knowledge based on the undamped system under a wide range of conditions. As expected, +the purely data-driven prediction does not accurately predict the system evolution outside the data +region, even as the amount of data increases (top right plot). When data is collected over the entire +region of interest, the quality of the purely data-driven estimate improves as the amount of data +increases (bottom row of Figure 2). Note that our proposed approach has sound performance even +while using only a small fraction of the data. We note the following important trends. +0 +1.0 +2.0 +3.0 +4.0 +5.0 +10−3 +10−2 +10−1 +100 +101 +Number of transition samples (·103) +Prediction Error +(Ours) Physics-informed kernel embedding +Kernel embedding w/o physics information +Figure 3: Physics-informed embeddings demon- +strate lower empirical error than purely +data-driven embeddings. +Approximate physics knowledge improves +out-of-distribution prediction accuracy. +As +shown in the top row of Figure 2, in contrast to +the purely data-driven embedding, the physics- +informed kernel embeddings generalize beyond +the training dataset. +Approximate physics knowledge improves +sample efficiency. +As seen in the bottom row +of Figure 2, when the observed transition data +encompasses the entire region of interest, our +approach is able to accurately predict the dy- +namics using only 10 data points. +Approximate physics knowledge reduces the prediction error. +Figure 3 compares the empir- +ical prediction error of the physics-informed kernel embedding ˆm0 against the purely data-driven +embedding ˆm. We randomly sample 100 initial states x0 uniformly in the region [−0.1, 0.1] × +[−0.1, 0.1], and use the learned embeddings to predict the evolution of the state 100 time steps into +the future. Figure 3 shows the median cumulative prediction error along these predicted trajectories +(measured as the Euclidean distance between the true state vector and the predicted state vector). +We observe that for small datasets (particularly for M smaller than 200) the physics-informed ker- +nel embedding enjoys prediction error values that are two orders of magnitude smaller than those of +the purely data-driven kernel embedding, and that the baseline method requires at least 5,000 data +points to achieve comparable levels of accuracy. +5.2. F-16 Aircraft +We consider a ground collision avoidance scenario for an F-16 aircraft at initial altitude, as de- +scribed in Djeumou and Topcu (2022); Heidlauf et al. (2018). The underlying nonlinear dynamics, +containing 13 states and 4 control inputs, capture the (6-DOF) motion via evolution of velocity vt, +angle of attack α, sideslip β, altitude h, attitude angles: roll φ, pitch θ, yaw ψ, and their correspond- +ing rates p, q, r, engine power and two more states pn, pe for translation along north and east, as +in Stevens et al. (2015). The plant is built on linearly interpolated lookup tables that incorporate +wind tunnel data describing the engine model, and other dynamic coefficients. We inject zero-mean +Gaussian noise with a standard deviation of 1% of the magnitude of each state, such that the noise +scales with the state magnitude. We consider the case where the true dynamics are unknown, but +presume that we have access to approximate dynamics with incorrect model parameters, including +8 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +0 +10 +500 +600 +Vt (ft/sec) +True dynamics +Approximate dynamics +Kernel embedding w/o physics information +(Ours) Physics-informed kernel embedding +0 +10 +0.0 +0.5 +1.0 +−θ(rad) +0 +10 +Time (s) +−0.7 +−0.6 +ψ (rad) +0 +10 +0 +2 +4 +6 +pn ( ·10−3 ft) +0 +10 +0 +1 +2 +h (·10−3 ft) +Figure 4: Our approach, physics-informed kernel embeddings (green), accurately predicts the dy- +namics of an F-16 aircraft. The purely data-driven approach (blue) is inaccurate due to +the system’s high-dimensional and nonlinear dynamics. +a gravitational constant of g = 7.0, and the interpolated lookup tables for the elevator control are +half of their original values. These changes significantly alter the response of the aircraft to pulling +up and avoiding collision with the ground. +We collect a sample S = {(x0,i, ξi)}M +i=1, consisting of M = 500 initial conditions x0,i taken +uniformly such that [vt, α, β, φ, θ, p, q, r, h, power]⊤ ∈ [490, 590]×[−0.01, 0.09]×[−0.05, 0.05]× +[0.55, 0.95]×[−1.2, −0.8]×[−0.2, 0.2]×[−0.2, 0.2]×[−0.2, 0.2]×[3800, 4200]×[8.7, 9.3], and the +resulting trajectories from those initial conditions ξi ∼ T(· | x0,i) using the true, nominal dynamics, +where T : BX N ×X → [0, 1] is a stochastic kernel that represents the LQR-controlled, closed-loop +system dynamics over N = 1500 time steps. Using trajectory data modifies the probability model +to be a stochastic kernel over state trajectories, but does not significantly alter the kernel estimate. +Modifications of our approach to accommodate trajectory data is described in Thorpe et al. (2022b). +Figure 4 shows the solution to (2) for state prediction. We see significant improvement in predic- +tion accuracy over the purely data-driven approach, in particular the altitude h and yaw angle Ψ. As +expected, the purely data-driven method fails to capture the F-16 system behavior with the limited +data due to the highly nonlinear and high-dimensional dynamics. Interestingly, the prediction of the +pitch angle θ using our approach shows oscillations due to the approximate dynamics. This raises +question of whether the addition of data can overcome unstable model effects in the approximate +dynamics. However, we leave this for future work. +5.3. Control of a Nonholonomic Vehicle System +We solve (3) for a target tracking control problem with a nonholonomic vehicle, as in Thorpe and +Oishi (2021). The dynamics are given by ˙x1 = u1 sin(x3), ˙x2 = u1 cos(x3), ˙x3 = u2, where x = +[x1, x2, x3]⊤ ∈ R3 is the state and u = [u1, u2]⊤ ∈ R2 is the control input, which we constrain to +be within the bounds [0.2, 1.5]×[−10.1, 10.1]. We discretize the system in time and apply an affine +disturbance with an exponential distribution wt ∼ Exp(0.1), with PDF f(x; α) = α exp(−αx) if +x ≥ 0 and f(x; α) = 0 if x < 0. We presume that the deterministic discrete-time dynamics are +given as approximate dynamical system knowledge, but that the stochastic dynamics are unknown +(i.e. we do not have prior knowledge of the disturbance). +We seek to solve (3), where we minimize the squared Euclidean distance to a moving target over +a time horizon of N = 60. We define a trajectory of target waypoints z0, z1, . . . , zN (shown in black +9 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +Sampled Transition Data +Target Trajectory +Data-Driven Trajectory +(Ours) Physics-Informed Trajectory +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +x1 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +x2 +Data Regime +Figure 5: Comparison of our proposed method against Thorpe and Oishi (2021). The solution via +physics-informed kernel embeddings (green) closely follows the target trajectory (black), +even outside the data regime, while the performance of the purely data-driven solution +(blue) degrades outside the region for which we have data. +in Figure 5). We consider the case where the future target position is unknown. Thus, we solve the +following (unconstrained) optimization problem at each time step: minπ E[∥xt+1 − zt∥2] as in (3). +See Thorpe and Oishi (2021) for more details. We collect a sample S = {(xi, ui, yi)}M +i=1 of size +M = 500, where the states xi are taken uniformly in the region shown in Figure 5. To compute +the control algorithm in (14), we generate a sample A = {˜uj}P +j=1 of P = 210 control actions taken +uniformly in the region [0.2, 1.2] × [−10.1, 10.1]. We then presume that the true dynamics are +unknown for the purpose of computing the control inputs. We then computed the physics informed +kernel embedding ˆm0 with σ = 0.75. Using ˆm0, we simulate the system from an initial condition +x0 = [−1, 0, π/2]⊤ and solve (14) at each time step to compute the stochastic policy. The total +computation time was approximately 0.272 seconds, and the results are shown in Figure 5. Using the +same sample size, the baseline method from Thorpe and Oishi (2021) fails to generate a meaningful +trajectory (not shown). To generate a comparable trajectory, we used a much larger sample size, +M = 5000, shown in blue in Figure 5, and the computation time was approximately 5.993 seconds. +This shows that our method demonstrates better empirical and computational performance, and +requires less data due to the inclusion of prior dynamics knowledge. +6. Conclusions & Future Work +In this paper, we presented physics-informed kernel embeddings, a novel technique for incorporat- +ing prior system knowledge in data-driven representations of system dynamics using kernel distribu- +tion embeddings. Numerical experiments demonstrate the effectiveness of the proposed method on +prediction tasks, including for systems with imperfect system knowledge on a spring-mass-damper +system and highly nonlinear dynamics on an F-16 system, and on control tasks via a nonholonomic +system target tracking problem. Results show that our approach generalizes well outside the data +regime, is computationally efficient, and is robust to common sampling issues. +An important direction for future work in this area involves an exploration of how to incorporate +other forms of prior knowledge, such as known system properties (e.g. symmetry, invariance) into +the learning problem. Additionally, of practical interest is a characterization of the effect that poor +or inaccurate approximate knowledge has on the learned representation. +10 + +PHYSICS-INFORMED KERNEL EMBEDDINGS +Acknowledgments +This material is based upon work supported by the National Science Foundation under NSF Grants +Number CNS-1836900 and NSF 1646522. Any opinions, findings, and conclusions or recom- +mendations expressed in this material are those of the authors and do not necessarily reflect the +views of the National Science Foundation. The NASA University Leadership initiative (Grant +#80NSSC20M0163) provided funds to assist the authors with their research, but this article solely +reflects the opinions and conclusions of its authors and not any NASA entity. This material is based +upon work supported by the Air Force Office of Scientific Research under award number FA9550- +19-1-0005. 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Extending Lagrangian and Hamil- +tonian neural networks with differentiable contact models, 2021. +15 + diff --git a/DtE1T4oBgHgl3EQf-QYz/content/tmp_files/load_file.txt b/DtE1T4oBgHgl3EQf-QYz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..513224870095ae687b3c68f1755840c1c3619c82 --- /dev/null +++ b/DtE1T4oBgHgl3EQf-QYz/content/tmp_files/load_file.txt @@ -0,0 +1,644 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf,len=643 +page_content='Proceedings of Machine Learning Research vol XX:1–15, 2023 Physics-Informed Kernel Embeddings: Integrating Prior System Knowledge with Data-Driven Control Adam J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Thorpe1 * AJTHOR@UNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='EDU Cyrus Neary2 * CNEARY@UTEXAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='EDU Franck Djeumou2 * FDJEUMOU@UTEXAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='EDU Meeko M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Oishi1 OISHI@UNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='EDU Ufuk Topcu2 UTOPCU@UTEXAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='EDU 1University of New Mexico 2University of Texas at Austin Abstract Data-driven control algorithms use observations of system dynamics to construct an implicit model for the purpose of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' However, in practice, data-driven techniques often require exces- sive sample sizes, which may be infeasible in real-world scenarios where only limited observations of the system are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Furthermore, purely data-driven methods often neglect useful a priori knowledge, such as approximate models of the system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We present a method to incor- porate such prior knowledge into data-driven control algorithms using kernel embeddings, a non- parametric machine learning technique based in the theory of reproducing kernel Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Our proposed approach incorporates prior knowledge of the system dynamics as a bias term in the kernel learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We formulate the biased learning problem as a least-squares problem with a regularization term that is informed by the dynamics, that has an efficiently computable, closed-form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Through numerical experiments, we empirically demonstrate the improved sample efficiency and out-of-sample generalization of our approach over a purely data-driven base- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We demonstrate an application of our method to control through a target tracking problem with nonholonomic dynamics, and on spring-mass-damper and F-16 aircraft state prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Introduction The practical deployment of autonomous systems demands algorithms that can account for stochas- ticity and unexpected events due to humans in the loop or dramatic changes in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Model-based approaches to stochastic optimal control (Bertsekas and Shreve, 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Bertsekas, 2012) offer an analytic representation that is highly generalizable, but often rely upon strict model assumptions, and can become inaccurate when deployed in new environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' They are particularly susceptible to model misspecifications, which can lead to inaccurate predictions that may lead to unpredictable or unsafe behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Data-driven control can account for poorly-characterized distur- bances, but typically neglect prior system knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Additionally, these methods (Mauroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Ansari and Murphey, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Rudy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2017) often exhibit poor data efficiency, meaning they require excessive sample sizes in order to adequately characterize the dynamical system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We present a method to incorporate (potentially) imperfect knowledge of the system dynamics in kernel embeddings in order to numerically estimate expectations in stochastic optimal control and These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' © 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Thorpe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Neary, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Djeumou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Oishi & U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Topcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='03565v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='SY] 9 Jan 2023 PHYSICS-INFORMED KERNEL EMBEDDINGS Empirical Distribution + Bias Bias Prior System Knowledge Data-Driven Kernel Embedding Physics-Informed Kernel Embedding Prior Knowledge Bias Term Reproducing Kernel Hilbert Space (RKHS) True Embedding Figure 1: Physics-informed kernel embeddings combine data and prior system knowledge to more accurately estimate the expectation operator in an RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' state prediction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Specifically, we propose physics-informed kernel embeddings, a non- parametric statistical learning technique based in reproducing kernel Hilbert spaces (RKHS) that incorporates prior knowledge of the dynamics as inductive bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As shown in Thorpe and Oishi (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Thorpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2022b,a), data-driven reformulations of stochastic optimal control problems using kernel embeddings can efficiently be solved as a linear program by exploiting the mathemati- cal properties of the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' However, despite the applicability to control, these techniques have thus far not seen widespread popularity, and presently do not take prior system knowledge into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We modify the regularized least-squares problem used to learn kernel embeddings with an addi- tional bias term that encodes prior knowledge of the dynamics (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We present a representer theorem, which provides a closed-form solution to the learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Finally, we describe how the proposed physics-informed kernel embeddings may be applied to solve approximate stochas- tic optimal control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We experimentally demonstrate our approach on state prediction and control tasks, including a spring-mass-damper system with a limited sample of system observations, a highly nonlinear F-16 aircraft, and a target tracking problem with nonholonomic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Related Work Many approaches in data-driven control construct implicit black-box representations using sparse regression over a library of nonlinear functions (Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2018), spectral properties of the col- lected data (Proctor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2016), Koopman theory (Abraham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Korda and Mezi´c, 2018), or Gaussian processes (Krause and Ong, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Gahlawat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' However, these approaches often suffer from high computational costs, expensive hyperparameter tuning, or nonconvexity of the surrogate functions, and are not readily amenable to incorporating prior dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Methods to incorporate a priori knowledge into learned models of physical systems have been studied extensively over the past several years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Djeumou and Topcu, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Djeumou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Ahmadi and Khadir, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' In particular, a number of recent works use neural networks to parametrize the unknown or unmodeled terms in differential equations (Djeumou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Rackauckas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' This approach allows for the inclusion of general forms physics knowledge into data-driven models , such as for so-called Lagrangian and Hamiltonian neu- ral networks (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Lutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Zhong and Leonard, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Allen-Blanchette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Greydanus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Matsubara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Toth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Finzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020), and it also enables learning control-oriented dynamics models (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Roehrl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2 PHYSICS-INFORMED KERNEL EMBEDDINGS 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Duong and Atanasov, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Menda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' However, although these methods take advantage of physics-based knowledge, they often require extensive training data and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Our proposed approach is based in the theory of kernel embeddings of distributions (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Smola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2007), which have been applied to Markov models (Gr¨unew¨alder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Nishiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2010a), statistical inference (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2009, 2010b), policy synthesis (Lever and Stafford, 2015) and recently used to solve stochastic optimal control problems (Thorpe and Oishi, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Thorpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2022b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' However, existing approaches to kernel-based control typically neglect prior knowledge of the system dynamics, or seek to encode structure di- rectly into the kernel (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2016) or learning prior (Geist and Trimpe, 2020), which yields a highly specialized solution that does not generalize well to all systems or problem domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Problem Formulation Let (X, BX ) be a Borel space called the state space and (U, BU) be a compact Borel space called the control or input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We consider discrete-time stochastic systems of the form xt+1 = f(xt, ut, θ, wt), t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' , N, (1) where xt ∈ X is the state of the system at time t, ut ∈ U is the control action, θ ∈ Θ are model parameters, and wt are independent random variables representing the stochastic disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The system evolves from an initial condition x0 ∈ X (which may be taken from an initial distribution P0 on X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' For notational convenience, we can represent the dynamics in (1) via a stochastic kernel Q : BX × X × U → [0, 1] that assigns a probability measure Q(· | x, u) to every (x, u) ∈ X × U on the measurable space (X, BX ), as shown in Bertsekas and Shreve (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We presume the dynamics in (1) are unknown, meaning we do not have direct knowledge of the system dynamics or the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Instead, we presume that a sample S, consisting of observations taken independently and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=') from the system evolution is available, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' observations of the system transitions S = {(x1, u1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' , (xM, uM, yM)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' where xi and ui are taken from X and U, respectively, and yi ∼ Q(· | xi, ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Such a sample may be collected from high-fidelity simulation or via observations of the system evolution from system trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' In addition, we presume that we have prior (potentially imperfect) knowledge of the dynamics, ˜f : X × U → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Such prior knowledge may be available, for instance, if we only have access to a first-order approximation of the dynamics, if the deterministic dynamics are available but the stochastic uncertainty is unknown, or if the model parameters θ are poorly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We solve, under the conditions above, two problems: 1) state prediction, where we seek to estimate the expected future state of the system after taking an action u in a given state x, Ey∼Q(·|x,u)[y], (2) and 2) unconstrained stochastic optimal control, which can generally be written as min u∈U Ey∼Q(·|x,u)[c(y)], (3) where c : X → R is a (well-posed) arbitrary cost function that could capture, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' LQR, MPC, or other typical control objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We focus on (2) and (3) because they are representative of common problems in controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As shown in Thorpe and Oishi (2021), by embedding the integral operator of 3 PHYSICS-INFORMED KERNEL EMBEDDINGS the stochastic kernel Q as an element in a high-dimensional space of functions known as a reproduc- ing kernel Hilbert space (RKHS), we can approximate the expected value as a linear operation in the RKHS, and the approximate kernel-based reformulation of (3) can be solved as a linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' This is important because it provides a data-driven approach that is potentially amenable to run-time implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The main challenges are twofold: kernel embeddings neglect important informa- tion about the dynamics and are therefore more susceptible to errors, and they are susceptible to common sampling issues such as limited sample information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The key contribution of this paper is a method to incorporate potentially imperfect knowledge of the system dynamics in the kernel embedding to numerically estimate (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We propose physics-informed kernel embeddings, that incorporates prior dynamics knowledge in kernel distribu- tion embeddings, and apply our proposed technique to the problem of state prediction and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Physics-Informed Kernel Embeddings of Distributions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Kernel Embeddings of Distributions Define the kernel k : X × X → R, which is a positive definite function (Steinwart and Christmann, 2008, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' According to the Moore-Aronszajn theorem (Aronszajn, 1950), given a positive definite kernel k, there exists a corresponding RKHS H of functions from X to R which satisfies the following properties: (i) For all x ∈ X, k(x, ·) ∈ H , and (ii) For all f ∈ H and x ∈ X, f(x) = ⟨f, k(x, ·)⟩H , which is known as the reproducing property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Similarly, let l : U × U → R be a reproducing kernel over U and let U be its associated RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Note that expectations Ey∼Q(·|x,u)[c(y)] are linear in the function argument c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As shown in Gr¨unew¨alder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2012b), assuming the kernel k is measurable and bounded, there exists an ele- ment m(x, u) ∈ H called the kernel distribution embedding, such that by the reproducing property, ⟨c, m(x, u)⟩H = Ey∼Q(·|x,u)[c(y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We can compute an empirical estimate ˆm(x, u) of the embed- ding m(x, u) using data S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As shown in Gr¨unew¨alder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2012a), the estimate ˆm(x, u) can be computed as the solution to a regularized least-squares (RLS) problem, ˆm = arg min f∈V 1 2λ M � i=1 ∥k(yi, ·) − f(xi, ui)∥2 H + 1 2∥f∥2 V, (4) where V is a vector-valued RKHS of functions from X × U to H (see Gr¨unew¨alder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=', 2012a and Micchelli and Pontil, 2005) and λ > 0 is the regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The solution to (4) is given by a well-known class of theorems known as representer theorems Sch¨olkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Incorporating Prior Knowledge of the Dynamics in the Kernel Embedding Following Sch¨olkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2001), we propose to learn a physics-informed kernel embedding esti- mate via the following biased RLS problem, ˆm0 = arg min f∈V 1 2λ M � i=1 ∥k(yi, ·) − f(xi, ui)∥2 H + 1 2∥f∥2 V − ⟨f, f0⟩V, (5) which differs from (4) in that it includes an additional penalty term ⟨f, f0⟩V, where f0 ∈ V is a user-specified bias term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As discussed in Sch¨olkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2001), this is a way to introduce bias into 4 PHYSICS-INFORMED KERNEL EMBEDDINGS the regularization, and penalizes the difference between the learned function and the bias f0, instead of only the RKHS norm ∥f∥2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The solution to (5) can be characterized via a representer theorem, which we present as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Theorem 1 If ˆf ∈ V minimizes the risk functional in (5), it is unique and has the form ˆf = M � i=1 βik(xi, ·)l(ui, ·) + f0, (6) where the coefficients βi ∈ H , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' , M, are the unique solution of the set of linear equations, M � j=1 (k(xi, xj)l(ui, uj) + λδij)βj = k(yi, ·) − f0(xi, ui), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (7) Proof The proof is similar to (Micchelli and Pontil, 2005, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Let f be any element of V such that f(xi, ui) = k(yi, ·), which minimizes the least-squared error of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Let g = f − ˆf, and note that 1 2∥f∥2 V can be expanded as 1 2∥f∥2 V = 1 2∥g + ˆf∥2 V = 1 2∥g∥2 V + ⟨g, ˆf⟩V + 1 2∥ ˆf∥2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Let E(f) be the risk functional, E(f) = 1 2λ � i ∥k(yi, ·) − f(xi, ui)∥2 H + 1 2∥f∥2 V − ⟨f, f0⟩V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (8) Taking the difference between the risk E(f) from (8) and the risk E( ˆf) using (6) , and using the above expansion, we obtain E(f) − E( ˆf) = 1 2λ � i ∥g(xi, ui)∥2 H − 2 � i ⟨k(yi, ·) − ˆf(xi, ui), g(xi, ui)⟩H + ⟨g, ˆf⟩V + 1 2∥g∥2 V − ⟨g, f0⟩V, (9) where the final term uses the fact that ⟨ ˆf, f0⟩V − ⟨f, f0⟩V = ⟨ ˆf − f, f0⟩V = −⟨g, f0⟩V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Using the fact that for any g ∈ V and f ∈ H , ⟨f, g(x, u)⟩H = ⟨g, k(xi, ·)l(ui, ·)f⟩V, which comes from well-known properties of the vector-valued RKHS V (Micchelli and Pontil, 2005, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1), and equations (6) and (7), we have that ⟨g, ˆf⟩V − 2 � i ⟨k(yi, ·) − ˆf(xi, ui), g(xi, ui)⟩H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (10) Then, using (10) in (9), we have that E(f) = E( ˆf) + 1 2λ � i ∥g(xi, ui)∥2 H + 1 2∥g∥2 V − ⟨g, f0⟩V ≥ E( ˆf), (11) from which we conclude that ˆf is the unique minimizer of E (uniqueness follows from the convexity of V), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' In practical terms, Theorem 1 shows that the solution ˆm0 to (5) can be represented as a combi- nation of two elements in the RKHS: a bias term f0, and a linear combination of kernel functions �M i=1 βik(xi, ·)l(ui, ·) that represents the data-driven part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 5 PHYSICS-INFORMED KERNEL EMBEDDINGS Now it remains to choose a bias f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' A natural choice for f0 is given by f0(x, u) = k( ˜f(x, u), ·), (12) such that for any c ∈ H , ⟨c, f0(x, u)⟩H = ⟨c, k( ˜f(x, u), ·)⟩H = c( ˜f(x, u)) by the reproducing property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' using the solution ˆm0 to the RLS problem in (5) given by Theorem 1 and the bias term f0 in (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' we have that for any function c ∈ H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Ey∼Q(·|x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='u)[c(y)] ≈ ⟨c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' ˆm0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' u)⟩H = c⊤WK(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' u) − ˜c⊤WK(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' u) + c( ˜f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' u)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (13) where c ∈ RM and ˜c ∈ RM are vectors with elements ci = c(yi) and ˜ci = c( ˜f(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' ui)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' re- spectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' W = (G + λI)−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' where G ∈ RM×M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' is a positive semi-definite matrix with elements Gij = k(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' xj)l(ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' uj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' and K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' u) ∈ RM is a vector that depends on x and u that has elements [K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' u)]i = k(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' x)l(ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The estimate in (13) has a simple interpretation via addition and subtraction of the cost over the approximate dynamics from the expected cost, Ey∼Q(·|x,u)[c(y) − c( ˜f(x, u))] + c( ˜f(x, u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Specif- ically, the first term c⊤WK(x, u) on the right-hand side of (13) corresponds to the purely data- driven kernel distribution embedding estimate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' the second term ˜c⊤WK(x, u) represents a kernel distribution embedding with the training data {(xi, ui, ˜f(xi, ui))}M i=1, where we substitute the ap- proximate dynamics over the data points ˜f(xi, ui) for the observations yi in the dataset S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' and the third term g( ˜f(x, u)) is a correction that shifts the estimate such that it is centered around ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Control Using Physics-Informed Kernel Embeddings In this section, we demonstrate how physics-informed kernel embeddings can be used to solve the kernel-based control problem in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' A stochastic policy π : BU × X → [0, 1] for the system in (1) is a stochastic kernel that assigns a probability measure π(· | x) to every x ∈ X on (U, BU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As shown in Thorpe and Oishi (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Thorpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2022b), we can represent the stochastic policy π as a kernel embedding p(x) in the RKHS U —a linear combination of kernels over a user-specified control set A = {˜uj}P j=1, given by p(x) = �P j=1 γj(x)l(˜uj, ·), where γ(x) ∈ RP are real coeffi- cients that depend on the value of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We use the physics-informed kernel embedding ˆm0 (in place of the embedding ˆm) as in (13) to estimate the expected cost with respect to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Using ˆm0 in (3), the policy embedding p(x) can be found as the solution to the following problem, min γ(x)∈RP c⊤WR(x)γ(x) − ˜c⊤WR(x)γ(x) + C(x)⊤γ(x) (14a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 1⊤γ(x) = 1, 0 ⪯ γ(x) (14b) where c ∈ RM and ˜c ∈ RM are as in (13), W ∈ RM×M is a real matrix as in (13), R(x) ∈ RM×P is a real matrix that depends on x, with elements [R(x)]ij = k(xi, x)l(ui, ˜uj), and C(x) ∈ RP is a vector with elements [C(x)]j = c( ˜f(x, ˜uj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Notably, the problem in (14) is a linear program, and can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' According to Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2004), since we seek to minimize a linear combination by choosing non-negative weights, it is immediately clear that we should allocate as much weight as possible to the smallest terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Thus, the solution is a vector γ∗(x) ∈ RP of all zeros except at the index corresponding to the control action in A that gives the lowest expected cost, where it is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' See Thorpe and Oishi (2021) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 6 PHYSICS-INFORMED KERNEL EMBEDDINGS (Ours) Physics-informed kernel embedding True dynamics Sampled transition data Kernel embedding w/o physics information Approximate dynamics −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 ˙q −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 q −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1 Figure 2: Phase-space trajectories predicted by the physics-informed kernel embedding (green) us- ing imperfect dynamics (gray dashed), and by a purely data-driven embedding (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Top row: our approach accurately predicts the system behavior, despite having imperfect system knowledge and data from a limited region of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Bottom row: our approach demonstrates better performance with smaller sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Numerical Results In all experiments, we use a Gaussian kernel function k(x, x′) = exp(−∥x − x′∥2/2σ2), σ > 0, and the hyperparameters σ and λ are chosen via cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' See Sch¨olkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2009) and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2022) for a detailed discussion of parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Code to reproduce all experiments is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='com/ajthor/socks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Spring-Mass-Damper System For the purpose of analysis, we first consider the prediction problem in (2) with an (uncontrolled) spring-mass-damper system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The equations of motion are given by m¨q = −b ˙q − kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We pre- sume that we have access to imperfect system dynamics ˜f(x) = −(k/m)q, corresponding to an undamped spring-mass system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We generate a synthetic dataset S = {(xi, yi)}M i=1 with varying sample sizes M = 10, 50, 100, 500, where the states xi are taken randomly from a bounded re- gion of X, and yi = f(xi) are corresponding next states at the subsequent timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We consider two cases for the sample: 1) the states xi are taken within the region [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='15] × [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='15], meaning we only have information within a limited operating regime, and 2) the states are taken within the region [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='15] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='15], which fully encompasses the operating region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Using the sample S, we then compute the physics-informed kernel embedding ˆm0 using (5) with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2, and use ˆm0 to predict the system evolution via (2) over N = 100 time steps from a fixed initial condition x0 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' To provide a baseline for comparison, we also use the purely data-driven embedding ˆm, computed using S via (4) (see Thorpe and Oishi, 2021) to compute (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 7 PHYSICS-INFORMED KERNEL EMBEDDINGS The top row of Figure 2 shows the performance of our approach for sample sizes M = 10, 50, 100, 500 when data is collected from a limited region of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Our approach demonstrates good empirical performance, and accurately predicts the evolution of the system despite having imperfect knowledge based on the undamped system under a wide range of conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As expected, the purely data-driven prediction does not accurately predict the system evolution outside the data region, even as the amount of data increases (top right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' When data is collected over the entire region of interest, the quality of the purely data-driven estimate improves as the amount of data increases (bottom row of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Note that our proposed approach has sound performance even while using only a small fraction of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We note the following important trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 10−3 10−2 10−1 100 101 Number of transition samples (·103) Prediction Error (Ours) Physics-informed kernel embedding Kernel embedding w/o physics information Figure 3: Physics-informed embeddings demon- strate lower empirical error than purely data-driven embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Approximate physics knowledge improves out-of-distribution prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As shown in the top row of Figure 2, in contrast to the purely data-driven embedding, the physics- informed kernel embeddings generalize beyond the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Approximate physics knowledge improves sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As seen in the bottom row of Figure 2, when the observed transition data encompasses the entire region of interest, our approach is able to accurately predict the dy- namics using only 10 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Approximate physics knowledge reduces the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Figure 3 compares the empir- ical prediction error of the physics-informed kernel embedding ˆm0 against the purely data-driven embedding ˆm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We randomly sample 100 initial states x0 uniformly in the region [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1], and use the learned embeddings to predict the evolution of the state 100 time steps into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Figure 3 shows the median cumulative prediction error along these predicted trajectories (measured as the Euclidean distance between the true state vector and the predicted state vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We observe that for small datasets (particularly for M smaller than 200) the physics-informed ker- nel embedding enjoys prediction error values that are two orders of magnitude smaller than those of the purely data-driven kernel embedding, and that the baseline method requires at least 5,000 data points to achieve comparable levels of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' F-16 Aircraft We consider a ground collision avoidance scenario for an F-16 aircraft at initial altitude, as de- scribed in Djeumou and Topcu (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Heidlauf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The underlying nonlinear dynamics, containing 13 states and 4 control inputs, capture the (6-DOF) motion via evolution of velocity vt, angle of attack α, sideslip β, altitude h, attitude angles: roll φ, pitch θ, yaw ψ, and their correspond- ing rates p, q, r, engine power and two more states pn, pe for translation along north and east, as in Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The plant is built on linearly interpolated lookup tables that incorporate wind tunnel data describing the engine model, and other dynamic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We inject zero-mean Gaussian noise with a standard deviation of 1% of the magnitude of each state, such that the noise scales with the state magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We consider the case where the true dynamics are unknown, but presume that we have access to approximate dynamics with incorrect model parameters, including 8 PHYSICS-INFORMED KERNEL EMBEDDINGS 0 10 500 600 Vt (ft/sec) True dynamics Approximate dynamics Kernel embedding w/o physics information (Ours) Physics-informed kernel embedding 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 −θ(rad) 0 10 Time (s) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='6 ψ (rad) 0 10 0 2 4 6 pn ( ·10−3 ft) 0 10 0 1 2 h (·10−3 ft) Figure 4: Our approach, physics-informed kernel embeddings (green), accurately predicts the dy- namics of an F-16 aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The purely data-driven approach (blue) is inaccurate due to the system’s high-dimensional and nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' a gravitational constant of g = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0, and the interpolated lookup tables for the elevator control are half of their original values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' These changes significantly alter the response of the aircraft to pulling up and avoiding collision with the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We collect a sample S = {(x0,i, ξi)}M i=1, consisting of M = 500 initial conditions x0,i taken uniformly such that [vt, α, β, φ, θ, p, q, r, h, power]⊤ ∈ [490, 590]×[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='09]×[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='05]× [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='95]×[−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='8]×[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2]×[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2]×[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2]×[3800, 4200]×[8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='7, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='3], and the resulting trajectories from those initial conditions ξi ∼ T(· | x0,i) using the true, nominal dynamics, where T : BX N ×X → [0, 1] is a stochastic kernel that represents the LQR-controlled, closed-loop system dynamics over N = 1500 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Using trajectory data modifies the probability model to be a stochastic kernel over state trajectories, but does not significantly alter the kernel estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Modifications of our approach to accommodate trajectory data is described in Thorpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Figure 4 shows the solution to (2) for state prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We see significant improvement in predic- tion accuracy over the purely data-driven approach, in particular the altitude h and yaw angle Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' As expected, the purely data-driven method fails to capture the F-16 system behavior with the limited data due to the highly nonlinear and high-dimensional dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Interestingly, the prediction of the pitch angle θ using our approach shows oscillations due to the approximate dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' This raises question of whether the addition of data can overcome unstable model effects in the approximate dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' However, we leave this for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Control of a Nonholonomic Vehicle System We solve (3) for a target tracking control problem with a nonholonomic vehicle, as in Thorpe and Oishi (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The dynamics are given by ˙x1 = u1 sin(x3), ˙x2 = u1 cos(x3), ˙x3 = u2, where x = [x1, x2, x3]⊤ ∈ R3 is the state and u = [u1, u2]⊤ ∈ R2 is the control input, which we constrain to be within the bounds [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='5]×[−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We discretize the system in time and apply an affine disturbance with an exponential distribution wt ∼ Exp(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1), with PDF f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' α) = α exp(−αx) if x ≥ 0 and f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' α) = 0 if x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We presume that the deterministic discrete-time dynamics are given as approximate dynamical system knowledge, but that the stochastic dynamics are unknown (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' we do not have prior knowledge of the disturbance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We seek to solve (3), where we minimize the squared Euclidean distance to a moving target over a time horizon of N = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We define a trajectory of target waypoints z0, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' , zN (shown in black 9 PHYSICS-INFORMED KERNEL EMBEDDINGS Sampled Transition Data Target Trajectory Data-Driven Trajectory (Ours) Physics-Informed Trajectory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 x1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='6 x2 Data Regime Figure 5: Comparison of our proposed method against Thorpe and Oishi (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The solution via physics-informed kernel embeddings (green) closely follows the target trajectory (black), even outside the data regime, while the performance of the purely data-driven solution (blue) degrades outside the region for which we have data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We consider the case where the future target position is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Thus, we solve the following (unconstrained) optimization problem at each time step: minπ E[∥xt+1 − zt∥2] as in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' See Thorpe and Oishi (2021) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We collect a sample S = {(xi, ui, yi)}M i=1 of size M = 500, where the states xi are taken uniformly in the region shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' To compute the control algorithm in (14), we generate a sample A = {˜uj}P j=1 of P = 210 control actions taken uniformly in the region [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='2] × [−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We then presume that the true dynamics are unknown for the purpose of computing the control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' We then computed the physics informed kernel embedding ˆm0 with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Using ˆm0, we simulate the system from an initial condition x0 = [−1, 0, π/2]⊤ and solve (14) at each time step to compute the stochastic policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The total computation time was approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='272 seconds, and the results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Using the same sample size, the baseline method from Thorpe and Oishi (2021) fails to generate a meaningful trajectory (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' To generate a comparable trajectory, we used a much larger sample size, M = 5000, shown in blue in Figure 5, and the computation time was approximately 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='993 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' This shows that our method demonstrates better empirical and computational performance, and requires less data due to the inclusion of prior dynamics knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Conclusions & Future Work In this paper, we presented physics-informed kernel embeddings, a novel technique for incorporat- ing prior system knowledge in data-driven representations of system dynamics using kernel distribu- tion embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Numerical experiments demonstrate the effectiveness of the proposed method on prediction tasks, including for systems with imperfect system knowledge on a spring-mass-damper system and highly nonlinear dynamics on an F-16 system, and on control tasks via a nonholonomic system target tracking problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Results show that our approach generalizes well outside the data regime, is computationally efficient, and is robust to common sampling issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' An important direction for future work in this area involves an exploration of how to incorporate other forms of prior knowledge, such as known system properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' symmetry, invariance) into the learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Additionally, of practical interest is a characterization of the effect that poor or inaccurate approximate knowledge has on the learned representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' 10 PHYSICS-INFORMED KERNEL EMBEDDINGS Acknowledgments This material is based upon work supported by the National Science Foundation under NSF Grants Number CNS-1836900 and NSF 1646522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Any opinions, findings, and conclusions or recom- mendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' The NASA University Leadership initiative (Grant #80NSSC20M0163) provided funds to assist the authors with their research, but this article solely reflects the opinions and conclusions of its authors and not any NASA entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550- 19-1-0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Any opinions, findings, conclusions and or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' This material is based upon work supported by the Department of the Navy, Office of Naval Research under award number N00014-22-1-2254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE1T4oBgHgl3EQf-QYz/content/2301.03565v1.pdf'} +page_content=' Any opinions, findings, and conclusions or recommenda- tions expressed in this material are those of the authors and do not necessarily reflect the views of the Office of Naval Research.' metadata={'source': 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sha256:80c0d6994080978aa7512683d7359361c47a1e0a0b1531bac9177130c6df77df +size 97205 diff --git a/ENFRT4oBgHgl3EQfyDiq/content/tmp_files/2301.13644v1.pdf.txt b/ENFRT4oBgHgl3EQfyDiq/content/tmp_files/2301.13644v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a1ef7def979bc62af61f8ce0405c24b8547bc0a --- /dev/null +++ b/ENFRT4oBgHgl3EQfyDiq/content/tmp_files/2301.13644v1.pdf.txt @@ -0,0 +1,1202 @@ +Dablander et al. +RESEARCH +Exploring QSAR Models for Activity-Cliff +Prediction +Markus Dablander1, Thierry Hanser2, Renaud Lambiotte1 and Garrett M. Morris3* +Abstract +Introduction and Methodology: Pairs of similar compounds that only differ by a small structural modification +but exhibit a large difference in their binding affinity for a given target are known as activity cliffs (ACs). It has +been hypothesised that QSAR models struggle to predict ACs and that ACs thus form a major source of predic- +tion error. However, a study to explore the AC-prediction power of modern QSAR methods and its relationship to +general QSAR-prediction performance is lacking. We systematically construct nine distinct QSAR models by com- +bining three molecular representation methods (extended-connectivity fingerprints, physicochemical-descriptor +vectors and graph isomorphism networks) with three regression techniques (random forests, k-nearest neighbours +and multilayer perceptrons); we then use each resulting model to classify pairs of similar compounds as ACs or +non-ACs and to predict the activities of individual molecules in three case studies: dopamine receptor D2, factor +Xa, and SARS-CoV-2 main protease. +Results and Conclusions: We observe low AC-sensitivity amongst the tested models when the activities of +both compounds are unknown, but a substantial increase in AC-sensitivity when the actual activity of one of +the compounds is given. Graph isomorphism features are found to be competitive with or superior to classical +molecular representations for AC-classification and can thus be employed as baseline AC-prediction models or +simple compound-optimisation tools. For general QSAR-prediction, however, extended-connectivity fingerprints +still consistently deliver the best performance. Our results provide strong support for the hypothesis that indeed +QSAR methods frequently fail to predict ACs. We propose twin-network training for deep learning models as a +potential future pathway to increase AC-sensitivity and thus overall QSAR performance. +Keywords: QSAR modelling; Activity cliffs; Activity cliff prediction; Machine learning; Deep learning; Molecular +representation; Physicochemical descriptors; Extended-connectivity fingerprints; Graph isomorphism networks; +Binding affinity prediction +*Correspondence: garrett.morris@dtc.ox.ac.uk +3Department of Statistics, University of Oxford, 24-29 St Giles’, OX1 3LB, +Oxford, United Kingdom +Full list of author information is available at the end of the article +arXiv:2301.13644v1 [cs.LG] 31 Jan 2023 + +Dablander et al. +Page 2 of 17 +Introduction +Activity cliffs (ACs) are pairs of small molecules that +exhibit high structural similarity but at the same +time show an unexpectedly large difference in their +binding affinity against a given pharmacological tar- +get [14, 47, 63, 64, 67–69]. The existence of ACs di- +rectly defies the intuitive idea that chemical com- +pounds with similar structures should have similar ac- +tivities, often referred to as the molecular similarity +principle. An example of an AC between two inhibitors +of blood coagulation factor Xa [43] is depicted in Fig- +ure 1; a small chemical modification involving the ad- +dition of a hydroxyl group leads to an increase in in- +hibition of almost three orders of magnitude. +For medicinal chemists, ACs can be puzzling and +confound their understanding of structure-activity re- +lationships (SARs) [19, 67, 77]. ACs reveal small +compound-modifications with large biological impact +and thus represent rich sources of pharmacological in- +formation. Mechanisms by which a small structural +transformation can give rise to an AC include a dras- +tic change in 3D-conformation and/or the switching +to a different binding mode or even binding site. +ACs form discontinuities in the SAR-landscape and +can therefore have a crucial impact on the success +of lead-optimisation programmes. While knowledge +about ACs can be powerful when trying to escape from +flat regions of the SAR-landscape, their presence can +be detrimental in later stages of the drug development +process, when multiple molecular properties beyond +mere activity need to be balanced carefully to arrive +at a safe and effective compound [14, 67]. +In the field of computational chemistry, ACs are sus- +pected to form one of the major roadblocks for success- +ful quantitative structure-activity relationship (QSAR) +modelling [14, 26, 47, 63]; abrupt changes in potency +are expected to negatively influence machine learn- +ing algorithms for pharmacological activity prediction. +During the development of QSAR models, ACs are +sometimes dismissed as measurement errors [49], but +simply removing ACs from a training data set can +result in a loss of precious SAR-information [15]. +Golbraikh et al. [26] developed the MODI metric +to quantify the smoothness of the SAR-landscape of +binary molecular classification data sets and showed +that the SAR-landscape smoothness is a strong de- +terminant for downstream QSAR-modelling perfor- +mance. In a related work, Sheridan et al. [63] found +that the density of ACs in a molecular data set is +strongly predictive of its overall modelability by clas- +sical descriptor- and fingerprint-based QSAR meth- +ods. Furthermore, they found that such methods in- +cur a significant drop in performance when the test +set is restricted to “cliffy” compounds that form a +large number of ACs. In a more extensive study, van +Tilborg et al. [75] observed a similar drop in perfor- +mance when testing classical and graph-based QSAR +techniques on compounds involved in ACs. Notably, +Figure 1 Example of an activity cliff (AC) for blood coagulation factor Xa. A small structural transformation in the upper compound +leads to an increase in inhibitory activity of almost three orders of magnitude. Both compounds were identified in the same ChEMBL +assay with ID 658338. + +Dablander et al. +Page 3 of 17 +in both studies this performance drop was also ob- +served for highly nonlinear and adaptive deep learning +models. In fact, van Tilborg reports that descriptor- +based QSAR methods even outperform more complex +deep learning models on “cliffy” compounds associ- +ated with ACs. This runs counter to earlier hopes ex- +pressed in the literature that the approximation power +of deep neural networks might ameliorate the problem +of ACs [79]. +While these works provide valuable insights into the +detrimental effects of SAR discontinuity on QSAR +models, they consider ACs mainly indirectly by fo- +cussing on individual compounds involved in ACs. Ar- +guably, a distinct and more natural approach would +be to investigate ACs directly at the level of com- +pound pairs. This approach has been followed in the +AC-prediction field which is concerned with developing +techniques to classify whether a pair of similar com- +pounds forms an AC or not. An effective AC-prediction +method would be of high value for drug development +with important applications in rational compound op- +timisation and automatic SAR-knowledge acquisition. +The AC-prediction literature is still very thin com- +pared to the QSAR-prediction literature. An attempt +to conduct an exhaustive literature review on AC- +prediction techniques revealed a total number of 15 +methods [4, 7, 10, 27, 29, 32, 34, 39, 44, 51, 52, 54, +57, 71], all of which have been published since 2012. +Current AC-prediction methods are often based on cre- +ative ways to extract features from pairs of molecular +compounds in a manner suitable for standard machine +learning pipelines. For example, Horvath et al. [29] +used condensed graphs of reactions [28, 35], a rep- +resentation technique originally introduced for mod- +elling of chemical reactions, to encode pairs of simi- +lar compounds and subsequently predict ACs. Another +method was recently described by Iqbal et al. [34] who +investigated the abilities of convolutional neural net- +works operating on 2D images of compound pairs to +distinguish between ACs and non-ACs. Interestingly, +none of the AC-prediction methods we identified em- +ploy feature extraction techniques built on modern +graph neural networks (GNNs) [20, 25, 40, 76, 81] +with the exception of Park et al. [54] who recently ap- +plied graph convolutional methods to compound-pairs +to predict ACs. +In spite of the existence of advanced AC-prediction +models there are significant gaps left in the current +AC-prediction literature. Note that any QSAR model +can immediately be repurposed as an AC-prediction +model by using it to individually predict the activ- +ities of two structurally similar compounds and then +thresholding the predicted absolute activity difference. +Nevertheless, at the moment there is no study that +uses this straightforward technique to investigate the +potential of current QSAR models to classify whether +a pair of compounds forms an AC or not. Impor- +tantly, this also entails that the most salient AC- +prediction models [27, 29, 34, 44, 71] have not been +compared to a simple QSAR-modelling baseline ap- +plied to compound pairs. It is thus an open question +to what extent (if at all) these tailored AC-prediction +techniques outcompete repurposed QSAR methods in +the detection of ACs. This is especially relevant in +light of the fact that several published AC-predict¸ion +models [27, 34, 44] are evaluated via compound-pair- +based data splits which incur a significant overlap be- +tween training set and test set at the level of individ- +ual molecules; this type of data split should strongly +favour standard QSAR models for AC-prediction, yet +a comparison to such baseline methods is lacking. +We address these gaps by systematically investigat- +ing the abilities of nine frequently used QSAR models +to classify pairs of similar compounds as ACs or non- +ACs within three pharmacological data sets: dopamine +receptor D2, factor Xa, and SARS-CoV-2 main pro- +tease. Each QSAR model is constructed by combining +a molecular representation method (physicochemical- +descriptor vectors (PDVs) [72], extended-connectivity +fingerprints (ECFPs) [59], or graph isomorphism net- +works (GINs) [81]) with a regression technique (ran- +dom forests (RFs), k-nearest neighbours (kNNs), or +multilayer perceptrons (MLPs)). All models are used +for two distinct prediction tasks: QSAR-prediction at +the level of individual molecules, and AC-classification +at the level of compound-pairs. The main contribution +of this study is to shed light on the following questions: +• What is the relationship between the ability of a +QSAR model to predict the activities of individual +compounds, versus its ability to classify whether +pairs of similar compounds form ACs? +• When (if at all) are common QSAR models capa- +ble of predicting ACs? +• When (if at all) are common QSAR models capa- +ble of predicting which of two similar compounds +is the more active one? +• Which QSAR model shows the strongest AC- +prediction performance, and should thus be used +as a baseline against which to compare tailored +AC-prediction models? +• Do differentiable GINs outperform classical non- +trainable ECFPs and PDVs as molecular repre- +sentations for QSAR- and/or AC-prediction? +• How could ACs potentially be used to improve +QSAR-modelling performance? + +Dablander et al. +Page 4 of 17 +Experimental Methodology +Molecular Data Sets +We built three binding affinity data sets of small- +molecule inhibitors of dopamine receptor D2, factor +Xa, and SARS-CoV-2 main protease. Factor Xa is an +enzyme in the coagulation cascade and a canonical tar- +get for blood-thinning drugs [43]. Dopamine receptor +D2 is the main site of action for classic antipsychotic +drugs which act as antagonists of the D2 receptor [62]. +SARS-CoV-2 main protease is one of the key enzymes +in the viral replication cycle of the SARS coronavirus +2, that recently caused the unprecedented COVID-19 +pandemic; it is one of the most promising targets for +antiviral drugs against this coronavirus [74]. +For dopamine receptor D2 and factor Xa, data was +extracted from the ChEMBL database [45] in the form +of SMILES strings with associated Ki [nM] values. +For SARS-CoV-2 main protease, data was obtained +from the COVID moonshot project [1] in the form +of SMILES strings with associated IC50 [µM] values. +SMILES strings were standardised and desalted via +the ChEMBL structure pipeline [8]. This step also re- +moved solvents and all isotopic information. Follow- +ing this, SMILES strings that produced error messages +when turned into an RDKit mol object were deleted. +Finally, a scan for duplicate molecules was performed: +If the activities in a set of duplicate molecules were +within the same order of magnitude then the set was +unified via geometric averaging. Otherwise, the mea- +surements were considered unreliable and the corre- +sponding set of duplicate molecules was removed. This +procedure reduced the data set for dopamine receptor +D2 / factor Xa / SARS-CoV-2 main protease from +8883 / 4116 / 1926 compounds to 6333 / 3605 / 1924 +unique compounds whereby 174 / 21 / 0 sets of dupli- +cate SMILES were removed and the rest was unified. +Activity Cliffs: Definition of Binary Classification Tasks +The exact definition of an AC hinges on two concepts: +structural similarity and large activity difference. An +elegant technique to measure structural similarity in +the context of AC analysis is given by the matched +molecular pair (MMP) formalism [31, 38]. An MMP +is a pair of compounds that share a common struc- +tural core but differ by a small chemical transforma- +tion at a specific site. Figure 1 depicts an example of an +MMP whose variable parts are formed by a hydrogen +atom and a hydroxyl group. To detect MMPs algorith- +mically, we used the mmpdb Python-package provided +by Dalke et al. [17]. We restricted ourselves to MMPs +with the following commonly used [27, 29, 71] size con- +straints: the MMP core was required to contain at least +twice as many heavy atoms as either of the two vari- +able parts; each variable part was required to contain +no more than 13 heavy atoms; the maximal size dif- +ference between both variable parts was set to eight +heavy atoms; and bond cutting was restricted to sin- +gle exocyclic bonds. To guarantee a well-defined map- +ping from each MMP to a unique structural core, we +canonically chose the core that contained the largest +number of heavy atoms whenever there was ambiguity. +Based on the ratio of the activity values of both MMP +compounds, each MMP was assigned to one of three +classes: “AC”, “non-AC” or “half-AC”. In accordance +with the literature [5, 27, 29, 52, 77] we assigned an +MMP to the “AC”-class if both activity values differed +by at least a factor of 100. If both activity values dif- +fered by no more than a factor of 10, then the MMP +was assigned to the “non-AC”-class. In the residual +case the MMP was assigned to the “half-AC”-class. To +arrive at a well-separated binary classification task, we +labelled all ACs as positives and all non-ACs as nega- +tives. The half-ACs were removed and not considered +further in our experiments. It is relevant to know the +direction of a potential activity cliff, i.e. which of the +compounds in the pair is the more active one. We thus +assigned a binary label to each MMP indicating its po- +tency direction (PD). PD-classification is a balanced +binary classification task. Table 1 gives an overview of +all our curated data sets. +Data Splitting Technique +ACs are molecular pairs rather than single molecules; +it is thus not obvious how best to split up a chemical +data set into non-overlapping training- and test sets +for the fair evaluation of an AC-prediction method. +There seems to be no consensus about which data +splitting strategy should be canonically used. Several +authors [27, 34, 44] have employed a random split +at the level of compound pairs. While this technique +is conceptually straightforward, it must be expected +to incur a significant overlap between training- and +test set at the level of individual molecules. For ex- +ample, randomly splitting up a set of three MMPs +{{s1, s2}, {s1, s3}, {s2, s3}} into a training- and a test +set might lead to {s1, s2} and {s1, s3} getting assigned +to the training- and {s2, s3} getting assigned to the +test set which leads to a full inclusion of the test set +in the training set at the level of individual molecules. +This molecular overlap is problematic for at least three +reasons: Firstly, it likely leads to overly optimistic re- +sults for AC-prediction methods since they will have +already encountered some of the test compounds dur- +ing training. Secondly, it does not model the natural +situation encountered by medicinal chemists who we +assume will not know the activity value of at least one + +Dablander et al. +Page 5 of 17 +Data Set +Dopamine Receptor D2 +Factor Xa +SARS-CoV-2 +Main Protease +Compounds +6333 +3605 +1924 +MMPs +35484 +21292 +12594 +ACs +461 +1896 +521 +Half-ACs +3804 +4693 +1762 +Non-ACs +31219 +14703 +10311 +ACs : Non-ACs +≈ 1 : 68 +≈ 1 : 8 +≈ 1 : 20 +Table 1 Sizes of our curated data sets and their respective numbers of matched molecular pairs (MMPs), activity cliffs (ACs), half- +activity-cliffs (half-ACs) and non-activity-cliffs (non-ACs). +compound in a test-set pair. Thirdly, the mentioned +molecular overlap should lead to strong AC-prediction +results for standard QSAR models, but to the best of +our knowledge, no such control experiments have been +run in the literature. +Horvath et al. [29] and Tamura et al. [71] have made +efforts to address the shortcomings of a compound- +pair-based random split. They came up with advanced +data splitting algorithms designed to mitigate the +molecular-overlap problem by either managing distinct +types of test sets according to compound membership +in the training set or by designing splitting techniques +based on the structural cores of MMPs. However, their +data splitting schemes exhibit a relatively high degree +of complexity which can make their implementation +and interpretation difficult. +We propose a novel data splitting method which rep- +resents a favourable trade-off between rigour, inter- +pretability and simplicity. Our technique shares some +of its concepts with the methods proposed by Horvath +et al. [29] and Tamura et al. [71] but might be simpler +to implement and interpret. We first split the data +into a training- and test set at the level of individual +molecules and then use this basic split to distinguish +several types of test sets at the level of compound pairs. +Let +D = {s1, s2, ...} +be the given data set of individual molecules. Further- +more, let +M ⊆ {{s, ˜s} | s ̸= ˜s and s, ˜s ∈ D} +be the set of all MMPs in D that have been labelled as +either ACs or non-ACs. Each MMP {s, ˜s} ∈ M shares +a common structural core denoted as core({s, ˜s}). We +use a random split to partition D into a training set +Dtrain and a test set Dtest and then define the following +MMP-sets: +Mtrain = {{s, ˜s} ∈ M | s, ˜s ∈ Dtrain}, +Minter = {{s, ˜s} ∈ M | s ∈ Dtrain, ˜s ∈ Dtest}, +Mtest = {{s, ˜s} ∈ M | s, ˜s ∈ Dtest}, +Mcores = {{s, ˜s} ∈ Mtest | core({s, ˜s}) /∈ Ctrain}. +Here, +Ctrain = {core({s, ˜s}) | {s, ˜s} ∈ Mtrain ∪ Minter}, +which describes the set of MMP-cores that appear in +Dtrain. +Note that Mtrain ∪ Minter ∪ Mtest = M. The pair +(Dtrain, Mtrain) describes the training space at the +level of individual molecules and MMPs, and can be +used to train a QSAR- or AC-prediction method. A +trained method can then classify MMPs in Mtest, +Minter and Mcores. Mtest models an AC-prediction +setting where the activities of both MMP-compounds +are unknown. Mcores represents the subset of MMPs +in Mtest whose structural cores do not appear in +Mtrain ∪ Minter; Mcores thus models the difficult task +of predicting ACs in a strucurally novel area of chemi- +cal space. Finally, Minter represents an AC-prediction +scenario where the activity of one MMP-compound is +given a priori; this can be interpreted as a compound- +optimisation task where one strives to predict small +AC-inducing modifications of a query compound with +known activity. An illustration of our data splitting +strategy is given in Figure 2. +We implemented our data splitting strategy within +a k-fold cross validation scheme repeated with m ran- +dom seeds. This generated data splits of the form +Sij = (Dij +train, Dij +test, Mij +train, Mij +test, Mij +inter, Mij +cores) +for i ∈ {1, ..., m} and j ∈ {1, ..., k} where (Dij +train, Dij +test) +represents the j-th split of D in the cross validation + +Dablander et al. +Page 6 of 17 +Figure 2 Illustration of our data splitting strategy. We distinguish between three MMP-sets, Mtrain, Minter and Mtest, depending on +whether both MMP-compounds are in Dtrain, one MMP-compound is in Dtrain and the other one is in Dtest, or both MMP-compounds +are in Dtest. We additionally consider a fourth MMP-set, Mcores, consisting of the MMPs in Mtest whose structural cores do not +appear in Mtrain ∪ Minter. +round with random seed i. The overall QSAR- and AC- +prediction performance of each model was recorded as +the average over the mk training- and test runs based +on all data splits S1,1, ..., Smk. We chose the config- +uration (k, m) = (2, 3) which gave a good trade-off +between computational costs and accuracy and rea- +sonable numbers of MMPs in the compound-pair-sets. +In particular, random cross-validation with k = 2 gave +expected relative sizes of: +|Mtrain| : |Minter| : |Mtest| = 1 : 2 : 1. +On average, 12.7 %, 11.91 %, and 6.84 % of MMPs +in Mtest were also in Mcores for dopamine receptor +D2, factor Xa, and SARS-CoV-2 main protease, re- +spectively. +Prediction Strategies and Performance Measures +In a data split of the form +S = (Dtrain, Dtest, Mtrain, Mtest, Minter, Mcores) +each individual compound, s ∈ Dtrain ∪ Dtest = D, +can be associated with an activity label a(s) ∈ R, de- +fined as the negative decadic logarithm of the exper- +imentally measured activity of s. We stuck with the +canonical units used in the ChEMBL database and +the COVID moonshot project ([nM] for Ki and [µM] +for IC50); each activity label a(s) thus represents a +standard pKi- or pIC50 value (with an additive shift +towards 0 caused by the units which might slightly +benefit prediction techniques initialised around the ori- +gin). We are interested in QSAR-prediction functions, +f : D → R, +that can map a chemical structure s ∈ D to an es- +timate of its binding affinity a(s). The mapping f is +found via an algorithmic training process on the la- +belled data set +{(s, a(s)) | s ∈ Dtrain} +and can then either be used to predict the activity la- +bels of compounds in Dtest, or it can be repurposed to +classify whether an MMP forms an activity cliff (AC- +classification) and what the potency direction of an +MMP is (PD-classification). If {s, ˜s} ∈ Minter, then +one can assume that the activity label of one of the +compounds, say a(s), is known; f is then used to clas- + +Dablander et al. +Page 7 of 17 +sify {s, ˜s} via: +{s, ˜s} �→ +� +Non-AC +if |a(s) − f(˜s)| ≤ dcrit, +AC +if |a(s) − f(˜s)| > dcrit. +Here dcrit ∈ R>0 is a critical threshold above which an +MMP is classified as an AC. Throughout this work we +use dcrit = 1.5 (in pKi- or pIC50 units) since this value +represents the middle point between the intervals [0, 1] +and [2, ∞) which correspond to absolute activity-label +differences associated with non-ACs and ACs respec- +tively. +If {s, ˜s} ∈ Mtest ∪ Mcores then the activities of both +compounds are unknown and we classify {s, ˜s} via: +{s, ˜s} �→ +� +Non-AC +if |f(s) − f(˜s)| ≤ dcrit, +AC +if |f(s) − f(˜s)| > dcrit. +PD-classification for MMPs is performed in a straight- +forward manner: the activity labels of both MMP- +compounds are predicted via f and then compared to +classify which compound is the more active one. +The performance of f for standard QSAR predic- +tion in Dtest is measured via the mean absolute er- +ror (MAE). For the balanced PD-classification prob- +lem we rely on accuracy as a suitable performance +measure. For the highly imbalanced task of AC- +classification, however, we use the Matthews corre- +lation coefficient (MCC), as well as sensitivity and +precision. For the relatively small SARS-CoV-2 main +protease data set we sometimes encountered the edge +case where there were no positive predictions; we then +set MCC = 0 and ignored ill-defined precision mea- +surements when averaging the performance metrics to +obtain the final results. +Molecular Representation- and Regression Techniques +We constructed nine QSAR models via a robust com- +binatorial methodology that systematically combines +three molecular representation methods with three re- +gression techniques. This setup allows, for example, to +compare the performance of molecular representation +methods across regression techniques, data sets and +predictions tasks. +For molecular representation, we used extended- +connectivity fingerprints [59] (ECFPs), physicochem- +ical molecular descriptor vectors [72] (PDVs), and +graph isomorphism networks (GINs) [81]. Both ECFPs +and PDVs were computed via RDKit [42]. The ECFPs +were chosen to use a radius of two, a length of 2048 +bits, and active chirality flags. The PDVs had a di- +mensionality of 200 and were constructed using the +general list of descriptors from the work of Fabian +et al. [21]. This list encompasses properties related +to druglikeness, logP, molecular refractivity, electro- +topological state, molecular graph-structure, fragment +profile, charge, and topological surface properties. The +GIN was implemented using PyTorch Geometric [23] +and consisted of a variable number of graph convolu- +tional layers, each with two internal hidden layers with +ReLU activations and batch normalisation [33]. We +further chose the maxpool operator which computes +the component-wise maximum over all atom feature +vectors in the final graph layer to obtain a graph-level +representation. +Each molecular representation was used as an input +featurisation for three regression techniques: random +forests (RFs), k-nearest neigbours (kNNs) and multi- +layer perceptrons (MLPs). The RF- and kNN-models +were implemented via scikit-learn [56] and the MLP- +models via PyTorch [55]. The MLPs used ReLU acti- +vations and batch normalisation at each hidden layer. +The GIN was combined with the regression tech- +niques as follows: For MLP regression, the GIN was +trained with the MLP as a projection head after the +pooling step in the usual end-to-end manner. For RF- +or kNN-regression, the GIN was first trained with a +single linear layer added after the global pooling step +that directly mapped the graph-level representation to +an activity prediction. After this training phase the +weights of the GIN were frozen and it was used as a +static feature extractor. The RF- or kNN-regressor was +then trained on the features extracted by the frozen +GIN. Figure 3 illustrates our combinatorial experimen- +tal methodology. +Model Training and Hyperparameter Optimisation +All models were trained using full inner hyperparame- +ter-optimisation loops. Hyperparameters of RFs and +kNNs were optimised in scikit-learn [56] by uniformly +random sampling of hyperparameters from a prede- +fined grid. The hyperparameters of MLPs and GINs +were sampled from a predefined grid via the tree- +structured Parzen estimator algorithm implemented in +Optuna [2]. Deep learning models were trained for 500 +epochs on a single NVIDIA GeForce RTX 3060 GPU +via the mean squared error loss function using AdamW +optimisation [46]. Weight decay, learning rate decay +and dropout [65] were employed at all hidden layers +for regularisation. Batch size, learning rate, learning +rate decay rate, weight decay rate, and dropout rate +were treated as hyperparameters and subsequently op- +timised. Note that the training length (i.e. the number +of gradient updates) was implicitly optimised by tun- +ing the batch size for the fixed number of 500 training + +Dablander et al. +Page 8 of 17 +Figure 3 Schematic showing the combinatorial experimental methodology used for the study. Each molecular representation method +is systematically combined with each regression technique, giving a total of nine QSAR models. Each QSAR model is trained and +evaluated for QSAR-prediction, AC-classification and PD-classification within a 2-fold cross validation scheme repeated with 3 random +seeds. For each of the 2 ∗ 3 = 6 trials, an extensive inner hyperparameter-optimisation loop on the training set is performed for each +QSAR model. +epochs. Further implementation details can be found +in our public code repository[1]. +Results and Discussion +The QSAR-prediction-, AC-classification- and PD- +classification results for all three data sets are depicted +in Figures 4 to 9. +QSAR-Prediction Performance +When considering the results depicted in Figures 4 +to 9 with respect to QSAR-prediction performance, +one can see that ECFPs tend to lead to better perfor- +mance (i.e. a lower QSAR-MAE) compared to GINs, +which in turn tend to lead to better performance com- +pared to PDVs. In particular, the combination MLP- +ECFP consistently produced the lowest QSAR-MAE +across all three targets. These observations reinforce a +growing corpus of literature that suggests that train- +able GNNs have not yet reached a level of techni- +cal maturity by which they consistently and defini- +tively outperform the much simpler non-differentiable +ECFPs at important molecular property prediction +tasks [13, 37, 48, 50, 60, 66, 80]. +[1]https://github.com/MarkusFerdinandDablander/ +QSAR-activity-cliff-experiments +AC-Classification Performance +The AC-MCC plots in Figures 4 to 6 reveal sur- +prisingly strong overall AC-classification results on +Minter. This type of MMP-set models a compound- +optimisation scenario where a researcher strives to +identify small structural modifications with a large im- +pact on the activity of query compounds with known +activities. For this task, a significant portion of our +QSAR models exhibit an AC-MCC value greater than +0.5 across targets, which appears impressive consider- +ing the simplicity of the approach. Exchanging Minter +with either Mtest or Mcores leads to a substantial drop +in the AC-MCC to approximately 0.3 that appears to +be mediated by a large drop in AC-sensitivity. +In most cases, GINs perform better than the other +molecular representation methods with respect to the +AC-MCC. Notably, kNN-regressors consistently per- +form best for AC-classification when combined with +GIN-features; this supports the idea that GINs might +have a heightened ability to resolve ACs by learning +an embedding of chemical space in which the distance +between two compounds is reflective of activity dif- +ference rather than structural difference. The combi- +nations GIN-MLP, GIN-RF and ECFP-MLP exhibit +particularly high AC-MCC values relative to the other +methods. We recommend using at least one of these +three models as a baseline against which to compare +tailored AC-prediction models; the practical utility of + +Dablander et al. +Page 9 of 17 +Figure 4 QSAR-prediction- and AC-classification results for dopamine receptor D2. For each plot, the x-axis corresponds to a +combination of MMP-set and AC-classification performance metric and the y-axis shows the QSAR-prediction performance on the +molecular test set Dtest. The total length of each error bar equals twice the standard deviation of the performance metric measured +over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models. For each plot, the lower right corner corresponds to strong performance +at both prediction tasks. +any AC-prediction technique that cannot outperform +these three common QSAR methods is questionable. +Across all three targets, AC-sensitivity is moder- +ately high on Minter but universally low on Mtest +and Mcores. This is consistent with the hypothesis +that ACs form one of the major sources of prediction +error for QSAR models. The weak AC-sensitivity on +Mtest and Mcores indicates that modern QSAR meth- +ods are largely blind to ACs in novel areas of chemi- +cal space and thus lack essential chemical knowledge. +GINs clearly outperform the other two more classi- +cal molecular representations across regression tech- +niques with respect to AC-sensitivity. In particular, +the GIN-MLP combination leads to the highest AC- + +Dablander et al. +Page 10 of 17 +Figure 5 QSAR-prediction- and AC-classification results for factor Xa. For each plot, the x-axis corresponds to a combination of +MMP-set and AC-classification performance metric and the y-axis shows the QSAR-prediction performance on the molecular test set +Dtest. The total length of each error bar equals twice the standard deviation of the performance metric measured over all mk = 3∗2 = 6 +hyperparameter-optimised models. For each plot, the lower right corner corresponds to strong performance at both prediction tasks. +sensitivity in all examined cases and thus discovers +the most ACs. The highly parametric nature of GINs +that makes them prone to overfitting could at the same +time enable them to better model jagged regions of the +SAR-landscape that contain ACs than classical task- +agnostic representations. +There is a wide gap between distinct prediction +techniques with respect to AC-precision: some models +achieve a considerable level of AC-precision such that +over 50% of positively predicted MMPs in Mtest and +Mcores are indeed actual ACs. Other QSAR models, +however, seem to fail almost entirely with respect to +this metric on Mtest and Mcores and only deliver mod- +est performance on Minter. RFs tend to exhibit the +strongest AC-precision and the weakest AC-sensitivity. +This might be as a result of their ensemble nature + +Dablander et al. +Page 11 of 17 +Figure 6 QSAR-prediction- and AC-classification results for SARS CoV-2 main protease. For each plot, the x-axis corresponds to +a combination of MMP-set and AC-classification performance metric and the y-axis shows the QSAR-prediction performance on the +molecular test set Dtest. The total length of each error bar equals twice the standard deviation of the performance metric measured +over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models. The precision of the AC-classification task is lacking for the ECFP + kNN +technique on Mtest and Mcores since this method produced only negative AC-predictions for all trials on this data set. For each plot, +the lower right corner corresponds to strong performance at both prediction tasks. +which should intuitively lead to conservative but trust- +worthy predictions of extreme effects such as ACs. +PD-Classification Performance +The abilities of the evaluated QSAR models to identify +which is the more active compound in an MMP is uni- +versally weak, with PD-accuracies clustering around +0.7 on Minter and around 0.6 on Mtest and Mcores, as +can be seen in the top rows of Figures 7 to 9. Predict- +ing the potency direction for two compounds with sim- +ilar structures and thus usually similar activity levels +must be considered a challenging task. The combina- +tion ECFP-MLP reaches the strongest PD-accuracy + +Dablander et al. +Page 12 of 17 +Figure 7 QSAR-prediction- and PD-classification results for dopamine receptor D2. Each column corresponds to an upper plot and +a lower plot for one of the MMP-sets Minter, Mtest or Mcores. The x-axis of each upper plot indicates the PD-classification accuracy +on the full MMP-set; the x-axis of each lower plot indicates the PD-classification accuracy on a restricted MMP-set only consisting +of MMP predicted to be ACs by the respective method. The y-axis of each plot shows the QSAR-prediction performance on the +molecular test set Dtest. The total length of each error bar equals twice the standard deviation of the performance metrics measured +over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models. For each plot, the lower right corner corresponds to strong performance +at both prediction tasks. +in the majority of cases and we recommend starting +with this model as a baseline for more advanced PD- +prediction methods. +One can argue that the activity order of two simi- +lar compounds is of little interest if the true activity +difference is small, as is often the case. We therefore +also restricted PD-classification to predicted ACs. The +three plots in the bottom rows of Figures 7 to 9 depict +the PD-accuracy of each QSAR model on the subset +of MMPs that were also predicted to be ACs by the +same model. In this practically more relevant scenario +PD-prediction accuracy tends to exceed 0.9 on Minter +and 0.8 on Mtest and Mcores. The QSAR models in- +vestigated here are thus able to identify the correct +activity order of MMPs if they also predict them to be +ACs. The relatively rare instances in which the PD of +a predicted AC is misclassified, however, reflect severe +QSAR-prediction errors. +Linear Relationship between QSAR-MAE and AC-MCC +Our experiments reveal a consistent linear relationship +between the QSAR-MAE and the AC-MCC as can +be seen in the left columns of Figures 4 to 6. A po- +tential mechanism driving this effect could be that as +the overall QSAR-MAE of a model improves, its ac- +curacy at predicting activity differences between sim- +ilar molecules might be expected to improve as well; +previously misclassified MMPs whose predicted abso- +lute activity differences were already close to the crit- +ical value dcrit = 1.5 might then gradually move to +the correct side of the decision boundary and increase +the AC-MCC. The results suggest that for real-world +QSAR models the AC-MCC and the QSAR-MAE are +strongly predictive of each other; while this observa- +tion only rests on nine models, it is highly consistent +across MMP-sets and pharmacological targets. + +Dablander et al. +Page 13 of 17 +Figure 8 QSAR-prediction- and PD-classification results for factor Xa. Each column corresponds to an upper plot and a lower plot +for one of the MMP-sets Minter, Mtest or Mcores. The x-axis of each upper plot indicates the PD-classification accuracy on the full +MMP-set; the x-axis of each lower plot indicates the PD-classification accuracy on a restricted MMP-set only consisting of MMP +predicted to be ACs by the respective method. The y-axis of each plot shows the QSAR-prediction performance on the molecular +test set Dtest. The total length of each error bar equals twice the standard deviation of the performance metrics measured over all +mk = 3 ∗ 2 = 6 hyperparameter-optimised models. For each plot, the lower right corner corresponds to strong performance at both +prediction tasks. +Future Research: Exploring Twin-Network Training +Schemes +ACs are rich in pharmacological information; at the +same time the experiments have shown that QSAR +models exhibit low AC-sensitivity and thus frequently +fail to predict ACs. In spite of this, to the best of +our knowledge so far no method has been described to +tackle this problem by attempting to increase the AC- +sensitivity of QSAR models. We propose twin-network +training of deep-learning models as a potential strat- +egy to increase AC-sensitivity. Comparatively little +work has been done to investigate twin neural net- +work architectures (also referred to as Siamese net- +works [9, 12, 41, 70]) in computational drug discov- +ery [3, 6, 11, 18, 22, 24, 36, 53, 58, 61, 73, 82]. How- +ever, twin networks provide a natural way to tackle +chemical prediction problems on compound pairs such +as AC-classification. +Instead of training a deep network, f, on an individ- +ual compound, s, with activity label, a(s), via a clas- +sical squared error loss, (a(s) − f(s))2, we suggest to +train f on compound pairs, {s, ˜s}, using a pair-based +loss: +w{s,˜s}[(a(s) − f(s))2 + (a(˜s) − f(˜s))2 ++ wdiff((a(s) − a(˜s)) − (f(s) − f(˜s)))2]. +The quantity w{s,˜s} is used to specify the weight put +on the compound pair {s, ˜s} during training; wdiff de- +termines the relative importance of predicting the in- +dividual activities of s and ˜s versus predicting the ac- +tivity difference associated with {s, ˜s}. Twin-network +training could be conducted in two phases: first on +general compound pairs in Dtrain × Dtrain and then +on MMPs in Mtrain. In the second phase, the weight +function w{s,˜s} could be used to assign training weights +to MMPs proportional to their associated activity dif- + +Dablander et al. +Page 14 of 17 +Figure 9 QSAR-prediction- and PD-classification results for SARS-CoV-2 main protease. Each column corresponds to an upper plot +and a lower plot for one of the MMP-sets Minter, Mtest or Mcores. The x-axis of each upper plot indicates the PD-classification +accuracy on the full MMP-set; the x-axis of each lower plot indicates the PD-classification accuracy on a restricted MMP-set only +consisting of MMP predicted to be ACs by the respective method. The y-axis of each plot shows the QSAR-prediction performance +on the molecular test set Dtest. The total length of each error bar equals twice the standard deviation of the performance metrics +measured over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models. The accuracy of the PD-classification task for predicted ACs is +lacking for the ECFP + kNN technique on Mtest and Mcores since this method produced only negative AC-predictions for all trials +on this data set. For each plot, the lower right corner corresponds to strong performance at both prediction tasks. +ferences; MMPs that represent larger activity differ- +ences might encode structural transformations that +are pharmacologically more relevant and thus should +receive more attention during training. This weight- +ing procedure could lead to increased AC-sensitivity +and the extraction of more chemical knowledge. Our +pair-based training strategy is depicted in Figure 10 +and is based on a twin neural network model for AC- +prediction with discrete outputs that we explored in +a previous research study [16]. We intend to evaluate +the proposed twin-network training scheme in a future +study. +Conclusions +To the best of our knowledge this is the first study +to investigate the AC-prediction capabilities of QSAR +models. It is also the first work to explore the quan- +titative relationship between QSAR-prediction (at the +level of individual molecules) and AC-prediction (at +the level of compound-pairs). As part of our method- +ology we have additionally introduced a simple, in- +terpretable, and rigorous data-splitting technique for +pair-based prediction tasks. +When the activities of both MMP-compounds are +unknown (i.e. absent from the training set) then com- +mon QSAR models exhibit low AC-sensitivity which +limits their utility for AC-prediction. This strongly +supports the hypothesis that QSAR methods do in- +deed regularly fail to predict ACs which might thus +form a major source of prediction errors in QSAR +modelling [14, 26, 47, 63]. However, if the activ- +ity of one MMP-compound is known (i.e. present in +the training set) then AC-sensitivity increases sub- +stantially; for query compounds with known activi- +ties, QSAR methods can therefore be used as simple +AC-prediction-, compound-optimisation- and SAR- + +Dablander et al. +Page 15 of 17 +Cl +F +Figure 10 Twin-network training strategy for deep-learning-based QSAR models that might increase AC-sensitivity. Twin-network +training could be conducted on general compound pairs and on MMPs, with larger weights given to MMPs associated with larger +activity differences. +knowledge-acquisition tools. Furthermore, based on +the observed potency-directon (PD) classification re- +sults we can expect the estimated activity direction of +predicted ACs to have a high degree of accuracy. +With respect to molecular representation, we have +found robust evidence that non-trainable task-agnostic +ECFPs still outcompete differentiable GINs at gen- +eral QSAR-prediction. This adds to a growing aware- +ness that standard message-passing GNNs might need +to be improved further to definitively beat classical +molecular featurisations such as ECFPs [13, 37, 48, +50, 60, 66, 80]. One potential angle to achieve this +could be self-supervised GNN-pretraining, which has +recently shown promising results in the molecular do- +main [30, 78]. However, while GINs appear to be infe- +rior to ECFPs for QSAR-prediction, they tend to be +advantageous for AC-classification; their highly para- +metric nature might simultaneously lead to increased +overfitting but to a better modelling of the more jagged +regions of the SAR-landscape. We thus recommend us- +ing GINs as an AC-classification baseline since such an +agreed-upon baseline is currently lacking. +Finally, the low AC-sensitivity of QSAR models +when the activites of both MMP-compounds are un- +known suggests that such methods are still lacking es- +sential SAR knowledge; on the flip side, it might be +possible to boost QSAR-modelling performance and +increase the amount of extracted SAR knowledge by +developing techniques to increase AC-sensitivity. To +this end, we propose an AC-sensitive twin-network [9, +12, 41, 70] training scheme for deep-learning models +that we intend to explore in the future. +Funding +This research was supported by the University of Oxford’s UK EPSRC Cen- +tre For Doctoral Training in Industrially Focused Mathematical Modelling +(EP/L015803/1) and by the not-for-profit organisation and educational char- +ity Lhasa Limited (https://www.lhasalimited.org/). +Abbreviations +• AC = Activity Cliff +• ECFP = Extended-Connectivity Fingerprint +• GIN = Graph Isomorphism Network +• GNN = Graph Neural Network +• kNN = k-Nearest Neighbour +• MAE = Mean Absolute Error +• MCC = Matthews Correlation Coefficient +• MLP = Multilayer Perceptron +• MMP = Matched Molecular Pair +• PD = Potency Direction +• PDV = Physicochemical-Descriptor Vector +• QSAR = Quantitative Structure-Activity Relationship +• RF = Random Forest +• SAR = Structure-Activity Relationship +Availability of data and materials +All used data sets, the code to reproduce and visualise the experimental +results, and the exact numerical results generated by the original exper- +iments are available in our public code repository https://github.com/ +MarkusFerdinandDablander/QSAR-activity-cliff-experiments. +Competing interests +The authors declare that they have no competing interests. +Authors’ contributions +The computational study was designed, implemented, conducted and in- +terpreted by the first author M.D. The research was supervised by R.L., +G.M.M., and T.H. who gave valuable scientific advice during weekly meet- +ings. The computer code was written by M.D. The paper manuscript was + +Dablander et al. +Page 16 of 17 +written by M.D. Feedback was provided by R.L., G.M.M. and T.H. during +the writing process. The novel data splitting technique for MMP-data, the +QSAR-modelling-based activity cliff prediction strategies and the proposed +twin-network training scheme were developed by M.D. All scientific figures +were designed by M.D., with input from G.M.M., R.L. and T.H. All chemical +data sets were gathered and cleaned by M.D. All authors read and approved +the final manuscript. +Author details +1Mathematical Institute, University of Oxford, Andrew Wiles Building, +Radcliffe Observatory Quarter (550), Woodstock Road, OX2 6GG, Oxford, +United Kingdom. +2Lhasa Limited, Granary Wharf House, 2 Canal Wharf, +LS11 5PS, Leeds, United Kingdom. +3Department of Statistics, University +of Oxford, 24-29 St Giles’, OX1 3LB, Oxford, United Kingdom. +References +1. 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Zhong Y, Chen X, Zhao Y, Chen X, Gao T, Weng Z (2019) +Graph-augmented convolutional networks on drug-drug interactions +prediction. arXiv:191203702 + diff --git a/ENFRT4oBgHgl3EQfyDiq/content/tmp_files/load_file.txt b/ENFRT4oBgHgl3EQfyDiq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6fa6a8aa78357327fb037572947475bf76de7825 --- /dev/null +++ b/ENFRT4oBgHgl3EQfyDiq/content/tmp_files/load_file.txt @@ -0,0 +1,617 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf,len=616 +page_content='Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' RESEARCH Exploring QSAR Models for Activity-Cliff Prediction Markus Dablander1, Thierry Hanser2, Renaud Lambiotte1 and Garrett M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Morris3* Abstract Introduction and Methodology: Pairs of similar compounds that only differ by a small structural modification but exhibit a large difference in their binding affinity for a given target are known as activity cliffs (ACs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' It has been hypothesised that QSAR models struggle to predict ACs and that ACs thus form a major source of predic- tion error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' However, a study to explore the AC-prediction power of modern QSAR methods and its relationship to general QSAR-prediction performance is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We systematically construct nine distinct QSAR models by com- bining three molecular representation methods (extended-connectivity fingerprints, physicochemical-descriptor vectors and graph isomorphism networks) with three regression techniques (random forests, k-nearest neighbours and multilayer perceptrons);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' we then use each resulting model to classify pairs of similar compounds as ACs or non-ACs and to predict the activities of individual molecules in three case studies: dopamine receptor D2, factor Xa, and SARS-CoV-2 main protease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Results and Conclusions: We observe low AC-sensitivity amongst the tested models when the activities of both compounds are unknown, but a substantial increase in AC-sensitivity when the actual activity of one of the compounds is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Graph isomorphism features are found to be competitive with or superior to classical molecular representations for AC-classification and can thus be employed as baseline AC-prediction models or simple compound-optimisation tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For general QSAR-prediction, however, extended-connectivity fingerprints still consistently deliver the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Our results provide strong support for the hypothesis that indeed QSAR methods frequently fail to predict ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We propose twin-network training for deep learning models as a potential future pathway to increase AC-sensitivity and thus overall QSAR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Keywords: QSAR modelling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Activity cliffs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Activity cliff prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Machine learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Molecular representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Physicochemical descriptors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Extended-connectivity fingerprints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Graph isomorphism networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Binding affinity prediction Correspondence: garrett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='morris@dtc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='uk 3Department of Statistics, University of Oxford, 24-29 St Giles’, OX1 3LB, Oxford, United Kingdom Full list of author information is available at the end of the article arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='13644v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='LG] 31 Jan 2023 Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 2 of 17 Introduction Activity cliffs (ACs) are pairs of small molecules that exhibit high structural similarity but at the same time show an unexpectedly large difference in their binding affinity against a given pharmacological tar- get [14, 47, 63, 64, 67–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The existence of ACs di- rectly defies the intuitive idea that chemical com- pounds with similar structures should have similar ac- tivities, often referred to as the molecular similarity principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' An example of an AC between two inhibitors of blood coagulation factor Xa [43] is depicted in Fig- ure 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' a small chemical modification involving the ad- dition of a hydroxyl group leads to an increase in in- hibition of almost three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For medicinal chemists, ACs can be puzzling and confound their understanding of structure-activity re- lationships (SARs) [19, 67, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' ACs reveal small compound-modifications with large biological impact and thus represent rich sources of pharmacological in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Mechanisms by which a small structural transformation can give rise to an AC include a dras- tic change in 3D-conformation and/or the switching to a different binding mode or even binding site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' ACs form discontinuities in the SAR-landscape and can therefore have a crucial impact on the success of lead-optimisation programmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' While knowledge about ACs can be powerful when trying to escape from flat regions of the SAR-landscape, their presence can be detrimental in later stages of the drug development process, when multiple molecular properties beyond mere activity need to be balanced carefully to arrive at a safe and effective compound [14, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In the field of computational chemistry, ACs are sus- pected to form one of the major roadblocks for success- ful quantitative structure-activity relationship (QSAR) modelling [14, 26, 47, 63];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' abrupt changes in potency are expected to negatively influence machine learn- ing algorithms for pharmacological activity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' During the development of QSAR models, ACs are sometimes dismissed as measurement errors [49], but simply removing ACs from a training data set can result in a loss of precious SAR-information [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Golbraikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [26] developed the MODI metric to quantify the smoothness of the SAR-landscape of binary molecular classification data sets and showed that the SAR-landscape smoothness is a strong de- terminant for downstream QSAR-modelling perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In a related work, Sheridan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [63] found that the density of ACs in a molecular data set is strongly predictive of its overall modelability by clas- sical descriptor- and fingerprint-based QSAR meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Furthermore, they found that such methods in- cur a significant drop in performance when the test set is restricted to “cliffy” compounds that form a large number of ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In a more extensive study, van Tilborg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [75] observed a similar drop in perfor- mance when testing classical and graph-based QSAR techniques on compounds involved in ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Notably, Figure 1 Example of an activity cliff (AC) for blood coagulation factor Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' A small structural transformation in the upper compound leads to an increase in inhibitory activity of almost three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Both compounds were identified in the same ChEMBL assay with ID 658338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 3 of 17 in both studies this performance drop was also ob- served for highly nonlinear and adaptive deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In fact, van Tilborg reports that descriptor- based QSAR methods even outperform more complex deep learning models on “cliffy” compounds associ- ated with ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This runs counter to earlier hopes ex- pressed in the literature that the approximation power of deep neural networks might ameliorate the problem of ACs [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' While these works provide valuable insights into the detrimental effects of SAR discontinuity on QSAR models, they consider ACs mainly indirectly by fo- cussing on individual compounds involved in ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Ar- guably, a distinct and more natural approach would be to investigate ACs directly at the level of com- pound pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This approach has been followed in the AC-prediction field which is concerned with developing techniques to classify whether a pair of similar com- pounds forms an AC or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' An effective AC-prediction method would be of high value for drug development with important applications in rational compound op- timisation and automatic SAR-knowledge acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The AC-prediction literature is still very thin com- pared to the QSAR-prediction literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' An attempt to conduct an exhaustive literature review on AC- prediction techniques revealed a total number of 15 methods [4, 7, 10, 27, 29, 32, 34, 39, 44, 51, 52, 54, 57, 71], all of which have been published since 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Current AC-prediction methods are often based on cre- ative ways to extract features from pairs of molecular compounds in a manner suitable for standard machine learning pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For example, Horvath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [29] used condensed graphs of reactions [28, 35], a rep- resentation technique originally introduced for mod- elling of chemical reactions, to encode pairs of simi- lar compounds and subsequently predict ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Another method was recently described by Iqbal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [34] who investigated the abilities of convolutional neural net- works operating on 2D images of compound pairs to distinguish between ACs and non-ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Interestingly, none of the AC-prediction methods we identified em- ploy feature extraction techniques built on modern graph neural networks (GNNs) [20, 25, 40, 76, 81] with the exception of Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [54] who recently ap- plied graph convolutional methods to compound-pairs to predict ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In spite of the existence of advanced AC-prediction models there are significant gaps left in the current AC-prediction literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Note that any QSAR model can immediately be repurposed as an AC-prediction model by using it to individually predict the activ- ities of two structurally similar compounds and then thresholding the predicted absolute activity difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Nevertheless, at the moment there is no study that uses this straightforward technique to investigate the potential of current QSAR models to classify whether a pair of compounds forms an AC or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Impor- tantly, this also entails that the most salient AC- prediction models [27, 29, 34, 44, 71] have not been compared to a simple QSAR-modelling baseline ap- plied to compound pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' It is thus an open question to what extent (if at all) these tailored AC-prediction techniques outcompete repurposed QSAR methods in the detection of ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This is especially relevant in light of the fact that several published AC-predict¸ion models [27, 34, 44] are evaluated via compound-pair- based data splits which incur a significant overlap be- tween training set and test set at the level of individ- ual molecules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' this type of data split should strongly favour standard QSAR models for AC-prediction, yet a comparison to such baseline methods is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We address these gaps by systematically investigat- ing the abilities of nine frequently used QSAR models to classify pairs of similar compounds as ACs or non- ACs within three pharmacological data sets: dopamine receptor D2, factor Xa, and SARS-CoV-2 main pro- tease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Each QSAR model is constructed by combining a molecular representation method (physicochemical- descriptor vectors (PDVs) [72], extended-connectivity fingerprints (ECFPs) [59], or graph isomorphism net- works (GINs) [81]) with a regression technique (ran- dom forests (RFs), k-nearest neighbours (kNNs), or multilayer perceptrons (MLPs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' All models are used for two distinct prediction tasks: QSAR-prediction at the level of individual molecules, and AC-classification at the level of compound-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The main contribution of this study is to shed light on the following questions: What is the relationship between the ability of a QSAR model to predict the activities of individual compounds, versus its ability to classify whether pairs of similar compounds form ACs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' When (if at all) are common QSAR models capa- ble of predicting ACs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' When (if at all) are common QSAR models capa- ble of predicting which of two similar compounds is the more active one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Which QSAR model shows the strongest AC- prediction performance, and should thus be used as a baseline against which to compare tailored AC-prediction models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Do differentiable GINs outperform classical non- trainable ECFPs and PDVs as molecular repre- sentations for QSAR- and/or AC-prediction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' How could ACs potentially be used to improve QSAR-modelling performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 4 of 17 Experimental Methodology Molecular Data Sets We built three binding affinity data sets of small- molecule inhibitors of dopamine receptor D2, factor Xa, and SARS-CoV-2 main protease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Factor Xa is an enzyme in the coagulation cascade and a canonical tar- get for blood-thinning drugs [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Dopamine receptor D2 is the main site of action for classic antipsychotic drugs which act as antagonists of the D2 receptor [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' SARS-CoV-2 main protease is one of the key enzymes in the viral replication cycle of the SARS coronavirus 2, that recently caused the unprecedented COVID-19 pandemic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' it is one of the most promising targets for antiviral drugs against this coronavirus [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For dopamine receptor D2 and factor Xa, data was extracted from the ChEMBL database [45] in the form of SMILES strings with associated Ki [nM] values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For SARS-CoV-2 main protease, data was obtained from the COVID moonshot project [1] in the form of SMILES strings with associated IC50 [µM] values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' SMILES strings were standardised and desalted via the ChEMBL structure pipeline [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This step also re- moved solvents and all isotopic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Follow- ing this, SMILES strings that produced error messages when turned into an RDKit mol object were deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Finally, a scan for duplicate molecules was performed: If the activities in a set of duplicate molecules were within the same order of magnitude then the set was unified via geometric averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Otherwise, the mea- surements were considered unreliable and the corre- sponding set of duplicate molecules was removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This procedure reduced the data set for dopamine receptor D2 / factor Xa / SARS-CoV-2 main protease from 8883 / 4116 / 1926 compounds to 6333 / 3605 / 1924 unique compounds whereby 174 / 21 / 0 sets of dupli- cate SMILES were removed and the rest was unified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Activity Cliffs: Definition of Binary Classification Tasks The exact definition of an AC hinges on two concepts: structural similarity and large activity difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' An elegant technique to measure structural similarity in the context of AC analysis is given by the matched molecular pair (MMP) formalism [31, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' An MMP is a pair of compounds that share a common struc- tural core but differ by a small chemical transforma- tion at a specific site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Figure 1 depicts an example of an MMP whose variable parts are formed by a hydrogen atom and a hydroxyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' To detect MMPs algorith- mically, we used the mmpdb Python-package provided by Dalke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We restricted ourselves to MMPs with the following commonly used [27, 29, 71] size con- straints: the MMP core was required to contain at least twice as many heavy atoms as either of the two vari- able parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' each variable part was required to contain no more than 13 heavy atoms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' the maximal size dif- ference between both variable parts was set to eight heavy atoms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' and bond cutting was restricted to sin- gle exocyclic bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' To guarantee a well-defined map- ping from each MMP to a unique structural core, we canonically chose the core that contained the largest number of heavy atoms whenever there was ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Based on the ratio of the activity values of both MMP compounds, each MMP was assigned to one of three classes: “AC”, “non-AC” or “half-AC”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In accordance with the literature [5, 27, 29, 52, 77] we assigned an MMP to the “AC”-class if both activity values differed by at least a factor of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' If both activity values dif- fered by no more than a factor of 10, then the MMP was assigned to the “non-AC”-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In the residual case the MMP was assigned to the “half-AC”-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' To arrive at a well-separated binary classification task, we labelled all ACs as positives and all non-ACs as nega- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The half-ACs were removed and not considered further in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' It is relevant to know the direction of a potential activity cliff, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' which of the compounds in the pair is the more active one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We thus assigned a binary label to each MMP indicating its po- tency direction (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' PD-classification is a balanced binary classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Table 1 gives an overview of all our curated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Data Splitting Technique ACs are molecular pairs rather than single molecules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' it is thus not obvious how best to split up a chemical data set into non-overlapping training- and test sets for the fair evaluation of an AC-prediction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' There seems to be no consensus about which data splitting strategy should be canonically used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Several authors [27, 34, 44] have employed a random split at the level of compound pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' While this technique is conceptually straightforward, it must be expected to incur a significant overlap between training- and test set at the level of individual molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For ex- ample, randomly splitting up a set of three MMPs {{s1, s2}, {s1, s3}, {s2, s3}} into a training- and a test set might lead to {s1, s2} and {s1, s3} getting assigned to the training- and {s2, s3} getting assigned to the test set which leads to a full inclusion of the test set in the training set at the level of individual molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This molecular overlap is problematic for at least three reasons: Firstly, it likely leads to overly optimistic re- sults for AC-prediction methods since they will have already encountered some of the test compounds dur- ing training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Secondly, it does not model the natural situation encountered by medicinal chemists who we assume will not know the activity value of at least one Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 5 of 17 Data Set Dopamine Receptor D2 Factor Xa SARS-CoV-2 Main Protease Compounds 6333 3605 1924 MMPs 35484 21292 12594 ACs 461 1896 521 Half-ACs 3804 4693 1762 Non-ACs 31219 14703 10311 ACs : Non-ACs ≈ 1 : 68 ≈ 1 : 8 ≈ 1 : 20 Table 1 Sizes of our curated data sets and their respective numbers of matched molecular pairs (MMPs), activity cliffs (ACs), half- activity-cliffs (half-ACs) and non-activity-cliffs (non-ACs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' compound in a test-set pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Thirdly, the mentioned molecular overlap should lead to strong AC-prediction results for standard QSAR models, but to the best of our knowledge, no such control experiments have been run in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Horvath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [29] and Tamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [71] have made efforts to address the shortcomings of a compound- pair-based random split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' They came up with advanced data splitting algorithms designed to mitigate the molecular-overlap problem by either managing distinct types of test sets according to compound membership in the training set or by designing splitting techniques based on the structural cores of MMPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' However, their data splitting schemes exhibit a relatively high degree of complexity which can make their implementation and interpretation difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We propose a novel data splitting method which rep- resents a favourable trade-off between rigour, inter- pretability and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Our technique shares some of its concepts with the methods proposed by Horvath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [29] and Tamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [71] but might be simpler to implement and interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We first split the data into a training- and test set at the level of individual molecules and then use this basic split to distinguish several types of test sets at the level of compound pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Let D = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='} be the given data set of individual molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Further- more, let M ⊆ {{s, ˜s} | s ̸= ˜s and s, ˜s ∈ D} be the set of all MMPs in D that have been labelled as either ACs or non-ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Each MMP {s, ˜s} ∈ M shares a common structural core denoted as core({s, ˜s}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We use a random split to partition D into a training set Dtrain and a test set Dtest and then define the following MMP-sets: Mtrain = {{s, ˜s} ∈ M | s, ˜s ∈ Dtrain}, Minter = {{s, ˜s} ∈ M | s ∈ Dtrain, ˜s ∈ Dtest}, Mtest = {{s, ˜s} ∈ M | s, ˜s ∈ Dtest}, Mcores = {{s, ˜s} ∈ Mtest | core({s, ˜s}) /∈ Ctrain}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Here, Ctrain = {core({s, ˜s}) | {s, ˜s} ∈ Mtrain ∪ Minter}, which describes the set of MMP-cores that appear in Dtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Note that Mtrain ∪ Minter ∪ Mtest = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The pair (Dtrain, Mtrain) describes the training space at the level of individual molecules and MMPs, and can be used to train a QSAR- or AC-prediction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' A trained method can then classify MMPs in Mtest, Minter and Mcores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Mtest models an AC-prediction setting where the activities of both MMP-compounds are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Mcores represents the subset of MMPs in Mtest whose structural cores do not appear in Mtrain ∪ Minter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Mcores thus models the difficult task of predicting ACs in a strucurally novel area of chemi- cal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Finally, Minter represents an AC-prediction scenario where the activity of one MMP-compound is given a priori;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' this can be interpreted as a compound- optimisation task where one strives to predict small AC-inducing modifications of a query compound with known activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' An illustration of our data splitting strategy is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We implemented our data splitting strategy within a k-fold cross validation scheme repeated with m ran- dom seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This generated data splits of the form Sij = (Dij train, Dij test, Mij train, Mij test, Mij inter, Mij cores) for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=', m} and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=', k} where (Dij train, Dij test) represents the j-th split of D in the cross validation Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 6 of 17 Figure 2 Illustration of our data splitting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We distinguish between three MMP-sets, Mtrain, Minter and Mtest, depending on whether both MMP-compounds are in Dtrain, one MMP-compound is in Dtrain and the other one is in Dtest, or both MMP-compounds are in Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We additionally consider a fourth MMP-set, Mcores, consisting of the MMPs in Mtest whose structural cores do not appear in Mtrain ∪ Minter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' round with random seed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The overall QSAR- and AC- prediction performance of each model was recorded as the average over the mk training- and test runs based on all data splits S1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=', Smk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We chose the config- uration (k, m) = (2, 3) which gave a good trade-off between computational costs and accuracy and rea- sonable numbers of MMPs in the compound-pair-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In particular, random cross-validation with k = 2 gave expected relative sizes of: |Mtrain| : |Minter| : |Mtest| = 1 : 2 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' On average, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='7 %, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='91 %, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='84 % of MMPs in Mtest were also in Mcores for dopamine receptor D2, factor Xa, and SARS-CoV-2 main protease, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Prediction Strategies and Performance Measures In a data split of the form S = (Dtrain, Dtest, Mtrain, Mtest, Minter, Mcores) each individual compound, s ∈ Dtrain ∪ Dtest = D, can be associated with an activity label a(s) ∈ R, de- fined as the negative decadic logarithm of the exper- imentally measured activity of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We stuck with the canonical units used in the ChEMBL database and the COVID moonshot project ([nM] for Ki and [µM] for IC50);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' each activity label a(s) thus represents a standard pKi- or pIC50 value (with an additive shift towards 0 caused by the units which might slightly benefit prediction techniques initialised around the ori- gin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We are interested in QSAR-prediction functions, f : D → R, that can map a chemical structure s ∈ D to an es- timate of its binding affinity a(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The mapping f is found via an algorithmic training process on the la- belled data set {(s, a(s)) | s ∈ Dtrain} and can then either be used to predict the activity la- bels of compounds in Dtest, or it can be repurposed to classify whether an MMP forms an activity cliff (AC- classification) and what the potency direction of an MMP is (PD-classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' If {s, ˜s} ∈ Minter, then one can assume that the activity label of one of the compounds, say a(s), is known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' f is then used to clas- Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 7 of 17 sify {s, ˜s} via: {s, ˜s} �→ � Non-AC if |a(s) − f(˜s)| ≤ dcrit, AC if |a(s) − f(˜s)| > dcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Here dcrit ∈ R>0 is a critical threshold above which an MMP is classified as an AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Throughout this work we use dcrit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='5 (in pKi- or pIC50 units) since this value represents the middle point between the intervals [0, 1] and [2, ∞) which correspond to absolute activity-label differences associated with non-ACs and ACs respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' If {s, ˜s} ∈ Mtest ∪ Mcores then the activities of both compounds are unknown and we classify {s, ˜s} via: {s, ˜s} �→ � Non-AC if |f(s) − f(˜s)| ≤ dcrit, AC if |f(s) − f(˜s)| > dcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' PD-classification for MMPs is performed in a straight- forward manner: the activity labels of both MMP- compounds are predicted via f and then compared to classify which compound is the more active one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The performance of f for standard QSAR predic- tion in Dtest is measured via the mean absolute er- ror (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For the balanced PD-classification prob- lem we rely on accuracy as a suitable performance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For the highly imbalanced task of AC- classification, however, we use the Matthews corre- lation coefficient (MCC), as well as sensitivity and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For the relatively small SARS-CoV-2 main protease data set we sometimes encountered the edge case where there were no positive predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' we then set MCC = 0 and ignored ill-defined precision mea- surements when averaging the performance metrics to obtain the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Molecular Representation- and Regression Techniques We constructed nine QSAR models via a robust com- binatorial methodology that systematically combines three molecular representation methods with three re- gression techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This setup allows, for example, to compare the performance of molecular representation methods across regression techniques, data sets and predictions tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For molecular representation, we used extended- connectivity fingerprints [59] (ECFPs), physicochem- ical molecular descriptor vectors [72] (PDVs), and graph isomorphism networks (GINs) [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Both ECFPs and PDVs were computed via RDKit [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The ECFPs were chosen to use a radius of two, a length of 2048 bits, and active chirality flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The PDVs had a di- mensionality of 200 and were constructed using the general list of descriptors from the work of Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This list encompasses properties related to druglikeness, logP, molecular refractivity, electro- topological state, molecular graph-structure, fragment profile, charge, and topological surface properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The GIN was implemented using PyTorch Geometric [23] and consisted of a variable number of graph convolu- tional layers, each with two internal hidden layers with ReLU activations and batch normalisation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We further chose the maxpool operator which computes the component-wise maximum over all atom feature vectors in the final graph layer to obtain a graph-level representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Each molecular representation was used as an input featurisation for three regression techniques: random forests (RFs), k-nearest neigbours (kNNs) and multi- layer perceptrons (MLPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The RF- and kNN-models were implemented via scikit-learn [56] and the MLP- models via PyTorch [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The MLPs used ReLU acti- vations and batch normalisation at each hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The GIN was combined with the regression tech- niques as follows: For MLP regression, the GIN was trained with the MLP as a projection head after the pooling step in the usual end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For RF- or kNN-regression, the GIN was first trained with a single linear layer added after the global pooling step that directly mapped the graph-level representation to an activity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' After this training phase the weights of the GIN were frozen and it was used as a static feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The RF- or kNN-regressor was then trained on the features extracted by the frozen GIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Figure 3 illustrates our combinatorial experimen- tal methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Model Training and Hyperparameter Optimisation All models were trained using full inner hyperparame- ter-optimisation loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Hyperparameters of RFs and kNNs were optimised in scikit-learn [56] by uniformly random sampling of hyperparameters from a prede- fined grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The hyperparameters of MLPs and GINs were sampled from a predefined grid via the tree- structured Parzen estimator algorithm implemented in Optuna [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Deep learning models were trained for 500 epochs on a single NVIDIA GeForce RTX 3060 GPU via the mean squared error loss function using AdamW optimisation [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Weight decay, learning rate decay and dropout [65] were employed at all hidden layers for regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Batch size, learning rate, learning rate decay rate, weight decay rate, and dropout rate were treated as hyperparameters and subsequently op- timised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Note that the training length (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' the number of gradient updates) was implicitly optimised by tun- ing the batch size for the fixed number of 500 training Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 8 of 17 Figure 3 Schematic showing the combinatorial experimental methodology used for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Each molecular representation method is systematically combined with each regression technique, giving a total of nine QSAR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Each QSAR model is trained and evaluated for QSAR-prediction, AC-classification and PD-classification within a 2-fold cross validation scheme repeated with 3 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each of the 2 ∗ 3 = 6 trials, an extensive inner hyperparameter-optimisation loop on the training set is performed for each QSAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Further implementation details can be found in our public code repository[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Results and Discussion The QSAR-prediction-, AC-classification- and PD- classification results for all three data sets are depicted in Figures 4 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' QSAR-Prediction Performance When considering the results depicted in Figures 4 to 9 with respect to QSAR-prediction performance, one can see that ECFPs tend to lead to better perfor- mance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' a lower QSAR-MAE) compared to GINs, which in turn tend to lead to better performance com- pared to PDVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In particular, the combination MLP- ECFP consistently produced the lowest QSAR-MAE across all three targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' These observations reinforce a growing corpus of literature that suggests that train- able GNNs have not yet reached a level of techni- cal maturity by which they consistently and defini- tively outperform the much simpler non-differentiable ECFPs at important molecular property prediction tasks [13, 37, 48, 50, 60, 66, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' [1]https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='com/MarkusFerdinandDablander/ QSAR-activity-cliff-experiments AC-Classification Performance The AC-MCC plots in Figures 4 to 6 reveal sur- prisingly strong overall AC-classification results on Minter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This type of MMP-set models a compound- optimisation scenario where a researcher strives to identify small structural modifications with a large im- pact on the activity of query compounds with known activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For this task, a significant portion of our QSAR models exhibit an AC-MCC value greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='5 across targets, which appears impressive consider- ing the simplicity of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Exchanging Minter with either Mtest or Mcores leads to a substantial drop in the AC-MCC to approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='3 that appears to be mediated by a large drop in AC-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In most cases, GINs perform better than the other molecular representation methods with respect to the AC-MCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Notably, kNN-regressors consistently per- form best for AC-classification when combined with GIN-features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' this supports the idea that GINs might have a heightened ability to resolve ACs by learning an embedding of chemical space in which the distance between two compounds is reflective of activity dif- ference rather than structural difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The combi- nations GIN-MLP, GIN-RF and ECFP-MLP exhibit particularly high AC-MCC values relative to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We recommend using at least one of these three models as a baseline against which to compare tailored AC-prediction models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' the practical utility of Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 9 of 17 Figure 4 QSAR-prediction- and AC-classification results for dopamine receptor D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the x-axis corresponds to a combination of MMP-set and AC-classification performance metric and the y-axis shows the QSAR-prediction performance on the molecular test set Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The total length of each error bar equals twice the standard deviation of the performance metric measured over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the lower right corner corresponds to strong performance at both prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' any AC-prediction technique that cannot outperform these three common QSAR methods is questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Across all three targets, AC-sensitivity is moder- ately high on Minter but universally low on Mtest and Mcores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This is consistent with the hypothesis that ACs form one of the major sources of prediction error for QSAR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The weak AC-sensitivity on Mtest and Mcores indicates that modern QSAR meth- ods are largely blind to ACs in novel areas of chemi- cal space and thus lack essential chemical knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' GINs clearly outperform the other two more classi- cal molecular representations across regression tech- niques with respect to AC-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In particular, the GIN-MLP combination leads to the highest AC- Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 10 of 17 Figure 5 QSAR-prediction- and AC-classification results for factor Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the x-axis corresponds to a combination of MMP-set and AC-classification performance metric and the y-axis shows the QSAR-prediction performance on the molecular test set Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The total length of each error bar equals twice the standard deviation of the performance metric measured over all mk = 3∗2 = 6 hyperparameter-optimised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the lower right corner corresponds to strong performance at both prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' sensitivity in all examined cases and thus discovers the most ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The highly parametric nature of GINs that makes them prone to overfitting could at the same time enable them to better model jagged regions of the SAR-landscape that contain ACs than classical task- agnostic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' There is a wide gap between distinct prediction techniques with respect to AC-precision: some models achieve a considerable level of AC-precision such that over 50% of positively predicted MMPs in Mtest and Mcores are indeed actual ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Other QSAR models, however, seem to fail almost entirely with respect to this metric on Mtest and Mcores and only deliver mod- est performance on Minter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' RFs tend to exhibit the strongest AC-precision and the weakest AC-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This might be as a result of their ensemble nature Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 11 of 17 Figure 6 QSAR-prediction- and AC-classification results for SARS CoV-2 main protease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the x-axis corresponds to a combination of MMP-set and AC-classification performance metric and the y-axis shows the QSAR-prediction performance on the molecular test set Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The total length of each error bar equals twice the standard deviation of the performance metric measured over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The precision of the AC-classification task is lacking for the ECFP + kNN technique on Mtest and Mcores since this method produced only negative AC-predictions for all trials on this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the lower right corner corresponds to strong performance at both prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' which should intuitively lead to conservative but trust- worthy predictions of extreme effects such as ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' PD-Classification Performance The abilities of the evaluated QSAR models to identify which is the more active compound in an MMP is uni- versally weak, with PD-accuracies clustering around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='7 on Minter and around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='6 on Mtest and Mcores, as can be seen in the top rows of Figures 7 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Predict- ing the potency direction for two compounds with sim- ilar structures and thus usually similar activity levels must be considered a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The combina- tion ECFP-MLP reaches the strongest PD-accuracy Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 12 of 17 Figure 7 QSAR-prediction- and PD-classification results for dopamine receptor D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Each column corresponds to an upper plot and a lower plot for one of the MMP-sets Minter, Mtest or Mcores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The x-axis of each upper plot indicates the PD-classification accuracy on the full MMP-set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' the x-axis of each lower plot indicates the PD-classification accuracy on a restricted MMP-set only consisting of MMP predicted to be ACs by the respective method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The y-axis of each plot shows the QSAR-prediction performance on the molecular test set Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The total length of each error bar equals twice the standard deviation of the performance metrics measured over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the lower right corner corresponds to strong performance at both prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' in the majority of cases and we recommend starting with this model as a baseline for more advanced PD- prediction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' One can argue that the activity order of two simi- lar compounds is of little interest if the true activity difference is small, as is often the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We therefore also restricted PD-classification to predicted ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The three plots in the bottom rows of Figures 7 to 9 depict the PD-accuracy of each QSAR model on the subset of MMPs that were also predicted to be ACs by the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In this practically more relevant scenario PD-prediction accuracy tends to exceed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='9 on Minter and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='8 on Mtest and Mcores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The QSAR models in- vestigated here are thus able to identify the correct activity order of MMPs if they also predict them to be ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The relatively rare instances in which the PD of a predicted AC is misclassified, however, reflect severe QSAR-prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Linear Relationship between QSAR-MAE and AC-MCC Our experiments reveal a consistent linear relationship between the QSAR-MAE and the AC-MCC as can be seen in the left columns of Figures 4 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' A po- tential mechanism driving this effect could be that as the overall QSAR-MAE of a model improves, its ac- curacy at predicting activity differences between sim- ilar molecules might be expected to improve as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' previously misclassified MMPs whose predicted abso- lute activity differences were already close to the crit- ical value dcrit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='5 might then gradually move to the correct side of the decision boundary and increase the AC-MCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The results suggest that for real-world QSAR models the AC-MCC and the QSAR-MAE are strongly predictive of each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' while this observa- tion only rests on nine models, it is highly consistent across MMP-sets and pharmacological targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 13 of 17 Figure 8 QSAR-prediction- and PD-classification results for factor Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Each column corresponds to an upper plot and a lower plot for one of the MMP-sets Minter, Mtest or Mcores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The x-axis of each upper plot indicates the PD-classification accuracy on the full MMP-set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' the x-axis of each lower plot indicates the PD-classification accuracy on a restricted MMP-set only consisting of MMP predicted to be ACs by the respective method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The y-axis of each plot shows the QSAR-prediction performance on the molecular test set Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The total length of each error bar equals twice the standard deviation of the performance metrics measured over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the lower right corner corresponds to strong performance at both prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Future Research: Exploring Twin-Network Training Schemes ACs are rich in pharmacological information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' at the same time the experiments have shown that QSAR models exhibit low AC-sensitivity and thus frequently fail to predict ACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In spite of this, to the best of our knowledge so far no method has been described to tackle this problem by attempting to increase the AC- sensitivity of QSAR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We propose twin-network training of deep-learning models as a potential strat- egy to increase AC-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Comparatively little work has been done to investigate twin neural net- work architectures (also referred to as Siamese net- works [9, 12, 41, 70]) in computational drug discov- ery [3, 6, 11, 18, 22, 24, 36, 53, 58, 61, 73, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' How- ever, twin networks provide a natural way to tackle chemical prediction problems on compound pairs such as AC-classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Instead of training a deep network, f, on an individ- ual compound, s, with activity label, a(s), via a clas- sical squared error loss, (a(s) − f(s))2, we suggest to train f on compound pairs, {s, ˜s}, using a pair-based loss: w{s,˜s}[(a(s) − f(s))2 + (a(˜s) − f(˜s))2 + wdiff((a(s) − a(˜s)) − (f(s) − f(˜s)))2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The quantity w{s,˜s} is used to specify the weight put on the compound pair {s, ˜s} during training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' wdiff de- termines the relative importance of predicting the in- dividual activities of s and ˜s versus predicting the ac- tivity difference associated with {s, ˜s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Twin-network training could be conducted in two phases: first on general compound pairs in Dtrain × Dtrain and then on MMPs in Mtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' In the second phase, the weight function w{s,˜s} could be used to assign training weights to MMPs proportional to their associated activity dif- Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 14 of 17 Figure 9 QSAR-prediction- and PD-classification results for SARS-CoV-2 main protease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Each column corresponds to an upper plot and a lower plot for one of the MMP-sets Minter, Mtest or Mcores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The x-axis of each upper plot indicates the PD-classification accuracy on the full MMP-set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' the x-axis of each lower plot indicates the PD-classification accuracy on a restricted MMP-set only consisting of MMP predicted to be ACs by the respective method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The y-axis of each plot shows the QSAR-prediction performance on the molecular test set Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The total length of each error bar equals twice the standard deviation of the performance metrics measured over all mk = 3 ∗ 2 = 6 hyperparameter-optimised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The accuracy of the PD-classification task for predicted ACs is lacking for the ECFP + kNN technique on Mtest and Mcores since this method produced only negative AC-predictions for all trials on this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' For each plot, the lower right corner corresponds to strong performance at both prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' ferences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' MMPs that represent larger activity differ- ences might encode structural transformations that are pharmacologically more relevant and thus should receive more attention during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This weight- ing procedure could lead to increased AC-sensitivity and the extraction of more chemical knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Our pair-based training strategy is depicted in Figure 10 and is based on a twin neural network model for AC- prediction with discrete outputs that we explored in a previous research study [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We intend to evaluate the proposed twin-network training scheme in a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Conclusions To the best of our knowledge this is the first study to investigate the AC-prediction capabilities of QSAR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' It is also the first work to explore the quan- titative relationship between QSAR-prediction (at the level of individual molecules) and AC-prediction (at the level of compound-pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' As part of our method- ology we have additionally introduced a simple, in- terpretable, and rigorous data-splitting technique for pair-based prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' When the activities of both MMP-compounds are unknown (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' absent from the training set) then com- mon QSAR models exhibit low AC-sensitivity which limits their utility for AC-prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This strongly supports the hypothesis that QSAR methods do in- deed regularly fail to predict ACs which might thus form a major source of prediction errors in QSAR modelling [14, 26, 47, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' However, if the activ- ity of one MMP-compound is known (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' present in the training set) then AC-sensitivity increases sub- stantially;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' for query compounds with known activi- ties, QSAR methods can therefore be used as simple AC-prediction-, compound-optimisation- and SAR- Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 15 of 17 Cl F Figure 10 Twin-network training strategy for deep-learning-based QSAR models that might increase AC-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Twin-network training could be conducted on general compound pairs and on MMPs, with larger weights given to MMPs associated with larger activity differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' knowledge-acquisition tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Furthermore, based on the observed potency-directon (PD) classification re- sults we can expect the estimated activity direction of predicted ACs to have a high degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' With respect to molecular representation, we have found robust evidence that non-trainable task-agnostic ECFPs still outcompete differentiable GINs at gen- eral QSAR-prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' This adds to a growing aware- ness that standard message-passing GNNs might need to be improved further to definitively beat classical molecular featurisations such as ECFPs [13, 37, 48, 50, 60, 66, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' One potential angle to achieve this could be self-supervised GNN-pretraining, which has recently shown promising results in the molecular do- main [30, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' However, while GINs appear to be infe- rior to ECFPs for QSAR-prediction, they tend to be advantageous for AC-classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' their highly para- metric nature might simultaneously lead to increased overfitting but to a better modelling of the more jagged regions of the SAR-landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' We thus recommend us- ing GINs as an AC-classification baseline since such an agreed-upon baseline is currently lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Finally, the low AC-sensitivity of QSAR models when the activites of both MMP-compounds are un- known suggests that such methods are still lacking es- sential SAR knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' on the flip side, it might be possible to boost QSAR-modelling performance and increase the amount of extracted SAR knowledge by developing techniques to increase AC-sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' To this end, we propose an AC-sensitive twin-network [9, 12, 41, 70] training scheme for deep-learning models that we intend to explore in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Funding This research was supported by the University of Oxford’s UK EPSRC Cen- tre For Doctoral Training in Industrially Focused Mathematical Modelling (EP/L015803/1) and by the not-for-profit organisation and educational char- ity Lhasa Limited (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='lhasalimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='org/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='Abbreviations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='AC = Activity Cliff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='ECFP = Extended-Connectivity Fingerprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='GIN = Graph Isomorphism Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='GNN = Graph Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='kNN = k-Nearest Neighbour ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='MAE = Mean Absolute Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='MCC = Matthews Correlation Coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='MLP = Multilayer Perceptron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='MMP = Matched Molecular Pair ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='PD = Potency Direction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='PDV = Physicochemical-Descriptor Vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='QSAR = Quantitative Structure-Activity Relationship ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='RF = Random Forest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='SAR = Structure-Activity Relationship ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='Availability of data and materials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='All used data sets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' the code to reproduce and visualise the experimental results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' and the exact numerical results generated by the original exper- iments are available in our public code repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='com/ MarkusFerdinandDablander/QSAR-activity-cliff-experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Authors’ contributions The computational study was designed, implemented, conducted and in- terpreted by the first author M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The research was supervised by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=', and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' who gave valuable scientific advice during weekly meet- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The computer code was written by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The paper manuscript was Dablander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Page 16 of 17 written by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Feedback was provided by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' during the writing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' The novel data splitting technique for MMP-data, the QSAR-modelling-based activity cliff prediction strategies and the proposed twin-network training scheme were developed by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' All scientific figures were designed by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=', with input from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' All chemical data sets were gathered and cleaned by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFRT4oBgHgl3EQfyDiq/content/2301.13644v1.pdf'} +page_content=' Author details 1Mathematical Institute, University of Oxford, Andrew Wiles 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a/G9FJT4oBgHgl3EQfEizA/content/tmp_files/2301.11438v1.pdf.txt b/G9FJT4oBgHgl3EQfEizA/content/tmp_files/2301.11438v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..47f787517b69639f14c0026377f269fc90271a92 --- /dev/null +++ b/G9FJT4oBgHgl3EQfEizA/content/tmp_files/2301.11438v1.pdf.txt @@ -0,0 +1,629 @@ +Study of the Uncertainties of the Galactic Radio +Background as a Calibration Source for Radio Arrays +Max Büsken,𝑎,𝑏,∗ Tomáš Fodran𝑐 and Tim Huege𝑑,𝑒 +𝑎Institute for Experimental Particle Physics (ETP), Karlsruhe Institute of Technology (KIT), +Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany +𝑏Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), +Av. General Paz 1555 (B1630KNA), San Martín, Buenos Aires, Argentina +𝑐Department of Astrophysics/IMAPP, Radboud University, PO Box 9010, 6500 GL, The Netherlands +𝑑Institute for Astroparticle Physics (IAP), Karlsruhe Institute of Technology (KIT), +Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany +𝑒Astrophysical Institute, Vrije Universiteit Brussels, +Pleinlaan 2, 1050 Brussels, Belgium +E-mail: max.buesken@kit.edu, t.fodran@science.ru.nl, tim.huege@kit.edu +The indirect detection of cosmic rays via the radio signal of extensive air showers is gaining a +lot of ground. Many new arrays of radio antennas are under construction or in the phase of +development. Calibrating these arrays is important for the reconstruction of observed events and +for the comparability between observatories. Using reference antennas in calibration campaigns +is not ideal because of uncertainties on their signal output strength that are large or difficult to +assess. In a different approach the arrays can be calibrated against the Galactic radio emission +as the dominant source of background. This so-called Galactic Calibration relies on predictions +of the diffuse Galactic radio emission, for which models are publicly available. We present a +comparison of these models in the frequency range from 10 to 408 MHz in order to estimate the +systematic uncertainties on the strength of the Galactic background. We do this comparison on a +global level as well as adapted for selected radio arrays and discuss implications for applying the +Galactic calibration method. Furthermore we study the influence of the quiet Sun as an additional +source of radio emission in the sky. +9th International Workshop on Acoustic and Radio EeV Neutrino Detection Activities - ARENA2022 +7-10 June 2022 +Santiago de Compostela, Spain +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.11438v1 [astro-ph.IM] 26 Jan 2023 + +Uncertainties of the Galactic Radio Background +Max Büsken +1. +Introduction +The radio detection technique in astroparticle physics – in particular, the detection of ultra- +high-energy cosmic rays – has become mature in recent years [1, 2]. Observing MHz radio signals +produced in cosmic-ray induced extensive air showers allows for assessing key observables like the +depth of shower maximum and the energy of the shower-initiating particle. Accurate measurements +of these observables require an accurate calibration of the antenna arrays. An emerging calibration +method utilizes the low-frequency Galactic radio emission as it provides the dominant background +signal in most arrays [3, 4]. By comparison of background data and predictions of the Galactic +signal strength with an understanding of the detector response, an absolute calibration can be made, +in principle, at any time and is much easier for large arrays than a calibration campaign with an +external reference source. +To evaluate the accuracy of the Galactic calibration, the systematic uncertainties on the prediction +of the Galactic emission have to be known. Therefore, we present here a comparison of sky models +that generate such predictions and estimate these systematic uncertainties for frequencies between +10 MHz and 408 MHz. After performing the comparison on a global scale, we adjust it to the cases +of a few selected radio antenna arrays and determine their individual uncertainties regarding the +prediction of the Galactic emission from their local skies. Furthermore, we investigate the influence +of the quiet Sun, which is another source of low-frequency radio emission in the sky. +2. +Reference maps and models for predicting the radio sky +We compare four publicly available models that predict the galactic radio emission and generate +sky maps of the brightness temperature at a given frequency: LFmap [5], the Global Sky Model +(GSM) in its original version from 2008 [6] and in its improved version from 2016 [7], and the +Low Frequency Sky Model (LFSM) [8]. The models take reference maps from radio-astronomical +surveys and interpolate between them. The reference maps used by the models for frequencies up +to 408 MHz and the uncertainties on their temperature scales as determined or estimated by the +respective authors are listed in Tab. 1. The models partly use the same reference maps. Uncertainties +on the observed brightness temperatures are within 20% for all of them. +The interpolation in the sky models is realized in different ways. In LFmap, spectral scaling +with a power law is applied to a single reference map from a survey at 408 MHz when evaluating +any other frequency. The spectral indices for scaling are taken from measurements in the respective +frequency range. In contrast, the other three models use principal component analysis to build a +model for predicting the radio sky. More details on the models can be found in Refs. [5–8]. +As an example, sky maps at 50 MHz are shown in Fig. 1, which are generated with the four +interpolation models. Differences between the outputs of the models regarding spatial structures +and the temperature scales are qualitatively visible. +3. +Comparison of the sky models +We aim to quantify the differences between the output of the sky models in order to estimate the +systematic uncertainty for predicting the radio sky. Therefore, we chose the average sky temperature +as a comparative variable which we define as +2 + +Uncertainties of the Galactic Radio Background +Max Büsken +map +No. +frequency +𝜈/MHz +covered +declination +relative +scale +uncert. 𝜎𝑘/% +zero-level +error 𝜎𝑇0/K +zero-level +error norm. +to average/% +used in +Ref. +1 +10 +−6° < 𝛿 < 74° +9* +2×104 +7.0 +1,2,3 +[9] +2 +22 +−28° < 𝛿 < 80° +16 +5×103 +11.5 +1,2,3,4 +[10] +3 +40 +−40° < 𝛿 < 90° +20 +10 +0.1 +3 +[8] +4 +45 +−90° < 𝛿 < 65° +10/15 +125† +1.5 +1,2,3,(4) +[11, 12] +5 +50 +−40° < 𝛿 < 90° +20 +10 +0.2 +3 +[8] +6 +60 +−40° < 𝛿 < 90° +20 +10 +0.2 +3 +[8] +7 +70 +−40° < 𝛿 < 90° +20 +10 +0.3 +3 +[8] +8 +80 +−40° < 𝛿 < 90° +20 +10 +0.5 +3 +[8] +9 +85 +−25° < 𝛿 < 25° +7 +120 +6.7 +2 +[13] +10 +150 +−25° < 𝛿 < 25° +5 +40 +9.2 +2 +[13] +11 +178 +−5° < 𝛿 < 88° +10 +15 +5.3 +(1,2,3) +[14] +12.a +408 (1982) +−90° < 𝛿 < 90° +10/5 +3 +8.8 +1 +[15] +12.b +408 (2003) +−90° < 𝛿 < 90° +10/5 +3 +8.8 +4 +[16] +12.c +408 (2015) +−90° < 𝛿 < 90° +10/5 +3 +8.8 +2,3 +[17] +Table 1: Overview of all reference maps used by the four sky models for frequencies up to 408 MHz. The +respective authors give the uncertainties of the surveys by considering a linear relation 𝑇true = 𝑘𝑇obs + 𝑇0 +between the true temperature 𝑇true and the observed temperature 𝑇obs. We list the quoted uncertainties on 𝑘 +and 𝑇0 (the latter as absolute values and normalized to the average temperature of the sky at the respective +frequency). Values of 𝑠𝑖𝑔𝑚𝑎𝑘 with a (*) are estimated by taking half of the smallest contour interval of +that map and dividing it by its minimum brightness temperature, because the authors gave no estimate. For +zero-level errors with a (†) half of the smallest contour interval of the map is taken as the estimate. The +usage of the reference maps in the individual models is denoted by the following notation: 1 = GSM, 2 = +GSM16, 3 = LFSM, 4 = LFmap. Numbers in brackets indicate that information from the reference map was +only used indirectly in the respective model. +¯𝑇(𝜈) = 1 +4𝜋 +∫ +𝜋 +−𝜋 +dℓ +∫ +𝜋 +2 +−𝜋 +2 +d𝑏 cos (𝑏) 𝑇(𝜈; ℓ, 𝑏). +(1) +Here, ℓ and 𝑏 are the Galactic spherical coordinates and 𝑇(𝜈; ℓ, 𝑏) is the brightness temperature +at a specific frequency 𝜈 and location in the sky from the output of one of the sky models. We use +the Python package PyGDSM [18] to obtain the output of the LFSM as well as the two versions +of the GSM in the HEALPix [19] format. LFmap is not included in PyGDSM and uses a different +output format, which we convert to HEALPix and separately feed into the analysis. +In the considered frequency range, the average sky temperature, in first order, drops exponen- +tially as a function of the frequency. Second-order differences between the models are shown in Fig. +2, where we plot the average sky temperature obtained from each of the four sky models as a function +of the frequency and normalize the curves to the one from the original GSM. Maximum deviations +between the models increase when going from 30 MHz to higher frequencies and decrease again +3 + +Uncertainties of the Galactic Radio Background +Max Büsken +120° +60° +0° +300° +240° +-60° +-30° +0° +30° +60° +103.5 +104 +104.5 +105 +T/K +(a) LFmap +120° +60° +0° +300° +240° +-60° +-30° +0° +30° +60° +103.5 +104 +104.5 +105 +T/K +(b) GSM +120° +60° +0° +300° +240° +-60° +-30° +0° +30° +60° +103.5 +104 +104.5 +105 +T/K +(c) GSM16 +120° +60° +0° +300° +240° +-60° +-30° +0° +30° +60° +103.5 +104 +104.5 +105 +T/K +(d) LFSM +Figure 1: Example sky maps at 50 MHz produced with each of the four models in galactic coordinates. +50 +100 +150 +200 +250 +300 +350 +400 +Frequency / MHz +0.95 +1.00 +1.05 +1.10 +1.15 +1.20 +¯T +¯TGSM(2008) +LFmap +LFSM +GSM (2008) +GSM (2016) +Figure 2: Average sky temperature from the output of the models plotted against the frequency after +normalization to the results of the GSM (2008). +towards 400 MHz, while staying within 20% overall. +We further quantify how any two of the models agree with each other by evaluating the quantity +𝑟m1,m2, which we define as +𝑟m1,m2 = 2 +∫ 408 MHz +30 MHz +[ ¯𝑇m1(𝜈) − ¯𝑇m2(𝜈)]d𝜈 +∫ 408 MHz +30 MHz +[ ¯𝑇m1(𝜈) + ¯𝑇m2(𝜈)]d𝜈 +. +(2) +The results of this evaluation are listed in Tab. 2. Over the whole frequency range, 𝑟m1,m2 is +smaller than 12%, which can therefore be considered a global measure for the level of agreement +between the sky models. In this evaluation, LFSM stands out against the other models, as it shows +the largest deviation when compared to the other three models. +Evaluation for astroparticle radio arrays +4 + +Uncertainties of the Galactic Radio Background +Max Büsken +m1 +m2 +LFmap +GSM +GSM16 +LFSM +LFmap +- +4.6% +0.7% +−7.3% +GSM +−4.6% +- +−3.8% +−11.9% +GSM16 +−0.7% +3.8% +- +−8.1% +LFSM +7.3% +11.9% +8.1% +- +Table 2: Resulting 𝑟m1,m2 tabulated for each model combination m1, m2. +In the next step, we switch from a global point of view to the specific cases of selected radio +arrays. The chosen arrays, which are – amongst other things – used in studies of astroparticle +physics, are RNO-G (surface antennas) [20], LOFAR [21], GRAND [22], SKA-low [23], the Pierre +Auger Observatory (AERA [24] and the AugerPrime Radio Detector [25]), and the radio antennas +of the IceCube surface array [26]. +We adapt the evaluation of the sky model comparison in a simple way to the specifications +of each radio array. First, we convert the output maps of the sky models from Galactic to local +coordinates (with the two angles azimuth 𝛼 and altitude 𝑎, the elevation angle above horizontal) +with the observer at the coordinates of the experiment (geographical latitude ℓexp) and evaluate the +average local sky temperature as +¯𝑇local(𝜈, ℓexp) = 1 +2𝜋 +∫ 24 h +0 h +d𝑡LST +∫ 90° +15° +cos (𝑎)d𝑎 +∫ +𝜋 +2 +−𝜋 +2 +𝑇(𝜈, ℓexp, 𝑡LST; 𝑎, 𝛼)d𝛼. +(3) +Here, we average the brightness temperature of the local sky 𝑇(𝜈, ℓexp, 𝑡LST; 𝑎, 𝛼) over the full +24 h of local sidereal time (LST) where we limit the contributing sky to everything above 15° to +mimic the typical directional sensitivity of an antenna. +Next, we adapt the frequency integral to the nominal frequency band of each array with the +boundaries 𝜈exp, lower and 𝜈exp, upper. With that, we calculate +T (ℓexp) = +∫ +𝜈exp, upper +𝜈exp, lower +¯𝑇local(𝜈, ℓexp) d𝜈. +(4) +Finally, we define 𝑟exp; m1, m2 to again measure the level of agreement between any two sky +models for a given experiment, +𝑟exp; m1, m2 = 2Tm1(ℓexp) − Tm2(ℓexp) +Tm1(ℓexp) + Tm2(ℓexp) . +(5) +The value of 𝑟exp; m1, m2 is listed for all arrays in Tab. 3, once when including all four sky models +and once when omitting LFSM. The maximum deviation is between 10% and 19%, depending on +the experiment. +These numbers reduce significantly to between 2% and 14% when excluding +LFSM, showing a discrepancy between the output of this model and the other three. +5 + +Uncertainties of the Galactic Radio Background +Max Büsken +experiment +geographic +latitude +frequency +band / MHz +maximum +relative +deviation +(incl. LFSM) +corresponding +sky models +maximum +relative +deviation +(excl. LFSM) +corresponding +sky models +RNO-G [20] +72.58° +100 to 408 +17.1% +LFSM/GSM16 +9.6% +LFmap/GSM16 +LOFAR low [21] +52.91° +30 to 80 +13.3% +LFSM/GSM +8.9% +GSM/GSM16 +LOFAR high [27] +52.91° +110 to 190 +18.4% +LFSM/GSM16 +12.8% +LFmap/GSM16 +GRAND [22] +42.93° +50 to 200 +16.3% +LFSM/GSM +2.0% +LFmap/GSM +SKA-low [23] +−26.70° +50 to 350 +13.6% +LFSM/GSM16 +7.5% +LFmap/GSM16 +Auger [28] +−35.21° +30 to 80 +10.5% +LFSM/GSM +4.8% +LFmap/GSM +IceCube [26] +−90.0° +70 to 350 +18.6% +LFSM/GSM16 +13.6% +LFmap/GSM16 +Table 3: +Levels of agreement of the sky models for each of the selected radio antenna arrays evaluated +from the comparisons based on the local skies and nominal frequency bands of the respective array. The +frequency band of RNO-G is capped at 408 MHz due to the limitations of LFmap and LFSM, although the +surface antennas will also cover higher frequencies. +4. +Influence of quiet Sun +In otherwise radio-quiet regions, the Galaxy is the dominant background contribution for radio +arrays. However, subdominant contributions can alter the accuracy of the Galactic calibration, e.g., +radio emission from the Sun [29]. Therefore, we study the influence of the quiet Sun, i.e., without +considering solar flares, by its relative strength compared to the Galaxy. We place a circularly +shaped spot of homogeneous brightness temperature on the output maps of the models for the local +skies of the selected radio arrays. The solar brightness temperature as a function of the frequency is +taken from Ref. [30]. We calculate the relative difference of the average sky brightness temperature +with and without the superimposed Sun. This relative difference is shown as a function of the +frequency for all experiments in Fig. 3. +In general, the relative solar contribution increases with frequency. +It is insignificant for +arrays operating at lower frequencies, like the Auger radio detectors or LOFAR in the low-band +configuration, while the contribution rises to around 20% at 400 MHz, which is relevant for RNO-G, +SKA-low, and IceCube. +5. +Discussion and conclusion +The comparison of four radio sky models conducted in this work allows for an estimation of the +uncertainty on how accurately the radio sky can be predicted at low frequencies. This uncertainty +is relevant for applying the Galactic calibration to radio antenna arrays. In literature research on +the reference surveys used in making the sky models, we find that uncertainties on the temperature +scales can be as large as 20%, which propagates into the accuracy of the models. We compare the +output of the models via the average temperature of the entire sky and find a level of agreement of +12%. Furthermore, we adapt the comparison to selected radio antenna arrays by observing the local +sky instead of the entire sky and using the operational frequency band of the arrays for cosmic-ray +detection. The comparison evaluation yields different results for the variety of arrays, ranging from +6 + +Uncertainties of the Galactic Radio Background +Max Büsken +50 +100 +150 +200 +Frequency / MHz +0.0% +1.0% +2.0% +3.0% +4.0% +5.0% +¯T sun +local− ¯Tlocal +¯Tlocal +GRAND +Auger +LOFAR low band +LOFAR high band +100 +200 +300 +400 +Frequency / MHz +0.0% +2.5% +5.0% +7.5% +10.0% +12.5% +15.0% +17.5% +20.0% +RNO-G +SKA-low +IceCube +Figure 3: Relative difference in the average temperature of the local sky induced by the quiet Sun plotted for +the selected radio arrays. The lines represent the average results from using the four sky models to produce +the maps, while the colored bands show any model’s maximum and minimum contribution. +10% for the Auger radio detectors up to 19% for the radio array of IceCube. +In addition, we study the contribution of the quiet Sun to the radio sky background and find it +to evolve from an insignificant influence at the lowest considered frequencies to around 20% at +400 MHz. Consequently, the quiet Sun can be neglected for arrays at the lowest frequencies, like +Auger and LOFAR in the low-band mode, while it should definitely be considered for the higher- +frequency arrays, like RNO-G, SKA-low, and IceCube. The latter may need to take measures in +that regard when applying the Galactic calibration, e.g., by restricting to night times. The relative +uncertainties on the brightness temperature determined in this work apply to the received power +in the antenna, which scales with the square of the electric-field amplitudes of radio signals from +detected particles and thus with the square of the energy scale of these particles. Therefore, un- +certainties on the energy scale are about half of the uncertainties on the temperature. In this light, +the Galactic calibration turns out to be on par with or better than typical calibration methods using +an external reference source [3, 31]. In the future, additional sky surveys and improved models +for predicting the Galactic radio emission will increase the accuracy and reduce the systematic +uncertainties associated with the strength of the Galactic radio background. This outlook will make +the Galactic calibration the future standard for the calibration of radio arrays in astroparticle physics. +References +[1] T. Huege, Phys. Rep. 620 (2016) 1 [1601.07426]. +[2] F.G. 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Zhang et al., ApJ 932 (2022) 17 [2205.00065]. +[31] Pierre Auger collaboration, JINST 12 (2017) T10005 [1702.01392]. +8 + diff --git a/HNAyT4oBgHgl3EQf5fqL/content/tmp_files/2301.00806v1.pdf.txt b/HNAyT4oBgHgl3EQf5fqL/content/tmp_files/2301.00806v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..598317b95b1b2947679b6b6cafa5f5eed99716d2 --- /dev/null +++ b/HNAyT4oBgHgl3EQf5fqL/content/tmp_files/2301.00806v1.pdf.txt @@ -0,0 +1,1343 @@ +arXiv:2301.00806v1 [math.GT] 2 Jan 2023 +THE CHARACTERIZATION OF (n − 1)-SPHERES WITH n + 4 VERTICES +HAVING MAXIMAL BUCHSTABER NUMBER +SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE +Abstract. We provide a GPU-friendly algorithm for obtaining all weak pseudo-manifolds +whose facets are all in an input set of facets satisfying given conditions. +We use it here to +completely list up toric colorable seed PL-spheres with a few vertices implying the complete +characterization of PL-spheres of dimension n − 1 with n + 4 vertices having maximal Buch- +staber numbers. +Contents +1. +Introduction +1 +1.1. +State-of-the-art known PL-spheres +1 +1.2. +State-of-the-art known toric manifolds +2 +1.3. +The goal of the paper +3 +2. +Classification of weak pseudo-manifolds by GPU computing +4 +2.1. +Enumerating weak pseudo-manifolds +4 +2.2. +Generalities about GPU programming +6 +2.3. +The GPU algorithm for classifying weak pseudo-manifolds +8 +3. +Preparation for applying the algorithm +8 +3.1. +Finiteness of the problem +8 +3.2. +Collecting PL-spheres among weak pseudo-manifolds +10 +4. +Toric colorable PL-spheres of Picard number 4 +11 +4.1. +Enumeration up to n ≤ 10 +11 +4.2. +Enumeration for n = 11 +13 +4.3. +Toric colorability +14 +Acknowledgements +14 +Appendix A. +A few algorithmic methods on simplicial complexes +14 +A.1. +Checking isomorphism using minimal non-faces +14 +A.2. +An inductive algorithm for checking PL-sphereness +15 +References +15 +1. Introduction +Our interest is located at the intersection of discrete mathematics, with the enumeration of +PL-spheres, and geometry, with the classification of toric manifolds. +1.1. State-of-the-art known PL-spheres. A PL-sphere is a pure simplicial complex which +has a subdivision piecewise linearly homeomorphic to the boundary of a standard simplex. A +PL-sphere is said to be polytopal if it is isomorphic to the boundary complex of a simplicial +polytope. Problems of enumerating specific classes of PL-spheres such as simplicial polytopes +Date: January 3, 2023. +2020 Mathematics Subject Classification. 57S12, 57Q15, 57M50, 52B70, 05E45, 51M20, 52B05. +Key words and phrases. PL sphere, simplicial sphere, toric manifold, Buchstaber number, real Buchstaber +number, Picard number, weak pseudo-manifold, characteristic map, binary matroid, parallel computing, GPU +programming. +This work was supported by the National Research Foundation of Korea Grant funded by the Korean Govern- +ment (NRF-2019R1A2C2010989). +1 + +2 +SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE +bring us back a few thousand years ago in the ancient Greece with Platonic solids. Let us fix a +dimension n − 1 ∈ Z>0 of a PL-sphere K, and its number of vertices m = n + p. We call the +number p the Picard number of K. Starting from the end of the 19th century, the first direction +for enumerating (polytopal) PL-spheres was to focus on small dimensions n, namely n ≤ 4: +n +m +Polytopal PL-sphere +General PL-sphere +2 +m ≥ 3 +Characterization: m-gon +3 +Characterization: Steinitz theorem, equivalent to 3-connected planar +graphs, 1922 +Enumeration: +m ≤ 13 +Br¨uckner by hand, 1897-1931 [9, 10] +m = 11 +Corrected by Grace, 1965 [21] +m = 12 +Corrected by Bowen and Fisk, 1967 [6] +m = 13 +Corrected by Royle, program plantri by Brinkmann and McKay, 1999 [7] +m ≤ 23 +Brinkmann, also using plantri, 2007 [8] +4 +Characterization: unknown +Characterization: unknown +Enumeration: +Enumeration: +m = 8 +Br¨uckner, 1909 [11], Gr¨unbaum and +Sreedharan, 1967 [24] +Non-polytopal sphere by Barnette, +1969 [5], Altshuler and Steinberg, +1985 [3] +m = 9 +Altshuler and Bokowkski and Stein- +berg, 1980 [1] +Altshuler and Steinberg, 1976 [2] +m = 10, +11 +Miyata and Padrol, 2015 [30], (neigh- +bourly polytopes), using oriented ma- +troids +Sulanke and Lutz, 2008-2009 [27, 33], +using lexicographic enumeration +Notice that for n ≤ 3, all PL-spheres are polytopal. +At the same time the complete characterization of PL-spheres with small p, namely p ≤ +3, was computed. To any polytopal PL-sphere K, one can associate a configuration of (p − +1) dimensional vectors which stores the combinatorial structure of K which is called a Gale +diagram. It is known that when p ≤ 3 then all PL-spheres are polytopal ([28]) and they are thus +characterized by their Gale diagram (see [23] for details): +p +Polytopal PL-spheres +1 +Characterization: The boundary of an n-simplex +2 +Characterization: Repeated pyramid over a free sum of two simplices, Gr¨unbaum [23] +3 +Characterization: Regular n-gonal Gale diagram, with n odd, Perles [23] +However, for p = 4, Gr¨unbaum and Sreedharan [24] gave an example of non-polytopal PL-sphere +making the use of 3-dimensional Gale diagrams to be pointless for enumerating PL-spheres with +p ≥ 4. +Characterizing or enumerating PL-spheres is important in toric geometry since they are cor- +nerstone combinatorial object for this theory. +1.2. State-of-the-art known toric manifolds. A toric variety of complex dimension n is a +normal algebraic variety over the field of complex numbers C which admits an effective algebraic +action of (C∗)n having a dense orbit. The fundamental theorem for toric geometry states that the +classification of toric varieties of complex dimension n is equivalent to the classification of fans +in Rn. Especially, compact smooth toric varieties, which are called toric manifolds, correspond +to complete non-singular fans. A complete non-singular fan Σ in Rn having m rays can be +described by a pair (K, λ), where: +• K is the underlying simplicial complex of Σ which is an (n − 1)-dimensional PL-sphere +on [m] = {1, . . . , m}, and + +TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 +3 +• λ: [m] → Zn is a fan-giving map that is bijectively assigning a vertex of K to the +primitive generator of a ray of Σ. +A fan-giving map λ should satisfy the following condition, known as the non-singularity +condition over K; for any simplex {i1, . . . , in} in K, λ(i1), . . . , λ(in) are unimodular. A map +λ: [m] → Zn is called a characteristic map over K if it satisfies the non-singularity condition +over K. We call a PL-sphere toric colorable if it supports a characteristic map. The existence of +a characteristic map over K is deeply related to the Buchstaber number s(K) of K, that is the +maximal dimension of subgroups of the canonical T m-action that act freely on the moment-angle +complex ZK (or the polyhedral product (D2, S1)K). It should be noted that s(K) ≤ m − n = p. +Indeed, K is toric colorable if and only if its Buchstaber number is maximal ([12]), that is, +s(K) = p . +Remark 1.1. Roughly speaking, a class of pairs of a PL-sphere K and a characteristic map λ +over K corresponds to a class of manifolds that all admit a well-behaved n-dimensional torus +action whose orbit space has its boundary complex isomorphic to K. +As a byproduct, the +combinatorial properties of K reflects on the geometry of the associated manifold. In particular, +if K is star-shaped, then the corresponding manifold is known as a topological toric manifold +introduced in [26]. If K is polytopal, then the corresponding manifold is known as a quasitoric +manifold introduced in [16]. +The following fundamental question appears. +Question 1.2. Which pairs (K, λ) are complete non-singular fans? +First of all, the simplicial complex K has to be a PL-sphere. However, not all PL-spheres are +toric colorable. It is well-known that all PL-spheres of Picard number ≤ 2 are toric colorable, and +they support toric manifolds as well. The ones of Picard number 3 may not be toric colorable; +a PL-sphere whose Gale-diagram is a regular (2k + 1)-gon is toric colorable if and only if k ≤ 3 +[17], and it supports a toric manifold if and only if k ≤ 2 [22]. There was no characterization for +higher Picard numbers since no combinatorial description exists in such cases and brute force +algorithms for obtaining the list of PL-spheres for big n and p ≥ 4 have an extremely high +complexity. +One remarkable step for solving Question 1.2 is a work of Choi and Park [14] that translated +this problem into a finite problem. The wedge of K at a vertex v is the simplicial complex +given by wedv(K) := (I ∗ LkK(v)) ∪ (∂I ∗ K \ {v}), where I is an interval (the details will +be given in Section 3). A seed is a PL-sphere that cannot be described by the wedge of any +lower dimensional PL-sphere. It is known that when one performs a wedge operation on a toric +colorable PL-sphere, then the resulting one is also toric colorable, see [19], and of same Picard +number. As a consequence, if we fix a Picard number p, then the complete characterization of +PL-spheres of Picard number p is simply given by the seeds of Picard number p. The result of +Choi and Park [14] is that there are only finitely many toric colorable seeds of Picard number p +whereas there are infinitely many seeds of Picard number p ≥ 3. More precisely, if an n − 1 +dimensional toric colorable seed is of Picard number p ≥ 3, then p and n must satisfy the +inequality +n + p ≤ 2p − 1. +In particular, if an (n − 1)-dimensional seed of Picard number 4 is toric colorable, then n ≤ 11. +The classification of toric manifolds of Picard number p = 1, 2, 3 have been entirely achieved, +and we thus take here one more step and complete the characterization problem for toric col- +orable PL-spheres of Picard number 4. +1.3. The goal of the paper. The essential part in this characterization problem is to find PL- +spheres satisfying specific conditions up to dimension 11. To check up all possible candidates, we +have to consider approximately 2(15 +11) ≈ 10410 cases that is too big, and we additionally have to +check their isomorphism classes. The complexity of the current fastest algorithm [34] is not good +enough for applying it to our cases. In order to obtain the complete list, we aggressively use the +finiteness of the problem. Our main approach is to construct an algorithm that enables us to + +4 +SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE +take profit of parallel computing, and then to use GPUs whose performances are skyrocketing +from their great development in the last decade, mainly from the popularity of machine learning +which requires a lot of linear algebra computation. In Section 2, we provide a GPU-friendly +algorithm (Algorithm 1) for obtaining all weak pseudo-manifolds whose facets are all in an +input set of facets satisfying given conditions written in terms of affine functions. +A PL-sphere is Zn +2-colorable if there is a map λR : [m] → Zn +2 such that for any simplex +{i1, . . . , in} in K, λR(i1), . . . , λR(in) is linearly independent over Z2. It can also be described +in terms of real Buchstaber number sR(K), see [18] for details. We have the inequality s(K) ≤ +sR(K) ≤ p, and K is Zn +2-colorable if and only if sR(K) = p. It should be noted that every +toric colorable seed is Zn +2-colorable. In order to find all Zn +2-colorable seed, one naive strategy +is to find all seeds up to n ≤ 11, and pick all Zn +2-colorable ones up. However, just as counting +PL-spheres, counting seeds still requires heavy computing powers. Therefore, we have to use +the Zn +2-colorability to obtain the candidates using Algorithm 1. +In Subsection 4.1, we see a mod 2 characteristic map as a binary matroid whose set of bases is +inputted to Algorithm 1 to obtain all Zn +2-colorable weak pseudo-manifolds, from which we select +all Zn +2-colorable seeds; our algorithm enables us to finish the enumeration in reasonable time for +n ≤ 10. We additionally show in Subsection 4.2 that a Zn +2-colorable seed of Picard number 4 +with n = 11 can be obtained from the list of seeds obtained before, enabling us to complete the +list of all Zn +2-colorable seeds of Picard number 4. Finally, in Subsection 4.3, we checked that +every Zn +2-colorable seed supports at least one characteristic map implying the following main +theorem. +Theorem 4.6. The number of toric (or Zn +2-)colorable seeds of dimension n − 1 with Picard +number p ≤ 4, up to isomorphism, is as follows: +p\ n +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +> 11 +total +1 +1 +1 +2 +1 +1 +3 +1 +1 +1 +3 +4 +1 +4 +21 +142 +733 +1190 +776 +243 +39 +4 +3141 +In the above table, the empty slots display zero. +Furthermore, the present result gives a complete characterization of toric (or Zn +2-)colorable +PL-spheres because they are obtained by consecutive wedge operations from a seed of the table of +Theorem 4.6. Furthermore, the result is useful for the classification of toric manifolds of Picard +number 4, or its topological analogues such as quasitoric manifolds or topological toric manifolds +whose second Betti number is 4. It should be noted that the real analogues of such spaces can +be computed easily since there is an efficient algorithm to obtain all mod 2 characteristic maps +over a wedged simplicial complex, see [15]. +2. Classification of weak pseudo-manifolds by GPU computing +In this section, we provide a general approach on how to use GPU parallel computing capa- +bility for classifying weak pseudo-manifolds with given properties. +Let K be a pure simplicial complex of dimension n − 1 on the vertex set [m]. We will call +facets any subset of size (n−1) of [m] and ridges any subset of size (n−2) of [m] not necessarily +in K. We denote by F(K) and R(K) the sets of facets and ridges of K, respectively. +We provide an algorithm as follows: +• Inputs: A set F of facets, and a collection G of affine functions on the subsets of F, +called properties. +• Output: The set of weak pseudo-manifolds K such that F(K) ⊂ F and g(F(K)) > 0 +for all g ∈ G, namely, K satisfies all the properties. +2.1. Enumerating weak pseudo-manifolds. In this subsection we give some computational +results which allows us to give an algorithmic description of how to enumerate weak pseudo- +manifolds. + +TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 +5 +Provided any set of facets F = {F1, . . . , FM}, we can compute the set R = {f1, . . . , fN} of +every ridges which come from these facets. We then construct the facets-ridges adjacency matrix +A(F) = (ai,j) of size N × M as follows: +ai,j = +� +1 +fi ⊂ Fj +0 +otherwise , +for i = 1, . . . , N and j = 1, . . . , M. A simplicial complex K whose facets are all included in some +set of facets F = {F1, . . . , FM} can be regarded as a vector K = (k1, . . . , kM)t ∈ ZM with +kj = +� +1 +Fj ∈ K +0 +Fj /∈ K , +for j = 1, . . . , M. The pure simplicial K is a weak pseudo-manifold if any ridge of K is included +in exactly two facets of K. This reflects as the following property: +Proposition 2.1. Let F be a set of facets, A = A(F) the facets-ridges adjacency matrix of F, +and K a pure simplicial complex whose facets are all in F. Then K is a weak pseudo-manifold +if and only if the coordinates of the product AK are all in {0, 2}. +As a consequence, weak pseudo-manifolds seen as vectors in ZM +2 +are all included in the Z2- +kernel of the matrix A, seen as a linear map A: ZM +2 → ZN +2 . +Let B = +� +K1 +· · · +Ks +� +be a matrix whose columns form a Z2-basis of kerZ2 A. Every weak +pseudo-manifold K is uniquely expressed as one of the 2s possible Z2-linear combinations of +K1, . . . , Ks, namely K = �s +i=1 xiKi (mod 2) = BX, for X = (x1, . . . , xs)t ∈ Zs +2. If we restrict +the set of facets F well enough so it removes many symmetries, we are hoping that s will be +small. Furthermore, we also can find a suitable basis ˜K1, . . . , ˜Ks to reduce the number of cases +to compute. +We first explain how to construct this basis in the case where the set F contains all possible +facets of [m], and R all possible ridges. There are +�m +n +� +facets and +� m +n−1 +� +ridges. For a ridge +f, we will write (AK)f the coordinate of AK corresponding to f. Let us denote by P(f) := +{j ∈ [M]: f ⊂ Fj} the set of the indexes in F of the facets containing f, called the parents of +f, which are the only facets contributing to (AK)f. In this first case any ridge has m − n + 1 +parents. For a kernel matrix B whose rows are indexed by F, let us denote by BP(f) the matrix +whose rows are the ones of B taken at indexes P(f). For every f ∈ R, for every t = 1, . . . , s, the +tth column of BP(f) has an even number of ones since the basis element Kt has an even number +of facets containing f. Performing a mod 2 Gaussian elimination on the columns of BP(f) leads +to a matrix of following form +BP(f)E = +�Zm−n +0� +, +with +Zk = + + +1 +0 +· · · +0 +0 +1 +... +... +... +... +... +0 +0 +· · · +0 +1 +1 +· · · +1 +1 + + +, +a (k + 1) × k-matrix, for k any positive integer, and E ∈ GL(s, Z2) the Z2-invertible matrix +corresponding to the operations performed in the Gaussian elimination. The columns of the new +matrix BE corresponds to another basis of the Z2-kernel of A with a convenient description about +which facets containing the ridge f each generators possess. In fact, only the first m−n ones have +facets contributing to (AK)f. Moreover, one can see that taking the mod 2 linear combination +of strictly more than two of them would lead to (AK)f being strictly greater than 2, which is a +case we want to avoid computing since we focus on weak pseudo-manifolds, see Proposition 2.1. +Thus this decreases the number of mod 2 combinations containing the first m−n new generators +that we need to compute from 2m−n to +�m−n +0 +� ++ +�m−n +1 +� ++ +�m−n +2 +� += 1 + (m − n) + +�m−n +2 +� +. + +6 +SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE +By writing f 1 := f and E1 := E, one can inductively repeat the latter process by taking care +at step k + 1 of: +• Choosing each time a new ridge f k+1 such that for all i = 1, . . . , k, P(f i) ∩ P(f k+1) = ∅, +• Starting the Gaussian pivot at columns index k(m − n) + 1 so that the structure of the +generators of previous columns is not lost. +This process terminates at some step kmax whenever one of the previous conditions cannot be +satisfied and we obtain a final matrix, whose columns are the new basis elements ˜K1, . . . , ˜Ks, +and which up to reordering the rows according to the sets P(f 1), . . . , P(f kmax) looks as follows: +BE1 · · · Ekmax = + + +Zm−n +0 +. . . +. . . +0 +0 +Zm−n +... +... +... +... +... +... +... +0 +. . . +0 +Zm−n +0 +⋆ +⋆ +⋆ +⋆ +⋆ + + += +� ˜K1 +· · · +˜Ks +� +. +In this case, we decrease the total number of mod 2 combinations from 2s to (1 + (m − n) + +�m−n +2 +� +)kmax2s−kmax(m−n) since we should take a maximum of 2 basis elements for each block +Zm−n. +As for the general case, there may be ridges which have less than m − n + 1 parents. In this +case, we try to wisely choose some ridges f 1, . . . , f kmax such that the blocks Zk are the biggest +possible so we minimize the number of mod 2 combinations BX of the generators we need to +compute. This provides a partition I1, . . . , Il of {1, . . . , s} such that no more than two basis +elements with indexes in Ik can be summed, otherwise we are sure not to obtain a weak pseudo- +manifold. We can split the vector X in the mod 2 combinations BX as blocks according to this +partition: X = �l +k=1 Xk, with Xk representing the part of X whose only nonzero coordinates +are in Ik. Let us denote by Xk the set of all such possible Xk, for k = 1, . . . , l. +If we recap our process, given a set of facets F, we constructed +• The facets-ridges adjacency matrix A whose Z2-kernel contains all weak pseudo-manifolds, +• A matrix B whose columns form a convenient basis ˜K1, . . . , ˜Ks of kerZ2(A), +• A partition I1, . . . , Il of {1, . . . , s}, +• A partition X1, . . . , Xl of the vectors of Zs +2 such that for all k = 1, . . . , l, Xk ∈ Xk has a +maximum of two nonzero coordinates which are all in Ik , +such that the weak pseudo-manifolds whose facets are in F are exactly the K = BX, with +X = �l +k=1 Xk for some (X1, . . . , Xl) ∈ X1 × · · · × Xl, satisfying that each coordinate of AK is +in {0, 2}. Moreover, given any affine function K �→ g(K), it is easy to check using computer +programming that g(K) > 0 is verified. We provide in the next subsection some concepts about +GPU programming. +2.2. Generalities about GPU programming. In this article we used Nvidia CUDA [31] so +one should notice that the syntax and vocabulary may differ from other GPU languages. +The general idea behind GPU computing is that it allows to parallelize tasks with two layers +of parallel programming without the need of using a super computer. Parallel programming +takes several forms, and the two we will use are the following: +• Data parallelism: one has a big list of elements X and wants to apply the same function +g to every element X ∈ X. In this case, each call of the function g is independent. +• Task parallelism: one has an element X and wants to apply a set of similar functions +g1, . . . , gk on X in order to obtain the result as a list (g1(X), . . . , gk(X)). The easiest +example is a matrix product AX, and if each row of A is denoted by ai, then the functions +gi are the inner products with the ais. +In all that follows, a thread (of execution) will be a processing unit computing machine +operations linearly, and a GPU will be a two-layered structure of threads. Namely a GPU will +be a set of p grids, and each grid will be a set of q threads. Therefore a GPU can be seen +as p × q threads organized for parallel programming, see Figure 1. The number p × q of GPU + +TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 +7 +threads which can run simultaneously is roughly the number of CUDA cores (if we consider +Nvidia GPUs) and is around seventeen thousands for the current architectures (as of 2022). +Thus a single GPU would be approximately equivalent to at least a thousand CPU threads. +· · · +q threads per grid +· · · +... +· · · +p grids +Figure 1. The two layered parallel structure of a GPU. +In CUDA programming one uses this two-layered structure as follows: +• First layer (blocks): Let X = {X1, . . . , XN} be the set of data on which we want +to apply the same function g, called the kernel. We create a big list of N blocks, each +one indexed by an integer i. Each block embodies the function call g(Xi). A block has +three possible states: on hold, active, and finished. At the beginning every blocks are on +hold. Then the p grids of the GPU are filled with some blocks which will be running, +these are active, the other ones are waiting to be launched on the grid and remain on +hold. Whenever an active block has finished, it will be replaced by a block on hold. The +program terminates when all blocks are finished. +• Second layer (threads): Whenever a block is sent to a grid, then the operations made +in the block are split into threads using task parallelism, any procedures in g is split into +q functions which will run simultaneously on all q threads of the grid. Notice that we +need every threads to finish their tasks in order to obtain the result, we can explicitly +require this condition by synchronizing the threads. +In all that follows, we will use such notations: +• A set X will be denoted as a list list X, +• A matrix A = [ai,j] will be represented as an array whose coefficient at index i, j is +A[i][j], +• A binary vector X ∈ Zk +2 will be represented as a binary variable x on k bits. +We will use the following processor instructions on binary variables [31]: +• The and operation x&y, 64 operations per cycle, +• The exclusive or operation x^y, 64 operations per cycle, +• The population count operation popcount(x) which counts the number of “1” bits in +the value of x, 16-32 operations per cycle. +• Atomic operations, which are useful since many threads may want to write at the same +memory location at the same time. The processor scheduler creates a queue of all atomic +operation calls so they do not overlap in order to avoid memory access error. +A cycle is the smallest time interval considered in a processor unit and is performed at the +frequency f of the processor unit (of each thread): if the frequency is 1GHz then 109 cycles are +done in one second. +The thread synchronization allows us managing how the threads behave in parallel as follows: +• The syncthreads() command asks all the threads to wait that all of them get to the +exact same line in the algorithm code of the kernel. + +8 +SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE +• For a local thread variable test, the syncthreads and(test) and syncthreads or(test) +command allows us apply the and or the or operation between all of the test variables +existing in each thread of a grid. For example if a thread encounters a condition which +should stop the current case in a loop, then all the threads should stop at once since this +case is not useful computing. +2.3. The GPU algorithm for classifying weak pseudo-manifolds. For simplifying the +explanation of the algorithm, we suppose that s = 64 and that we can write the product +X1 ×· · ·×Xl as Xa ×Xb such that Xa and Xb describe the 32 first or last generators, respectively. +We thus decompose K as Ka+Kb, with Ka = BXa and Kb = BXb for every (Xa, Xb) ∈ Xa×Xb. +Both vectors Xa and Xb are binary vectors whose nonzero coordinates are in the 32 first or last +coordinates, respectively, they are then stored as 32 bits variables xa and xb, namely as unsigned +integers. +The dot product in Z of two integers x, y is nothing more than the number of active bits +of the XOR operation: |x ∧ y|. Its mod 2 reduction is just the value of its least significant bit: +|x ∧ y|&1. +The main idea of the algorithm is to use M threads to compute each coordinate of K ∈ ZM +2 , +with M being the number of facets in F, as provided in Algorithm 1: +Remark 2.2. When we say “using the threads”, this means that we evenly distribute the +operations to perform among the threads, for example to reinitialize the array r, we use the fact +that we have q threads which can set to zero q coordinates at the same time until all coordinates +are reset. Thus, it requires ⌈N +q ⌉ loops, if N is the number of ridges. The same process is applied +for calculating the image by the affine functions g ∈ G. +Remark 2.3. We use the atomic add operation for incrementing values in r since many threads +may write at the same memory location r[k]. +The global complexity of this algorithm is: +O +�|Xa| +p +× |Xb| × N +q × (α|G| + 1) +� += O +�|X| × N × (α|G| + 1) +pq +� +, +with α the average complexity of the atomic operation when called multiple times for a given +g ∈ G. +3. Preparation for applying the algorithm +We will regard a simplicial complex either as a binary vector with entries corresponding to +facets or as a set of facets. +3.1. Finiteness of the problem. Let K be an n − 1 dimensional simplicial complex on [m] = +{1, 2, . . . , m}. The join K ∗ L of two simplicial complexes K and L is the simplicial complex +{σ ∪ τ | σ ∈ K, τ ∈ L}. The link LkK(σ) of a face σ in K is the simplicial complex {τ \ σ | +σ ⊂ τ ∈ K}. +For the sake of simplicity, we denote the simplicial complex consisting of a +single maximal simplex σ by just σ. The (simplicial) wedge wedv(K) of K at a vertex v is +(I ∗LkK(v))∪(∂I ∗K \v), where I is a 1-simplex, and K \v is the simplicial complex consisting +of the facets of K which do not contain v. We call K a seed if K cannot be represented as a +wedge of L for some simplicial complex L. +A PL-manifold is a simplicial complex such that the link of each of its vertex is a PL-sphere. +It is well known that a PL-sphere is a PL-manifold ([25, Lemma 1.17]). By the definition of +wedge, wedv(K) contains an isomorphic copy of K as the link of each of the new vertices. This +observation implies that if wedv(K) is a PL-sphere, then so is K. In fact, the converse is also +true. +Proposition 3.1. Let K be a PL-sphere and v a vertex of K. Then wedv(K) is a PL-sphere. +Proof. Suppose that K is an n − 1 dimensional PL-sphere. +It can be easily seen that the +suspension K ∗ ∂{w1, w2} of K is isomorphic to an edge subdivision of wedv(K), so they have + +TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 +9 +Algorithm 1 The GPU algorithm for classifying weak pseudo-manifolds whose facets are all in +a facets set F. +Input: The list list F, corresponding to the set of facets F, and the list list G, correspond- +ing to the set of affine functions G +Output: The list list K of weak pseudo-manifolds K with facets in list F and which satisfy +g(K)>0 for every g in list G. +Initialization: +– Compute the facets-ridges adjacency matrix A = A(F) ∈ ZN×M +2 +and store it in A, a +column sparse matrix: A[k][i] represents the index of the kth nonzero coordinate of the +ith column of A. +– Compute B = +� ˜K1 +· · · +˜K64 +� += + + +a1 +b1 +... +... +aM +bM + + and store it as two lists list a and list b +of integers, where list a[k] and list b[k] represents the binary value of the row vectors +ak and bk, respectively. +– Enumerate Xa and Xb, and store them as two lists list Xa and list Xb +– Create a list list Ka of all the Kas: for all xa in list Xa, for all k= 1, . . . M, +Ka[k]←popcount(a[k]^xa)&1. +Shared memory: r integer array of size N, such that r[k] stores the kth coefficient of the +product AK. +Kernel(xa,Ka) +– Let i be the local thread index. +– b←list b[i] +– ka←list Ka[i] +– For every xb in list Xb: +∗ skip←False +∗ Ki←popcount(b^xb)^ka and use syncthreads(). +∗ For every g in list G: compute g(K) using the thread values Ki and if g(K) ≤ 0, +then skip←True and exit this loop +∗ If syncthreads or(skip) is verified then go to the next xb +∗ Reinitialize each value of r to 0 using the threads +∗ If Ki is equal to 1 then for k= 1, . . . , n: add 1 to r[A[k][i]] using the atomic add +operation and if the value of r[A[k][i]] is greater than 3 then skip←True and +exit the loop +∗ If syncthreads or(skip) is verified then go to the next xb +∗ Add K to the list of results list K +Main: Launch the |Xa| blocks which correspond to all the pairs (xa,Ka) on the kernel. +the same PL-structure. Moreover, K ∗ w1 is a subdivision of an n-simplex since K is a PL- +sphere. Hence K ∗ ∂{w1, w2} = (K ∗ w1) ∪K (K ∗ w2) is a subdivision of the boundary of an +n-simplex. +□ +We recall that K is toric colorable if K admits a characteristic map over K that is a map +satisfying the non-singularity condition over K. One can consider its mod 2 analogue as well. A +mod 2 characteristic map over K is a map λ: [m] −→ Zn +2 satisfying the non-singularity condition +that for each facet σ of K, λ(σ) forms a basis of Zn +2. Similarly, We say that K is Zn +2-colorable +if K admits a mod 2 characteristic map. +Proposition 3.2 ([19], [13]). Let K be a PL-sphere and v a vertex of K. Then K is toric +colorable if and only if so is wedv(K). In addition, K is Zn +2-colorable if and only if wedv(K) is +Zn+1 +2 +-colorable. +Notice that the composition of a characteristic map over K and mod 2 reduction Zn → Zn +2 +becomes a mod 2 characteristic map over K. As a consequence, we can focus only on Zn +2-colorable +seeds. + +10 +SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE +We often see a mod 2 characteristic map λ as a matrix +� +λ(1) +λ(2) +· · · +λ(m) +� +. +Up to +simplicial isomorphisms, we may assume that the facet {1, 2, . . . , n} is contained in K. With +this assumption, to check Zn +2-colorability, it is enough to consider characteristic maps of the +form λ = +� +In +M +� +since the non-singularity on its facets is preserved by the left multipli- +cation of an element of GL(n, Z2). +Let us define dual characteristic maps (DCM) over K. +For λ = +�In +M� +, the DCM associated with λ is a map ¯λ: [m] −→ Zm−n +2 +such that ¯λt = +�¯λ(1) +¯λ(2) +· · · +¯λ(m)�t = +� +M +Im−n +� +. Moreover, the term injective DCM will be shortened to +IDCM. +Theorem 3.3 ([14]). Let K be an n−1 dimensional PL-sphere with m vertices and v, w distinct +vertices of K. Then the following are true. +(1) If every facet of K contains v or w, then K is a wedge or a suspension. +(2) If K is a seed which is not a suspension then every DCM over K must be an IDCM. +Statements (2) and (3) both imply: +(3) If K is a seed and m − n ≥ 3 then m ≤ 2m−n − 1. +We will call a seed which is not a suspension a regular seed. We conclude from Statement (3) +of Theorem 3.3 that there are only finitely many Zn +2-colorable seeds to enumerate for a fixed +integer p := m−n, that we match here with the Picard number of K written Pic(K). For p ≤ 3, +it is known that PL-spheres are all boundaries of polytopes [28] and were already completely +enumerated by Perles [23]. The Zn +2-colorability can easily be checked by an algorithm described +in [20], and seedness is verified from the following lemma. +Lemma 3.4 (seedness). A PL-sphere K is a seed if and only if it has no edge {v, w} such that +every facet of K has either v or w. +Proof. The sufficiency of the condition is from the definition of wedge. Suppose that K has +an edge {v, w} such that every facet of K has either v or w. +Then by Statement (1) and +Statement (2) of Theorem 3.3, K is a wedge. +□ +We now focus on the case p = 4. By Statement (3) of Theorem 3.3 we have n ≤ 11, implying +it is enough to enumerate PL-spheres of dimension up to 10 (n = 11). However, recall that +PL-spheres of Picard number ≥ 4 are not necessarily to be polytopal [5]. We thus need to put +our concern on checking PL-sphereness. +3.2. Collecting PL-spheres among weak pseudo-manifolds. We need a criterion for a +weak pseudo-manifold to be a PL-sphere. Fortunately, we have a powerful criterion for a PL- +manifold to be a PL-sphere when its Picard number is small enough. +Theorem 3.5 ([4]). Let K be a PL-manifold with Pic(K) ≤ 7 vertices. If K is a Z2-homology +sphere, then K is a PL-sphere. +By using the above theorem, we obtain the following lemma. +Lemma 3.6 (PL-sphereness). A weak pseudo-manifold K of Picard number ≤ 7 is a PL-sphere +if and only if the link of any face (possibly the empty face) of K is a Z2-homology sphere. +Proof. The sufficiency is obvious, so it is enough to show the necessity. Suppose that the link +of any face of K is a Z2-homology sphere. By applying Theorem 3.5, let us prove that the link +of each face of K is a PL-sphere. +We use an induction on the dimension of the link of a face. We remark that the link of each +(n − 2)-face of K is the 0-sphere by the definition of weak pseudo-manifold, in particular, it is +a PL-sphere. For k ≤ n − 3, let σ be a k-face of K, and L = LkK(σ) its link. Note that LkL(v) +for a vertex v of L is equal LkK({v} ∪ σ). Therefore, if the link of any (k + 1)-face of K is a +PL-sphere, then L is a PL-manifold. Since L is a Z2-homology sphere by the assumption and +Pic L ≤ Pic K ≤ 7, and L is a PL-sphere by Theorem 3.5. By induction, the link of each face is +a PL-sphere. +□ + +TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 +11 +We now have the tools for: +• Checking the PL-sphereness of a weak pseudo-manifold of Picard number 4 with Lemma 3.6 +• Checking the seedness of a PL-sphere with Lemma 3.4. +4. Toric colorable PL-spheres of Picard number 4 +We devote this section to enumerate all n − 1 dimensional toric colorable seeds of Picard +number 4. +4.1. Enumeration up to n ≤ 10. In this subsection, we firstly enumerate all n−1 dimensional +Zn +2-colorable seeds on [m] with Picard number 4 up to n ≤ 10. One could intuitively try to input +Algorithm 1 with all n subsets of [m], and then check the PL-sphereness, the Zn +2-colorability and +the seedness. However, it is hopeless when we consider high dimensions. Indeed, we manage to +obtain results up to n = 6, but the program takes too long time to finish with n = 7. +On the other hand, Theorem 3.3 states that there exists only two kind of Zn +2-colorable seeds: +regular or suspended. We will first consider how to enumerate the first ones as they can only +support IDCM. To do so we rephrase the fact that a pure simplicial complex K supports a +mod 2 characteristic map λ as K is included in the binary matroid associated to λ. We recall +here some matroid theory definitions. A matroid M is a simplicial complex with the so-called +augmentation property: for any τ, σ ∈ M with |τ| < |σ|, there exists x ∈ σ \ τ such that +τ ∪ {x} ∈ M. Although the facets of a matroid are called the bases, we will keep the simplicial +complex terminology here and call them facets. The dual matroid M of a matroid M is the +matroid on the same vertex set as M and whose facets, called cofacets, are the complement of +each facets of M. For an n × m matrix λ over Z2 of full row rank n, the simplicial complex Mλ +whose facets are the sets of column indexes of n mod 2 independent columns of λ is a matroid, +called the binary matroid associated to λ. By linear Gale duality [19], the dual Mλ is equal to +M¯λt. The following proposition is easily verified by the definition of Mλ and M¯λt. +Proposition 4.1. Let K be an n−1 dimensional simplicial complex on [m] and ¯λ a full column +rank m × (m − n) matrix over Z2 of rank m − n. Then K supports ¯λ as a DCM if and only if +it is a subcomplex of M¯λt = Mλ. +Recall that the more we reduce the number of facets in the input of Algorithm 1, the smaller +the dimension of the mod 2 kernel of the ridges-facets adjacency matrix will be and thus the +faster the algorithm will run. Proposition 4.1 gives us exactly what we want: a finer set of facets. +In addition, we take advantage of the upper bound theorem ([32]): the number of facets of a +simplicial spheres is less than of equal to the one of the cyclic n-polytope C(m, n) with m vertices. +This condition is embodied by the following affine function: g(K) = fn−1(C(m, n))−∥K∥1. Fix +an injective map ¯λ: [m] −→ Z4 +2 and set F(λ) = F(Mλ). Algorithm 1 with inputs being the +set of facets F(λ) and the affine function g outputs the set of all weak pseudo-manifolds which +support ¯λ and satisfy the upper bound theorem. +At first sight, it seems that we need to run the algorithm on each of the +�11 +n +� +× (n!) injective +maps ¯λ even if we fix +�¯λ(n + 1) +¯λ(n + 2) +¯λ(n + 3) +¯λ(n + 4)� += I4. However, we will drasti- +cally reduce this large number of cases to compute by noticing that many injective maps provide +the same outputs up to simplicial isomorphism, this can be understood from the fact that the +inclusion of K into a binary matroid Mλ depends on the choice of the labelling on the set of +vertices, if we state that we label the binary matroid and the vertices of a given facet of K, we +do not need to consider all the cases. +Let Λ(n, p) be the set of all (n + p) × p matrices over Z2 of the form +� +M +Ip +� +such that each +matrix has no repeated rows. Consider the product of symmetric groups Sn × Sp. This group +gives a group action on Λ(n, p) by +�� +M +Ip +� +, (s, t) +� +�→ +� +P t +sMPt +Ip +� +, where Ps and Pt are column +permutation matrices corresponding to s and t. Let us call each element of Λ(n, p)/Sn × Sp an +IDCM orbit. + +12 +SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE +Proposition 4.2. For (s, t) ∈ Sn × Sp, there is a simplicial isomorphism between matroids +associated to ¯λ ∈ Λ(n, p) and ¯λ ◦ (s, t) ∈ Λ(n, p). +Proof. Divide the vertex set [m] as V1 = [n] and V2 = {n + 1, n + 2, . . . , n + p}. +First, let +us consider s. It is easy to observe that the matrix +� +P t +sM +Ip +� +stores the same non-singularity +information on cofacets with relabeling V1 using s. +Because Pt is invertible, multiplying Pt at right side of a DCM does not affect non-singularity +on cofacets of the DCM. With the equation +� +P t +sMPt +Ip +� += +� +P t +sM +P t +t Ip +� +Pt, +applying t to V2 gives the same non-singularity information between +� +P t +sM +Ip +� +and +� +P t +sMPt +Ip +� +. +□ +Let Λ◦(n, 4) ⊆ Λ(n, 4) be a set containing one representative of each IDCM orbit. By Propo- +sition 4.2, it is enough to input Algorithm 1 with F(λ), for all λ ∈ Λ◦(n, 4). Find the number +of IDCM orbits and the computation time of Algorithm 1 when n < 11 in Table 1. +n +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +Number of IDCM orbits +7 +16 +28 +35 +35 +28 +16 +7 +3 +1 +max¯λ(dim ker A(F(λ))) +7 +13 +21 +24 +28 +34 +42 +48 +56 +64 +max¯λ |X(¯λ)| +56 +3e3 +5e5 +1e6 +2e7 +9e8 +1e11 +3e12 +4.4e14 +4.2e16 +Time spent for one orbit +1ms +10ms +0.1s +0.6s +1.3s +3m +15m +2h +12d +3y +Table 1. Data for Picard number 4, and n = 2, . . . , 11. The time spent is the +one taken by Algorithm 1 running on an Nvidia Quadro A5000. The time written +in bold in the case n = 11 is an estimation. +We provide now the global strategy for enumerating all Zn +2-colorable seeds of Picard number 4 +by recaping the case of regular seeds and explaining the case of suspended seeds. +Strategy. CASE I : Regular seeds For every representative λ ∈ Λ◦(n, 4), run Algorithm 1 +with inputs being the set of facets F(λ) and the affine function g(K) = fn−1(C(m, n)) − ∥K∥1. +After reducing isomorphic ones, we obtain the list of Zn +2-colorable weak pseudo-manifolds on +[m] satisfying the upper bound theorem, up to isomorphism. We then apply Lemma 3.4 and +Lemma 3.6 to collect the seeds up to isomorphism. +CASE II : Suspended seeds. From Theorem 3.3, a seed without IDCM is a suspension. It +can be easily seen that from the definition of wedge, the suspension of a wedge is again a wedge, +and the suspension operation increases the Picard number by one. Therefore, it is enough to +consider suspensions of seeds of Picard number 3. +Let L = ∂[v, w]∗K for an n−2 dimensional simplicial complex K, and λ a characteristic map +over L. Without loss of generality, we may assume v = 1 so that λ(v) = +�1 +0 +· · · +0�t. Then +for any facet {v}∪{v1, . . . , vn−1} of L, the (1, 1) minor of the matrix +� +λ(v) +λ(v1) +· · · +λ(vn−1) +� +is equal to 1. This implies that LkL(1) = K is Zn−1 +2 +-colorable. +We know there are three Zn +2-colorable seed PL-spheres of the Picard number 3; 5-gon, 3-cube, +and the cyclic polytope C(7, 4). The suspension of the 3-cube does not support any IDCM but +does support a DCM, while the suspensions of the others support IDCM. +The algorithm provides us the following table: +Theorem 4.3. Let n ≥ 1 be a positive integer. The number of Zn +2-colorable seed PL-spheres of +dimension n − 1 with Picard number p ≤ 4, up to isomorphism, is as follows: + +TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 +13 +p\n +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +> 11 +total +1 +1 +1 +2 +1 +1 +3 +1 +1 +1 +3 +4 +1 +4 +21 +142 +733 +1190 +776 +243 +39 +? +3141 +4.2. Enumeration for n = 11. As we can see in Table 1, the time complexity of the extremal +case n = 11 is still too long. To remedy this, let us use the results we obtained from dimension +just below to construct the seeds of this extremal case. +Let K be a Z1 +21-colorable seed on +{1, 2, . . . , 15} of dimension 10 (n = 11). We know that the link of the vertex 15 has Picard +number at most 4 and is a Z1 +20-colorable seed, which we already have enumerated before. The +idea is then to build all Z1 +21-colorable seeds from the Z1 +20-colorable seeds. If K has only vertices +whose links have Picard numbers at most 2, K would be the boundary of a product of simplices +[23], that is, not a seed. Suppose that the link of 15 has Picard number 3. Recall that there is +no 9-dimensional seed PL-sphere with Picard number 3 implying that the link of 15 is a wedge. +By the following lemma, we find another vertex of K whose link has Picard number 4. +Lemma 4.4. Let K be a seed PL-sphere with Pic(K) = 4. Assume that K has a vertex v such +that Pic(LkK(v)) = 3 and there exist two vertices v1 and v2 of LkK(v) such that every facets of +LkK(v) contain either v1 or v2. Then there is a vertex of K whose link has Picard number 4. +Proof. Let {v1} ∪ σ be a facet without v2. Since σ is a ridge of the PL-sphere LkK(v), there is +one more facet containing σ. By the assumption, it must be {v2} ∪ σ. This verifies that every +vertex in LkK(v) forms an edge with both v1 and v2. +Let w be the vertex not in LkK(v). If v1 ∈ LkK(w), then LkK(v1) has Picard number 4. If +both v1, v2 ̸∈ LkK(w), then LkK(w) is an (n − 2)-dimensional PL-sphere with n vertices. This +means that LkK(w) = ∂∆n−1. Then if w′ is a vertex of LkK(w), then Pic(LkK(w′)) = 4. +□ +This implies that any Z1 +21-colorable seed has a vertex of link being of Picard number 4, that +we relabel as 15. +Before we apply Algorithm 1 for this case, we need some preparation as +follows. We firstly select an injective map ¯µ : [14] → Z4 +2 and choose a 9-dimensional PL-sphere +L which supports ¯λ. We see L as the link of the vertex 15 in some Z11 +2 -colorable seed PL-sphere +K supporting some IDCM ¯λ with the restriction ¯λ|[14] = ¯µ. Since +��Z4 +2 +�� \ {0} = 11 , once ¯µ is +chosen, ¯λ is uniquely determined. Our computation shows there exist 13 wedge cases(checked +by computing). +Let ˆL be the simplicial complex {σ ∪ {15} | σ ∈ L}. All PL-spheres K having its vertex +15 whose link is L contains ˆL. This provides the following conditions on the components of +K ∈ ZM: +(1) for all ˆLj = 1, Kj = 1 +(2) for all ˆLj = 0 with ˆLj ∋ {15}, Kj = 0. +The set of the indexes of the facets satisfying Condition (1), respectively Condition (2), will +be denoted by I, respectively by J. After reordering the rows of B, the two conditions are +illustrated as follows +BX = + + +BI +BJ +B[M]\(I∪J) + + X = + + +1 +0 +⋆ + + . +(4.1) +A Gaussian elimination process on the columns of B will lead to a column reduced echelon form +˜B. Denote by sI, respectively sJ, the maximal index of non-zero column of ˜BI, respectively +of ˜BJ. In order to respect the conditions (1) and (2), we need xt = + + + + + +1 +t = 1, . . . , sI +0 +t = sI + 1, . . . , sJ +⋆ +otherwise +, +where X = (x1, . . . , xM)t. If this setting on X contradicts to (4.1) with ˜B instead of B, then it + +14 +SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE +would mean that there is no such Z11 +2 -colorable seed PL-sphere K supporting ¯λ whose link of +the vertex 15 is L. +The same strategy as in Subsection 4.1 applied to the rest of the entries of X, Lemma 3.4 +and Lemma 3.6 gives the following. +Theorem 4.5. There are exactly four Z11 +2 -colorable 10-dimensional seed PL-spheres of the Pi- +card number 4. +4.3. Toric colorability. We note that all toric colorable seeds are Zn +2-colorable. In order to +obtain the list of toric colorable seeds of Picard number 4, it is enough to check whether each +seed in the list of Zn +2-colorable seeds supports a characteristic map or not. +First, one may regard each vector in Zn +2 as a (0, 1)-vector in Zn. This may produce singular +facet in K. Then we change some 1’s in λ to −1’s until getting toric colorable one. Bruteforcing +this method provided at least one characteristic map supported by every Zn +2-colorable seed we +enumerated. The toric colorability thus is equivalent to the Zn +2-colorability for PL-spheres of +Picard number 4. +Theorem 4.6. The number of toric (or Zn +2-)colorable seeds of dimension n − 1 with Picard +number p ≤ 4, up to isomorphism, is as follows: +p\ n +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +> 11 +total +1 +1 +1 +2 +1 +1 +3 +1 +1 +1 +3 +4 +1 +4 +21 +142 +733 +1190 +776 +243 +39 +4 +3141 +In the above table, the empty slots display zero. +Acknowledgements +The authors are very grateful to Dr. Axel Bacher who introduced the last named author to +CUDA programming allowing Algorithm 1 to be adapted to GPU computing and thus computed +in reasonable time. +Appendix A. A few algorithmic methods on simplicial complexes +In this appendix, we will provide the readers a few algorithmic methods which were used in +order to obtain our results. +A.1. Checking isomorphism using minimal non-faces. One very hard problem when enu- +merating simplicial complexes is to deal with isomorphisms. If a simplicial complex K has m +vertices, then there exists m! possible relabellings for K. Provided two simplicial complexes K1 +and K2 one the respective vertex sets V and W one wants to find if they are isomorphic. One +solution is to use McKay graph isomorphism on the face posets of K1 and K2 ([29]). We provide +here a different solution for testing if they are isomorphic by mainly using their sets of minimal +non-faces (MNF), but also some combinatorial information such as their f-vectors. +Let K be a simplicial complex on a vertex set V . Our main motivation to use the MNF sets +is that the simplicial complexes K we are considering are seeds and thus satisfy some property +in their MNF set. We say that a pair of distinct vertices (v, w) of K form a couple if they share +the exact same “neighbourhood” in the MNF set of K, more explicitly, if for every minimal non +faces σ of K we have either {v, w} ⊆ σ or v, w /∈ σ. A simplicial complex K is a seed if and +only if there is no couple of vertices K. Expecting that the vertices of a seed will share distinct +neighbourhoods we define a color sequence cK(v), for each vertex v of K, which is invariant up to +relabeling, such that the color sequence cK(v) represents the increasing sizes of the minimal non +faces of K containing v. For example if MNF(K) = {123, 34, 456, 26, 16} then cK(1) = (2, 3), +cK(5) = (3) and c(6) = (2, 2, 3). +A relabeling φ: V → W between K1 and K2 may satisfy +cK2(φ(v)) = cK1(v) for every v. The procedure is the following: +(1) Check if K1 and K2 have the same dimension and the same number of vertices. + +TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 +15 +(2) Check if K1 and K2 have the same f-vector. +(3) Check if {cK1(v): v ∈ [m]} = {cK2(v): v ∈ [m]}, by counting repetitions, using their +MNF sets. +(4) We give partitions V1, . . . , Vk and W1, . . . , Wk of V and W with respect to color sequences +in K1 or K2, respectively. +We compute every relabeling φi : Vi → Wi for every i = +1, . . . , k. They provide every relabeling φ = φ1 × · · · × φk : V → W preserving the color +sequences. If one φ sends one-to-one the minimal non faces of K1 to the ones of K2 then +K1 is isomorphic to K2. +The number of relabeling that are computed is (|V1|!) × · · · × (|Vk|!) instead of |V |! which is a +nice improvement when there are many different color sequences and only few vertices having +the same color sequence. +A.2. An inductive algorithm for checking PL-sphereness. Denote by CRSP(p, n) the set +of Zn +2-colorable seed PL-spheres of Picard number p and of dimension n − 1 up to isomorphism. +Let us suppose that we have obtained all CRSP(p, k) for p ≤ 4 and k < n. The output of the +CUDA algorithms is the collection of all weak pseudo-manifolds compatible with all possible +IDCM ¯λ from which we select only the seeds. In order to check if they are PL-spheres, we use +Lemma 3.6. 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MR 2552676 +Department of mathematics, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon 16499, +Republic of Korea +Email address: schoi@ajou.ac.kr +Department of mathematics, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon 16499, +Republic of Korea +Email address: a24325@ajou.ac.kr +Universit´e Sorbonne Paris Nord, LIPN, CNRS UMR 7030, F-93430, Villetaneuse, France +Email address: vallee@lipn.fr + diff --git a/HNAyT4oBgHgl3EQf5fqL/content/tmp_files/load_file.txt b/HNAyT4oBgHgl3EQf5fqL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5474d10176a314bd9180cbd61c2c54a496bf49cd --- /dev/null +++ b/HNAyT4oBgHgl3EQf5fqL/content/tmp_files/load_file.txt @@ -0,0 +1,913 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf,len=912 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='00806v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='GT] 2 Jan 2023 THE CHARACTERIZATION OF (n − 1)-SPHERES WITH n + 4 VERTICES HAVING MAXIMAL BUCHSTABER NUMBER SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We provide a GPU-friendly algorithm for obtaining all weak pseudo-manifolds whose facets are all in an input set of facets satisfying given conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We use it here to completely list up toric colorable seed PL-spheres with a few vertices implying the complete characterization of PL-spheres of dimension n − 1 with n + 4 vertices having maximal Buch- staber numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' State-of-the-art known PL-spheres 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' State-of-the-art known toric manifolds 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The goal of the paper 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Classification of weak pseudo-manifolds by GPU computing 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Enumerating weak pseudo-manifolds 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Generalities about GPU programming 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The GPU algorithm for classifying weak pseudo-manifolds 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Preparation for applying the algorithm 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Finiteness of the problem 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Collecting PL-spheres among weak pseudo-manifolds 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Toric colorable PL-spheres of Picard number 4 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Enumeration up to n ≤ 10 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Enumeration for n = 11 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Toric colorability 14 Acknowledgements 14 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A few algorithmic methods on simplicial complexes 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Checking isomorphism using minimal non-faces 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' An inductive algorithm for checking PL-sphereness 15 References 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Introduction Our interest is located at the intersection of discrete mathematics, with the enumeration of PL-spheres, and geometry, with the classification of toric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' State-of-the-art known PL-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A PL-sphere is a pure simplicial complex which has a subdivision piecewise linearly homeomorphic to the boundary of a standard simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A PL-sphere is said to be polytopal if it is isomorphic to the boundary complex of a simplicial polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Problems of enumerating specific classes of PL-spheres such as simplicial polytopes Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 57S12, 57Q15, 57M50, 52B70, 05E45, 51M20, 52B05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' PL sphere, simplicial sphere, toric manifold, Buchstaber number, real Buchstaber number, Picard number, weak pseudo-manifold, characteristic map, binary matroid, parallel computing, GPU programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This work was supported by the National Research Foundation of Korea Grant funded by the Korean Govern- ment (NRF-2019R1A2C2010989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1 2 SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE bring us back a few thousand years ago in the ancient Greece with Platonic solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let us fix a dimension n − 1 ∈ Z>0 of a PL-sphere K, and its number of vertices m = n + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We call the number p the Picard number of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Starting from the end of the 19th century,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' the first direction for enumerating (polytopal) PL-spheres was to focus on small dimensions n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' namely n ≤ 4: n m Polytopal PL-sphere General PL-sphere 2 m ≥ 3 Characterization: m-gon 3 Characterization: Steinitz theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' equivalent to 3-connected planar graphs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1922 Enumeration: m ≤ 13 Br¨uckner by hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1897-1931 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 10] m = 11 Corrected by Grace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1965 [21] m = 12 Corrected by Bowen and Fisk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1967 [6] m = 13 Corrected by Royle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' program plantri by Brinkmann and McKay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1999 [7] m ≤ 23 Brinkmann,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' also using plantri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 2007 [8] 4 Characterization: unknown Characterization: unknown Enumeration: Enumeration: m = 8 Br¨uckner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1909 [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Gr¨unbaum and Sreedharan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1967 [24] Non-polytopal sphere by Barnette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1969 [5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Altshuler and Steinberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1985 [3] m = 9 Altshuler and Bokowkski and Stein- berg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1980 [1] Altshuler and Steinberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1976 [2] m = 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 11 Miyata and Padrol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 2015 [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' (neigh- bourly polytopes),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' using oriented ma- troids Sulanke and Lutz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 2008-2009 [27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' using lexicographic enumeration Notice that for n ≤ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' all PL-spheres are polytopal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' At the same time the complete characterization of PL-spheres with small p, namely p ≤ 3, was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' To any polytopal PL-sphere K, one can associate a configuration of (p − 1) dimensional vectors which stores the combinatorial structure of K which is called a Gale diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It is known that when p ≤ 3 then all PL-spheres are polytopal ([28]) and they are thus characterized by their Gale diagram (see [23] for details): p Polytopal PL-spheres 1 Characterization: The boundary of an n-simplex 2 Characterization: Repeated pyramid over a free sum of two simplices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Gr¨unbaum [23] 3 Characterization: Regular n-gonal Gale diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' with n odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Perles [23] However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' for p = 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Gr¨unbaum and Sreedharan [24] gave an example of non-polytopal PL-sphere making the use of 3-dimensional Gale diagrams to be pointless for enumerating PL-spheres with p ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Characterizing or enumerating PL-spheres is important in toric geometry since they are cor- nerstone combinatorial object for this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' State-of-the-art known toric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A toric variety of complex dimension n is a normal algebraic variety over the field of complex numbers C which admits an effective algebraic action of (C∗)n having a dense orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The fundamental theorem for toric geometry states that the classification of toric varieties of complex dimension n is equivalent to the classification of fans in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Especially, compact smooth toric varieties, which are called toric manifolds, correspond to complete non-singular fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A complete non-singular fan Σ in Rn having m rays can be described by a pair (K, λ), where: K is the underlying simplicial complex of Σ which is an (n − 1)-dimensional PL-sphere on [m] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , m}, and TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 3 λ: [m] → Zn is a fan-giving map that is bijectively assigning a vertex of K to the primitive generator of a ray of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A fan-giving map λ should satisfy the following condition, known as the non-singularity condition over K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' for any simplex {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , in} in K, λ(i1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , λ(in) are unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A map λ: [m] → Zn is called a characteristic map over K if it satisfies the non-singularity condition over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We call a PL-sphere toric colorable if it supports a characteristic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The existence of a characteristic map over K is deeply related to the Buchstaber number s(K) of K, that is the maximal dimension of subgroups of the canonical T m-action that act freely on the moment-angle complex ZK (or the polyhedral product (D2, S1)K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It should be noted that s(K) ≤ m − n = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Indeed, K is toric colorable if and only if its Buchstaber number is maximal ([12]), that is, s(K) = p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Roughly speaking, a class of pairs of a PL-sphere K and a characteristic map λ over K corresponds to a class of manifolds that all admit a well-behaved n-dimensional torus action whose orbit space has its boundary complex isomorphic to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' As a byproduct, the combinatorial properties of K reflects on the geometry of the associated manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In particular, if K is star-shaped, then the corresponding manifold is known as a topological toric manifold introduced in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If K is polytopal, then the corresponding manifold is known as a quasitoric manifold introduced in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The following fundamental question appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Which pairs (K, λ) are complete non-singular fans?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' First of all, the simplicial complex K has to be a PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' However, not all PL-spheres are toric colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It is well-known that all PL-spheres of Picard number ≤ 2 are toric colorable, and they support toric manifolds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The ones of Picard number 3 may not be toric colorable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' a PL-sphere whose Gale-diagram is a regular (2k + 1)-gon is toric colorable if and only if k ≤ 3 [17], and it supports a toric manifold if and only if k ≤ 2 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' There was no characterization for higher Picard numbers since no combinatorial description exists in such cases and brute force algorithms for obtaining the list of PL-spheres for big n and p ≥ 4 have an extremely high complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' One remarkable step for solving Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2 is a work of Choi and Park [14] that translated this problem into a finite problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The wedge of K at a vertex v is the simplicial complex given by wedv(K) := (I ∗ LkK(v)) ∪ (∂I ∗ K \\ {v}), where I is an interval (the details will be given in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A seed is a PL-sphere that cannot be described by the wedge of any lower dimensional PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It is known that when one performs a wedge operation on a toric colorable PL-sphere, then the resulting one is also toric colorable, see [19], and of same Picard number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' As a consequence, if we fix a Picard number p, then the complete characterization of PL-spheres of Picard number p is simply given by the seeds of Picard number p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The result of Choi and Park [14] is that there are only finitely many toric colorable seeds of Picard number p whereas there are infinitely many seeds of Picard number p ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' More precisely, if an n − 1 dimensional toric colorable seed is of Picard number p ≥ 3, then p and n must satisfy the inequality n + p ≤ 2p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In particular, if an (n − 1)-dimensional seed of Picard number 4 is toric colorable, then n ≤ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The classification of toric manifolds of Picard number p = 1, 2, 3 have been entirely achieved, and we thus take here one more step and complete the characterization problem for toric col- orable PL-spheres of Picard number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The goal of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The essential part in this characterization problem is to find PL- spheres satisfying specific conditions up to dimension 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' To check up all possible candidates, we have to consider approximately 2(15 11) ≈ 10410 cases that is too big, and we additionally have to check their isomorphism classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The complexity of the current fastest algorithm [34] is not good enough for applying it to our cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In order to obtain the complete list, we aggressively use the finiteness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Our main approach is to construct an algorithm that enables us to 4 SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE take profit of parallel computing, and then to use GPUs whose performances are skyrocketing from their great development in the last decade, mainly from the popularity of machine learning which requires a lot of linear algebra computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In Section 2, we provide a GPU-friendly algorithm (Algorithm 1) for obtaining all weak pseudo-manifolds whose facets are all in an input set of facets satisfying given conditions written in terms of affine functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A PL-sphere is Zn 2-colorable if there is a map λR : [m] → Zn 2 such that for any simplex {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , in} in K, λR(i1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , λR(in) is linearly independent over Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It can also be described in terms of real Buchstaber number sR(K), see [18] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We have the inequality s(K) ≤ sR(K) ≤ p, and K is Zn 2-colorable if and only if sR(K) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It should be noted that every toric colorable seed is Zn 2-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In order to find all Zn 2-colorable seed, one naive strategy is to find all seeds up to n ≤ 11, and pick all Zn 2-colorable ones up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' However, just as counting PL-spheres, counting seeds still requires heavy computing powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Therefore, we have to use the Zn 2-colorability to obtain the candidates using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1, we see a mod 2 characteristic map as a binary matroid whose set of bases is inputted to Algorithm 1 to obtain all Zn 2-colorable weak pseudo-manifolds, from which we select all Zn 2-colorable seeds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' our algorithm enables us to finish the enumeration in reasonable time for n ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We additionally show in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2 that a Zn 2-colorable seed of Picard number 4 with n = 11 can be obtained from the list of seeds obtained before, enabling us to complete the list of all Zn 2-colorable seeds of Picard number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Finally, in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3, we checked that every Zn 2-colorable seed supports at least one characteristic map implying the following main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The number of toric (or Zn 2-)colorable seeds of dimension n − 1 with Picard number p ≤ 4, up to isomorphism, is as follows: p\\ n 1 2 3 4 5 6 7 8 9 10 11 > 11 total 1 1 1 2 1 1 3 1 1 1 3 4 1 4 21 142 733 1190 776 243 39 4 3141 In the above table, the empty slots display zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Furthermore, the present result gives a complete characterization of toric (or Zn 2-)colorable PL-spheres because they are obtained by consecutive wedge operations from a seed of the table of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Furthermore, the result is useful for the classification of toric manifolds of Picard number 4, or its topological analogues such as quasitoric manifolds or topological toric manifolds whose second Betti number is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It should be noted that the real analogues of such spaces can be computed easily since there is an efficient algorithm to obtain all mod 2 characteristic maps over a wedged simplicial complex, see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Classification of weak pseudo-manifolds by GPU computing In this section, we provide a general approach on how to use GPU parallel computing capa- bility for classifying weak pseudo-manifolds with given properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be a pure simplicial complex of dimension n − 1 on the vertex set [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We will call facets any subset of size (n−1) of [m] and ridges any subset of size (n−2) of [m] not necessarily in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We denote by F(K) and R(K) the sets of facets and ridges of K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We provide an algorithm as follows: Inputs: A set F of facets, and a collection G of affine functions on the subsets of F, called properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Output: The set of weak pseudo-manifolds K such that F(K) ⊂ F and g(F(K)) > 0 for all g ∈ G, namely, K satisfies all the properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Enumerating weak pseudo-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In this subsection we give some computational results which allows us to give an algorithmic description of how to enumerate weak pseudo- manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 5 Provided any set of facets F = {F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , FM}, we can compute the set R = {f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , fN} of every ridges which come from these facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We then construct the facets-ridges adjacency matrix A(F) = (ai,j) of size N × M as follows: ai,j = � 1 fi ⊂ Fj 0 otherwise , for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , N and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A simplicial complex K whose facets are all included in some set of facets F = {F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , FM} can be regarded as a vector K = (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , kM)t ∈ ZM with kj = � 1 Fj ∈ K 0 Fj /∈ K , for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The pure simplicial K is a weak pseudo-manifold if any ridge of K is included in exactly two facets of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This reflects as the following property: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let F be a set of facets, A = A(F) the facets-ridges adjacency matrix of F, and K a pure simplicial complex whose facets are all in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then K is a weak pseudo-manifold if and only if the coordinates of the product AK are all in {0, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' As a consequence, weak pseudo-manifolds seen as vectors in ZM 2 are all included in the Z2- kernel of the matrix A, seen as a linear map A: ZM 2 → ZN 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let B = � K1 · · Ks � be a matrix whose columns form a Z2-basis of kerZ2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Every weak pseudo-manifold K is uniquely expressed as one of the 2s possible Z2-linear combinations of K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , Ks, namely K = �s i=1 xiKi (mod 2) = BX, for X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , xs)t ∈ Zs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If we restrict the set of facets F well enough so it removes many symmetries, we are hoping that s will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Furthermore, we also can find a suitable basis ˜K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , ˜Ks to reduce the number of cases to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We first explain how to construct this basis in the case where the set F contains all possible facets of [m], and R all possible ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' There are �m n � facets and � m n−1 � ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For a ridge f, we will write (AK)f the coordinate of AK corresponding to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let us denote by P(f) := {j ∈ [M]: f ⊂ Fj} the set of the indexes in F of the facets containing f, called the parents of f, which are the only facets contributing to (AK)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In this first case any ridge has m − n + 1 parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For a kernel matrix B whose rows are indexed by F, let us denote by BP(f) the matrix whose rows are the ones of B taken at indexes P(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For every f ∈ R, for every t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , s, the tth column of BP(f) has an even number of ones since the basis element Kt has an even number of facets containing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Performing a mod 2 Gaussian elimination on the columns of BP(f) leads to a matrix of following form BP(f)E = �Zm−n 0� , with Zk = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 0 · · 0 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 0 0 · · 0 1 1 · · 1 1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , a (k + 1) × k-matrix, for k any positive integer, and E ∈ GL(s, Z2) the Z2-invertible matrix corresponding to the operations performed in the Gaussian elimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The columns of the new matrix BE corresponds to another basis of the Z2-kernel of A with a convenient description about which facets containing the ridge f each generators possess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In fact, only the first m−n ones have facets contributing to (AK)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Moreover, one can see that taking the mod 2 linear combination of strictly more than two of them would lead to (AK)f being strictly greater than 2, which is a case we want to avoid computing since we focus on weak pseudo-manifolds, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Thus this decreases the number of mod 2 combinations containing the first m−n new generators that we need to compute from 2m−n to �m−n 0 � + �m−n 1 � + �m−n 2 � = 1 + (m − n) + �m−n 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 6 SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE By writing f 1 := f and E1 := E, one can inductively repeat the latter process by taking care at step k + 1 of: Choosing each time a new ridge f k+1 such that for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , k, P(f i) ∩ P(f k+1) = ∅, Starting the Gaussian pivot at columns index k(m − n) + 1 so that the structure of the generators of previous columns is not lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This process terminates at some step kmax whenever one of the previous conditions cannot be satisfied and we obtain a final matrix, whose columns are the new basis elements ˜K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , ˜Ks, and which up to reordering the rows according to the sets P(f 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , P(f kmax) looks as follows: BE1 · · · Ekmax = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 Zm−n 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 0 0 Zm−n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 0 Zm−n 0 ⋆ ⋆ ⋆ ⋆ ⋆ \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = � ˜K1 · · ˜Ks � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In this case, we decrease the total number of mod 2 combinations from 2s to (1 + (m − n) + �m−n 2 � )kmax2s−kmax(m−n) since we should take a maximum of 2 basis elements for each block Zm−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' As for the general case, there may be ridges which have less than m − n + 1 parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In this case, we try to wisely choose some ridges f 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , f kmax such that the blocks Zk are the biggest possible so we minimize the number of mod 2 combinations BX of the generators we need to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This provides a partition I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , Il of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , s} such that no more than two basis elements with indexes in Ik can be summed, otherwise we are sure not to obtain a weak pseudo- manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We can split the vector X in the mod 2 combinations BX as blocks according to this partition: X = �l k=1 Xk, with Xk representing the part of X whose only nonzero coordinates are in Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let us denote by Xk the set of all such possible Xk, for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If we recap our process, given a set of facets F, we constructed The facets-ridges adjacency matrix A whose Z2-kernel contains all weak pseudo-manifolds, A matrix B whose columns form a convenient basis ˜K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , ˜Ks of kerZ2(A), A partition I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , Il of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , s}, A partition X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , Xl of the vectors of Zs 2 such that for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , l, Xk ∈ Xk has a maximum of two nonzero coordinates which are all in Ik , such that the weak pseudo-manifolds whose facets are in F are exactly the K = BX, with X = �l k=1 Xk for some (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , Xl) ∈ X1 × · · · × Xl, satisfying that each coordinate of AK is in {0, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Moreover, given any affine function K �→ g(K), it is easy to check using computer programming that g(K) > 0 is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We provide in the next subsection some concepts about GPU programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Generalities about GPU programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In this article we used Nvidia CUDA [31] so one should notice that the syntax and vocabulary may differ from other GPU languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The general idea behind GPU computing is that it allows to parallelize tasks with two layers of parallel programming without the need of using a super computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Parallel programming takes several forms, and the two we will use are the following: Data parallelism: one has a big list of elements X and wants to apply the same function g to every element X ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In this case, each call of the function g is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Task parallelism: one has an element X and wants to apply a set of similar functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , gk on X in order to obtain the result as a list (g1(X), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , gk(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The easiest example is a matrix product AX, and if each row of A is denoted by ai, then the functions gi are the inner products with the ais.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In all that follows, a thread (of execution) will be a processing unit computing machine operations linearly, and a GPU will be a two-layered structure of threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Namely a GPU will be a set of p grids, and each grid will be a set of q threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Therefore a GPU can be seen as p × q threads organized for parallel programming, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The number p × q of GPU TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 7 threads which can run simultaneously is roughly the number of CUDA cores (if we consider Nvidia GPUs) and is around seventeen thousands for the current architectures (as of 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Thus a single GPU would be approximately equivalent to at least a thousand CPU threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' · · q threads per grid · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' · · p grids Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The two layered parallel structure of a GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In CUDA programming one uses this two-layered structure as follows: First layer (blocks): Let X = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , XN} be the set of data on which we want to apply the same function g, called the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We create a big list of N blocks, each one indexed by an integer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Each block embodies the function call g(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A block has three possible states: on hold, active, and finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' At the beginning every blocks are on hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then the p grids of the GPU are filled with some blocks which will be running, these are active, the other ones are waiting to be launched on the grid and remain on hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Whenever an active block has finished, it will be replaced by a block on hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The program terminates when all blocks are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Second layer (threads): Whenever a block is sent to a grid, then the operations made in the block are split into threads using task parallelism, any procedures in g is split into q functions which will run simultaneously on all q threads of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Notice that we need every threads to finish their tasks in order to obtain the result, we can explicitly require this condition by synchronizing the threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In all that follows, we will use such notations: A set X will be denoted as a list list X, A matrix A = [ai,j] will be represented as an array whose coefficient at index i, j is A[i][j], A binary vector X ∈ Zk 2 will be represented as a binary variable x on k bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We will use the following processor instructions on binary variables [31]: The and operation x&y, 64 operations per cycle, The exclusive or operation x^y, 64 operations per cycle, The population count operation popcount(x) which counts the number of “1” bits in the value of x, 16-32 operations per cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Atomic operations, which are useful since many threads may want to write at the same memory location at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The processor scheduler creates a queue of all atomic operation calls so they do not overlap in order to avoid memory access error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A cycle is the smallest time interval considered in a processor unit and is performed at the frequency f of the processor unit (of each thread): if the frequency is 1GHz then 109 cycles are done in one second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The thread synchronization allows us managing how the threads behave in parallel as follows: The syncthreads() command asks all the threads to wait that all of them get to the exact same line in the algorithm code of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 8 SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE For a local thread variable test, the syncthreads and(test) and syncthreads or(test) command allows us apply the and or the or operation between all of the test variables existing in each thread of a grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For example if a thread encounters a condition which should stop the current case in a loop, then all the threads should stop at once since this case is not useful computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The GPU algorithm for classifying weak pseudo-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For simplifying the explanation of the algorithm, we suppose that s = 64 and that we can write the product X1 ×· · ·×Xl as Xa ×Xb such that Xa and Xb describe the 32 first or last generators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We thus decompose K as Ka+Kb, with Ka = BXa and Kb = BXb for every (Xa, Xb) ∈ Xa×Xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Both vectors Xa and Xb are binary vectors whose nonzero coordinates are in the 32 first or last coordinates, respectively, they are then stored as 32 bits variables xa and xb, namely as unsigned integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The dot product in Z of two integers x, y is nothing more than the number of active bits of the XOR operation: |x ∧ y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Its mod 2 reduction is just the value of its least significant bit: |x ∧ y|&1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The main idea of the algorithm is to use M threads to compute each coordinate of K ∈ ZM 2 , with M being the number of facets in F, as provided in Algorithm 1: Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' When we say “using the threads”, this means that we evenly distribute the operations to perform among the threads, for example to reinitialize the array r, we use the fact that we have q threads which can set to zero q coordinates at the same time until all coordinates are reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Thus, it requires ⌈N q ⌉ loops, if N is the number of ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The same process is applied for calculating the image by the affine functions g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We use the atomic add operation for incrementing values in r since many threads may write at the same memory location r[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The global complexity of this algorithm is: O �|Xa| p × |Xb| × N q × (α|G| + 1) � = O �|X| × N × (α|G| + 1) pq � , with α the average complexity of the atomic operation when called multiple times for a given g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Preparation for applying the algorithm We will regard a simplicial complex either as a binary vector with entries corresponding to facets or as a set of facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Finiteness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be an n − 1 dimensional simplicial complex on [m] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The join K ∗ L of two simplicial complexes K and L is the simplicial complex {σ ∪ τ | σ ∈ K, τ ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The link LkK(σ) of a face σ in K is the simplicial complex {τ \\ σ | σ ⊂ τ ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For the sake of simplicity, we denote the simplicial complex consisting of a single maximal simplex σ by just σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The (simplicial) wedge wedv(K) of K at a vertex v is (I ∗LkK(v))∪(∂I ∗K \\v), where I is a 1-simplex, and K \\v is the simplicial complex consisting of the facets of K which do not contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We call K a seed if K cannot be represented as a wedge of L for some simplicial complex L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A PL-manifold is a simplicial complex such that the link of each of its vertex is a PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It is well known that a PL-sphere is a PL-manifold ([25, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By the definition of wedge, wedv(K) contains an isomorphic copy of K as the link of each of the new vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This observation implies that if wedv(K) is a PL-sphere, then so is K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In fact, the converse is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be a PL-sphere and v a vertex of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then wedv(K) is a PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Suppose that K is an n − 1 dimensional PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It can be easily seen that the suspension K ∗ ∂{w1, w2} of K is isomorphic to an edge subdivision of wedv(K), so they have TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 9 Algorithm 1 The GPU algorithm for classifying weak pseudo-manifolds whose facets are all in a facets set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Input: The list list F, corresponding to the set of facets F, and the list list G, correspond- ing to the set of affine functions G Output: The list list K of weak pseudo-manifolds K with facets in list F and which satisfy g(K)>0 for every g in list G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Initialization: – Compute the facets-ridges adjacency matrix A = A(F) ∈ ZN×M 2 and store it in A, a column sparse matrix: A[k][i] represents the index of the kth nonzero coordinate of the ith column of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' – Compute B = � ˜K1 · · ˜K64 � = \uf8ee \uf8ef\uf8f0 a1 b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' aM bM \uf8f9 \uf8fa\uf8fb and store it as two lists list a and list b of integers, where list a[k] and list b[k] represents the binary value of the row vectors ak and bk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' – Enumerate Xa and Xb, and store them as two lists list Xa and list Xb – Create a list list Ka of all the Kas: for all xa in list Xa, for all k= 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' M, Ka[k]←popcount(a[k]^xa)&1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Shared memory: r integer array of size N, such that r[k] stores the kth coefficient of the product AK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Kernel(xa,Ka) – Let i be the local thread index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' – b←list b[i] – ka←list Ka[i] – For every xb in list Xb: ∗ skip←False ∗ Ki←popcount(b^xb)^ka and use syncthreads().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' ∗ For every g in list G: compute g(K) using the thread values Ki and if g(K) ≤ 0, then skip←True and exit this loop ∗ If syncthreads or(skip) is verified then go to the next xb ∗ Reinitialize each value of r to 0 using the threads ∗ If Ki is equal to 1 then for k= 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , n: add 1 to r[A[k][i]] using the atomic add operation and if the value of r[A[k][i]] is greater than 3 then skip←True and exit the loop ∗ If syncthreads or(skip) is verified then go to the next xb ∗ Add K to the list of results list K Main: Launch the |Xa| blocks which correspond to all the pairs (xa,Ka) on the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' the same PL-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Moreover, K ∗ w1 is a subdivision of an n-simplex since K is a PL- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Hence K ∗ ∂{w1, w2} = (K ∗ w1) ∪K (K ∗ w2) is a subdivision of the boundary of an n-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' □ We recall that K is toric colorable if K admits a characteristic map over K that is a map satisfying the non-singularity condition over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' One can consider its mod 2 analogue as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A mod 2 characteristic map over K is a map λ: [m] −→ Zn 2 satisfying the non-singularity condition that for each facet σ of K, λ(σ) forms a basis of Zn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Similarly, We say that K is Zn 2-colorable if K admits a mod 2 characteristic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2 ([19], [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be a PL-sphere and v a vertex of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then K is toric colorable if and only if so is wedv(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In addition, K is Zn 2-colorable if and only if wedv(K) is Zn+1 2 colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Notice that the composition of a characteristic map over K and mod 2 reduction Zn → Zn 2 becomes a mod 2 characteristic map over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' As a consequence, we can focus only on Zn 2-colorable seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 10 SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE We often see a mod 2 characteristic map λ as a matrix � λ(1) λ(2) · · λ(m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Up to simplicial isomorphisms, we may assume that the facet {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , n} is contained in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' With this assumption, to check Zn 2-colorability, it is enough to consider characteristic maps of the form λ = � In M � since the non-singularity on its facets is preserved by the left multipli- cation of an element of GL(n, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let us define dual characteristic maps (DCM) over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For λ = �In M� , the DCM associated with λ is a map ¯λ: [m] −→ Zm−n 2 such that ¯λt = �¯λ(1) ¯λ(2) · · ¯λ(m)�t = � M Im−n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Moreover, the term injective DCM will be shortened to IDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3 ([14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be an n−1 dimensional PL-sphere with m vertices and v, w distinct vertices of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then the following are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' (1) If every facet of K contains v or w, then K is a wedge or a suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' (2) If K is a seed which is not a suspension then every DCM over K must be an IDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Statements (2) and (3) both imply: (3) If K is a seed and m − n ≥ 3 then m ≤ 2m−n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We will call a seed which is not a suspension a regular seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We conclude from Statement (3) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3 that there are only finitely many Zn 2-colorable seeds to enumerate for a fixed integer p := m−n, that we match here with the Picard number of K written Pic(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For p ≤ 3, it is known that PL-spheres are all boundaries of polytopes [28] and were already completely enumerated by Perles [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The Zn 2-colorability can easily be checked by an algorithm described in [20], and seedness is verified from the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='4 (seedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A PL-sphere K is a seed if and only if it has no edge {v, w} such that every facet of K has either v or w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The sufficiency of the condition is from the definition of wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Suppose that K has an edge {v, w} such that every facet of K has either v or w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then by Statement (1) and Statement (2) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3, K is a wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' □ We now focus on the case p = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By Statement (3) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3 we have n ≤ 11, implying it is enough to enumerate PL-spheres of dimension up to 10 (n = 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' However, recall that PL-spheres of Picard number ≥ 4 are not necessarily to be polytopal [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We thus need to put our concern on checking PL-sphereness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Collecting PL-spheres among weak pseudo-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We need a criterion for a weak pseudo-manifold to be a PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Fortunately, we have a powerful criterion for a PL- manifold to be a PL-sphere when its Picard number is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='5 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be a PL-manifold with Pic(K) ≤ 7 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If K is a Z2-homology sphere, then K is a PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By using the above theorem, we obtain the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6 (PL-sphereness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A weak pseudo-manifold K of Picard number ≤ 7 is a PL-sphere if and only if the link of any face (possibly the empty face) of K is a Z2-homology sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The sufficiency is obvious, so it is enough to show the necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Suppose that the link of any face of K is a Z2-homology sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='5, let us prove that the link of each face of K is a PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We use an induction on the dimension of the link of a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We remark that the link of each (n − 2)-face of K is the 0-sphere by the definition of weak pseudo-manifold, in particular, it is a PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For k ≤ n − 3, let σ be a k-face of K, and L = LkK(σ) its link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Note that LkL(v) for a vertex v of L is equal LkK({v} ∪ σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Therefore, if the link of any (k + 1)-face of K is a PL-sphere, then L is a PL-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Since L is a Z2-homology sphere by the assumption and Pic L ≤ Pic K ≤ 7, and L is a PL-sphere by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By induction, the link of each face is a PL-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' □ TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 11 We now have the tools for: Checking the PL-sphereness of a weak pseudo-manifold of Picard number 4 with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6 Checking the seedness of a PL-sphere with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Toric colorable PL-spheres of Picard number 4 We devote this section to enumerate all n − 1 dimensional toric colorable seeds of Picard number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Enumeration up to n ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In this subsection, we firstly enumerate all n−1 dimensional Zn 2-colorable seeds on [m] with Picard number 4 up to n ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' One could intuitively try to input Algorithm 1 with all n subsets of [m], and then check the PL-sphereness, the Zn 2-colorability and the seedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' However, it is hopeless when we consider high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Indeed, we manage to obtain results up to n = 6, but the program takes too long time to finish with n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' On the other hand, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3 states that there exists only two kind of Zn 2-colorable seeds: regular or suspended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We will first consider how to enumerate the first ones as they can only support IDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' To do so we rephrase the fact that a pure simplicial complex K supports a mod 2 characteristic map λ as K is included in the binary matroid associated to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We recall here some matroid theory definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A matroid M is a simplicial complex with the so-called augmentation property: for any τ, σ ∈ M with |τ| < |σ|, there exists x ∈ σ \\ τ such that τ ∪ {x} ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Although the facets of a matroid are called the bases, we will keep the simplicial complex terminology here and call them facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The dual matroid M of a matroid M is the matroid on the same vertex set as M and whose facets, called cofacets, are the complement of each facets of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For an n × m matrix λ over Z2 of full row rank n, the simplicial complex Mλ whose facets are the sets of column indexes of n mod 2 independent columns of λ is a matroid, called the binary matroid associated to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By linear Gale duality [19], the dual Mλ is equal to M¯λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The following proposition is easily verified by the definition of Mλ and M¯λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be an n−1 dimensional simplicial complex on [m] and ¯λ a full column rank m × (m − n) matrix over Z2 of rank m − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then K supports ¯λ as a DCM if and only if it is a subcomplex of M¯λt = Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Recall that the more we reduce the number of facets in the input of Algorithm 1, the smaller the dimension of the mod 2 kernel of the ridges-facets adjacency matrix will be and thus the faster the algorithm will run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1 gives us exactly what we want: a finer set of facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In addition, we take advantage of the upper bound theorem ([32]): the number of facets of a simplicial spheres is less than of equal to the one of the cyclic n-polytope C(m, n) with m vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This condition is embodied by the following affine function: g(K) = fn−1(C(m, n))−∥K∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Fix an injective map ¯λ: [m] −→ Z4 2 and set F(λ) = F(Mλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Algorithm 1 with inputs being the set of facets F(λ) and the affine function g outputs the set of all weak pseudo-manifolds which support ¯λ and satisfy the upper bound theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' At first sight, it seems that we need to run the algorithm on each of the �11 n � × (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=') injective maps ¯λ even if we fix �¯λ(n + 1) ¯λ(n + 2) ¯λ(n + 3) ¯λ(n + 4)� = I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' However, we will drasti- cally reduce this large number of cases to compute by noticing that many injective maps provide the same outputs up to simplicial isomorphism, this can be understood from the fact that the inclusion of K into a binary matroid Mλ depends on the choice of the labelling on the set of vertices, if we state that we label the binary matroid and the vertices of a given facet of K, we do not need to consider all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let Λ(n, p) be the set of all (n + p) × p matrices over Z2 of the form � M Ip � such that each matrix has no repeated rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Consider the product of symmetric groups Sn × Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This group gives a group action on Λ(n, p) by �� M Ip � , (s, t) � �→ � P t sMPt Ip � , where Ps and Pt are column permutation matrices corresponding to s and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let us call each element of Λ(n, p)/Sn × Sp an IDCM orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 12 SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For (s, t) ∈ Sn × Sp, there is a simplicial isomorphism between matroids associated to ¯λ ∈ Λ(n, p) and ¯λ ◦ (s, t) ∈ Λ(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Divide the vertex set [m] as V1 = [n] and V2 = {n + 1, n + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , n + p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' First, let us consider s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It is easy to observe that the matrix � P t sM Ip � stores the same non-singularity information on cofacets with relabeling V1 using s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Because Pt is invertible, multiplying Pt at right side of a DCM does not affect non-singularity on cofacets of the DCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' With the equation � P t sMPt Ip � = � P t sM P t t Ip � Pt, applying t to V2 gives the same non-singularity information between � P t sM Ip � and � P t sMPt Ip � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' □ Let Λ◦(n, 4) ⊆ Λ(n, 4) be a set containing one representative of each IDCM orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2, it is enough to input Algorithm 1 with F(λ), for all λ ∈ Λ◦(n, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Find the number of IDCM orbits and the computation time of Algorithm 1 when n < 11 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' n 2 3 4 5 6 7 8 9 10 11 Number of IDCM orbits 7 16 28 35 35 28 16 7 3 1 max¯λ(dim ker A(F(λ))) 7 13 21 24 28 34 42 48 56 64 max¯λ |X(¯λ)| 56 3e3 5e5 1e6 2e7 9e8 1e11 3e12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='4e14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2e16 Time spent for one orbit 1ms 10ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3s 3m 15m 2h 12d 3y Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Data for Picard number 4, and n = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The time spent is the one taken by Algorithm 1 running on an Nvidia Quadro A5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The time written in bold in the case n = 11 is an estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We provide now the global strategy for enumerating all Zn 2-colorable seeds of Picard number 4 by recaping the case of regular seeds and explaining the case of suspended seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' CASE I : Regular seeds For every representative λ ∈ Λ◦(n, 4), run Algorithm 1 with inputs being the set of facets F(λ) and the affine function g(K) = fn−1(C(m, n)) − ∥K∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' After reducing isomorphic ones, we obtain the list of Zn 2-colorable weak pseudo-manifolds on [m] satisfying the upper bound theorem, up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We then apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='4 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6 to collect the seeds up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' CASE II : Suspended seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3, a seed without IDCM is a suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' It can be easily seen that from the definition of wedge, the suspension of a wedge is again a wedge, and the suspension operation increases the Picard number by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Therefore, it is enough to consider suspensions of seeds of Picard number 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let L = ∂[v, w]∗K for an n−2 dimensional simplicial complex K, and λ a characteristic map over L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Without loss of generality, we may assume v = 1 so that λ(v) = �1 0 · · 0�t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then for any facet {v}∪{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , vn−1} of L, the (1, 1) minor of the matrix � λ(v) λ(v1) · · λ(vn−1) � is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This implies that LkL(1) = K is Zn−1 2 colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We know there are three Zn 2-colorable seed PL-spheres of the Picard number 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 5-gon, 3-cube, and the cyclic polytope C(7, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The suspension of the 3-cube does not support any IDCM but does support a DCM, while the suspensions of the others support IDCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The algorithm provides us the following table: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let n ≥ 1 be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The number of Zn 2-colorable seed PL-spheres of dimension n − 1 with Picard number p ≤ 4, up to isomorphism, is as follows: TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 13 p\\n 1 2 3 4 5 6 7 8 9 10 11 > 11 total 1 1 1 2 1 1 3 1 1 1 3 4 1 4 21 142 733 1190 776 243 39 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 3141 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Enumeration for n = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' As we can see in Table 1, the time complexity of the extremal case n = 11 is still too long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' To remedy this, let us use the results we obtained from dimension just below to construct the seeds of this extremal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be a Z1 21-colorable seed on {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , 15} of dimension 10 (n = 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We know that the link of the vertex 15 has Picard number at most 4 and is a Z1 20-colorable seed, which we already have enumerated before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The idea is then to build all Z1 21-colorable seeds from the Z1 20-colorable seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If K has only vertices whose links have Picard numbers at most 2, K would be the boundary of a product of simplices [23], that is, not a seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Suppose that the link of 15 has Picard number 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Recall that there is no 9-dimensional seed PL-sphere with Picard number 3 implying that the link of 15 is a wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By the following lemma, we find another vertex of K whose link has Picard number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be a seed PL-sphere with Pic(K) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Assume that K has a vertex v such that Pic(LkK(v)) = 3 and there exist two vertices v1 and v2 of LkK(v) such that every facets of LkK(v) contain either v1 or v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then there is a vertex of K whose link has Picard number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let {v1} ∪ σ be a facet without v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Since σ is a ridge of the PL-sphere LkK(v), there is one more facet containing σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' By the assumption, it must be {v2} ∪ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This verifies that every vertex in LkK(v) forms an edge with both v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let w be the vertex not in LkK(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If v1 ∈ LkK(w), then LkK(v1) has Picard number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If both v1, v2 ̸∈ LkK(w), then LkK(w) is an (n − 2)-dimensional PL-sphere with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This means that LkK(w) = ∂∆n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then if w′ is a vertex of LkK(w), then Pic(LkK(w′)) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' □ This implies that any Z1 21-colorable seed has a vertex of link being of Picard number 4, that we relabel as 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Before we apply Algorithm 1 for this case, we need some preparation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We firstly select an injective map ¯µ : [14] → Z4 2 and choose a 9-dimensional PL-sphere L which supports ¯λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We see L as the link of the vertex 15 in some Z11 2 -colorable seed PL-sphere K supporting some IDCM ¯λ with the restriction ¯λ|[14] = ¯µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Since ��Z4 2 �� \\ {0} = 11 , once ¯µ is chosen, ¯λ is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Our computation shows there exist 13 wedge cases(checked by computing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let ˆL be the simplicial complex {σ ∪ {15} | σ ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' All PL-spheres K having its vertex 15 whose link is L contains ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This provides the following conditions on the components of K ∈ ZM: (1) for all ˆLj = 1, Kj = 1 (2) for all ˆLj = 0 with ˆLj ∋ {15}, Kj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The set of the indexes of the facets satisfying Condition (1), respectively Condition (2), will be denoted by I, respectively by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' After reordering the rows of B, the two conditions are illustrated as follows BX = \uf8ee \uf8f0 BI BJ B[M]\\(I∪J) \uf8f9 \uf8fb X = \uf8ee \uf8f0 1 0 ⋆ \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1) A Gaussian elimination process on the columns of B will lead to a column reduced echelon form ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Denote by sI, respectively sJ, the maximal index of non-zero column of ˜BI, respectively of ˜BJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In order to respect the conditions (1) and (2), we need xt = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , sI 0 t = sI + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , sJ ⋆ otherwise , where X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , xM)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If this setting on X contradicts to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1) with ˜B instead of B, then it 14 SUYOUNG CHOI, HYEONTAE JANG, AND MATHIEU VALL´EE would mean that there is no such Z11 2 -colorable seed PL-sphere K supporting ¯λ whose link of the vertex 15 is L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The same strategy as in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1 applied to the rest of the entries of X, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='4 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6 gives the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' There are exactly four Z11 2 -colorable 10-dimensional seed PL-spheres of the Pi- card number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Toric colorability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We note that all toric colorable seeds are Zn 2-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In order to obtain the list of toric colorable seeds of Picard number 4, it is enough to check whether each seed in the list of Zn 2-colorable seeds supports a characteristic map or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' First, one may regard each vector in Zn 2 as a (0, 1)-vector in Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' This may produce singular facet in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Then we change some 1’s in λ to −1’s until getting toric colorable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Bruteforcing this method provided at least one characteristic map supported by every Zn 2-colorable seed we enumerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The toric colorability thus is equivalent to the Zn 2-colorability for PL-spheres of Picard number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The number of toric (or Zn 2-)colorable seeds of dimension n − 1 with Picard number p ≤ 4, up to isomorphism, is as follows: p\\ n 1 2 3 4 5 6 7 8 9 10 11 > 11 total 1 1 1 2 1 1 3 1 1 1 3 4 1 4 21 142 733 1190 776 243 39 4 3141 In the above table, the empty slots display zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Acknowledgements The authors are very grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Axel Bacher who introduced the last named author to CUDA programming allowing Algorithm 1 to be adapted to GPU computing and thus computed in reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A few algorithmic methods on simplicial complexes In this appendix, we will provide the readers a few algorithmic methods which were used in order to obtain our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Checking isomorphism using minimal non-faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' One very hard problem when enu- merating simplicial complexes is to deal with isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If a simplicial complex K has m vertices, then there exists m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' possible relabellings for K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Provided two simplicial complexes K1 and K2 one the respective vertex sets V and W one wants to find if they are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' One solution is to use McKay graph isomorphism on the face posets of K1 and K2 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We provide here a different solution for testing if they are isomorphic by mainly using their sets of minimal non-faces (MNF), but also some combinatorial information such as their f-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let K be a simplicial complex on a vertex set V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Our main motivation to use the MNF sets is that the simplicial complexes K we are considering are seeds and thus satisfy some property in their MNF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We say that a pair of distinct vertices (v, w) of K form a couple if they share the exact same “neighbourhood” in the MNF set of K, more explicitly, if for every minimal non faces σ of K we have either {v, w} ⊆ σ or v, w /∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A simplicial complex K is a seed if and only if there is no couple of vertices K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Expecting that the vertices of a seed will share distinct neighbourhoods we define a color sequence cK(v), for each vertex v of K, which is invariant up to relabeling, such that the color sequence cK(v) represents the increasing sizes of the minimal non faces of K containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' For example if MNF(K) = {123, 34, 456, 26, 16} then cK(1) = (2, 3), cK(5) = (3) and c(6) = (2, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A relabeling φ: V → W between K1 and K2 may satisfy cK2(φ(v)) = cK1(v) for every v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The procedure is the following: (1) Check if K1 and K2 have the same dimension and the same number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' TORIC COLORABLE PL-SPHERES OF PICARD NUMBER 4 15 (2) Check if K1 and K2 have the same f-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' (3) Check if {cK1(v): v ∈ [m]} = {cK2(v): v ∈ [m]}, by counting repetitions, using their MNF sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' (4) We give partitions V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , Vk and W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , Wk of V and W with respect to color sequences in K1 or K2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We compute every relabeling φi : Vi → Wi for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' They provide every relabeling φ = φ1 × · · · × φk : V → W preserving the color sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' If one φ sends one-to-one the minimal non faces of K1 to the ones of K2 then K1 is isomorphic to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The number of relabeling that are computed is (|V1|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=') × · · · × (|Vk|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=') instead of |V |!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' which is a nice improvement when there are many different color sequences and only few vertices having the same color sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' An inductive algorithm for checking PL-sphereness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Denote by CRSP(p, n) the set of Zn 2-colorable seed PL-spheres of Picard number p and of dimension n − 1 up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Let us suppose that we have obtained all CRSP(p, k) for p ≤ 4 and k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' The output of the CUDA algorithms is the collection of all weak pseudo-manifolds compatible with all possible IDCM ¯λ from which we select only the seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' In order to check if they are PL-spheres, we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We thus apply the following procedure for testing the PL-sphereness of K: (1) Check if the Z2-Betti numbers of K are the ones of a sphere, namely (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' , 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' (2) For every vertex v of K, let Kv = LkK(v), and let Lv be the seed of Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Since the PL-sphereness property is invariant under wedging operation, we simply need to check for every v that Lv is in CRSP(p, k), for some p ≤ 4 and k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' We expect Lv to be of low dimension so the isomorphism checking algorithm finishes faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Altshuler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Bokowski, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Steinberg, The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' 31 (1980), 115–124 (English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Altshuler and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content=' Steinberg, An enumeration of combinatorial 3-manifolds with nine vertices, Discrete Math.' metadata={'source': 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Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon 16499, Republic of Korea Email address: a24325@ajou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='kr Universit´e Sorbonne Paris Nord, LIPN, CNRS UMR 7030, F-93430, Villetaneuse, France Email address: vallee@lipn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAyT4oBgHgl3EQf5fqL/content/2301.00806v1.pdf'} diff --git a/IdAzT4oBgHgl3EQfx_4Y/content/tmp_files/2301.01745v1.pdf.txt b/IdAzT4oBgHgl3EQfx_4Y/content/tmp_files/2301.01745v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..45ece255d276fa917cfe4d704da5471d0204648b --- /dev/null +++ b/IdAzT4oBgHgl3EQfx_4Y/content/tmp_files/2301.01745v1.pdf.txt @@ -0,0 +1,2563 @@ +Two-Point Functions of Composite Twist Fields in the Ising Field Theory +Olalla A. Castro-Alvaredo and Michele Mazzoni +Department of Mathematics, City, University of London, 10 Northampton Square EC1V 0HB, UK +All standard measures of bipartite entanglement in one-dimensional quantum field theories +can be expressed in terms of correlators of branch point twist fields, here denoted by T and +T :. These are symmetry fields associated to cyclic permutation symmetry in a replica theory +and having the smallest conformal dimension at the critical point. Recently, other twist fields +(composite twist fields), typically of higher dimension, have been shown to play a role in the +study of a new measure of entanglement known as the symmetry resolved entanglement entropy. +In this paper we give an exact expression for the two-point function of a composite twist field +that arises in the Ising field theory. In doing so we extend the techniques originally developed +for the standard branch point twist field in free theories as well as an existing computation +due to Horv´ath and Calabrese of the same two-point function which focused on the leading +large-distance contribution. We study the ground state two-point function of the composite +twist field Tµ and its conjugate T : +µ . At criticality, this field can be defined as the leading field +in the operator product expansion of T and the disorder field µ. We find a general formula +for logxTµp0qT : +µ prqy and for (the derivative of) its analytic continuation to positive real replica +numbers greater than 1. We check our formula for consistency by showing that at short distances +it exactly reproduces the expected conformal dimension +Keywords: Integrable Quantum Field theory, Ising model, Branch Point Twist Fields, Sym- +metry Resolved Entanglement Entropy, Form Factor Expansion, Correlation Functions. +o.castro-alvaredo@city.ac.uk +michele.mazzoni.2@city.ac.uk +January 5, 2023 +arXiv:2301.01745v1 [hep-th] 4 Jan 2023 + +1 +Introduction +It is well-known that the Ising field theory has an internal Z2 symmetry, associated to which we +can define two fields: σ, the spin field (order operator), and µ (disorder operator)1. The theory +also contains a free Majorana fermion field Ψ so that, in the disordered phase of the theory, the +three fields can be characterised by their mutual equal-time exchange relations [1–3]: +Ψpxqσpyq “ +# +σpyqΨpxq +y ą x +σpyqΨpxq +y ă x +and +Ψpxqµpyq “ +# +´µpyqΨpxq +y ą x +µpyqΨpxq +y ă x +(1) +Furthermore, in the context of the investigation of entanglement measures it is often convenient +to consider a “replica” version of the theory, namely a model consisting of n non-interacting, +identical copies of the original. In this model, the fields above acquire an index tµj, σj, Ψju with +j “ 1, . . . , n, running over the copy numbers. The resulting model possesses a large amount of +symmetry, namely, not only Z2 symmetry on each copy, but symmetry under the exchange of +any copies. Cyclic permutation symmetry is one of these many symmetries and, as it turns out, +it is the symmetry that plays the most fundamental role in computations of the entanglement +entropy and other measures of entanglement [4–6]. +In [6] the branch point twist fields T and its conjugate T : (called ˜T in the original paper) +were defined as the symmetry fields associated with cyclic permutation symmetry of copies in +a replica theory. These fields too are characterised by their exchange relations with respect to +the fermions: +ΨjpxqT pyq “ +# +T pyqΨj`1pxq +y ą x +T pyqΨjpxq +y ă x +and +ΨjpxqT :pyq “ +# +T :pyqΨj´1pxq +y ą x +T :pyqΨjpxq +y ă x +(2) +for j “ 1, . . . , n and j ” j ` n. These relations can be written for any 1+1D quantum field +theory, integrable or not, however, in the context of massive integrable quantum field theory +(IQFT), they provide, together with the two-body scattering matrix, all the information needed +to compute correlation functions and matrix elements of T [6]. These computations have now +been carried out for many theories and entanglement measures (see e.g. [7–10]) revealing many +new insights into the universal properties of entanglement at near-critical points. +In recent years, it has been shown that also the fields resulting from the conformal OPE of T +with other fields of the Ising field theory can be of interest in the context of entanglement [11–16]. +In particular, the correlation functions of the leading field in the OPE of T and ř +j µj, denoted +by Tµ, are related to an entanglement measure known as the symmetry resolved entanglement +entropy [15–17]. Tµ satisfies exchange relations which combine those for T and µ as seen above: +ΨjpxqTµpyq “ +# +´TµpyqΨj`1pxq +y ą x +TµpyqΨjpxq +y ă x +and +ΨjpxqT : +µ pyq “ +# +´T : +µ pyqΨj´1pxq +y ą x +T : +µ pyqΨjpxq +y ă x +(3) +1In this paper we use the conventions of [18], which corresponds to choosing the disordered phase of the model, +where the fields σpµq are odd (even) with respect to the Majorana fermion Ψ. +1 + +In general, such measures can always be defined for theories that possess an internal symmetry +(such as Z2 in the Ising case), and it gives access to information about the amount of entangle- +ment that is stored in each symmetry sector. +The computation of the symmetry resolved entanglement provides strong motivation to study +correlators of Tµ and this will be the focus of this paper. Our aim is finding an exact analytic +expression for the two-point function xTµp0qT : +µ prqy using IQFT techniques. The applications of +such a result in the context of entanglement will not be discussed here, but they follow quite +straightforwardly from existing literature. In particular, the form factors of Tµ and the leading +contribution to its two-point function were computed in [17]. The present work is an extension of +those results to include higher particle contributions and to show how non-trivial resummation +identities allow for relatively simple closed formulae for all correlation function cumulants. +Correlation functions of composite twist fields have been studied in a number of works +both for the Ising field theory and other, interacting models. Most of these results build upon +the form factor program for the matrix elements of T [6] and its extension to composite twist +fields [17]. In [20–22] free theories were studied, whereas interacting IQFTs such as the Ising and +sinh-Gordon models (both with discrete Z2 symmetry), the sine-Gordon model (with continuous +Up1q symmetry) and the 3-state Potts model (with discrete Z3 symmetry) were studied in [17,23] +and [24], respectively. +It is also possible to study composite twist fields where T is composed with a local field +not associated with an internal symmetry. +Such composite fields are associated with cyclic +permutation symmetry too and have a conformal dimension which is distinct from that of T . +In particular, for theories whose UV fixed point is described by a non-unitary conformal field +theory (CFT), it is possible to construct composite twist fields whose dimension is lower than +that of T and they play a critical role in describing the usual measures of entanglement [13]. +This happens for instance for the Lee-Yang theory both at and away from criticality. The form +factors and two-point functions of the branch point twist field and composite twist field for this +theory were studied in [14]. The expectation values of composite twist fields involving the energy +field in the Ising field theory were studied in [11,12]. +This paper is organised as follows: In Section 2 we review form factor results for the order +and disorder fields in the Ising field theory as well as for T and Tµ. We review the cumulant +expansion of two-point functions and introduce an example of the type of convergence issues that +arise in the cumulant expansion of xTµp0qT : +µ prqy{xTµy2. In Section 3 we find closed formulae for +all higher cumulants, leading to a close expression for the two-point function. In Section 4 we test +this expression by obtaining the exact conformal dimension of Tµ from resummation of leading +terms in the short-distance expansion of the cumulants. We show that the normalised two- +point function xTµp0qT : +µ prqy{xTµy2 is in fact proportional to the normalised two-point function +xµp0qµprqy{xµy2, thus it factorises into n-dependent and n-independent parts. In Section 5 we +show how to analytically continue the cumulant expansion from n integer and greater than 1 +to n real. This allows us to write a formula for the n-derivative of the two-point function at +n “ 1, a quantity that typically plays a role in entanglement measures. We conclude in Section +6. Appendix A provides proofs of new useful resummation formulae for the form factors of Tµ. +2 + +2 +Field Content of the Ising Model and Form Factors +The correlation functions and form factors of the fields σ, µ defined by (1) can be obtained +via form factor bootstrap [25, 26] and where studied in detail by Yurov and Zamolodchikov in +their seminal paper [18]. Form factors of descendent fields (in the CFT sense) of the energy +field ε were studied in [19] and shown to match in number and spin the field content of the +corresponding Verma module in the underlying Ising CFT. +Starting from the relations (1) the form factor equations can be written and solved for matrix +elements of σ, µ and these were found to take an extremely simple form [18], namely (the factor +ik is needed to satisfy the kinematic residue equation): +F µ +2kpθ1, . . . , θ2kq “ ikxµy +ź +1ďiăjď2k +tanh θij +2 , +(4) +F σ +2k`1pθ1, . . . , θ2k`1q “ ikF σ +1 +ź +1ďiăjď2k`1 +tanh θij +2 , +(5) +with θij :“ θi ´ θj and xµy and F σ +1 normalisation constants which can be identified with the +vacuum expectation value of µ and the one-particle form factor of σ, respectively. More generally, +a k-particle form factor is defined as +F O +k pθ1, . . . , θkq :“ x0|Op0q|θ1 . . . θk|0y +(6) +that is, a matrix element of a local or quasi-local field between the ground state |0y and a k- +particle state characterised by rapidities tθiuk. In general, particles will also be characterised +by their quantum numbers, but in the Ising model there is a single particle type so these do not +need to be specified. +For the field µ the products above can be rewritten as a Pfaffian of a 2k ˆ 2k antisymmetric +matrix A with entries Aij “ tanh θij +2 . In particular this means that in the disordered phase the +vacuum expectation value of µ is non-vanishing whereas it is vanishing for σ. +In the context of the study of entanglement we know also that branch point twist fields T +and T : play a prominent role. Their form factors in the (replica) Ising model have been known +for some time [6,8] and due to the free nature of the model they can also be expressed in terms +of a Pfaffian +F T |11...1 +2k +pθ1, . . . , θ2k; nq “ xT yPfpKq , +PfpKq “ +? +det K , +(7) +where n labels the number of replicas, +Kij :“ kpθijq “ +sin π +n +2n sinh +´ +iπ´θij +2n +¯ +sinh +´ +iπ`θij +2n +¯ sinh θij +2n +sinh iπ +2n +with +i, j “ 1, . . . , 2k , +(8) +and the superindices 11 . . . 1 indicate that all particles are in the same copy 1. +From this +representation we also see that all form factors are functions of rapidity differences only, a +property that holds for all spinless fields in relativistic quantum field theory. The two-particle +3 + +form factor is simply F T |11 +2 +pθ1, θ2; nq “ xT ykpθ12q. Form factors for particles in copies j1 . . . j2k +can be obtained from the above using the standard form factor equations presented in [6] +F T |j1...j2k +2k +pθ1, . . . , θ2k; nq “ F T |1...1 +2k +pθj1´1 +1 +, . . . , θjk´1 +2k +; nq , +for +j1 ě j2 ¨ ¨ ¨ ě j2k , +(9) +with +θj :“ θ ` 2πij . +(10) +The form factors of the composite twist field Tµ where first obtained in [17] and have again the +Pfaffian structure typical of the Ising model, that is +F Tµ|11...1 +2k +pθ1, . . . , θ2k; nq “ xTµyPfpWq +(11) +with +Wij :“ wpθijq “ +sin π +n +2n sinh +´ +iπ´θij +2n +¯ +sinh +´ +iπ`θij +2n +¯ sinh θij +n +sinh iπ +n +. +(12) +As we can see, this differs from kpθq above only because n is replaced by n{2 in the minimal part +of the form factor (i.e. the part that does not contain kinematic poles). However, this small +change leads to some important differences, the main one being the asymptotic properties +lim +θÑ8 kpθq “ 0 +and +lim +θÑ˘8 wpθq “ ˘ i +n , +(13) +as well as +lim +nÑ1 kpθq “ 0 +and +lim +nÑ1 wpθq “ i tanh θ +2 . +(14) +Note that the last equality simply shows that the two-particle form factor of Tµ reduces to that +of µ for n “ 1, as expected. This extends to higher-particle form factors too. It is known from +the study of many models and arguments such as those presented in [27] that the asymptotics +of two particle form factors should be related to the value of a one-particle form factor. This +is a consequence of so-called cluster decomposition in momentum space. In simple theories, as +assumed in [27], this one-particle form factor would be that of the same field. However, in the +Ising model, due to Z2 symmetry there is a mixing between form factors of µ and σ and also +those of Tσ (defined as the composition of T and ř +j σj) and Tµ in such a way that: +lim +θÑ˘8 wpθq “ ˘τ 2 . +(15) +where τ :“ F Tσ|1 +1 +is the one-particle form factor of Tσ, which by relativistic invariance is θ +independent. Combining (15) with (13) we have that +|F Tσ|1 +1 +|2 “ |τ|2 “ 1 +n . +(16) +Higher form factors of Tσ can also be related to Pfaffians by employing a more general version +of the cluster decomposition property. Namely +lim +θ2k`2Ñ8xTµy´1F Tµ +2k`2pθ1, . . . , θ2k`2; nq “ τF Tσ +2k`1pθ1, . . . , θ2k`1; nq. +(17) +4 + +Note that the prefactor xTµy´1 ensures that when k “ 0 both sides of the equation become τ 2. +In this way, the form factors F Tσ +2k`1pθ1, . . . , θ2k`1; nq can be computed systematically and it is +easy to show that they can be written as a sum of Pfaffians involving 2k variables. In fact, we +can show that +F Tσ +2k`1pθ1, . . . , θ2k`1; nq “ +τ +xTµy +2k`1 +ÿ +j“1 +p´1qj`1F Tµ +2k pθ1, . . . , ¯θj, . . . θ2k`1; nq , +(18) +where the sign depends on the position of the variable θj and can be worked out by counting +Wick contractions. Similarly, the symbol ¯θj means that this variable is removed, hence this is +a sum over 2k-particle form factors depending on a subset of the variables tθ1, . . . , θ2k`1u. For +instance +F Tσ +3 pθ1, θ2, θ3; nq +“ +τpwpθ12q ´ wpθ13q ` wpθ23qq +“ +τ +xTµypF Tµ +2 pθ1, θ2; nq ´ F Tµ +2 pθ1, θ3; nq ` F Tµ +2 pθ2, θ3; nqq , +(19) +so that each “contraction” θij where |i ´ j| is even produces one minus sign. The formula (18) +is, as far as we know, new and first presented here. However, this structure is the same as for +the form factors of the field σ which are obtained in the limit n “ 1, for which the function +wpθq reduces to a tanh (see Eq. (14)). The special properties of the tanh function mean that +formulae such as (18) and (19) can be shown to factorise also as products of tanh functions. +In [17] it was also shown that the form factor (12) gives the correct conformal dimension of +Tµ via the ∆-sum rule [27]. This dimension is [5,11,15,28,29] +∆Tµ “ ∆T ` ∆µ +n “ n +48 ` +1 +24n +with +∆T “ 1 +48 +ˆ +n ´ 1 +n +˙ +, +∆µ “ 1 +16 . +(20) +In fact ∆Tµ “ ∆Tσ since ∆µ “ ∆σ even if, for symmetry reasons, ∆σ cannot be obtained from +the ∆-sum rule. +2.1 +Two-Point Function and Cluster Expansion +In this paper we are interested in properties of the ground state two-point function of the field +Tµ. In general we would like to write down an expansion of the form +log +˜ +xTµp0qT : +µ prqy +xTµy2 +¸ +“ +8 +ÿ +ℓ“1 +cTµ +ℓ pr; nq +mr!1 +« +´4∆Tµ logpmrq ´ KTµ , +(21) +where the sum is over functions cTµ +ℓ pr; nq known as cumulants, ∆Tµ is the conformal dimension +of the field Tµ and KTµ is a constant that depends on the vacuum expectation value xTµy. +These cumulants are multiple integrals of linear combinations of products of form factors. More +precisely, we have the following structure +cTµ +ℓ pr; nq “ +1 +ℓ!p2πqℓ +nÿ +j1,...,jℓ“1 +ˆ 8 +´8 +dθ1 ¨ ¨ ¨ +ˆ 8 +´8 +dθℓ hTµ|j1...jℓ +ℓ +pθ1, ¨ ¨ ¨ , θℓ, nqe´mr řℓ +i“1 cosh θi, +(22) +5 + +where the functions hO|j1...jk +k +pθ1, ¨ ¨ ¨ , θk, nq are given in terms of the form factors of the field +involved, and ji represent the particle’s quantum numbers which in our examples will be also +the copy numbers. For example: +hTµ|j1j2 +2 +pθ1, θ2, nq +“ +xTµy´2 ˇˇˇF Tµ|j1j2 +2 +pθ1, θ2, nq +ˇˇˇ +2 +, +hTµ|j1j2j3j4 +4 +pθ1, θ2, θ3, θ4, nq +“ +xTµy´2 ˇˇˇF Tµ|j1j2j3j4 +4 +pθ1, θ2, θ3, θ4, nq +ˇˇˇ +2 +´hTµ|j1j2 +2 +pθ1, θ2, nqhTµ|j3j4 +2 +pθ3, θ4, nq +´hTµ|j1j3 +2 +pθ1, θ3, nqhTµ|j2j4 +2 +pθ2, θ4, nq +´hTµ|j1j4 +2 +pθ1, θ4, nqhTµ|j2j3 +2 +pθ2, θ3, nq, +(23) +and so on, whereas all odd particle terms are vanishing. Similar formulae can be written for the +cumulants of Tσ where only odd particle cumulants are non-vanishing. A generic combinato- +rial/diagramatic construction of these functions can be found for instance in [30]. For a generic +local field O, it is standard to require +hO|j1...jℓ +ℓ +pθ1, ¨ ¨ ¨ , θℓq „ e´θi +for +θi P R +and +θi Ñ 8 . +(24) +Given the properties of the form factors presented in the previous section, we see that this +property is not satisfied for the cumulants of the two-point function of Tµ, or indeed for the +cumulants of the two-point function of µ as shown in [18]. In fact, the cumulant expansion is +still convergent in both cases, but the leading behaviour for small mr is harder to extract than +in theories where (24) holds. +2.2 +Two-Particle Contribution +The aim of this work is to find a general, compact form, for all terms in the expansion of +xTµp0qT : +µ prqy{xTµy2. One way to check the two-point function expansion is to recover the con- +formal dimension of the field by exact resummation of all terms which are proportional to +logpmrq for mr ! 1, that is the first term in (21). +Let us start by considering the simplest contribution to the connected part of the two-point +function xTµp0qT : +µ prqy{xTµy2, which has already been studied in the literature [17]. The first +non-vanishing contribution to the cumulant expansion comes from hTµ|j1j2 +2 +pθ1, θ2, nq, which is +nothing but the normalised squared modulus of the two-particle form factor. Using (9) the +latter can be rewritten as +nÿ +i,j“1 +ˇˇˇF Tµ|ij +2 +pθ1, θ2q +ˇˇˇ +2 +“ n +n´1 +ÿ +j“0 +ˇˇˇF Tµ|11 +2 +pθ1 ` 2πij, θ2q +ˇˇˇ +2 +“ nxTµy2 +n´1 +ÿ +j“0 +wpp´θ12qjqwpθj +12q , +(25) +where the superindex j is defined as in (10). Thus we have +cTµ +2 pr; nq +“ +n +n´1 +ÿ +j“0 +ˆ 8 +´8 +ˆ 8 +´8 +dθ1dθ2 +2p2πq2 wpp´θ12qjqwpθj +12q e´mr cosh θ1´mr cosh θ2 +“ +n +p2πq2 +n´1 +ÿ +j“0 +ˆ 8 +´8 +dθ wpp´θqjqwpθjq K0p2mr cosh θ +2q . +(26) +6 + +The sum above has been computed in [17] by using the cotangent trick and is given by +n´1 +ÿ +j“0 +wpp´θqjqwpθjq “ ´i tanh θ +2pwp2θ ` iπq ` wp2θ ´ iπqq ´ 1 +n . +(27) +This function tends asymptotically to the value +1 +n for |θ| Ñ 8. This means that the usual +procedure consisting of expanding the Bessel function for mr ! 1 and isolating the logpmrq +leading term, thus effectively removing the Bessel function from the integrand (26), now leads +to a divergent integral. The integral is however not divergent, it simply needs to be done with +care. We can rewrite (26) as +cTµ +2 pr; nq +“ +n +p2πq2 +ˆ 8 +´8 +dθ +» +– +n´1 +ÿ +j“0 +wpp´θqjqwpθjq ´ 1 +n +fi +fl K0p2mr cosh θ +2q +` +1 +p2πq2 +ˆ 8 +´8 +dθ K0p2mr cosh θ +2q . +(28) +In this form, the integral in the first line can be approximated for mr ! 1 by expanding the +Bessel function, giving a leading contribution which is proportional to logpmrq. The integral in +the second line can be computed exactly to +ˆ 8 +´8 +dθ K0p2mr cosh θ +2q “ 2K0pmrq2 mr!1 +« +´2plogpmrqq2 , +(29) +so that, in this case, the leading small mr contribution diverges as plogpmrqq2 rather than +logpmrq. Thus, although the cumulant (26) is still well-defined, its leading small mr behaviour +is now dominated by plogpmrqq2 instead of logpmrq. +This is a consequence of the property +(24) not holding in this case. Nonetheless, terms of order plogpmrqq2 should cancel out when +including further contributions in the form factor series as one expects to recover the 1{r4∆Tµ +behaviour of the two-point function at short distances. In the next sections we will show one +particular way to recover the expected scaling (21) from our cumulant expansion. +3 +Higher Particle Contributions: Closed Formulae +Existing studies of the branch point twist field two-point function for free fermions [8] and +bosons [9] have revealed that the form of higher cumulants can be considerably simplified. This +is because under sum over particle types and integration over the rapidities, many of the terms +in the cumulant either cancel each other out or can be shown to be identical. In fact, it is +possible to show that just as for the standard branch point twist field, and for the same reasons +already discussed in [8,9] the cumulants of the two-point function of Tµ take the generic form +cTµ +2ℓ pr; nq “ +n +2ℓp2πq2ℓ +n´1 +ÿ +j1,...,j2ℓ´1“0 +» +– +2ℓ +ź +i“1 +ˆ `8 +´8 +dθi e´mr cosh θi +fi +fl +ˆ p´1qℓ +¨ +˝wpθ´j1 +12 q +ℓ´1 +ź +k“1 +wpθj2k´j2k`1 +2k`1 2k`2qq +˛ +‚ +¨ +˝wpθj2ℓ´1 +1 2ℓ q +ℓ´1 +ź +k“1 +wpθ´j2k´1`j2k +2k 2k`1 +q +˛ +‚. +(30) +7 + +By using the fact that wpθ´jq “ ´wpp´θqjq, we can change the sign of half of the factors in the +second line, cancelling out the factor p´1qℓ. The integrand becomes: +n´1 +ÿ +j1,...,j2ℓ´1“0 +wpp´θ12qj +1qwpθj2ℓ´1 +1 2ℓ q +ℓ´1 +ź +k“1 +wpθj2k´j2k`1 +2k`1 2k`2qwpp´θ2k 2k`1qj2k´1´j2kq . +(31) +In order to evaluate the integrals (30) it is convenient to perform a change of variables whereby +we first change the sign of all the even rapidities, without any change in the integration measure. +Then, defining ˆθij ” θi ` θj the integrand becomes a function of rapidity sums only: +n´1 +ÿ +j1,...,j2ℓ´1“0 +wpp´ˆθ12qj1qwpˆθ j1´j2 +23 +qwpˆθ j2´j3 +34 +q . . . wpˆθ j2ℓ´2´j2ℓ´1 +2ℓ´1 2ℓ +qwpˆθ j2ℓ´1 +1 2ℓ +q . +(32) +We will refer to this as a fully connected sum, meaning that all terms are cyclicly “connected” +both at the level of the rapidities and the summation indices. The sum (32) and others of a +similar type can be computed recursively as shown below and in Appendix A. +3.1 +Recursive Formulae +The sum (32) can be carried out leading to generalisations of the following result +f1px, y; nq :“ +n´1 +ÿ +j“0 +wpp´xqjqwpyjq “ ´ i +2 +sinh +´ +x`y +2 +¯ +cosh x +2 cosh y +2 +rwpx ` y ` iπq ` wpx ` y ´ iπqs ´ 1 +n , (33) +which is presented here for the first time, although the case x “ y was obtained in [17] and has +already been reported in (27). It is also useful to know that +n´1 +ÿ +j“0 +wpxjq “ i tanh x +2 . +(34) +A derivation of formulae (33), (34) and their generalisations to multiple sums (see below) is +presented in Appendix A. +For the branch point twist field of free fermions and bosons [8,9] a formula almost identical +to (33) also holds, albeit without the term ´ 1 +n. This term in fact makes the generalisation of this +sum to multiple sums more complex for Tµ than for T . It can nonetheless be done as follows. +Let us consider, as an example, the next sum in the series, namely a sum of the form +n´1 +ÿ +j1,j2“0 +wpp´xqj1qwpyj1´j2qwpzj2q “ +n´1 +ÿ +j“0 +f1px, y´j, nqwpzjq . +(35) +Repeated use of (33) and (34) to simplify (35) leads to +n´1 +ÿ +j1,j2“0 +wpp´xqj1qwpyj1´j2qwpzj2q “ ´ i +n +ˆ +tanh x +2 ` tanh y +2 ` tanh z +2 +˙ +`1 +4 +cosh +´ +x`y`z +2 +¯ +cosh x +2 cosh y +2 cosh z +2 +“ +2wpx ` y ` zq ` wpx ` y ` z ` 2iπq ` wpx ` y ` z ´ 2iπq +‰ +.(36) +8 + +This special case gives a good indication of the kind of structures that emerge. We observe that +the contribution in the second line has exactly the same structure as found for the branch point +twist field in the free fermion theory [8]. The terms in the first line form a symmetric polynomial +on the variables tanh x +2, tanh y +2, tanh z +2. The general structure for higher sums goes as follows: +let us define +fℓpx1, . . . , x2ℓ, nq :“ 2ip´1qℓ sinh x +2 +ś2ℓ +i“1 2 cosh xi +2 +Fℓpx; nq , +gℓpx1, . . . , x2ℓ`1, nq :“ 2p´1qℓ`1 cosh x +2 +ś2ℓ`1 +i“1 2 cosh xi +2 +Gℓpx; nq +(37) +where x :“ ř +i xi in both cases, and +Fℓpx; nq +:“ +ℓÿ +j“1 +ˆ2ℓ ´ 1 +ℓ ´ j +˙ ” +wpxj´ 1 +2 q ` wpx´j` 1 +2 q +ı +(38) +Gℓpx; nq +:“ +ˆ2ℓ +ℓ +˙ +wpxq ` +ℓÿ +j“1 +ˆ 2ℓ +ℓ ´ j +˙ ” +wpxjq ` wpx´jq +ı +(39) +with +lim +|x|Ñ8 Fℓ px ; nq “ sgnpxq i +n22ℓ´1 , +lim +|x|Ñ8 Gℓ px ; nq “ sgnpxq i +n22ℓ +(40) +and +Fℓ px ; 1q “ 22ℓ´1i coth x +2 , +Gℓ px ; 1q “ 22ℓ´1i tanh x +2 . +(41) +We can then compute the sum (32) to +n´1 +ÿ +j1,...,j2ℓ´1“0 +wpp´x1qjqwpx j1´j2 +2 +q . . . wpx j2ℓ´2´j2ℓ´1 +2ℓ´1 +qwpxj2ℓ´1 +2ℓ +q +“ fℓpx1, . . . , x2ℓ, nq ` p´1qℓ +n +ℓ´1 +ÿ +j“0 +σp2ℓq +2j +ˆ +tanh x1 +2 , . . . , tanh x2ℓ +2 +˙ +, +(42) +whereas a similar sum involving an even number of indices can be evaluated to +n´1 +ÿ +j1,...,j2ℓ“0 +wpp´x1qjqwpx j1´j2 +2 +q . . . wpxj2ℓ´1´j2ℓ +2ℓ +qwpxj2ℓ +2ℓ`1q +“ gℓpx1, . . . , x2ℓ`1, nq ` ip´1qℓ +n +ℓ´1 +ÿ +j“0 +σp2ℓ`1q +2j`1 +ˆ +tanh x1 +2 , . . . , tanh x2ℓ`1 +2 +˙ +. +(43) +In both formulae, σpℓq +j pa1, . . . , aℓq is the elementary symmetric polynomial of order j in ℓ vari- +ables, defined as +σpℓq +0 pa1, . . . , aℓq “ 1 +and +σpℓq +j pa1, . . . , aℓq “ +ÿ +1ďi1ăi2㨨¨ăijďℓ +ai1ai2 ¨ ¨ ¨ aij . +(44) +9 + +These formulae can be proven by induction, similar to computations presented in [8, 9]. The +proofs are presented in Appendix A. +An interesting property of the formula (42) and a consistency check of its validity is the fact +that we must recover the cumulant expansion of xµp0qµprqy{xµy2 for n “ 1. Indeed, from (41) +it follows that +fℓpx1, . . . , x2ℓ, 1q “ p´1qℓ`1 cosh x +2 +ś2ℓ +i“1 cosh xi +2 +“ p´1qℓ`1 +ℓÿ +j“0 +σp2ℓq +2j ptanh x1 +2 , . . . , tanh x2ℓ +2 q . +(45) +Then, in the limit n Ñ 1 the only term remaining from the sum (42) is the symmetric polynomial +σp2ℓq +2ℓ ptanh x1 +2 , . . . , tanh x2ℓ +2 q which is just the product of its arguments. This agrees exactly with +the cumulant expansion of logxµp0qµprqy given in [18], formula (3.12a). Similarly, it can be +shown that +gℓpx1, . . . , x2ℓ`1, 1q “ ip´1qℓ`1 sinh x +2 +ś2ℓ`1 +i“1 cosh xi +2 +“ ip´1qℓ`1 +ℓÿ +j“0 +σp2ℓ`1q +2j`1 ptanh x1 +2 , . . . , tanh x2ℓ`1 +2 +q . +(46) +3.2 +Main Result from this Section +In summary, putting together the cumulant expansion (30) with the sum formula (42) we have +the following exact formula for the logarithm of the correlation function of composite twist fields +in the Ising model: +log +˜ +xTµp0qT : +µ prqy +xTµy2 +¸ +“ +8 +ÿ +ℓ“1 +cTµ +2ℓ pr; nq +“ +8 +ÿ +ℓ“1 +n +2ℓp2πq2ℓ +» +– +2ℓ +ź +i“1 +ˆ `8 +´8 +dθi e´mr cosh θi +fi +fl +” +fℓpˆθ12, . . . , ˆθ2ℓ´1 2ℓ, ˆθ1 2ℓ, nq +` p´1qℓ +n +ℓ´1 +ÿ +j“0 +σp2ℓq +2j +˜ +tanh +ˆθ12 +2 , . . . , tanh +ˆθ2ℓ´1 2ℓ +2 +, tanh +ˆθ1 2ℓ +2 +¸fi +fl . +(47) +We now proceed to check this expression for consistency by examining its leading short-distance +behaviour. +4 +Conformal Dimensions from the Cumulant Expansion +One possible way to test the cumulant expansion of the previous section is to obtain the correct +conformal dimension of the field Tµ by identifying the leading short-distance contributions to the +sum over cumulants. Note that this dimension was already recovered by ∆-sum rule in [17], but +the computation in that case only involved the two-particle form factor of Tµ, whereas our study +below involves all cumulants, thus providing a more extensive test of all the form factors and +cumulants. Each cumulant is expected to contain a leading contribution which is proportional +to log mr so that the overall sum gives (21) with dimension given by (20). +10 + +First, let us return to our sum (32) and change variables once more. We define +xi “ ˆθi,i`1 +for +i “ 1, . . . , 2ℓ ´ 1 +, +x2ℓ “ θ2ℓ , +(48) +so that: +θi “ +2ℓ +ÿ +j“i +p´1qj´ixj +, +2ℓ +ÿ +i“1 +θi “ +ℓÿ +i“1 +x2i´1 +, +ˆθ1,2ℓ “ +2ℓ´1 +ÿ +i“1 +p´1qi´1xi . +(49) +The Jacobian of the transformation from the θ variables to the x variables is an upper triangular +matrix with the diagonal terms being all `1, so the measure acquires no extra factor. Applying +this change of variables to (32) and expressing the result in the new variables (48) we get: +n´1 +ÿ +j1,...,j2ℓ´1“0 +wpp´x1qjqwpx j1´j2 +2 +q . . . wpx j2ℓ´2´j2ℓ´1 +2ℓ´1 +qwpp +2ℓ´1 +ÿ +i“1 +p´1qi´1xiq j2ℓ´1q +“ p´1qℓ +2i sinh +´řℓ +i“1 x2i´1 +¯ +2 cosh +ˆř2ℓ´1 +i“1 p´1qi´1xi +2 +˙ ś2ℓ´1 +i“1 2 cosh +` xi +2 +˘Fℓ +¨ +˝2 +ℓÿ +i“1 +x2i´1 ; n +˛ +‚ +` p´1qℓ +n +ℓ´1 +ÿ +j“0 +σp2ℓq +2j +˜ +tanh x1 +2 , . . . , tanh x2ℓ´1 +2 +, tanh +ř2ℓ´1 +i“1 p´1qi´1xi +2 +¸ +, +(50) +with Fℓpx; nq the function defined by (38), with the asymptotics (40). Recalling (45) it is possible +to also express the sum over symmetric polynomials in terms of products of hyperbolic functions +as +ℓ´1 +ÿ +j“0 +σp2ℓq +2j +˜ +tanh x1 +2 , . . . , tanh x2ℓ´1 +2 +, tanh +ř2ℓ´1 +i“1 p´1qi´1xi +2 +¸ +“ +cosh +´řℓ +i“1 x2i´1 +¯ +cosh +ˆř2ℓ´1 +i“1 p´1qi´1xi +2 +˙ ś2ℓ´1 +i“1 cosh +` xi +2 +˘ ´ tanh +ř2ℓ´1 +i“1 p´1qi´1xi +2 +2ℓ´1 +ź +i“1 +tanh xi +2 . (51) +This rewriting will prove useful later on. +4.1 +Exponential Factors +Now let us look at the exponential factors in the integrand of (30) and see what they look like +in terms of the new variables xi. From the first relation in (49) one has: +2ℓ´1 +ÿ +j“i +p´1qj´ixj “ +# +θi ´ θ2ℓ +for i even +θi ` θ2ℓ +for i odd , +(52) +11 + +so that +2ℓ +ÿ +i“1 +cosh θi “ cosh θ2ℓ ` +2ℓ´1 +ÿ +i“1 +coshpθi ´ θ2ℓ ` θ2ℓq +“ cosh θ2ℓ ` cosh θ2ℓ +¨ +˝ ÿ +i even +coshpθi ´ θ2ℓq ` +ÿ +i odd +coshpθi ` θ2ℓq +˛ +‚ +` sinh θ2ℓ +¨ +˝ ÿ +i even +sinhpθi ´ θ2ℓq ´ +ÿ +i odd +sinhpθi ` θ2ℓq +˛ +‚ +“ cosh x2ℓ +» +—–1 ` +2ℓ´1 +ÿ +i“1 +cosh +¨ +˝ +2ℓ´1 +ÿ +j“i +p´1qj´ixj +˛ +‚ +fi +ffifl ` sinh x2ℓ +» +—– +2ℓ´1 +ÿ +i“1 +p´1qi sinh +¨ +˝ +2ℓ´1 +ÿ +j“i +p´1qj´ixj +˛ +‚ +fi +ffifl . +(53) +Therefore, since none of the functions in (50) depends on x2ℓ the integral on this variable can +be carried out by making use of the identity +ˆ `8 +´8 +dt expp´A cosh t ´ B sinh tq “ 2K0 +´a +A2 ´ B2 +¯ +, +(54) +giving +ˆ `8 +´8 +dx2ℓ e´mr ř2ℓ +i“1 cosh θi “ 2K0pmrd2ℓ´1q , +(55) +with +d2 +2ℓ´1 “ +» +—–1 ` +2ℓ´1 +ÿ +i“1 +cosh +¨ +˝ +2ℓ´1 +ÿ +j“i +p´1qj´ixj +˛ +‚ +fi +ffifl +2 +´ +» +—– +2ℓ´1 +ÿ +i“1 +p´1qi sinh +¨ +˝ +2ℓ´1 +ÿ +j“i +p´1qj´ixj +˛ +‚ +fi +ffifl +2 +. +(56) +The mr ! 1 expansion of the modified Bessel function is: +K0pmrd2ℓ´1q “ ´ log mr ` log 2 ´ ln d2ℓ´1 ´ γ ` opmrd2ℓ´1q , +(57) +from which the leading short distance contributions to the cumulant expansion can be obtained. +It is worth mentioning that one could also resum contributions proportional to the constant term +log 2 ´ γ in (57) and those should contribute to the KTµ-term in (21), that is to the logarithm +of xTµy. A similar computation was carried out in [9] for xT y in the free boson theory. +4.2 +Short-Distance Behaviour of the Cumulant Expansion +Putting together (50), (51) and (55) in (30) we can split the cumulant into three contributions +cTµ +2ℓ pr; nq “ cp1q +2ℓ pr; nq ` cp2q +2ℓ prq ` cµ +2ℓprq . +(58) +12 + +We will define these contributions as follows. First: +cp1q +2ℓ pr; nq “ 2p´1qℓin +ℓp4πq2ℓ +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dx2ℓ´1 K0pmrd2ℓ´1q +ˆ +sinh +´řℓ +i“1 x2i´1 +¯ +cosh +ˆř2ℓ´1 +i“1 p´1qi´1xi +2 +˙ ś2ℓ´1 +i“1 cosh +` xi +2 +˘ ˆFℓ +¨ +˝2 +ℓÿ +i“1 +x2i´1 ; n +˛ +‚, +(59) +with +ˆFℓpx; nq :“ Fℓpx; nq ´ sgnpxq i +n22ℓ´1 . +(60) +This shift is motivated by the asymptotics (40) and ensures that the function ˆFℓpx; nq goes to +zero for |x| large. This in turn ensures the convergence of the integrals even when the Bessel +function is approximated by its leading short-distance contribution ´ logpmrq. +The next contribution is then a combination of the first term in (51) and the term introduced +by the shift (60): +cp2q +2ℓ prq +“ +p´1qℓ +ℓp2πq2ℓ +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dx2ℓ´1 K0pmrd2ℓ´1q +ˆ +» +———– +cosh +´řℓ +i“1 x2i´1 +¯ +´ sinh +´řℓ +i“1 x2i´1 +¯ +sgnpřℓ +i“1 x2i´1q +cosh +ˆř2ℓ´1 +i“1 p´1qi´1xi +2 +˙ ś2ℓ´1 +i“1 cosh +` xi +2 +˘ +fi +ffiffiffifl . +(61) +Note that this contribution is n-independent. Finally, the contribution cµ +2ℓprq is nothing but the +cumulant of the expansion of xµp0qµprqy{xµy2 resulting from the last term (the product of tanh +functions) in (51): +cµ +2ℓprq +“ +p´1qℓ`1 +ℓp2πq2ℓ +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dx2ℓ´1 K0pmrd2ℓ´1q +ˆtanh +˜ř2ℓ´1 +i“1 p´1qi´1xi +2 +¸ 2ℓ´1 +ź +i“1 +tanh +ˆxi +2 +˙ +. +(62) +This may look a bit different from the cumulant presented in [18] but this is simply due to +the change of variables. Note also that in [18] they implicitly take xµy “ 1 in the cumulant +expansion. +4.3 +Leading Contribution to cp1q +2ℓ pr; nq +In order to evaluate the integral (59) we can perform yet another (and final!) change of variables: +y “ +ℓÿ +i“1 +x2i´1 +ñ +x2ℓ´1 “ y ´ +ℓ´1 +ÿ +i“1 +x2i´1 +, +2ℓ´1 +ÿ +i“1 +p´1qi´1xi “ y ´ +ℓ´1 +ÿ +i“1 +x2i . +(63) +13 + +so that, at short distances ř +ℓ cp1q +2ℓ pr; nq « ´zn logpmrq with +zn “ +8 +ÿ +ℓ“1 +p´1qℓ2ni +ℓp4πq2ℓ +ˆ `8 +´8 +dy sinh y ˆFℓp2y; nq +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dx2ℓ´2 +» +–sech +˜ +y ´ řℓ´1 +i“1 x2i´1 +2 +¸ ℓ´1 +ź +i“1 +sech +ˆx2i´1 +2 +˙fi +fl +» +–sech +˜ +y ´ řℓ´1 +i“1 x2i +2 +¸ ℓ´1 +ź +i“1 +sech +ˆx2i +2 +˙fi +fl +“ +8 +ÿ +ℓ“1 +p´1qℓ2ni +ℓp4πq2ℓ +ˆ `8 +´8 +dy sinh y ˆFℓp2y; nqG2 +ℓpyq , +(64) +where, exactly as in [8,9]: +Gℓpyq “ +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dxℓ´1 +» +–sech +˜ +y ´ řℓ´1 +i“1 xi +2 +¸ ℓ´1 +ź +i“1 +sech +ˆxi +2 +˙fi +fl “ +ˆ `8 +´8 +dap2πqℓ´1eiay +coshℓ πa +, +(65) +and the functions Gℓpyq can be evaluated explicitly to +Gℓpyq “ p2πqℓ´1 +pℓ ´ 1q! +$ +’ +& +’ +% +y +π sinh y +2 +ś ℓ +2 ´1 +j“1 p y2 +π2 ` p2jq2q +for ℓ even +1 +cosh y +2 +ś ℓ´1 +2 +j“1p y2 +π2 ` p2j ´ 1q2q +for ℓ odd +. +(66) +By replacing Fℓpx; nq by ˆFℓpx; nq in (59), we have ensured that the integrals (64) are convergent +since Gℓpyq sinh y is asymptotically polynomial in y and ˆFℓp2y; nq is exponentially decaying. +They can be evaluated with great precision and fitted to the function +zn “ 1 +12 +ˆ +n ´ 1 +n +˙ +` 1 +4n ` z1 , +(67) +with z1 “ ´0.217p4q. This gives 4∆Tµ plus an additional constant z1 which should be cancelled +by contributions coming from cp2q +2ℓ prq ` cµ +2ℓprq. +Numerical results for zn are shown in Fig. 1. It is interesting to observe that there is very +good agreement with the formula (67) for n integer and also for n not integer, greater than +2. However for 1 ă n ă 2 the numerical data differ from (67) suggesting that the analytic +continuation of (64) to n “ 1 from n real greater than 1 is non-trivial. This is in agreement +with results found in [17] where the limit n Ñ 1 of the two-particle form factor contribution +produced a delta-function term. +4.4 +Leading Contribution to cp2q +2ℓ prq ` cµ +2ℓprq +Consider now the leading contribution to the second term in the cumulant. This is independent +of n and employing the same change of variables as above, it is easy to write an expression which +14 + +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +0 +2 +4 +6 +8 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +n +zn +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +n +zn +Figure 1: Left: The function zn evaluated numerically through the sum (64) for integer values +of n “ 1, . . . , 10 (red squares) against the formula (67) (blue solid line). +Right: The same +comparison for n P r1, 3s including non integer values. When evaluating the sum (64) numerically +we truncate at some value of ℓ. This value of ℓ is different for each value of n and is chosen so +that the sum is stable up to 5 decimal digits. +is given by a convergent integral involving the functions Gℓpyq. Letting ř +ℓ cp2q +2ℓ prq « ´z2 logpmrq +we have that +z2 “ +8 +ÿ +ℓ“1 +2p´1qℓ +ℓp2πq2ℓ +ˆ `8 +0 +dy e´y G2 +ℓpyq “ ´0.0326p1q . +(68) +Note that +z1 ` z2 “ ´0.250p0q « ´1 +4 . +(69) +Remarkably, this value is precisely what we need to recover the correct dimension of the field +Tµ. This is because we know that +8 +ÿ +ℓ“1 +cµ +2ℓprq « ´1 +4 logpmrq , +(70) +as this is the sum over cumulants corresponding to the two-point function xµp0qµprqy{xµy2 and +µ has dimension 4∆µ “ 1{4. Therefore, the overall leading short distance behaviour of the +xTµp0qT : +µ prqy{xTµy2 cumulants correctly predicts the conformal dimension (20). +This highly +non-trivial result provides strong support for the formula (47). In addition, the structure of the +cumulants means that we can also write +xTµp0qT : +µ prqy +xTµy2 +“ Rpr; nq xµp0qµprqy +xµy2 +, +(71) +where Rpr; nq “ ś8 +ℓ“1 ecp1q +2ℓ pr;nqecp2q +2ℓ prq has the property Rpr; 1q “ 1. +15 + +Recalling the observation of Section 2.2, namely that the cumulant expansion of Tµ posed +some convergence issues, we note that those issues did not feature in the computations of this +section. This is because by writing the cumulant as we have done, all convergence issues have +been “hidden” in the contribution cµ +2ℓprq. Indeed, a naive expansion of the Bessel function in +(62) leads to a divergent integral. Nonetheless, as shown in [18], the short distance limit of this +quantity can be obtained via a semiclassical approach and it ultimately leads to the expected +result (70). +5 +Analytic Continuation to n P Rě1 +All results obtained so far are valid for n P Z`. This is always the case in the replica picture +where n represents a replica number. However, the entanglement measures that our two-point +function describes are typically defined for generic positive n. Therefore it is interesting to try +and write an expression for the correlation function which is valid for n P Rě1. Let us start by +studying the analytic continuation of the leading short-distance terms. +5.1 +Analytic Continuation of Leading Short-Distance Contributions +Fig. 1 (right) strongly suggests that our formula needs to be analytically continued in the region +1 ă n ă 2. A similar problem was addressed in [8,9], where is was shown that as n approaches +1 from n " 1 some of the poles of the cumulants will cross or pinch the real line and provide +additional contributions to the cumulant expansion which are non-vanishing for n P R and +need to be added. The correct analytic continuation is obtained when these contributions are +correctly accounted for. The discussion is nearly identical as for the free boson case [9], albeit +involving different functions. +As we have seen, only the contribution cp1q +2ℓ pr; nq to the cumulant is n-dependent. Therefore +we only need to analytically continue the coefficient of the leading short-distance contribution +to this term, that is zn defined in (64). For non-integer n larger than 1, zn picks up additional +contributions which account for the residues of the poles of ˆFℓp2y; nq that cross the real axis as +n Ñ 1`. The sum (38) in the function ˆFℓp2y, nq has kinematic poles at2 +2y ˘ p2j ´ 1qiπ “ p2kn ` 1qiπ +and +2y ˘ p2j ´ 1qiπ “ p2kn ´ 1qiπ +for +k P Z. +(72) +These poles result are due to the kinematic poles of the two-particle form factor (12) at θ “ iπ +and θ “ iπp2n´1q, together with those resulting from the periodicity property wpθq “ ´wp´θ` +2πinq. This gives rise to four families of poles +y1 +“ +pkn ` 1 ´ jqiπ, +y2 “ pkn ´ jqiπ, +k P Z +(73) +y3 +“ +pkn ´ 1 ` jqiπ, +y4 “ pkn ` jqiπ, +k P Z, +(74) +2The twist field approach assumes n integer larger than 1 (since n is a copy number). For that reason it is +natural to look for an analytic continuation to n “ 1 from n ą 1. However, once found, the analytic continuation +should be unique, thus valid for all n. +16 + +with corresponding residues of the function inside the sum (64) given by: +R1pℓ, j, k, nq +“ +np´1qℓ`j +ℓp4πq2ℓ +˜ +2ℓ ´ 1 +ℓ ´ j +¸ +sinhpiπknqG2 +ℓppnk ´ j ` 1qiπq, +(75) +R2pℓ, j, k, nq +“ +´np´1qℓ`j +ℓp4πq2ℓ +˜ +2ℓ ´ 1 +ℓ ´ j +¸ +sinhpiπknqG2 +ℓppnk ´ jqiπq, +(76) +R3pℓ, j, k, nq +“ +np´1qℓ`j +ℓp4πq2ℓ +˜ +2ℓ ´ 1 +ℓ ´ j +¸ +sinhpiπknqG2 +ℓppnk ` j ´ 1qiπq, +(77) +R4pℓ, j, k, nq +“ +´np´1qℓ`j +ℓp4πq2ℓ +˜ +2ℓ ´ 1 +ℓ ´ j +¸ +sinhpiπknqG2 +ℓppnk ` jqiπq. +(78) +These functions are all zero for n integer but they contribute for non-integer n. Let us now +investigate which of these poles cross the real line in the limit n Ñ 1`. +Since there are many indices involved, let us start by considering just one example: n “ 4 +3 +and ℓ “ 3 in the sum (64). According to the formula (20) 4∆Tµ “ 0.236111 in this case but the +numerical evaluation of (64), after subtracting the constant z1, gives the value 0.243211 which +slightly overestimates the result. The disagreement is not simply due to numerical imprecision. +The function ˆF3py, 4{3q has poles that cross the integration line as n Ñ 4{3. From (74) and the +definition (38) we see that for ℓ “ 1 the sum runs only over the value j “ 1. For j “ 1 the four +families of poles labeled by the integer k are: +y1 +“ +iknπ, +y2 “ pkn ´ 1qiπ, +k P Z +(79) +y3 +“ +iknπ, +y4 “ pkn ` 1qiπ, +k P Z. +(80) +It is clear that all these poles are always above the real line (for k ą 0) or below the real line +(for k ă 0), that is they never cross the real line, as n approaches 4 +3. Therefore there is no +correction coming from the ℓ “ 1 contribution. Let us consider ℓ “ 2. Now j “ 1, 2. For j “ 1 +the poles are the same as above and never cross the real line. For j “ 2 we have the following +four families: +y1 +“ +ipkn ´ 1qπ, +y2 “ pkn ´ 2qiπ, +k P Z +(81) +y3 +“ +ipkn ` 1qπ, +y4 “ pkn ` 2qiπ, +k P Z. +(82) +We have already seen above that the poles y1 and y3 never cross the real line, so we can only +have some contributions from y2 and y4. For k ą 0 and n positive and large both families of +poles are above the real line. However, for n “ 4 +3 we see that the pole pkn ´ 2qiπ crosses the +real line for k “ 1. Similarly, for k ă 0 and n positive and large all poles are in the lower half +plane but the pole pkn ` 2qiπ crosses the real line for n “ 4 +3 and k “ ´1. +In summary, there are two poles for j “ 2 located at ˘2πi +3 . +The corresponding residue +contributions are +2πipR2p2, 2, 1, 4{3q ´ R4p2, 2, ´1, 4{3qq “ ´0.00680653 . +(83) +17 + +Therefore, the addition of the residua of these two poles improves the estimate of the conformal +dimension from 4∆Tµ “ 0.243211 to 4∆µ “ 0.243211 ´ 0.00680653 “ 0.236404 which is much +closer to the exact value (note that the formula (64) gives -4∆Tµ, hence the minus sign of (83)). +The addition of poles for higher values of j will bring this value ever closer to formula (20) as +shown in Fig 2. In the general n case, in order to fully identify those poles that will cross the +◆ ◆ ◆ ◆ ◆ ◆ ◆ +◆ +◆ +◆ +◆ +▲ +▲ ▲ ▲ ▲ ▲ ▲ +▲ +▲ +▲ +▲ +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +n +zn +Figure 2: The function zn evaluated numerically through the sum (64) for n P r1, 3s (red squares) +against the formula (67) (green dashed line) and its analytically continued values (blue triangles) +given by (85). +real line we find once more four cases: +y1 : kn ` 1 ´ j ă 0 +ñ +1 ď k ă j ´ 1 +n +, +y2 : kn ´ j ă 0 +ñ +1 ď k ă j +n, +y3 : kn ´ 1 ` j ă 0 +ñ +´j ´ 1 +n +ă k ď ´1, +y4 : kn ` j ă 0 +ñ +´ j +n ă k ď ´1, +(84) +This gives the analytically continued values ˆzn +ˆzn +“ +zn ` +8 +ÿ +ℓ“1 +ℓÿ +j“1 +r j´1 +n s´q1 +ÿ +k“1 +inp´1qℓ`j`1 +ℓp4πq2ℓ´1 +˜ +2ℓ ´ 1 +ℓ ´ j +¸ +sinh piπnkq G2 +ℓ +` +pnk ´ j ` 1q iπ +˘ +` +8 +ÿ +ℓ“1 +ℓÿ +j“1 +r j +n s´q2 +ÿ +k“1 +inp´1qℓ`j`1 +ℓp4πq2ℓ´1 +˜ +2ℓ ´ 1 +ℓ ´ j +¸ +sinh piπnkq G2 +ℓ +` +pnk ´ jq iπ +˘ +, +(85) +where we used the fact that the residues R2pℓ, j, k, nq “ ´R4pℓ, j, k, nq and R1pℓ, j, k, nq “ +´R3pℓ, j, k, nq (which produces a factor 2) and multiplied by 2πi as required by the residue +18 + +theorem. The shifts q1, q2 take the value 1 when nr j´1 +n s “ j ´ 1 and nr j +ns “ j, respectively and +are zero otherwise (they can be removed by requiring n to be non-integer). Here the symbol +r.s represents the integer part. Fig. 2 shows the same functions as in Fig. 1 (right) plus an +additional set of values, which are the analytically continued values of zn (in blue). As we can +see these now agree perfectly with the fit (67), even for non-integer n between 1 and 2. +5.2 +Analytic Continuation of the n-Derivative +Applications of the correlation function (47) in the context of entanglement measures frequently +requires the computation of its derivative with respect to n followed by the limit n Ñ 1. As +discussed in [6,8] and [17] the derivative with respect to n of the function (37) has a discontinuity. +More precisely, as n approaches 1 and poles cross the real line, the derivative is not uniformly +convergent as a function of θ and this leads to terms involving δ-functions. The simplest examples +of this phenomenon are seen for the two-particle contribution to the two-point function of T [6] +and of Tµ [17]. Here we show how this generalises to the whole cumulant sum. Notice that we +only need to consider the contribution from the function cp1q +2ℓ pr; nq in (59) since all other terms +are independent of n and so the derivative is zero. For this term, we actually only need to +consider Fpx; nq as the additional term in the “hatted” version is also n-independent. So, we +define +sTµ +2ℓ prq +:“ +´ lim +nÑ1 +d +dncp1q +2ℓ pr; nq +“ +2p´1qℓ`1i +ℓp4πq2ℓ +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dx2ℓ´1 K0pmrd2ℓ´1q +ˆ +sinh +´řℓ +i“1 x2i´1 +¯ +cosh +ˆř2ℓ´1 +i“1 p´1qi´1xi +2 +˙ ś2ℓ´1 +i“1 cosh +` xi +2 +˘ lim +nÑ1 +d +dn +» +—–nFℓ +¨ +˝2 +ℓÿ +i“1 +x2i´1 ; n +˛ +‚ +fi +ffifl . (86) +One way to treat the derivative is to recall the ℓ “ 1 result that was derived in [17], namely +lim +nÑ1 +d +dnnf1px, x, nq +“ +´ i +2 +sinh x +cosh2 x +2 +lim +nÑ1 +d +dnnrwp2x ` iπq ` wp2x ´ iπqs +“ +x +cosh2 x +2 sinh x ´ π2 +2 δpxq , +(87) +that is, there is a finite part and a distribution part that accounts for the behaviour around +x “ 0. Recall that the function f1px, y, nq is defined in (33). This extends to higher cumulants +in similar ways, so that we can write +sTµ +2ℓ prq “ sfin +2ℓ prq ` sδ +2ℓprq , +(88) +where the two contributions represent the “finite” and δ-function contributions. The finite part +can be easily computed by noting that +lim +nÑ1 +d +dnn sinh xFp2x; nq “ i22ℓ´1x +sinh x , +(89) +19 + +Therefore +sfin +2ℓ prq +“ +p´1qℓ +ℓp2πq2ℓ +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dx2ℓ´1 K0pmrd2ℓ´1q +ˆ +řℓ +i“1 x2i´1 +sinh +´řℓ +i“1 x2i´1 +¯ +cosh +ˆř2ℓ´1 +i“1 p´1qi´1xi +2 +˙ ś2ℓ´1 +i“1 cosh +` xi +2 +˘ . +(90) +The δ-function contribution is a generalisation of the ℓ “ 1 case seen above and can be obtained +by identical arguments as those presented in [8]. In fact, the result is also identical to formula +(4.6) in [8], that is, +sδ +2ℓprq +“ +π2p´1qℓ +ℓp4πq2ℓ +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dx2ℓ´1 δp +ℓÿ +i“1 +x2i´1q +ˆ +» +———– +˜ +2ℓ ´ 2 +ℓ ´ 1 +¸ +2K0p2mrd2ℓ´1q +cosh +ˆř2ℓ´1 +i“1 p´1qi´1xi +2 +˙ ś2ℓ´1 +i“1 cosh +` xi +2 +˘ +fi +ffiffiffifl +´π2p´1qℓ +ℓp4πq2ℓ +ˆ `8 +´8 +dx1 . . . +ˆ `8 +´8 +dx2ℓ δp +ℓÿ +i“1 +x2i´1q +ˆ +ℓÿ +j“1 +j´1 +ÿ +k“1 +ÿ +q“˘ +» +———– +˜ +2ℓ ´ 1 +ℓ ´ j +¸ +p´1qj +ś2ℓ +i“1 e´rm cosh +´ř2ℓ +j“1p´1qj´ixi`iπq j´k +2ℓ +¯ +cosh +ˆř2ℓ´1 +i“1 p´1qi´1xi +2 +˙ ś2ℓ´1 +i“1 cosh +` xi +2 +˘ +fi +ffiffiffifl . (91) +6 +Conclusion +In this paper we have studied the normalised two-point function xTµp0qT : +µ prqy{xTµy2 of the +composite twist field Tµ and its conjugate. The motivation to study this object comes from recent +investigations of a measure of entanglement known as symmetry resolved entanglement [15– +17]. More fundamentally, our work contributes to developing the understanding of correlation +functions in the replica Ising field theory, a theory that, although free and seemingly simple, +contains a large number of symmetry fields or twist fields which are not present in the standard, +non-replicated, model. +The current work uses traditional IQFT techniques, mainly the form factor bootstrap pro- +gram adapted to composite twist fields [17], to expand the logarithm of the correlation function +into a series of cumulants. The main result of the paper is finding simplified expressions for +these cumulants which result from proving a number of multiple sum formulae, presented in +Appendix A, involving the two-particle form factors of the field Tµ. +Employing this cumulant expansion we have found the following structure +xTµp0qT : +µ prqy +xµp0qµprqy +xµy2 +xTµy2 “ Rpr; nq +with +Rpr; 1q “ 1 , +(92) +20 + +where µ is the disorder field of the Ising field theory, and Rpr; nq has an explicit form given in +terms of a multiple integral representations which we describe in detail in this paper. By exact +resummation of leading contributions to the cumulant expansion, we have shown that at short +distances this two-point function scales as a power law in r with the power (20) which is exactly +predicted by CFT. Showing that the two-point correlation function of the field µ factors out of +the correlator (92) has been a crucial ingredient in recovering the correct scaling dimension. We +have also shown how our formulae may be analytically continued to real replica number and +how non-analytic, delta-function terms emerge for all cumulants when computing the derivative +w.r.t. to n. This generalises results found in [17] and [6]. +As a byproduct of our investigation, we have also shown that the form factors of the composite +field Tσ can be obtained from those of Tµ via clustering in momentum space, in much the same +way as the form factors of the fields σ and µ are related. We have also fixed the normalisation of +the Tσ form factors by fixing the one-particle form factor from the asymptotics of the two-particle +form factor of Tµ. +As mentioned above, our result has applications in the context of the symmetry resolved +entanglement entropy and directly leads to a more complete formula for the latter in the Ising +model. We further expect the results of this investigation to apply with some modifications +to other composite fields, at least for free theories, for instance those associated with Up1q +symmetry in doubled free models which were studied in [20–22]. +Acknowledgments: The authors thank Benjamin Doyon and D´avid X. Horv´ath for useful +discussions. Michele Mazzoni is grateful for funding under the EPSRC Mathematical Sciences +Doctoral Training Partnership EP/W524104/1. Olalla A. Castro-Alvaredo thanks EPSRC for +financial support under Small Grant EP/W007045/1 and the Kavli Institute for Theoretical +Physics (Santa Barbara) for financial support from the National Science Foundation under +Grant No. NSF PHY-1748958, and hospitality during the conference “Talking Integrability: +Spins, Fields and Strings”, August 29-September 1 (2022). +A +Summation formulae +Let us first prove the identities (34) and (33), which are both obtained via the cotangent trick. +To show (34), consider the integral: +1 +2πi +˛ +C +dz π cotpπzqwpxzq +(93) +where C is the rectangular contour with vertices ´ϵ ` iL, ´ϵ ´ iL, n ´ ϵ ´ iL, n ´ ϵ ` iL. The +vertical contributions cancel off because the integrand is invariant under the shift z Ñ z ` n. +The same holds for the horizontal contributions in the large L limit, as +lim +LÑ8 cot πpt ˘ iLq “ ¯i , +lim +LÑ8 w +´ +xt˘iL¯ +“ ¯ i +n +(94) +and therefore the sum of the residues must vanish. Within the integration contour, the function +π cotpπzq has simple poles at z “ 0, 1, . . . n´1 with unite residue. The kinematic poles of wpxzq +21 + +are at z “ 1 +2 ´ +x +2πi, z “ n ´ 1 +2 ´ +x +2πi, with residue +1 +2π. At both these points, cotpπzq “ ´i tanh x +2. +Putting all the pieces together, one has therefore: +0 “ +n´1 +ÿ +j“0 +wpxjq ´ i tanh x +2 . +(95) +Using the very same strategy, one can prove (33). The integral to evaluate is now +1 +2πi +˛ +C +dz π cotpπzqwpp´xqzqwpyzq , +(96) +along the same contour C as before. In this case, however, the horizontal contributions do not +cancel off, as +lim +LÑ8 wpp´xqt˘iLqqwpyt˘iLq “ ´ 1 +n2 , +(97) +and thus the integral evaluates to ´ 1 +n in the large L limit. +Summing over the residues of +the poles of π cot pπzq gives the left-hand side of (33), while the kinematic poles are now at +z “ 1 +2 ` +x +2πi , ´1 +2 ` n ` +x +2πi and z “ 1 +2 ´ +y +2πi , ´1 +2 ` n ´ +y +2πi, with residues: +Res +z“ 1 +2 ` x +2πi +wpp´xqzqwpyzq “ 1 +2πwpx ` y ` iπq , +Res +z“n´ 1 +2 ` x +2πi +wpp´xqzqwpyzq “ 1 +2πwpx ` y ´ iπq +Res +z“ 1 +2 ´ y +2πi +wpp´xqzqwpyzq “ ´ 1 +2πwpx ` y ´ iπq , +Res +z“n´ 1 +2 ´ y +2πi +wpp´xqzqwpyzq “ ´ 1 +2πwpx ` y ` iπq . +Evaluating the cotangent at the kinematic poles and putting all the pieces together, one has +´ 1 +n “ +ÿ +Resrπ cotpπzqwpp´xqzqwpyzq +“ +n´1 +ÿ +j“0 +wpp´xqjqwpyjq ` i +2 +sinh +´ +x`y +2 +¯ +cosh +` x +2 +˘ +cosh +` y +2 +˘rwpx ` y ` iπq ` wpx ` y ´ iπqs , +which is indeed (33). +Using this result, it is possible to prove (42) and (43) by induction. It is useful to observe +beforehand that the following expansions hold: +sinh +´řk +i“1 xi +¯ +śk +i“1 cosh xi +“ +r k´1 +2 s +ÿ +j“0 +σpkq +2j`1ptanh x1, . . . , tanh xkq +(98) +cosh +´řk +i“1 xi +¯ +śk +i“1 cosh xi +“ +r k +2 s +ÿ +j“0 +σpkq +2j ptanh x1, . . . , tanh xkq . +(99) +Following the procedure employed in [8, 9] we will prove that (42) implies (43). If (42) holds, +than we can shift x2ℓ by ´2iπp, multiply the left-hand side by a factor wpxp +2ℓ`1q and sum over +22 + +p to obtain: +n´1 +ÿ +j1,...,j2ℓ´1,p“0 +wpp´x1qjqwpx j1´j2 +2 +q . . . wpx j2ℓ´2´j2ℓ´1 +2ℓ´1 +qwpxj2ℓ´1´p +2ℓ +qwpxp +2ℓ`1q +“ +n´1 +ÿ +p“0 +fℓpx1, . . . , x´p +2ℓ , nqwpxp +2ℓ`1q ` +n´1 +ÿ +p“0 +p´1qℓ +n +ℓ´1 +ÿ +j“0 +σp2ℓq +2j +˜ +tanh x1 +2 , . . . , tanh x´p +2ℓ +2 +¸ +wpxp +2ℓ`1q , +(100) +Let us focus on the first term in the second line, which yields two contributions due to the +presence of a constant term in the right-hand side of (33). Defining x “ ř2ℓ +i“1 xi, this reads: +2ip´1qℓ sinh x +2 +ś2ℓ +i“1 2 cosh xi +2 +n´1 +ÿ +p“0 +ℓÿ +j“1 +ˆ2ℓ ´ 1 +ℓ ´ j +˙ „ +w +´ +xj´p ´ iπ +¯ +` w +´ +x´j´p ` iπ +¯ȷ +wpxp +2ℓ`1q +“ gℓpx1, . . . , x2ℓ`1; nq ` 4ip´1qℓ sinh x +2 +n ś2ℓ +i“1 2 cosh xi +2 +ℓÿ +j“1 +ˆ2ℓ ´ 1 +ℓ ´ j +˙ +“ gℓpx1, . . . , x2ℓ`1; nq ` ip´1qℓ +n +ℓ´1 +ÿ +j“0 +σp2ℓq +2j`1ptanh x1 +2 , . . . , tanh x2ℓ +2 q +(101) +The emergence of the function gℓ was already proved in Appendix C of [9]. In going from the +second to the third line we used the first identity in (98) and řℓ +j“1 +`2ℓ´1 +ℓ´j +˘ +“ 22ℓ´2. Consider +now the second term in the second line of (100): using the fact that tanh x˘p +2 +“ tanh x +2 and (34) +we have: +n´1 +ÿ +p“0 +p´1qℓ +n +ℓ´1 +ÿ +j“0 +σp2ℓq +2j +˜ +tanh x1 +2 , . . . , tanh x´p +2ℓ +2 +¸ +wpxp +2ℓ`1q +“ ip´1qℓ +n +ℓ´1 +ÿ +j“0 +σp2ℓq +2j +ˆ +tanh x1 +2 , . . . , tanh x2ℓ +2 +˙ +tanh x2ℓ`1 +2 +(102) +Now we observe that the elementary symmetric polynomial of degree j in ℓ variables can be +decomposed as σpℓq +j pa1, . . . , aℓq “ σpℓ´1q +j +pa1, . . . , aℓ´1q ` σpℓ´1q +j´1 pa1, . . . , aℓ´1q aℓ, hence +ℓ´1 +ÿ +j“0 +« +σp2ℓq +2j`1ptanh x1 +2 , . . . , tanh x2ℓ +2 q ` σp2ℓq +2j +ˆ +tanh x1 +2 , . . . , tanh x2ℓ +2 +˙ +tanh x2ℓ`1 +2 +ff +“ +ℓ´1 +ÿ +j“0 +σp2ℓ`1q +2j`1 ptanh x1 +2 , . . . , tanh x2ℓ`1 +2 +q . +(103) +Thus the sum of (101) and (102) yields (43). In an analogous way it is possible to prove (42) +starting from (43). +23 + +References +[1] B. Zuber and C. Itzykson, Quantum field theory and the two-dimensional Ising model, +Phys. Rev. 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A 19S2, 34–49 (2004). +25 + diff --git a/IdAzT4oBgHgl3EQfx_4Y/content/tmp_files/load_file.txt b/IdAzT4oBgHgl3EQfx_4Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbf56bec2c2534e433c5496fc31c4b8124c98e57 --- /dev/null +++ b/IdAzT4oBgHgl3EQfx_4Y/content/tmp_files/load_file.txt @@ -0,0 +1,890 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf,len=889 +page_content='Two-Point Functions of Composite Twist Fields in the Ising Field Theory Olalla A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Castro-Alvaredo and Michele Mazzoni Department of Mathematics, City, University of London, 10 Northampton Square EC1V 0HB, UK All standard measures of bipartite entanglement in one-dimensional quantum field theories can be expressed in terms of correlators of branch point twist fields, here denoted by T and T :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' These are symmetry fields associated to cyclic permutation symmetry in a replica theory and having the smallest conformal dimension at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Recently, other twist fields (composite twist fields), typically of higher dimension, have been shown to play a role in the study of a new measure of entanglement known as the symmetry resolved entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In this paper we give an exact expression for the two-point function of a composite twist field that arises in the Ising field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In doing so we extend the techniques originally developed for the standard branch point twist field in free theories as well as an existing computation due to Horv´ath and Calabrese of the same two-point function which focused on the leading large-distance contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We study the ground state two-point function of the composite twist field Tµ and its conjugate T : µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' At criticality, this field can be defined as the leading field in the operator product expansion of T and the disorder field µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We find a general formula for logxTµp0qT : µ prqy and for (the derivative of) its analytic continuation to positive real replica numbers greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We check our formula for consistency by showing that at short distances it exactly reproduces the expected conformal dimension Keywords: Integrable Quantum Field theory, Ising model, Branch Point Twist Fields, Sym- metry Resolved Entanglement Entropy, Form Factor Expansion, Correlation Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='castro-alvaredo@city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='uk michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='mazzoni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='2@city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='uk January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='01745v1 [hep-th] 4 Jan 2023 1 Introduction It is well-known that the Ising field theory has an internal Z2 symmetry, associated to which we can define two fields: σ, the spin field (order operator), and µ (disorder operator)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The theory also contains a free Majorana fermion field Ψ so that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' in the disordered phase of the theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' the three fields can be characterised by their mutual equal-time exchange relations [1–3]: Ψpxqσpyq “ # σpyqΨpxq y ą x σpyqΨpxq y ă x and Ψpxqµpyq “ # ´µpyqΨpxq y ą x µpyqΨpxq y ă x (1) Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' in the context of the investigation of entanglement measures it is often convenient to consider a “replica” version of the theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' namely a model consisting of n non-interacting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' identical copies of the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In this model, the fields above acquire an index tµj, σj, Ψju with j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , n, running over the copy numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The resulting model possesses a large amount of symmetry, namely, not only Z2 symmetry on each copy, but symmetry under the exchange of any copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Cyclic permutation symmetry is one of these many symmetries and, as it turns out, it is the symmetry that plays the most fundamental role in computations of the entanglement entropy and other measures of entanglement [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In [6] the branch point twist fields T and its conjugate T : (called ˜T in the original paper) were defined as the symmetry fields associated with cyclic permutation symmetry of copies in a replica theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' These fields too are characterised by their exchange relations with respect to the fermions: ΨjpxqT pyq “ # T pyqΨj`1pxq y ą x T pyqΨjpxq y ă x and ΨjpxqT :pyq “ # T :pyqΨj´1pxq y ą x T :pyqΨjpxq y ă x (2) for j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , n and j ” j ` n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' These relations can be written for any 1+1D quantum field theory, integrable or not, however, in the context of massive integrable quantum field theory (IQFT), they provide, together with the two-body scattering matrix, all the information needed to compute correlation functions and matrix elements of T [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' These computations have now been carried out for many theories and entanglement measures (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' [7–10]) revealing many new insights into the universal properties of entanglement at near-critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In recent years, it has been shown that also the fields resulting from the conformal OPE of T with other fields of the Ising field theory can be of interest in the context of entanglement [11–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In particular, the correlation functions of the leading field in the OPE of T and ř j µj, denoted by Tµ, are related to an entanglement measure known as the symmetry resolved entanglement entropy [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Tµ satisfies exchange relations which combine those for T and µ as seen above: ΨjpxqTµpyq “ # ´TµpyqΨj`1pxq y ą x TµpyqΨjpxq y ă x and ΨjpxqT : µ pyq “ # ´T : µ pyqΨj´1pxq y ą x T : µ pyqΨjpxq y ă x (3) 1In this paper we use the conventions of [18], which corresponds to choosing the disordered phase of the model, where the fields σpµq are odd (even) with respect to the Majorana fermion Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1 In general, such measures can always be defined for theories that possess an internal symmetry (such as Z2 in the Ising case), and it gives access to information about the amount of entangle- ment that is stored in each symmetry sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The computation of the symmetry resolved entanglement provides strong motivation to study correlators of Tµ and this will be the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Our aim is finding an exact analytic expression for the two-point function xTµp0qT : µ prqy using IQFT techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The applications of such a result in the context of entanglement will not be discussed here, but they follow quite straightforwardly from existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In particular, the form factors of Tµ and the leading contribution to its two-point function were computed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The present work is an extension of those results to include higher particle contributions and to show how non-trivial resummation identities allow for relatively simple closed formulae for all correlation function cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Correlation functions of composite twist fields have been studied in a number of works both for the Ising field theory and other, interacting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Most of these results build upon the form factor program for the matrix elements of T [6] and its extension to composite twist fields [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In [20–22] free theories were studied, whereas interacting IQFTs such as the Ising and sinh-Gordon models (both with discrete Z2 symmetry), the sine-Gordon model (with continuous Up1q symmetry) and the 3-state Potts model (with discrete Z3 symmetry) were studied in [17,23] and [24], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' It is also possible to study composite twist fields where T is composed with a local field not associated with an internal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Such composite fields are associated with cyclic permutation symmetry too and have a conformal dimension which is distinct from that of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In particular, for theories whose UV fixed point is described by a non-unitary conformal field theory (CFT), it is possible to construct composite twist fields whose dimension is lower than that of T and they play a critical role in describing the usual measures of entanglement [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This happens for instance for the Lee-Yang theory both at and away from criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The form factors and two-point functions of the branch point twist field and composite twist field for this theory were studied in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The expectation values of composite twist fields involving the energy field in the Ising field theory were studied in [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This paper is organised as follows: In Section 2 we review form factor results for the order and disorder fields in the Ising field theory as well as for T and Tµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We review the cumulant expansion of two-point functions and introduce an example of the type of convergence issues that arise in the cumulant expansion of xTµp0qT : µ prqy{xTµy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In Section 3 we find closed formulae for all higher cumulants, leading to a close expression for the two-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In Section 4 we test this expression by obtaining the exact conformal dimension of Tµ from resummation of leading terms in the short-distance expansion of the cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We show that the normalised two- point function xTµp0qT : µ prqy{xTµy2 is in fact proportional to the normalised two-point function xµp0qµprqy{xµy2, thus it factorises into n-dependent and n-independent parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In Section 5 we show how to analytically continue the cumulant expansion from n integer and greater than 1 to n real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This allows us to write a formula for the n-derivative of the two-point function at n “ 1, a quantity that typically plays a role in entanglement measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Appendix A provides proofs of new useful resummation formulae for the form factors of Tµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 2 2 Field Content of the Ising Model and Form Factors The correlation functions and form factors of the fields σ, µ defined by (1) can be obtained via form factor bootstrap [25, 26] and where studied in detail by Yurov and Zamolodchikov in their seminal paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Form factors of descendent fields (in the CFT sense) of the energy field ε were studied in [19] and shown to match in number and spin the field content of the corresponding Verma module in the underlying Ising CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Starting from the relations (1) the form factor equations can be written and solved for matrix elements of σ, µ and these were found to take an extremely simple form [18], namely (the factor ik is needed to satisfy the kinematic residue equation): F µ 2kpθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2kq “ ikxµy ź 1ďiăjď2k tanh θij 2 , (4) F σ 2k`1pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k`1q “ ikF σ 1 ź 1ďiăjď2k`1 tanh θij 2 , (5) with θij :“ θi ´ θj and xµy and F σ 1 normalisation constants which can be identified with the vacuum expectation value of µ and the one-particle form factor of σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' More generally, a k-particle form factor is defined as F O k pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θkq :“ x0|Op0q|θ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' θk|0y (6) that is, a matrix element of a local or quasi-local field between the ground state |0y and a k- particle state characterised by rapidities tθiuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In general, particles will also be characterised by their quantum numbers, but in the Ising model there is a single particle type so these do not need to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For the field µ the products above can be rewritten as a Pfaffian of a 2k ˆ 2k antisymmetric matrix A with entries Aij “ tanh θij 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In particular this means that in the disordered phase the vacuum expectation value of µ is non-vanishing whereas it is vanishing for σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In the context of the study of entanglement we know also that branch point twist fields T and T : play a prominent role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Their form factors in the (replica) Ising model have been known for some time [6,8] and due to the free nature of the model they can also be expressed in terms of a Pfaffian F T |11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 2k pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ xT yPfpKq , PfpKq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' det K , (7) where n labels the number of replicas, Kij :“ kpθijq “ sin π n 2n sinh ´ iπ´θij 2n ¯ sinh ´ iπ`θij 2n ¯ sinh θij 2n sinh iπ 2n with i, j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , 2k , (8) and the superindices 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1 indicate that all particles are in the same copy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' From this representation we also see that all form factors are functions of rapidity differences only, a property that holds for all spinless fields in relativistic quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The two-particle 3 form factor is simply F T |11 2 pθ1, θ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ xT ykpθ12q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Form factors for particles in copies j1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' j2k can be obtained from the above using the standard form factor equations presented in [6] F T |j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='j2k 2k pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ F T |1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 2k pθj1´1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θjk´1 2k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq , for j1 ě j2 ¨ ¨ ¨ ě j2k , (9) with θj :“ θ ` 2πij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (10) The form factors of the composite twist field Tµ where first obtained in [17] and have again the Pfaffian structure typical of the Ising model, that is F Tµ|11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 2k pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ xTµyPfpWq (11) with Wij :“ wpθijq “ sin π n 2n sinh ´ iπ´θij 2n ¯ sinh ´ iπ`θij 2n ¯ sinh θij n sinh iπ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (12) As we can see, this differs from kpθq above only because n is replaced by n{2 in the minimal part of the form factor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' the part that does not contain kinematic poles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' However, this small change leads to some important differences, the main one being the asymptotic properties lim θÑ8 kpθq “ 0 and lim θÑ˘8 wpθq “ ˘ i n , (13) as well as lim nÑ1 kpθq “ 0 and lim nÑ1 wpθq “ i tanh θ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (14) Note that the last equality simply shows that the two-particle form factor of Tµ reduces to that of µ for n “ 1, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This extends to higher-particle form factors too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' It is known from the study of many models and arguments such as those presented in [27] that the asymptotics of two particle form factors should be related to the value of a one-particle form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This is a consequence of so-called cluster decomposition in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In simple theories, as assumed in [27], this one-particle form factor would be that of the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' However, in the Ising model, due to Z2 symmetry there is a mixing between form factors of µ and σ and also those of Tσ (defined as the composition of T and ř j σj) and Tµ in such a way that: lim θÑ˘8 wpθq “ ˘τ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (15) where τ :“ F Tσ|1 1 is the one-particle form factor of Tσ, which by relativistic invariance is θ independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Combining (15) with (13) we have that |F Tσ|1 1 |2 “ |τ|2 “ 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (16) Higher form factors of Tσ can also be related to Pfaffians by employing a more general version of the cluster decomposition property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Namely lim θ2k`2Ñ8xTµy´1F Tµ 2k`2pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k`2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ τF Tσ 2k`1pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (17) 4 Note that the prefactor xTµy´1 ensures that when k “ 0 both sides of the equation become τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In this way, the form factors F Tσ 2k`1pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq can be computed systematically and it is easy to show that they can be written as a sum of Pfaffians involving 2k variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In fact, we can show that F Tσ 2k`1pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ τ xTµy 2k`1 ÿ j“1 p´1qj`1F Tµ 2k pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , ¯θj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' θ2k`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq , (18) where the sign depends on the position of the variable θj and can be worked out by counting Wick contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Similarly, the symbol ¯θj means that this variable is removed, hence this is a sum over 2k-particle form factors depending on a subset of the variables tθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , θ2k`1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For instance F Tσ 3 pθ1, θ2, θ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ τpwpθ12q ´ wpθ13q ` wpθ23qq “ τ xTµypF Tµ 2 pθ1, θ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq ´ F Tµ 2 pθ1, θ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq ` F Tµ 2 pθ2, θ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nqq , (19) so that each “contraction” θij where |i ´ j| is even produces one minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The formula (18) is, as far as we know, new and first presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' However, this structure is the same as for the form factors of the field σ which are obtained in the limit n “ 1, for which the function wpθq reduces to a tanh (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The special properties of the tanh function mean that formulae such as (18) and (19) can be shown to factorise also as products of tanh functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In [17] it was also shown that the form factor (12) gives the correct conformal dimension of Tµ via the ∆-sum rule [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This dimension is [5,11,15,28,29] ∆Tµ “ ∆T ` ∆µ n “ n 48 ` 1 24n with ∆T “ 1 48 ˆ n ´ 1 n ˙ , ∆µ “ 1 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (20) In fact ∆Tµ “ ∆Tσ since ∆µ “ ∆σ even if, for symmetry reasons, ∆σ cannot be obtained from the ∆-sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 Two-Point Function and Cluster Expansion In this paper we are interested in properties of the ground state two-point function of the field Tµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In general we would like to write down an expansion of the form log ˜ xTµp0qT : µ prqy xTµy2 ¸ “ 8 ÿ ℓ“1 cTµ ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq mr!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 « ´4∆Tµ logpmrq ´ KTµ , (21) where the sum is over functions cTµ ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq known as cumulants, ∆Tµ is the conformal dimension of the field Tµ and KTµ is a constant that depends on the vacuum expectation value xTµy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' These cumulants are multiple integrals of linear combinations of products of form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' More precisely, we have the following structure cTµ ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ 1 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='p2πqℓ nÿ j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=',jℓ“1 ˆ 8 ´8 dθ1 ¨ ¨ ¨ ˆ 8 ´8 dθℓ hTµ|j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='jℓ ℓ pθ1, ¨ ¨ ¨ , θℓ, nqe´mr řℓ i“1 cosh θi, (22) 5 where the functions hO|j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='jk k pθ1, ¨ ¨ ¨ , θk, nq are given in terms of the form factors of the field involved, and ji represent the particle’s quantum numbers which in our examples will be also the copy numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For example: hTµ|j1j2 2 pθ1, θ2, nq “ xTµy´2 ˇˇˇF Tµ|j1j2 2 pθ1, θ2, nq ˇˇˇ 2 , hTµ|j1j2j3j4 4 pθ1, θ2, θ3, θ4, nq “ xTµy´2 ˇˇˇF Tµ|j1j2j3j4 4 pθ1, θ2, θ3, θ4, nq ˇˇˇ 2 ´hTµ|j1j2 2 pθ1, θ2, nqhTµ|j3j4 2 pθ3, θ4, nq ´hTµ|j1j3 2 pθ1, θ3, nqhTµ|j2j4 2 pθ2, θ4, nq ´hTµ|j1j4 2 pθ1, θ4, nqhTµ|j2j3 2 pθ2, θ3, nq, (23) and so on, whereas all odd particle terms are vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Similar formulae can be written for the cumulants of Tσ where only odd particle cumulants are non-vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' A generic combinato- rial/diagramatic construction of these functions can be found for instance in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For a generic local field O, it is standard to require hO|j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='jℓ ℓ pθ1, ¨ ¨ ¨ , θℓq „ e´θi for θi P R and θi Ñ 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (24) Given the properties of the form factors presented in the previous section, we see that this property is not satisfied for the cumulants of the two-point function of Tµ, or indeed for the cumulants of the two-point function of µ as shown in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In fact, the cumulant expansion is still convergent in both cases, but the leading behaviour for small mr is harder to extract than in theories where (24) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='2 Two-Particle Contribution The aim of this work is to find a general, compact form, for all terms in the expansion of xTµp0qT : µ prqy{xTµy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' One way to check the two-point function expansion is to recover the con- formal dimension of the field by exact resummation of all terms which are proportional to logpmrq for mr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1, that is the first term in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Let us start by considering the simplest contribution to the connected part of the two-point function xTµp0qT : µ prqy{xTµy2, which has already been studied in the literature [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The first non-vanishing contribution to the cumulant expansion comes from hTµ|j1j2 2 pθ1, θ2, nq, which is nothing but the normalised squared modulus of the two-particle form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Using (9) the latter can be rewritten as nÿ i,j“1 ˇˇˇF Tµ|ij 2 pθ1, θ2q ˇˇˇ 2 “ n n´1 ÿ j“0 ˇˇˇF Tµ|11 2 pθ1 ` 2πij, θ2q ˇˇˇ 2 “ nxTµy2 n´1 ÿ j“0 wpp´θ12qjqwpθj 12q , (25) where the superindex j is defined as in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Thus we have cTµ 2 pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ n n´1 ÿ j“0 ˆ 8 ´8 ˆ 8 ´8 dθ1dθ2 2p2πq2 wpp´θ12qjqwpθj 12q e´mr cosh θ1´mr cosh θ2 “ n p2πq2 n´1 ÿ j“0 ˆ 8 ´8 dθ wpp´θqjqwpθjq K0p2mr cosh θ 2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (26) 6 The sum above has been computed in [17] by using the cotangent trick and is given by n´1 ÿ j“0 wpp´θqjqwpθjq “ ´i tanh θ 2pwp2θ ` iπq ` wp2θ ´ iπqq ´ 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (27) This function tends asymptotically to the value 1 n for |θ| Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This means that the usual procedure consisting of expanding the Bessel function for mr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1 and isolating the logpmrq leading term, thus effectively removing the Bessel function from the integrand (26), now leads to a divergent integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The integral is however not divergent, it simply needs to be done with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We can rewrite (26) as cTµ 2 pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ n p2πq2 ˆ 8 ´8 dθ » – n´1 ÿ j“0 wpp´θqjqwpθjq ´ 1 n fi fl K0p2mr cosh θ 2q ` 1 p2πq2 ˆ 8 ´8 dθ K0p2mr cosh θ 2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (28) In this form, the integral in the first line can be approximated for mr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1 by expanding the Bessel function, giving a leading contribution which is proportional to logpmrq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The integral in the second line can be computed exactly to ˆ 8 ´8 dθ K0p2mr cosh θ 2q “ 2K0pmrq2 mr!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 « ´2plogpmrqq2 , (29) so that, in this case, the leading small mr contribution diverges as plogpmrqq2 rather than logpmrq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Thus, although the cumulant (26) is still well-defined, its leading small mr behaviour is now dominated by plogpmrqq2 instead of logpmrq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This is a consequence of the property (24) not holding in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Nonetheless, terms of order plogpmrqq2 should cancel out when including further contributions in the form factor series as one expects to recover the 1{r4∆Tµ behaviour of the two-point function at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In the next sections we will show one particular way to recover the expected scaling (21) from our cumulant expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 3 Higher Particle Contributions: Closed Formulae Existing studies of the branch point twist field two-point function for free fermions [8] and bosons [9] have revealed that the form of higher cumulants can be considerably simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This is because under sum over particle types and integration over the rapidities, many of the terms in the cumulant either cancel each other out or can be shown to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In fact, it is possible to show that just as for the standard branch point twist field, and for the same reasons already discussed in [8,9] the cumulants of the two-point function of Tµ take the generic form cTµ 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ n 2ℓp2πq2ℓ n´1 ÿ j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=',j2ℓ´1“0 » – 2ℓ ź i“1 ˆ `8 ´8 dθi e´mr cosh θi fi fl ˆ p´1qℓ ¨ ˝wpθ´j1 12 q ℓ´1 ź k“1 wpθj2k´j2k`1 2k`1 2k`2qq ˛ ‚ ¨ ˝wpθj2ℓ´1 1 2ℓ q ℓ´1 ź k“1 wpθ´j2k´1`j2k 2k 2k`1 q ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (30) 7 By using the fact that wpθ´jq “ ´wpp´θqjq, we can change the sign of half of the factors in the second line, cancelling out the factor p´1qℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The integrand becomes: n´1 ÿ j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=',j2ℓ´1“0 wpp´θ12qj 1qwpθj2ℓ´1 1 2ℓ q ℓ´1 ź k“1 wpθj2k´j2k`1 2k`1 2k`2qwpp´θ2k 2k`1qj2k´1´j2kq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (31) In order to evaluate the integrals (30) it is convenient to perform a change of variables whereby we first change the sign of all the even rapidities, without any change in the integration measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Then, defining ˆθij ” θi ` θj the integrand becomes a function of rapidity sums only: n´1 ÿ j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=',j2ℓ´1“0 wpp´ˆθ12qj1qwpˆθ j1´j2 23 qwpˆθ j2´j3 34 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' wpˆθ j2ℓ´2´j2ℓ´1 2ℓ´1 2ℓ qwpˆθ j2ℓ´1 1 2ℓ q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (32) We will refer to this as a fully connected sum, meaning that all terms are cyclicly “connected” both at the level of the rapidities and the summation indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The sum (32) and others of a similar type can be computed recursively as shown below and in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 Recursive Formulae The sum (32) can be carried out leading to generalisations of the following result f1px, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq :“ n´1 ÿ j“0 wpp´xqjqwpyjq “ ´ i 2 sinh ´ x`y 2 ¯ cosh x 2 cosh y 2 rwpx ` y ` iπq ` wpx ` y ´ iπqs ´ 1 n , (33) which is presented here for the first time, although the case x “ y was obtained in [17] and has already been reported in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' It is also useful to know that n´1 ÿ j“0 wpxjq “ i tanh x 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (34) A derivation of formulae (33), (34) and their generalisations to multiple sums (see below) is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For the branch point twist field of free fermions and bosons [8,9] a formula almost identical to (33) also holds, albeit without the term ´ 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This term in fact makes the generalisation of this sum to multiple sums more complex for Tµ than for T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' It can nonetheless be done as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Let us consider, as an example, the next sum in the series, namely a sum of the form n´1 ÿ j1,j2“0 wpp´xqj1qwpyj1´j2qwpzj2q “ n´1 ÿ j“0 f1px, y´j, nqwpzjq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (35) Repeated use of (33) and (34) to simplify (35) leads to n´1 ÿ j1,j2“0 wpp´xqj1qwpyj1´j2qwpzj2q “ ´ i n ˆ tanh x 2 ` tanh y 2 ` tanh z 2 ˙ `1 4 cosh ´ x`y`z 2 ¯ cosh x 2 cosh y 2 cosh z 2 “ 2wpx ` y ` zq ` wpx ` y ` z ` 2iπq ` wpx ` y ` z ´ 2iπq ‰ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (36) 8 This special case gives a good indication of the kind of structures that emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We observe that the contribution in the second line has exactly the same structure as found for the branch point twist field in the free fermion theory [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The terms in the first line form a symmetric polynomial on the variables tanh x 2, tanh y 2, tanh z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The general structure for higher sums goes as follows: let us define fℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x2ℓ, nq :“ 2ip´1qℓ sinh x 2 ś2ℓ i“1 2 cosh xi 2 Fℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq , gℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x2ℓ`1, nq :“ 2p´1qℓ`1 cosh x 2 ś2ℓ`1 i“1 2 cosh xi 2 Gℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq (37) where x :“ ř i xi in both cases, and Fℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq :“ ℓÿ j“1 ˆ2ℓ ´ 1 ℓ ´ j ˙ ” wpxj´ 1 2 q ` wpx´j` 1 2 q ı (38) Gℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq :“ ˆ2ℓ ℓ ˙ wpxq ` ℓÿ j“1 ˆ 2ℓ ℓ ´ j ˙ ” wpxjq ` wpx´jq ı (39) with lim |x|Ñ8 Fℓ px ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ sgnpxq i n22ℓ´1 , lim |x|Ñ8 Gℓ px ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ sgnpxq i n22ℓ (40) and Fℓ px ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1q “ 22ℓ´1i coth x 2 , Gℓ px ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1q “ 22ℓ´1i tanh x 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (41) We can then compute the sum (32) to n´1 ÿ j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=',j2ℓ´1“0 wpp´x1qjqwpx j1´j2 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' wpx j2ℓ´2´j2ℓ´1 2ℓ´1 qwpxj2ℓ´1 2ℓ q “ fℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x2ℓ, nq ` p´1qℓ n ℓ´1 ÿ j“0 σp2ℓq 2j ˆ tanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ 2 ˙ , (42) whereas a similar sum involving an even number of indices can be evaluated to n´1 ÿ j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=',j2ℓ“0 wpp´x1qjqwpx j1´j2 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' wpxj2ℓ´1´j2ℓ 2ℓ qwpxj2ℓ 2ℓ`1q “ gℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x2ℓ`1, nq ` ip´1qℓ n ℓ´1 ÿ j“0 σp2ℓ`1q 2j`1 ˆ tanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ`1 2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (43) In both formulae, σpℓq j pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , aℓq is the elementary symmetric polynomial of order j in ℓ vari- ables, defined as σpℓq 0 pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , aℓq “ 1 and σpℓq j pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , aℓq “ ÿ 1ďi1ăi2㨨¨ăijďℓ ai1ai2 ¨ ¨ ¨ aij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (44) 9 These formulae can be proven by induction, similar to computations presented in [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The proofs are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' An interesting property of the formula (42) and a consistency check of its validity is the fact that we must recover the cumulant expansion of xµp0qµprqy{xµy2 for n “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Indeed, from (41) it follows that fℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x2ℓ, 1q “ p´1qℓ`1 cosh x 2 ś2ℓ i“1 cosh xi 2 “ p´1qℓ`1 ℓÿ j“0 σp2ℓq 2j ptanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (45) Then, in the limit n Ñ 1 the only term remaining from the sum (42) is the symmetric polynomial σp2ℓq 2ℓ ptanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ 2 q which is just the product of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This agrees exactly with the cumulant expansion of logxµp0qµprqy given in [18], formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='12a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Similarly, it can be shown that gℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x2ℓ`1, 1q “ ip´1qℓ`1 sinh x 2 ś2ℓ`1 i“1 cosh xi 2 “ ip´1qℓ`1 ℓÿ j“0 σp2ℓ`1q 2j`1 ptanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ`1 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (46) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='2 Main Result from this Section In summary, putting together the cumulant expansion (30) with the sum formula (42) we have the following exact formula for the logarithm of the correlation function of composite twist fields in the Ising model: log ˜ xTµp0qT : µ prqy xTµy2 ¸ “ 8 ÿ ℓ“1 cTµ 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ 8 ÿ ℓ“1 n 2ℓp2πq2ℓ » – 2ℓ ź i“1 ˆ `8 ´8 dθi e´mr cosh θi fi fl ” fℓpˆθ12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , ˆθ2ℓ´1 2ℓ, ˆθ1 2ℓ, nq ` p´1qℓ n ℓ´1 ÿ j“0 σp2ℓq 2j ˜ tanh ˆθ12 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh ˆθ2ℓ´1 2ℓ 2 , tanh ˆθ1 2ℓ 2 ¸fi fl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (47) We now proceed to check this expression for consistency by examining its leading short-distance behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 4 Conformal Dimensions from the Cumulant Expansion One possible way to test the cumulant expansion of the previous section is to obtain the correct conformal dimension of the field Tµ by identifying the leading short-distance contributions to the sum over cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Note that this dimension was already recovered by ∆-sum rule in [17], but the computation in that case only involved the two-particle form factor of Tµ, whereas our study below involves all cumulants, thus providing a more extensive test of all the form factors and cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Each cumulant is expected to contain a leading contribution which is proportional to log mr so that the overall sum gives (21) with dimension given by (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 10 First, let us return to our sum (32) and change variables once more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We define xi “ ˆθi,i`1 for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , 2ℓ ´ 1 , x2ℓ “ θ2ℓ , (48) so that: θi “ 2ℓ ÿ j“i p´1qj´ixj , 2ℓ ÿ i“1 θi “ ℓÿ i“1 x2i´1 , ˆθ1,2ℓ “ 2ℓ´1 ÿ i“1 p´1qi´1xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (49) The Jacobian of the transformation from the θ variables to the x variables is an upper triangular matrix with the diagonal terms being all `1, so the measure acquires no extra factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Applying this change of variables to (32) and expressing the result in the new variables (48) we get: n´1 ÿ j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=',j2ℓ´1“0 wpp´x1qjqwpx j1´j2 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' wpx j2ℓ´2´j2ℓ´1 2ℓ´1 qwpp 2ℓ´1 ÿ i“1 p´1qi´1xiq j2ℓ´1q “ p´1qℓ 2i sinh ´řℓ i“1 x2i´1 ¯ 2 cosh ˆř2ℓ´1 i“1 p´1qi´1xi 2 ˙ ś2ℓ´1 i“1 2 cosh ` xi 2 ˘Fℓ ¨ ˝2 ℓÿ i“1 x2i´1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' n ˛ ‚ ` p´1qℓ n ℓ´1 ÿ j“0 σp2ℓq 2j ˜ tanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ´1 2 , tanh ř2ℓ´1 i“1 p´1qi´1xi 2 ¸ , (50) with Fℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq the function defined by (38), with the asymptotics (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Recalling (45) it is possible to also express the sum over symmetric polynomials in terms of products of hyperbolic functions as ℓ´1 ÿ j“0 σp2ℓq 2j ˜ tanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ´1 2 , tanh ř2ℓ´1 i“1 p´1qi´1xi 2 ¸ “ cosh ´řℓ i“1 x2i´1 ¯ cosh ˆř2ℓ´1 i“1 p´1qi´1xi 2 ˙ ś2ℓ´1 i“1 cosh ` xi 2 ˘ ´ tanh ř2ℓ´1 i“1 p´1qi´1xi 2 2ℓ´1 ź i“1 tanh xi 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (51) This rewriting will prove useful later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 Exponential Factors Now let us look at the exponential factors in the integrand of (30) and see what they look like in terms of the new variables xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' From the first relation in (49) one has: 2ℓ´1 ÿ j“i p´1qj´ixj “ # θi ´ θ2ℓ for i even θi ` θ2ℓ for i odd ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (52) 11 so that 2ℓ ÿ i“1 cosh θi “ cosh θ2ℓ ` 2ℓ´1 ÿ i“1 coshpθi ´ θ2ℓ ` θ2ℓq “ cosh θ2ℓ ` cosh θ2ℓ ¨ ˝ ÿ i even coshpθi ´ θ2ℓq ` ÿ i odd coshpθi ` θ2ℓq ˛ ‚ ` sinh θ2ℓ ¨ ˝ ÿ i even sinhpθi ´ θ2ℓq ´ ÿ i odd sinhpθi ` θ2ℓq ˛ ‚ “ cosh x2ℓ » —–1 ` 2ℓ´1 ÿ i“1 cosh ¨ ˝ 2ℓ´1 ÿ j“i p´1qj´ixj ˛ ‚ fi ffifl ` sinh x2ℓ » —– 2ℓ´1 ÿ i“1 p´1qi sinh ¨ ˝ 2ℓ´1 ÿ j“i p´1qj´ixj ˛ ‚ fi ffifl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (53) Therefore, since none of the functions in (50) depends on x2ℓ the integral on this variable can be carried out by making use of the identity ˆ `8 ´8 dt expp´A cosh t ´ B sinh tq “ 2K0 ´a A2 ´ B2 ¯ , (54) giving ˆ `8 ´8 dx2ℓ e´mr ř2ℓ i“1 cosh θi “ 2K0pmrd2ℓ´1q , (55) with d2 2ℓ´1 “ » —–1 ` 2ℓ´1 ÿ i“1 cosh ¨ ˝ 2ℓ´1 ÿ j“i p´1qj´ixj ˛ ‚ fi ffifl 2 ´ » —– 2ℓ´1 ÿ i“1 p´1qi sinh ¨ ˝ 2ℓ´1 ÿ j“i p´1qj´ixj ˛ ‚ fi ffifl 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (56) The mr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1 expansion of the modified Bessel function is: K0pmrd2ℓ´1q “ ´ log mr ` log 2 ´ ln d2ℓ´1 ´ γ ` opmrd2ℓ´1q , (57) from which the leading short distance contributions to the cumulant expansion can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' It is worth mentioning that one could also resum contributions proportional to the constant term log 2 ´ γ in (57) and those should contribute to the KTµ-term in (21), that is to the logarithm of xTµy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' A similar computation was carried out in [9] for xT y in the free boson theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='2 Short-Distance Behaviour of the Cumulant Expansion Putting together (50), (51) and (55) in (30) we can split the cumulant into three contributions cTµ 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ cp1q 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq ` cp2q 2ℓ prq ` cµ 2ℓprq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (58) 12 We will define these contributions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' First: cp1q 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ 2p´1qℓin ℓp4πq2ℓ ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dx2ℓ´1 K0pmrd2ℓ´1q ˆ sinh ´řℓ i“1 x2i´1 ¯ cosh ˆř2ℓ´1 i“1 p´1qi´1xi 2 ˙ ś2ℓ´1 i“1 cosh ` xi 2 ˘ ˆFℓ ¨ ˝2 ℓÿ i“1 x2i´1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' n ˛ ‚, (59) with ˆFℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq :“ Fℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq ´ sgnpxq i n22ℓ´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (60) This shift is motivated by the asymptotics (40) and ensures that the function ˆFℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq goes to zero for |x| large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This in turn ensures the convergence of the integrals even when the Bessel function is approximated by its leading short-distance contribution ´ logpmrq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The next contribution is then a combination of the first term in (51) and the term introduced by the shift (60): cp2q 2ℓ prq “ p´1qℓ ℓp2πq2ℓ ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dx2ℓ´1 K0pmrd2ℓ´1q ˆ » ———– cosh ´řℓ i“1 x2i´1 ¯ ´ sinh ´řℓ i“1 x2i´1 ¯ sgnpřℓ i“1 x2i´1q cosh ˆř2ℓ´1 i“1 p´1qi´1xi 2 ˙ ś2ℓ´1 i“1 cosh ` xi 2 ˘ fi ffiffiffifl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (61) Note that this contribution is n-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Finally, the contribution cµ 2ℓprq is nothing but the cumulant of the expansion of xµp0qµprqy{xµy2 resulting from the last term (the product of tanh functions) in (51): cµ 2ℓprq “ p´1qℓ`1 ℓp2πq2ℓ ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dx2ℓ´1 K0pmrd2ℓ´1q ˆtanh ˜ř2ℓ´1 i“1 p´1qi´1xi 2 ¸ 2ℓ´1 ź i“1 tanh ˆxi 2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (62) This may look a bit different from the cumulant presented in [18] but this is simply due to the change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Note also that in [18] they implicitly take xµy “ 1 in the cumulant expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='3 Leading Contribution to cp1q 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq In order to evaluate the integral (59) we can perform yet another (and final!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=') change of variables: y “ ℓÿ i“1 x2i´1 ñ x2ℓ´1 “ y ´ ℓ´1 ÿ i“1 x2i´1 , 2ℓ´1 ÿ i“1 p´1qi´1xi “ y ´ ℓ´1 ÿ i“1 x2i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (63) 13 so that, at short distances ř ℓ cp1q 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq « ´zn logpmrq with zn “ 8 ÿ ℓ“1 p´1qℓ2ni ℓp4πq2ℓ ˆ `8 ´8 dy sinh y ˆFℓp2y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dx2ℓ´2 » –sech ˜ y ´ řℓ´1 i“1 x2i´1 2 ¸ ℓ´1 ź i“1 sech ˆx2i´1 2 ˙fi fl » –sech ˜ y ´ řℓ´1 i“1 x2i 2 ¸ ℓ´1 ź i“1 sech ˆx2i 2 ˙fi fl “ 8 ÿ ℓ“1 p´1qℓ2ni ℓp4πq2ℓ ˆ `8 ´8 dy sinh y ˆFℓp2y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nqG2 ℓpyq , (64) where, exactly as in [8,9]: Gℓpyq “ ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dxℓ´1 » –sech ˜ y ´ řℓ´1 i“1 xi 2 ¸ ℓ´1 ź i“1 sech ˆxi 2 ˙fi fl “ ˆ `8 ´8 dap2πqℓ´1eiay coshℓ πa , (65) and the functions Gℓpyq can be evaluated explicitly to Gℓpyq “ p2πqℓ´1 pℓ ´ 1q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' $ ’ & ’ % y π sinh y 2 ś ℓ 2 ´1 j“1 p y2 π2 ` p2jq2q for ℓ even 1 cosh y 2 ś ℓ´1 2 j“1p y2 π2 ` p2j ´ 1q2q for ℓ odd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (66) By replacing Fℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq by ˆFℓpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq in (59), we have ensured that the integrals (64) are convergent since Gℓpyq sinh y is asymptotically polynomial in y and ˆFℓp2y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq is exponentially decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' They can be evaluated with great precision and fitted to the function zn “ 1 12 ˆ n ´ 1 n ˙ ` 1 4n ` z1 , (67) with z1 “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='217p4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This gives 4∆Tµ plus an additional constant z1 which should be cancelled by contributions coming from cp2q 2ℓ prq ` cµ 2ℓprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Numerical results for zn are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' It is interesting to observe that there is very good agreement with the formula (67) for n integer and also for n not integer, greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' However for 1 ă n ă 2 the numerical data differ from (67) suggesting that the analytic continuation of (64) to n “ 1 from n real greater than 1 is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This is in agreement with results found in [17] where the limit n Ñ 1 of the two-particle form factor contribution produced a delta-function term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='4 Leading Contribution to cp2q 2ℓ prq ` cµ 2ℓprq Consider now the leading contribution to the second term in the cumulant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This is independent of n and employing the same change of variables as above, it is easy to write an expression which 14 ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='7 n zn ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='10 n zn Figure 1: Left: The function zn evaluated numerically through the sum (64) for integer values of n “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , 10 (red squares) against the formula (67) (blue solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Right: The same comparison for n P r1, 3s including non integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' When evaluating the sum (64) numerically we truncate at some value of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This value of ℓ is different for each value of n and is chosen so that the sum is stable up to 5 decimal digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' is given by a convergent integral involving the functions Gℓpyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Letting ř ℓ cp2q 2ℓ prq « ´z2 logpmrq we have that z2 “ 8 ÿ ℓ“1 2p´1qℓ ℓp2πq2ℓ ˆ `8 0 dy e´y G2 ℓpyq “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='0326p1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (68) Note that z1 ` z2 “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='250p0q « ´1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (69) Remarkably, this value is precisely what we need to recover the correct dimension of the field Tµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This is because we know that 8 ÿ ℓ“1 cµ 2ℓprq « ´1 4 logpmrq , (70) as this is the sum over cumulants corresponding to the two-point function xµp0qµprqy{xµy2 and µ has dimension 4∆µ “ 1{4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Therefore, the overall leading short distance behaviour of the xTµp0qT : µ prqy{xTµy2 cumulants correctly predicts the conformal dimension (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This highly non-trivial result provides strong support for the formula (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In addition, the structure of the cumulants means that we can also write xTµp0qT : µ prqy xTµy2 “ Rpr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq xµp0qµprqy xµy2 , (71) where Rpr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ ś8 ℓ“1 ecp1q 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='nqecp2q 2ℓ prq has the property Rpr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1q “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 15 Recalling the observation of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='2, namely that the cumulant expansion of Tµ posed some convergence issues, we note that those issues did not feature in the computations of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This is because by writing the cumulant as we have done, all convergence issues have been “hidden” in the contribution cµ 2ℓprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Indeed, a naive expansion of the Bessel function in (62) leads to a divergent integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Nonetheless, as shown in [18], the short distance limit of this quantity can be obtained via a semiclassical approach and it ultimately leads to the expected result (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 5 Analytic Continuation to n P Rě1 All results obtained so far are valid for n P Z`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This is always the case in the replica picture where n represents a replica number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' However, the entanglement measures that our two-point function describes are typically defined for generic positive n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Therefore it is interesting to try and write an expression for the correlation function which is valid for n P Rě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Let us start by studying the analytic continuation of the leading short-distance terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='1 Analytic Continuation of Leading Short-Distance Contributions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1 (right) strongly suggests that our formula needs to be analytically continued in the region 1 ă n ă 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' A similar problem was addressed in [8,9], where is was shown that as n approaches 1 from n " 1 some of the poles of the cumulants will cross or pinch the real line and provide additional contributions to the cumulant expansion which are non-vanishing for n P R and need to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The correct analytic continuation is obtained when these contributions are correctly accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The discussion is nearly identical as for the free boson case [9], albeit involving different functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' As we have seen, only the contribution cp1q 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq to the cumulant is n-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Therefore we only need to analytically continue the coefficient of the leading short-distance contribution to this term, that is zn defined in (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For non-integer n larger than 1, zn picks up additional contributions which account for the residues of the poles of ˆFℓp2y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq that cross the real axis as n Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The sum (38) in the function ˆFℓp2y, nq has kinematic poles at2 2y ˘ p2j ´ 1qiπ “ p2kn ` 1qiπ and 2y ˘ p2j ´ 1qiπ “ p2kn ´ 1qiπ for k P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (72) These poles result are due to the kinematic poles of the two-particle form factor (12) at θ “ iπ and θ “ iπp2n´1q, together with those resulting from the periodicity property wpθq “ ´wp´θ` 2πinq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This gives rise to four families of poles y1 “ pkn ` 1 ´ jqiπ, y2 “ pkn ´ jqiπ, k P Z (73) y3 “ pkn ´ 1 ` jqiπ, y4 “ pkn ` jqiπ, k P Z, (74) 2The twist field approach assumes n integer larger than 1 (since n is a copy number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For that reason it is natural to look for an analytic continuation to n “ 1 from n ą 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' However, once found, the analytic continuation should be unique, thus valid for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 16 with corresponding residues of the function inside the sum (64) given by: R1pℓ, j, k, nq “ np´1qℓ`j ℓp4πq2ℓ ˜ 2ℓ ´ 1 ℓ ´ j ¸ sinhpiπknqG2 ℓppnk ´ j ` 1qiπq, (75) R2pℓ, j, k, nq “ ´np´1qℓ`j ℓp4πq2ℓ ˜ 2ℓ ´ 1 ℓ ´ j ¸ sinhpiπknqG2 ℓppnk ´ jqiπq, (76) R3pℓ, j, k, nq “ np´1qℓ`j ℓp4πq2ℓ ˜ 2ℓ ´ 1 ℓ ´ j ¸ sinhpiπknqG2 ℓppnk ` j ´ 1qiπq, (77) R4pℓ, j, k, nq “ ´np´1qℓ`j ℓp4πq2ℓ ˜ 2ℓ ´ 1 ℓ ´ j ¸ sinhpiπknqG2 ℓppnk ` jqiπq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (78) These functions are all zero for n integer but they contribute for non-integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Let us now investigate which of these poles cross the real line in the limit n Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Since there are many indices involved, let us start by considering just one example: n “ 4 3 and ℓ “ 3 in the sum (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' According to the formula (20) 4∆Tµ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='236111 in this case but the numerical evaluation of (64), after subtracting the constant z1, gives the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='243211 which slightly overestimates the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The disagreement is not simply due to numerical imprecision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The function ˆF3py, 4{3q has poles that cross the integration line as n Ñ 4{3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' From (74) and the definition (38) we see that for ℓ “ 1 the sum runs only over the value j “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For j “ 1 the four families of poles labeled by the integer k are: y1 “ iknπ, y2 “ pkn ´ 1qiπ, k P Z (79) y3 “ iknπ, y4 “ pkn ` 1qiπ, k P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (80) It is clear that all these poles are always above the real line (for k ą 0) or below the real line (for k ă 0), that is they never cross the real line, as n approaches 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Therefore there is no correction coming from the ℓ “ 1 contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Let us consider ℓ “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Now j “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For j “ 1 the poles are the same as above and never cross the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For j “ 2 we have the following four families: y1 “ ipkn ´ 1qπ, y2 “ pkn ´ 2qiπ, k P Z (81) y3 “ ipkn ` 1qπ, y4 “ pkn ` 2qiπ, k P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (82) We have already seen above that the poles y1 and y3 never cross the real line, so we can only have some contributions from y2 and y4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For k ą 0 and n positive and large both families of poles are above the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' However, for n “ 4 3 we see that the pole pkn ´ 2qiπ crosses the real line for k “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Similarly, for k ă 0 and n positive and large all poles are in the lower half plane but the pole pkn ` 2qiπ crosses the real line for n “ 4 3 and k “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In summary, there are two poles for j “ 2 located at ˘2πi 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The corresponding residue contributions are 2πipR2p2, 2, 1, 4{3q ´ R4p2, 2, ´1, 4{3qq “ ´0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='00680653 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (83) 17 Therefore, the addition of the residua of these two poles improves the estimate of the conformal dimension from 4∆Tµ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='243211 to 4∆µ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='243211 ´ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='00680653 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='236404 which is much closer to the exact value (note that the formula (64) gives -4∆Tµ, hence the minus sign of (83)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The addition of poles for higher values of j will bring this value ever closer to formula (20) as shown in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In the general n case, in order to fully identify those poles that will cross the ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='10 n z\uf111n Figure 2: The function zn evaluated numerically through the sum (64) for n P r1, 3s (red squares) against the formula (67) (green dashed line) and its analytically continued values (blue triangles) given by (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' real line we find once more four cases: y1 : kn ` 1 ´ j ă 0 ñ 1 ď k ă j ´ 1 n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' y2 : kn ´ j ă 0 ñ 1 ď k ă j n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' y3 : kn ´ 1 ` j ă 0 ñ ´j ´ 1 n ă k ď ´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' y4 : kn ` j ă 0 ñ ´ j n ă k ď ´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (84) This gives the analytically continued values ˆzn ˆzn “ zn ` 8 ÿ ℓ“1 ℓÿ j“1 r j´1 n s´q1 ÿ k“1 inp´1qℓ`j`1 ℓp4πq2ℓ´1 ˜ 2ℓ ´ 1 ℓ ´ j ¸ sinh piπnkq G2 ℓ ` pnk ´ j ` 1q iπ ˘ ` 8 ÿ ℓ“1 ℓÿ j“1 r j n s´q2 ÿ k“1 inp´1qℓ`j`1 ℓp4πq2ℓ´1 ˜ 2ℓ ´ 1 ℓ ´ j ¸ sinh piπnkq G2 ℓ ` pnk ´ jq iπ ˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (85) where we used the fact that the residues R2pℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ ´R4pℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq and R1pℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ ´R3pℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq (which produces a factor 2) and multiplied by 2πi as required by the residue 18 theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The shifts q1, q2 take the value 1 when nr j´1 n s “ j ´ 1 and nr j ns “ j, respectively and are zero otherwise (they can be removed by requiring n to be non-integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Here the symbol r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='s represents the integer part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 2 shows the same functions as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1 (right) plus an additional set of values, which are the analytically continued values of zn (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' As we can see these now agree perfectly with the fit (67), even for non-integer n between 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='2 Analytic Continuation of the n-Derivative Applications of the correlation function (47) in the context of entanglement measures frequently requires the computation of its derivative with respect to n followed by the limit n Ñ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' As discussed in [6,8] and [17] the derivative with respect to n of the function (37) has a discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' More precisely, as n approaches 1 and poles cross the real line, the derivative is not uniformly convergent as a function of θ and this leads to terms involving δ-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The simplest examples of this phenomenon are seen for the two-particle contribution to the two-point function of T [6] and of Tµ [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Here we show how this generalises to the whole cumulant sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Notice that we only need to consider the contribution from the function cp1q 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq in (59) since all other terms are independent of n and so the derivative is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' For this term, we actually only need to consider Fpx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq as the additional term in the “hatted” version is also n-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' So, we define sTµ 2ℓ prq :“ ´ lim nÑ1 d dncp1q 2ℓ pr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ 2p´1qℓ`1i ℓp4πq2ℓ ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dx2ℓ´1 K0pmrd2ℓ´1q ˆ sinh ´řℓ i“1 x2i´1 ¯ cosh ˆř2ℓ´1 i“1 p´1qi´1xi 2 ˙ ś2ℓ´1 i“1 cosh ` xi 2 ˘ lim nÑ1 d dn » —–nFℓ ¨ ˝2 ℓÿ i“1 x2i´1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' n ˛ ‚ fi ffifl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (86) One way to treat the derivative is to recall the ℓ “ 1 result that was derived in [17], namely lim nÑ1 d dnnf1px, x, nq “ ´ i 2 sinh x cosh2 x 2 lim nÑ1 d dnnrwp2x ` iπq ` wp2x ´ iπqs “ x cosh2 x 2 sinh x ´ π2 2 δpxq , (87) that is, there is a finite part and a distribution part that accounts for the behaviour around x “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Recall that the function f1px, y, nq is defined in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This extends to higher cumulants in similar ways, so that we can write sTµ 2ℓ prq “ sfin 2ℓ prq ` sδ 2ℓprq , (88) where the two contributions represent the “finite” and δ-function contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The finite part can be easily computed by noting that lim nÑ1 d dnn sinh xFp2x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq “ i22ℓ´1x sinh x , (89) 19 Therefore sfin 2ℓ prq “ p´1qℓ ℓp2πq2ℓ ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dx2ℓ´1 K0pmrd2ℓ´1q ˆ řℓ i“1 x2i´1 sinh ´řℓ i“1 x2i´1 ¯ cosh ˆř2ℓ´1 i“1 p´1qi´1xi 2 ˙ ś2ℓ´1 i“1 cosh ` xi 2 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (90) The δ-function contribution is a generalisation of the ℓ “ 1 case seen above and can be obtained by identical arguments as those presented in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In fact, the result is also identical to formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='6) in [8], that is, sδ 2ℓprq “ π2p´1qℓ ℓp4πq2ℓ ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dx2ℓ´1 δp ℓÿ i“1 x2i´1q ˆ » ———– ˜ 2ℓ ´ 2 ℓ ´ 1 ¸ 2K0p2mrd2ℓ´1q cosh ˆř2ℓ´1 i“1 p´1qi´1xi 2 ˙ ś2ℓ´1 i“1 cosh ` xi 2 ˘ fi ffiffiffifl ´π2p´1qℓ ℓp4πq2ℓ ˆ `8 ´8 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' ˆ `8 ´8 dx2ℓ δp ℓÿ i“1 x2i´1q ˆ ℓÿ j“1 j´1 ÿ k“1 ÿ q“˘ » ———– ˜ 2ℓ ´ 1 ℓ ´ j ¸ p´1qj ś2ℓ i“1 e´rm cosh ´ř2ℓ j“1p´1qj´ixi`iπq j´k 2ℓ ¯ cosh ˆř2ℓ´1 i“1 p´1qi´1xi 2 ˙ ś2ℓ´1 i“1 cosh ` xi 2 ˘ fi ffiffiffifl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (91) 6 Conclusion In this paper we have studied the normalised two-point function xTµp0qT : µ prqy{xTµy2 of the composite twist field Tµ and its conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The motivation to study this object comes from recent investigations of a measure of entanglement known as symmetry resolved entanglement [15– 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' More fundamentally, our work contributes to developing the understanding of correlation functions in the replica Ising field theory, a theory that, although free and seemingly simple, contains a large number of symmetry fields or twist fields which are not present in the standard, non-replicated, model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The current work uses traditional IQFT techniques, mainly the form factor bootstrap pro- gram adapted to composite twist fields [17], to expand the logarithm of the correlation function into a series of cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The main result of the paper is finding simplified expressions for these cumulants which result from proving a number of multiple sum formulae, presented in Appendix A, involving the two-particle form factors of the field Tµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Employing this cumulant expansion we have found the following structure xTµp0qT : µ prqy xµp0qµprqy xµy2 xTµy2 “ Rpr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq with Rpr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 1q “ 1 , (92) 20 where µ is the disorder field of the Ising field theory, and Rpr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq has an explicit form given in terms of a multiple integral representations which we describe in detail in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' By exact resummation of leading contributions to the cumulant expansion, we have shown that at short distances this two-point function scales as a power law in r with the power (20) which is exactly predicted by CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Showing that the two-point correlation function of the field µ factors out of the correlator (92) has been a crucial ingredient in recovering the correct scaling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We have also shown how our formulae may be analytically continued to real replica number and how non-analytic, delta-function terms emerge for all cumulants when computing the derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' This generalises results found in [17] and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' As a byproduct of our investigation, we have also shown that the form factors of the composite field Tσ can be obtained from those of Tµ via clustering in momentum space, in much the same way as the form factors of the fields σ and µ are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We have also fixed the normalisation of the Tσ form factors by fixing the one-particle form factor from the asymptotics of the two-particle form factor of Tµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' As mentioned above, our result has applications in the context of the symmetry resolved entanglement entropy and directly leads to a more complete formula for the latter in the Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' We further expect the results of this investigation to apply with some modifications to other composite fields, at least for free theories, for instance those associated with Up1q symmetry in doubled free models which were studied in [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Acknowledgments: The authors thank Benjamin Doyon and D´avid X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Horv´ath for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Michele Mazzoni is grateful for funding under the EPSRC Mathematical Sciences Doctoral Training Partnership EP/W524104/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Olalla A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Castro-Alvaredo thanks EPSRC for financial support under Small Grant EP/W007045/1 and the Kavli Institute for Theoretical Physics (Santa Barbara) for financial support from the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' NSF PHY-1748958, and hospitality during the conference “Talking Integrability: Spins, Fields and Strings”, August 29-September 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' A Summation formulae Let us first prove the identities (34) and (33), which are both obtained via the cotangent trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' To show (34), consider the integral: 1 2πi ˛ C dz π cotpπzqwpxzq (93) where C is the rectangular contour with vertices ´ϵ ` iL, ´ϵ ´ iL, n ´ ϵ ´ iL, n ´ ϵ ` iL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The vertical contributions cancel off because the integrand is invariant under the shift z Ñ z ` n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The same holds for the horizontal contributions in the large L limit, as lim LÑ8 cot πpt ˘ iLq “ ¯i , lim LÑ8 w ´ xt˘iL¯ “ ¯ i n (94) and therefore the sum of the residues must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Within the integration contour, the function π cotpπzq has simple poles at z “ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' n´1 with unite residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The kinematic poles of wpxzq 21 are at z “ 1 2 ´ x 2πi, z “ n ´ 1 2 ´ x 2πi, with residue 1 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' At both these points, cotpπzq “ ´i tanh x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Putting all the pieces together, one has therefore: 0 “ n´1 ÿ j“0 wpxjq ´ i tanh x 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (95) Using the very same strategy, one can prove (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' The integral to evaluate is now 1 2πi ˛ C dz π cotpπzqwpp´xqzqwpyzq , (96) along the same contour C as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In this case, however, the horizontal contributions do not cancel off, as lim LÑ8 wpp´xqt˘iLqqwpyt˘iLq “ ´ 1 n2 , (97) and thus the integral evaluates to ´ 1 n in the large L limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Summing over the residues of the poles of π cot pπzq gives the left-hand side of (33), while the kinematic poles are now at z “ 1 2 ` x 2πi , ´1 2 ` n ` x 2πi and z “ 1 2 ´ y 2πi , ´1 2 ` n ´ y 2πi, with residues: Res z“ 1 2 ` x 2πi wpp´xqzqwpyzq “ 1 2πwpx ` y ` iπq , Res z“n´ 1 2 ` x 2πi wpp´xqzqwpyzq “ 1 2πwpx ` y ´ iπq Res z“ 1 2 ´ y 2πi wpp´xqzqwpyzq “ ´ 1 2πwpx ` y ´ iπq , Res z“n´ 1 2 ´ y 2πi wpp´xqzqwpyzq “ ´ 1 2πwpx ` y ` iπq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Evaluating the cotangent at the kinematic poles and putting all the pieces together, one has ´ 1 n “ ÿ Resrπ cotpπzqwpp´xqzqwpyzq “ n´1 ÿ j“0 wpp´xqjqwpyjq ` i 2 sinh ´ x`y 2 ¯ cosh ` x 2 ˘ cosh ` y 2 ˘rwpx ` y ` iπq ` wpx ` y ´ iπqs , which is indeed (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Using this result, it is possible to prove (42) and (43) by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' It is useful to observe beforehand that the following expansions hold: sinh ´řk i“1 xi ¯ śk i“1 cosh xi “ r k´1 2 s ÿ j“0 σpkq 2j`1ptanh x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh xkq (98) cosh ´řk i“1 xi ¯ śk i“1 cosh xi “ r k 2 s ÿ j“0 σpkq 2j ptanh x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh xkq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (99) Following the procedure employed in [8, 9] we will prove that (42) implies (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' If (42) holds, than we can shift x2ℓ by ´2iπp, multiply the left-hand side by a factor wpxp 2ℓ`1q and sum over 22 p to obtain: n´1 ÿ j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=',j2ℓ´1,p“0 wpp´x1qjqwpx j1´j2 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' wpx j2ℓ´2´j2ℓ´1 2ℓ´1 qwpxj2ℓ´1´p 2ℓ qwpxp 2ℓ`1q “ n´1 ÿ p“0 fℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x´p 2ℓ , nqwpxp 2ℓ`1q ` n´1 ÿ p“0 p´1qℓ n ℓ´1 ÿ j“0 σp2ℓq 2j ˜ tanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x´p 2ℓ 2 ¸ wpxp 2ℓ`1q , (100) Let us focus on the first term in the second line, which yields two contributions due to the presence of a constant term in the right-hand side of (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Defining x “ ř2ℓ i“1 xi, this reads: 2ip´1qℓ sinh x 2 ś2ℓ i“1 2 cosh xi 2 n´1 ÿ p“0 ℓÿ j“1 ˆ2ℓ ´ 1 ℓ ´ j ˙ „ w ´ xj´p ´ iπ ¯ ` w ´ x´j´p ` iπ ¯ȷ wpxp 2ℓ`1q “ gℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x2ℓ`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq ` 4ip´1qℓ sinh x 2 n ś2ℓ i“1 2 cosh xi 2 ℓÿ j“1 ˆ2ℓ ´ 1 ℓ ´ j ˙ “ gℓpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , x2ℓ`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' nq ` ip´1qℓ n ℓ´1 ÿ j“0 σp2ℓq 2j`1ptanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ 2 q (101) The emergence of the function gℓ was already proved in Appendix C of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In going from the second to the third line we used the first identity in (98) and řℓ j“1 `2ℓ´1 ℓ´j ˘ “ 22ℓ´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' Consider now the second term in the second line of (100): using the fact that tanh x˘p 2 “ tanh x 2 and (34) we have: n´1 ÿ p“0 p´1qℓ n ℓ´1 ÿ j“0 σp2ℓq 2j ˜ tanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x´p 2ℓ 2 ¸ wpxp 2ℓ`1q “ ip´1qℓ n ℓ´1 ÿ j“0 σp2ℓq 2j ˆ tanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ 2 ˙ tanh x2ℓ`1 2 (102) Now we observe that the elementary symmetric polynomial of degree j in ℓ variables can be decomposed as σpℓq j pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , aℓq “ σpℓ´1q j pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , aℓ´1q ` σpℓ´1q j´1 pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , aℓ´1q aℓ, hence ℓ´1 ÿ j“0 « σp2ℓq 2j`1ptanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ 2 q ` σp2ℓq 2j ˆ tanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ 2 ˙ tanh x2ℓ`1 2 ff “ ℓ´1 ÿ j“0 σp2ℓ`1q 2j`1 ptanh x1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' , tanh x2ℓ`1 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' (103) Thus the sum of (101) and (102) yields (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' In an analogous way it is possible to prove (42) starting from (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfx_4Y/content/2301.01745v1.pdf'} +page_content=' 23 References [1] B.' metadata={'source': 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a/JtE2T4oBgHgl3EQfpQiY/content/tmp_files/2301.04027v1.pdf.txt b/JtE2T4oBgHgl3EQfpQiY/content/tmp_files/2301.04027v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f0aaf01cea8e007c6d0260363125adf63c2d4b7 --- /dev/null +++ b/JtE2T4oBgHgl3EQfpQiY/content/tmp_files/2301.04027v1.pdf.txt @@ -0,0 +1,1768 @@ +1 + +Differentiable modeling to unify machine learning and physical models and +advance Geosciences + +Chaopeng Shen1*, Alison P. Appling2, Pierre Gentine3, Toshiyuki Bandai4, Hoshin Gupta5, Alexandre +Tartakovsky6, Marco Baity-Jesi7, Fabrizio Fenicia7, Daniel Kifer8, Li Li1, Xiaofeng Liu1, Wei Ren9, Yi +Zheng10, Ciaran J. Harman11, Martyn Clark12, Matthew Farthing13, Dapeng Feng1, Praveen Kumar6,14, Doaa +Aboelyazeed1, Farshid Rahmani1, Hylke E. Beck15, Tadd Bindas1, Dipankar Dwivedi16, Kuai Fang17, +Marvin Höge7, Chris Rackauckas18, Tirthankar Roy19, Chonggang Xu20, Kathryn Lawson1 + +1 Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, USA. +2 U.S. Geological Survey, Water Mission Area, Integrated Modeling and Prediction Division, Reston, VA, +USA +3 National Science Foundation Science and Technology Center for Learning the Earth with Artificial +Intelligence and Physics (LEAP), Columbia University, New York, NY USA +4 Life and Environmental Science Department, University of California, Merced, CA, USA +5 Hydrology and Atmospheric Sciences, The University of Arizona, Tucson, AZ, USA. +6 Civil and Environmental Engineering, University of Illinois, Urbana Champaign, IL, USA +7 Eawag: Swiss Federal Institute of Aquatic Science and Technology, Dübendorf, Switzerland +8 Computer Science and Engineering, The Pennsylvania State University, University Park, PA, USA +9 Department of Natural Resources and the Environment, University of Connecticut, Storrs, CT, USA +10 Southern University of Science and Technology, Shenzhen, Guangdong Province, China +11 Department of Environmental Health and Engineering, Johns Hopkins University, Baltimore, MD, USA +12 Global Institute for Water Security, University of Saskatchewan, Canmore, Alberta, Canada +13 US Army Engineer Research and Development Center, Vicksburg, MS, USA +14 Prairie Research Institute, University of Illinois, Urbana Champaign, IL, USA +15 Physical Science and Engineering Division, King Abdullah University of Science and Technology, +Thuwal, Saudi Arabia +16 Lawrence Berkeley National Laboratory, Berkeley, CA, USA +17 Department of Earth System Science, Stanford University, Stanford, CA, USA +18 Computer Science and Artificial Intelligence Laboratory (CSAIL), Massachusetts Institute of +Technology, Massachusetts, USA +19 Civil and Environmental Engineering, University of Nebraska-Lincoln, NE, USA +20 Earth and Environmental Divisions, Los Alamos National Laboratory, NM, USA +* Corresponding author, email cshen@engr.psu.edu + + + +2 + + +Abstract +Process-Based Modeling (PBM) and Machine Learning (ML) are often perceived as distinct paradigms +in the geosciences. Here we present differentiable geoscientific modeling as a powerful pathway toward +dissolving the perceived barrier between them and ushering in a paradigm shift. For decades, PBM offered +benefits in interpretability and physical consistency but struggled to efficiently leverage large datasets. ML +methods, especially deep networks, presented strong predictive skills yet lacked the ability to answer +specific scientific questions. While various methods have been proposed for ML-physics integration, an +important underlying theme — differentiable modeling — is not sufficiently recognized. Here we outline +the concepts, applicability, and significance of differentiable geoscientific modeling (DG). “Differentiable” +refers to accurately and efficiently calculating gradients with respect to model variables, critically enabling +the learning of high-dimensional unknown relationships. DG refers to a range of methods connecting +varying amounts of prior knowledge to neural networks and training them together, capturing a different +scope than physics-guided machine learning and emphasizing first principles. Preliminary evidence +suggests DG offers better interpretability and causality than ML, improved generalizability and +extrapolation capability, and strong potential for knowledge discovery, while approaching the performance +of purely data-driven ML. DG models require less training data while scaling favorably in performance and +efficiency with increasing amounts of data. With DG, geoscientists may be better able to frame and +investigate questions, test hypotheses, and discover unrecognized linkages. + +Introduction +Geoscientific models encompass a wide range of domains, with evolving scopes and ever-increasing +societal importance, especially in the face of rapid climate change. For example, hydrologic models help +us manage water resources1,2 and plan for extremes such as floods and droughts3; vegetation models can +help predict the fate of carbon and other key biogeochemical cycles on land4 or in the ocean5; agricultural +models estimate crop yields and also their environmental impacts6; geophysical models aim to predict land +surface changes via processes like landslides7, land subsidence8, and earthquakes; biogeochemical reactive +transport models aim to understand and predict surface and subsurface water chemistry and quality9,10. +Combining many such components, Earth System Models11–13 and integrated assessment models14–16 +provide crucial guidance for resource managers and policy makers17,18. The uses of such models go beyond +making predictions of the future to also facilitating communication with the stakeholders and aiding in the +policy-making process18. +Geoscientific models often share some commonalities as they describe the dynamic responses of +systems to time-dependent forcings as modulated by semi-static attributes. Many such problems can be +described as systems of nonlinear equations, algebraic differential equations, or ordinary and/or partial +differential equations (ODE/PDEs), along with parameterizations (empirical representations) of physical +processes with spatially-varying parameters. The overall system can contain multiple processes chained +together, some of which are well understood while others are not19,20. Further, many of these process +representations and parameterizations are subject to considerable uncertainty, some of which is related to +scale, and thus has significant room for improvement. Here we argue that differentiable implementations +of geoscientific models offer a transformative approach to simultaneously advancing process +representations, parameter estimation, and predictive accuracy. In particular, differentiable +implementations provide an unprecedentedly seamless connection between process-based and machine- +learning-based model components, potentially enabling us to realize the value and minimize the limitations +of each. + + +3 + +Value and limitations of process-based geoscientific modeling +The traditional process-based modeling (PBM) approach has served the geosciences well in helping +to improve our understanding of system functions and behaviors. Due to their physical basis, they can be +leveraged in hypothesis testing to assess system responses, and cause-effect relationships (see the Physical +Laws row in Table 1), e.g., the impacts of land use changes on flooding trends21 and future warming on +glacial melt22. Further, they can simulate a wide variety of observed (e.g., discharge or leaf area index) and +unobserved variables (e.g., groundwater recharge or fine-root carbon). Such an ability is critical to both +advancing scientific understanding and to providing a narrative when communicating with the public and +stakeholders, who are engaged in the decision-making process23. It is possible to ask and examine specific +questions regarding processes within the modelled system, by progressively improving the representations +of processes 24–27 and evaluating them using controlled experiments. +Despite these benefits, there remain important challenges with PBMs. +(1) Process-based models often cannot rapidly evolve with and fully exploit the information in “big +data” due to the time needed to develop and test process representations and parameterizations28,29. +Traditionally, the differences between model predictions and observations are first reconciled by +parameter calibration, which adds significant uncertainty (more about this later)30. For model errors +beyond parameter adjustments, modelers then hypothesize different causes, implement structural +changes to the model, and iteratively confront the updated model hypotheses with the data24. This +iterative process is highly expensive (in both labor and time) and dauntingly complex, and is +dependent on developer intuition and legacy31. Consequently, it is common for the structural +representation of a specific process in a geoscientific model to stagnate, with years or decades +passing between structural updates32–35. +(2) Process-based models are limited by knowledge gaps. Extensive physical, biological, and +socioeconomic knowledge is required to achieve adequate representations and updates for +processes in a geoscientific model, and any deficiencies can amplify errors and ambiguity. Another +major challenge is the interactions of processes across disciplinary boundaries36. For instance, +vegetation, human management, and socioeconomic systems all interact with each other and affect +the water and carbon and other biogeochemical cycles37–40. While the intersections of these domains +will continue to stimulate scientific discovery, a new paradigm could enable us to make faster +progress despite knowledge gaps. + +Potential and limitations of machine-learning-based geoscientific modeling +Irrespective of the domain of application, one cannot help but notice the “Cambrian explosion” of +purely data-driven machine learning (ML) approaches, especially deep neural networks (NNs), applied to +a wide range of scientific applications36,41 (see Discussion A in S1, Supplementary Discussion). In +geosciences, NNs have shown strong accuracy in predicting crop production42,43, precipitation fields44,45 +and clouds46, water quality variables47,48 such as water temperature49–52, dissolved oxygen53, phosphorous54, +and nitrogen55,56, and the full hydrologic cycle57 including soil moisture58–60, streamflow61–64, +evapotranspiration65–67, groundwater levels68, and snow69, etc. Deep networks like long short-term memory +(LSTM) networks70, graph neural networks63, and convolutional neural networks (CNNs)71,72 have become +widely known in geosciences. Many such studies reported noticeably better performance than conventional +approaches, revealing that the latter did not fully exploit the information in the data28 (Table S1 in +Supplementary Information S1). +Nevertheless, there remain important challenges with purely data-driven ML: + +4 + +(1) Deep networks are data hungry. The success of deep networks relies on the availability of "big +data”, which can, unfortunately, be sparse for many geoscientific pr oblems56,73, where many +variables are measured at dozens, hundreds, or thousands of sites only. For example, water quality +data are sparse and inconsistent in temporal, spatial, and chemical coverage74,75. For rare and +extreme events such as mega floods, droughts, and earthquakes, available data is even scarcer. +(2) ML has difficulties with errors, incompleteness, or bias in the inputs or observations. The quality +of ML models is limited by the quantity, diversity, and quality of training data52,76. Since a purely +data-driven model can, at best, nearly-perfectly replicate the patterns in the training data, it +invariably inherits various issues from the training data including implicit or explicit biases, +inadequate spatiotemporal resolutions (e.g., with satellite-based observations), and the inability to +account for non-stationarity in time series due to the short data record. +(3) Neural Networks remain challenging to interpret. Although explainable AI methods such as +layerwise relevance backpropagation77–79 can be highly helpful in revealing some of the internal +workings of a network and should be pursued, they are not designed to flexibly query a model or +identify missing physics. +(4) Purely data-driven ML models cannot predict untrained variables (those not provided as training +targets). Due to their very nature, ML-based models are designed to only output the training targets. +They cannot provide an account of how events unfolded, e.g., the ability to state that “the flood +occurred because the soil was saturated” in a study where soil moisture is unobserved. This hinders +both formation of hypotheses and communication with stakeholders. +(5) Most geoscientific ML algorithms capture correlations and not causality regarding both attributes +and temporal changes. There are always confounding covarying factors in data, so that ML models +can produce the “right” results for the wrong reasons, potentially making projections less reliable +when circumstances are changed. + +The root of deep network’s success – Differentiable Programming +Considering both the exceptional successes and limitations of ML and especially NNs, one can ask: +What are the foundational strengths of NNs? +How can we maximize these strengths while overcoming the limitations associated with data? +How can we extract knowledge in an interpretable form while maintaining ML-level performance? +In answering these questions, we argue that differentiable programming (explained below) is the +computing paradigm that supports the efficient training of NNs which, in turn, can deliver many +philosophically and practically transformative outcomes. Traditional modeling has been dealing with +optimization problems for decades (see the Similarity block in Table 1). However, it is argued here that +only by exploiting the power of parallelized gradient-based optimization have we been able to learn from +big data and train the large numbers of weights (parameters) necessary to approximate complex unknown +functions. +The ability of generic NN architectures such as CNNs, LSTMs, and attention mechanisms to +approximate unknown functions has achieved desirable outcomes (Figure 1 & Table 1). First, the cost of +learning a few generic architectures is lower than the significant domain expertise required by traditional +models, making NNs suitable for widening access to usable predictions. Second, NNs can help in the +identification of previously unrecognized linkages. Third, NN training can scale up favorably with the +amount of data (in terms of accuracy, generalizability, and efficiency)76,80, in contrast with traditional +modeling where the learning may saturate after some limited calibration of parameters or functions52. + +5 + +All of these features are possible only because we can now train NNs with a large number of weights, +providing a large learnable function space81,82. The number of weights easily exceeds the optimization +capabilities of conventional algorithms. The most recent computer vision model contains two billion +weights83 and LSTM models widely employed in hydrology can contain ~500,000 weights. In contrast, +traditional evolutionary84–86, or genetic87 or particle swarm optimization methods88 can hardly handle more +than a few dozen independent parameters (Table 1). +The computing paradigm that enables efficient training of so many parameters is Differentiable +Programming89,90 (Figure 1), where accurate derivatives of the model outputs with respect to inputs and/or +intermediate variables can be efficiently computed. Without getting into details, this paradigm is often (but +not always) enabled by ML platforms, which support reverse- or forward-mode automatic differentiation +(AD)89–91 using various approaches. Models written on these platforms can, often without much effort, be +programmatically differentiable – even where certain operations are mathematically indifferentiable (e.g., +thresholding or if statements), the fact that they are piecewise differentiable enables gradient computations +to be performed. The chain rule can be applied to efficiently accumulate the derivatives in a process called +“backpropagation”92. Note that differentiability is normally only needed for training, not when running the +model in forward mode. +Here we expand the scope and use the term differentiable modeling to include any method that can +produce the gradients rapidly and accurately at scale. A non-AD example is that of adjoint methods, which +solve accompanying equations (called adjoint equations)93–95 for the derivatives. AD differentiates through +the code in an automated manner and is independent of the problem, while adjoint methods differentiate +through the mathematical model equations and thus require manual derivations of adjoint equations for +each problem96. Many alternative gradient estimation methods, e.g., finite differences, are intractable for +any reasonably-sized NNs (10,000 weights would require 10,001 forward model evaluations) and can be +challenged by stiffness. Cheaply obtained gradients allow for parameter updates via various first-order +gradient-descent methods97. Second-order methods, such as Newton Raphson, have not gained popularity +for the training of NNs due to the cost of computing the Hessian matrix. The vast majority of NNs are +implemented on platforms supporting differentiable programming, while most existing PBMs are not. +Historical differences in the training of geoscientists vs ML practitioners (Education row in Table 1) +may give the impression that ML and process-based modeling are fundamentally unrelated, but the +perceived divide is more of a legacy issue now that differentiable modeling is broadly accessible. In reality, +both ML and parametric physical models can be expressed in nearly identical mathematical forms +(Mathematical form row in Table 1), and the code forms also converge when both process-based and ML +components are implemented within the differentiable programming paradigm. +This leads us to the conclusion of this section: differentiable programming is the core distinguishing +feature of neural networks, and differentiable modeling can serve as the basis for unifying NN and +process-based geoscientific modeling. As we will discuss in the following sections, this unification requires +only minor modifications to our conceptual modeling and implementation strategies, but it opens new doors +to scientific discovery. +Table 1. Similarities and differences between purely data-driven neural networks and purely process- +based models. [Pro] annotates the comparative strengths, also shown in green text. In the equations, W +stands for weights of the neural network ������������; ������������ stands for the physical parameters of the process-based +model f; x, u and A are dynamic forcings, state variables, and semi-static attributes, respectively; and L +represents the loss function which quantifies the difference between simulation outputs and observations. + +Purely data-driven neural networks +Purely process-based models + +Similarities + +6 + +Mathematical +form +������������ = ������������������������(������������, ������������, ������������) +������������ = ������������������������������������������������������������������������(������������(������������, ������������∗)) +������������ = ������������������������(������������, ������������, ������������) +������������ = ������������������������������������������������������������������������(������������(������������, ������������∗)) +Programmatically +differentiable +Yes +Traditionally no, but could be reimplemented on +differentiable platforms or supported by new +libraries + +Differences +Ease of use +[Pro] Generic model architecture – Easy to +develop even without domain expertise. +Specialized domain knowledge +Architecture +[Pro] Generic structure with a large number of +weights that allow the model to approximate a +wide range of functions. +Specific structural priors representing human +understanding of physics, with a small number of +parameters +Data +[Pro] Capable of efficiently learning from and +obtaining scaling benefits from big data. +Typically calibrated at a few sites, or a few +parameters are calibrated in a regionalization +equation. Learning saturates at a small data +quantity. +[Pro] The potential to overcome data limitations in +accuracy, resolution, and availability. +Training/ +Calibration +[Pro] +Trained +using +gradient +descent, +supported by differentiable programming. +Calibrated using various small-scale algorithms. +Normally code does not support differentiable +programming. +Unknown +processes +[Pro] Data can be used to make up for +processes we are not certain about. This also +means we can learn unrecognized connections +and expand knowledge. +We must specify the processes to be used in the +model, even if they are only assumptions. +Outputs +Output trained variables only. + +[Pro] Output many intermediate variables that +facilitate providing an interpretable full narrative. +Physical laws +May not fully respect physical laws. +[Pro] Respect physical laws. Help us to assess +cause-effect relationships. +Interpretation +Difficult to interpret +[Pro] Elucidate physical processes, allowing us to +ask specific science questions. +Education +Taught in computer science or data science +curricula. +Taught in engineering or science curricula. + + + +7 + + + +Figure 1. (a) ML (blue boxes) gives us great results with easy-to-use models, resulting from the +complexity of neural networks (many parameters) and the technologies that make it feasible to train such +complex models. The most fundamental of these technologies is differentiable programming. (b) In the +DG paradigm, which incorporates differentiable non-ML model components (physically based structural +priors), we can now obtain additional great features (plum boxes) while retaining and augmenting the old +ones (blue and blue-plum boxes, respectively). + +Differentiable Geosciences: Absorbing the core power of scientific ML into geoscientific domains +What is differentiable modeling in the geosciences? +Here we advocate for a new modeling paradigm: “Differentiable Geoscientific modeling”, or simply +“Differentiable Geosciences” (DG). DG refers to the use of models intermingling process-based +descriptions and NNs to simulate geoscientific processes, update our physical process representations, learn +physically meaningful parameters, quantify uncertainty, etc. DG allows us to replace poorly-understood or +low-accuracy process-based model components with ML components that may be more accurate, while + +a) Machine learning (ML) paradigm +Technology +Scientific Benefits +KnowledgeOutcomes +ay +Differentiable programming +TechnicalBenefits +Represent complex& +ap +Highlyaccurate predictions +unknownprocesses +Efficient convergence +despite many parameters +Gain accuracy & +Progress despite knowledge +Data-driven training methods +generalizability with big data +gaps +b) Differentiable geosciences (DG) paradigm +Scientific Benefits +Represent complex& +unknownprocesses +Technology +Gain accuracy,efficiency& +KnowledgeOutcomes +generalizability with big data +ay +Differentiableprogramming +TechnicalBenefits +Highlyaccuratepredictions +ap +Efficient convergence +Interpret &narrate model +despite many parameters +behavior +Progress despiteknowledge +Data-driventrainingmethods +gaps +亚 +Narrowersearchspace,still +目目 +Learn causalityfrom precise +including true function +physical questions +Physicallybased structural +Reduction of knowledge +priors +gaps +Rapidexperimentation with +Predict meaningful yet +process representations +untrained variables +Overcome data quality& +quantity limitations8 + +retaining those process-based model components that we already trust or want to improve. DG may also +exploit gradients for other purposes such as sensitivity analysis or trajectory optimization. A distinct feature +of DG is its full programmatical differentiability – that is, the whole model needs to support gradient +calculation from the start to the end of the workflow – to ensure that we can incorporate neural network +units that can adapt to and evolve from data. The process-based descriptions retained in the model can be +called the structural priors. DG seeks to marry the core of NN models – their optimizing and learning +capabilities – to geoscientific process descriptions. +DG can be considered a branch of scientific machine learning98,99 that emphasizes improving process +representations and understanding. With DG, we trade the model genericity for physical interpretability, +with minimal compromises to accuracy. DG reduces the cost (in terms of data) of finding good solutions +because the structural priors serve to constrain the model. Meanwhile it also scales well with data quantity +and can reap the benefits of big data, just as does purely data-driven ML. There are two perspectives from +which we can view DG models (Figure 2): +(a) they are ML models constrained to a smaller searchable space by the structural priors. +(b) they are PBMs augmented with learnable and adaptable components (and thus an expanded +searchable space) provided by NNs. +In DG, NNs can be commissioned in a wide variety of ways, ranging from learning parameters100 to +updating assumptions used in the model (e.g., process representations)76, and from estimating time- +dependent forcing terms to describing the whole space-time solution101. The next section provides some +forms of use cases, and examples are provided in Classes of DG methods with examples section below. DG +is different from previous concepts of physics-guided machine learning (PGML) or not-fully-differentiable +models in the methodology (must be fully differentiable), mission (to advance process understanding), and +philosophy (whether treating physical law as truth or not). Please see Supplementary Information S1, +Discussion C. + +From technical breakthrough to philosophical change – why will DG be transformative? +While efficient gradient calculation may appear to be merely a technical change, it is likely to +transform our modeling philosophy and scientific objectives. First, the ability to approximate complex, +unknown functions greatly broadens the type of questions we can ask, by enabling us to treat trusted +components as priors and focus on improving uncertain model components, one at a time. To explain this +idea in concise mathematical terms, let us consider a physics-based model y=g(u, x, θ) where u, x, θ +represent state variables, dynamic forcings, and physical parameters, respectively (This representation +encompasses differential equations, i.e., ∂u/∂t= g(u, x, θ), but is more generic). Traditional inversion +algorithms only estimate the parameters, i.e., asking “θ =?”) while requiring that the functional form g be +assumed a priori (except for some rigid methods, e.g., nonparametric regression, which require complicated +derivations and specialized training algorithms, and thus have not gained popularity) []. However, +differentiable models allow us to ask questions about the functional form, i.e., “������������ =?”, by training a neural +network (NN) (or parameterized functions) to replace ������������: y = NNW(u, x, θ) where W is the high-dimensional +weights (see examples later). Hence, with DG, we now can place our question mark precisely in the model. +The functions to estimate could be +(i) a parameterization scheme, as done in differentiable parameter learning100: ������������ = ������������(������������, ������������, ������������ = +������������������������������������(������������)); +(ii) a module in a model, e.g., where we can replace ������������3 in ������������ = ������������(������������1 , ������������2, ������������3(������������, ������������, ������������)) with NN: ������������ = + ������������(������������1 , ������������2, ������������������������������������(������������, ������������, ������������)), as Feng et al.102 optionally replaced the runoff function; or + +9 + +(iii) a part of a governing equation or constitutive laws, e.g., we can estimate ������������������������������������ in ������������������������/������������������������ = +������������(������������1 , ������������2, ������������������������������������(������������, ������������, ������������))103,104. +In the above equations, physical process equations provide a backbone for the overall model; e.g., in (i) the +physical backbone is ������������; in (ii) and (iii) the physical backbone is ������������, ������������1, ������������2 and ������������3. The unchanged parts +(structural priors), i.e., ������������, ������������1, ������������2 in (ii) and (iii), critically serve as physical constraints, allowing us to +isolate and focus our attention (and data) on the most unknown model components. We may gain insights +by simply visualizing the relationships learned by NNW 63,105 or applying knowledge distillation methods106. +We are also able to evolve better process representations for some model components like ������������1 or ������������2 +mentioned above, e.g., the relation between soil moisture and effective rainfall in conceptual hydrologic +models, without needing a full understanding of all the processes. This precision of questioning is opposed +to some popular off-the-shelf interpretive AI approaches, e.g., layerwise relevance propagation77,107, +Shapley additive explanations108, or local surrogate methods109, that are limited to only asking a few fixed +questions, e.g., which parts of the inputs caused this result? Moreover, in geoscientific modeling, directly +interpreting the trained sensitivities may be risky – with only limited measurement sites, the trained +relationship related to the spatial attributes tends to be overfitted. +DG provides a framework for combining deductive reasoning and inductive learning. Purely data- +driven models are inductive and seek to derive almost all relationships from data, whereas process-based +models first posit hypotheses and then test those hypothesis using data, albeit facing many challenges in +doing so. The DG paradigm posits a user-defined number of structural priors, and then identifies many +other parts of the model from data. This design follows the traditional scientific approach that identifies +parsimonious models to reflect the general properties of the phenomenon, along with a quantification of the +predictable aspects that are not yet understood110. Moreover, differentiable, learnable models can and have +obtained state-of-the-art performance that can match fully data-driven models (Supplementary Information +S1, Discussion B). + + +Figure 2. Differentiable models can be viewed as (A) machine learning models guided into a smaller +searchable space by structural priors or (B) process-based models with expanded search space supported +by learnable units. The background fill colors indicate model optimality, related to the cost function if we +had infinite data. + +X +Process-Based +optimal +B +Differentiable Geosciences +Machine Learning +searchablefunctionspace10 + + +Why is differentiable modeling particularly valuable for the geosciences? +First of all, geoscientific data are strongly imbalanced in spatial extent, temporal coverage, and in +terms of variables of interest. While satellites can measure leaf area index111 or coarse-resolution surface +soil moisture112,113 all over the world, there are a limited number of sites measuring photosynthesis rates114 +or streamflow, especially in Africa and Asia115; and there is very limited knowledge of subsurface properties. +Purely data-driven ML may be biased or stymied by these data limitations, which may be overcome by the +inclusion of physics. Indeed, preliminary analysis shows that differentiable models with a physical model +as the backbone can outperform LSTM in regional extrapolation116. +The second major motivation is system nonstationarity induced by climate change, which could drive +many systems out of the previously observed range of variability117. Data-driven methods are tailored to +the training data and may not maintain accuracy in the face of strongly changing conditions (this is a +nuanced statement as models like LSTM may have highly competitive scores even in long-term projection +tests62,116, but nonetheless experience large declines in accuracy when faced with nonstationary processes). +Careful testing suggests adding stronger priors may lead to better future projections116. +As DG models can also output any diagnostic (latent) variable available from the process-based +equations within the DG model, we can perform model conditioning and/or data assimilation operations +with sparse and scattered data. By conditioning, we mean constraining the model using an observable to +learn more realistic parameters or processes so that the overall model dynamics are better. For example, +satellite-based soil moisture data can condition a hydrologic model to better predict vegetation water use100 +or primary productivity; streamflow can constrain a model to better simulate snow water equivalent118. For +data assimilation, the model can use recent observations of B to improve the short-term forecast of A, as B +can also help to update our model state variables. +Physical parameters play key roles in geoscientific models in modulating the behaviors of the system. +Parameter estimation transfers information from either (i) raw observable physiographical variables or (ii) +fine-scale dynamics to parameters. Quite often, we have no ground truth information for the parameters and +they require inversion using observations or high-resolution simulations. Parameter estimation has, for +decades, been fraught with uncertainty, ambiguity, and frustration. Due to different parameters producing +very similar output and their sensitivity to spatiotemporal resolutions, calibration at a geographic location +can often lead to nonunique inference (sometimes referred to as “equifinality”)119–121. Extending parameters +to unmonitored locations requires “regionalization”, which also introduces uncertainty. Because of +increasing geospatial data availability, parameter estimation is an area where machine learning is well- +poised to make significant progress. A novel aspect is that, as with purely data-driven ML, DG methods +provide favorable scaling relationships – more training data leads to improved performance, efficiency, and +generalizability100 (discussed in Supplementary Information Text S1). + +What are the promises of differentiable modeling in geosciences? +We hope to evolve differentiable models so that we can gain process knowledge while improving the +model predictions. Success can be claimed if we obtain models with the following features: +(i) +Predictive accuracy and transferability equal to or superseding purely data-driven models for +extensively measured variables; +(ii) +Models capable of structural evolution, i.e., we can improve the parameterization and +formulation of the processes; +(iii) +Accurate generalizability to data-sparse regions or into long-term future; +(iv) +Conservation of mass/energy/momentum; + +11 + +(v) +Consistency of internal physical fluxes and states that can provide a full narrative of the events +and full support to downstream processes; +(vi) +Permits efficient isolation of one uncertain model component at a time to learn physics with +less ambiguity. +This wish list is ambitious and yet partial. However, as shown below, some examples already +demonstrated the plausibility of these goals. + +Motivating questions for DG +With differentiable geosciences models, we hope to ask and answer the following types of questions: +a. +What is the relationship between variable x and variable y? +b. +What is the missing physics as part of the differential equation? +c. +What should have been the assumption or function here? +d. +How does factor A influence parameter β? +e. +Which process is causing phenomenon P? +f. +What will happen in new environmental conditions? +g. +What is the information content of datasets, either input or target data for training? +Most domains in geosciences could benefit from DG (Figure 3). To provide more concrete motivating +examples, we now list one example question that DG is primed to answer from each of the domains below +(ordered alphabetically): + +Agriculture: Can we predict crop phenology dynamics (e.g., planting, shooting, flowering, harvesting) and +assess potential production risk under future climate change (type f), which involves interconnected biotic, +abiotic, and human influences? DG can optimize model representations of more and less +understood components of this interconnected system for accuracy even in climate +extrapolations. + +Climate: Can we predict cloud processes and ocean eddies and their impact on climate sensitivity? PBM- +NN hybrids implemented with DG can help to improve cloud representations. + +Ecosystem: Should we parameterize ecosystem models regarding carbon and nutrient cycles on the plant +functional type level or the trait level (type c)? Testing the configurations of differentiable parameter +learning schemes could answer this question. + +Coastal: Can we better leverage emerging sensing platforms while improving our model representations of +sediment transport and nonlinear wave-wave interactions in order to infer nearshore bathymetry at large +scales (type g)? +Cryosphere: Can we leverage both physics and data to create more accurate models for ice dynamics within +the cryosphere and better constrain its fate under climate change(type f)? For example, the plumbing +system for melted water and its influence on ice-basal bed rock friction are two of the key components for +ice mass movement121,122, with increasingly available data. + + +12 + +Coastal: Can we better leverage emerging sensing platforms while improving our model representations of +sediment transport and nonlinear wave-wave interactions in order to infer nearshore bathymetry at large +scales (type g)? + +Geohazards: Can we use space-based observations of geohazards, e.g., landslides122, to quantify subsurface +properties (type d) so we can better predict future events (type f)? Space-based observations and combined +with differentiable parameter learning provides an opportunity to inversely estimate properties like soil +cohesion and friction angle which are challenging and expensive to measure. + +Hydraulics: How do we estimate floodplain hydraulic parameter values efficiently at large scales using +new sensing data (type a, d)? Estimation and inversion are most difficult problems facing the hydraulics +research community, e.g., Manning’s n for flow resistance and sediment transport rate. Another example is +bathymetry which is required to run any hydraulics model but hard to observe. + +Hydrology: How does global groundwater-dominated baseflow respond to climate change (type a)? What +is a proper, scale-appropriate way to parameterize groundwater storage and flow at the global scale (type +c)? For this question, we cannot answer it using a purely data-driven method, but could leverage +differentiable models for the diagnosis. + +Soil science: Can we find functional forms to express soil hydraulic properties (water retention and +hydraulic conductivity function) that describes non-equilibrium flow (type c or b)? + +Water quality: How and to what extent do denitrification rates vary across gradients of climate, vegetation, +land use, and geology conditions (type d) and thus how do they change under different climates. Nitrate is +one of the most widespread and persistent contaminants. Denitrification removes nitrate from water but the +rates and extent of denitrification however depend on an array of entangled environmental factors. + +13 + + +Figure 3. Differentiable Geosciences can help almost all geoscientific domains in knowledge discovery +and improving simulation quality. Green and blue highlighting is used to show how there can be multiple +uses for neural networks within a single model. + +Classes of DG methods with examples. +DG is a young modeling paradigm that could benefit from wider participation. This section briefly +describes early explorations of DG, categorized by how gradients are computed and employed. This section +also gives examples, which are by no means exhaustive, to explain the concepts and to inspire more +innovation. +I. +Directly differentiating through numerical models and connecting them to NNs. +Among the several options, directly differentiating numerical models is the most straightforward +method and is most similar to traditional models. Utilizing the AD functionality provided by modern ML +platforms, one can reimplement an existing model to obtain a differentiable model version (and ensure +reproducibility). Then the differentiable model is connected to NNs as discussed in the section “Why will +DG be transformative”. Because the model being trained is the same one for the forward simulation, the +physics is clearly enforced, and the user can apply the forward simulator for any initial, boundary and +forcing conditions. They can also migrate the learned relationships to existing implementations, e.g., the +national water model, to immediately support operations. However, reimplementing a model does incur +non-trivial initial development cost. Mathematical changes may be required to adapt previously non- +differentiable mathematical operations to be mathematically differentiable, e.g., by replacing indexing with +convolutions, and to improve parallel efficiency. While DG models may not always have to run on +Graphical Process Units (GPUs), enabling GPUs will improve the computational efficiency by orders of +magnitude, notwithstanding some current challenges (described in the Challenges to address for DG + +Water +Soil science +Cryosphere +quality +Hydraulics +Hydrology +Coastal +Ecosystem +Differentiable process-based mode + Geohazards +Climate +f(u, x, o, NNw(u, x, 0) ...) = y +Agriculture + Other +replacinga moduleina model +Forcings +parameterization +Observations +Attributes14 + +section). Our position is that in most cases, the cost is well justified due to the potential to interrogate into +the model, make changes, and learn physics. The reimplementation may provide a “reset” opportunity to +reexamine many of the habitually-made assumptions. +As an example, Feng et al.102 implemented the conceptual hydrologic model HBV (a system of ODEs) +on PyTorch and used coupled NNs for parameterization and optionally replaced processes with NNs (Figure +4a). Strikingly, they approached the performance level of LSTM, giving a median Nash Sutcliffe model +Efficiency coefficient (NSE) of 0.732 for the CAMELS streamflow benchmark, compared to LSTM’s 0.748 +for the same dataset, or 0.715 vs. 0.722 for another forcing dataset (Figure 4b). They also output untrained +variables such as evapotranspiration and baseflow, which agreed well with alternative estimates (Figure +4e). Moreover, in spatial extrapolation test cases, the differentiable model outperformed LSTM with respect +to daily metrics and decadal trends116 (Figure 4 c-d) due to the structural constraints, demonstrating its +potential for global hydrologic modeling. Similarly, Jiang et al.118 encoded the hydrologic model EXP- +HYDRO as a recurrent NN architecture and coupled it with fully connected NNs which served as the +parameterization pipeline as well as postprocessor to improve runoff. They showed that a symbiotic +integration between NN and physics led to robust transferability and that snow water equivalent was well +captured. In the Biogeosciences or ecosystem modeling, differentiable models found improved parameters +for photosynthesis123 at large scales. + +Apart from models similar to ODEs, direct differentiation can also be applied to models operating on +graphs representing the natural systems such as river networks. Bindas et al.124 created a differentiable river +routing model that was trained on daily discharge at a gauge downstream of a river network (with pretrained +LSTM producing runoff as inputs to the graph) to learn a parameterization scheme for Manning’s roughness +coefficient (n). They obtained a power-law-like curve between n and catchment area that was consistent +with the expected n behavior. Similarly, Bao et al.125 implemented an advective dispersion equation on the +river graph to simulate stream water temperature and found that the model performed better in data-sparse +situations. + + +15 + + +Figure 4. (From Feng et al.102,116. Reprint permission obtained). (a) Sketch of a differentiable hydrologic +model using process-based hydrologic model HBV as a backbone (b) For temporal test using NLDAS +forcings, δ models can approach the performance of LSTM and greatly outperform traditional +approaches; (c) For prediction in ungauged regions (PUR; train in some regions and test in another large +ungauged region), δ models can surpass the performance of LSTM; (d) For the PUR test, δ models can +predict long-term trends of annual flow percentiles more reliably than LSTM. (d) δ models can predict +high-quality evapotranspiration estimate (not used in training) compared to a satellite product for both +in-sample and spatial generalization tests. + +ADDITIONAL CLASSES, CHALLENGES TO DG, AND CONCLUDING REMARKS REDACTED +BEFORE PAPER ACCEPTANCE. + +Acknowledgements +We attribute many ideas of the paper to a discussion in the HydroML symposium, University Park, +PA, May 2022, https://bit.ly/3g3DQNX, sponsored by National Science Foundation EAR #2015680 and +Penn State Institute for Computational and Data Sciences. Content related to this paper was also presented +in some presentations, including Artificial Intelligence for Earth System Processes (AI4ESP) talk online +https://bit.ly/3etm5aI in Nov 2021. We thank Jordan Read and James McCreight for valuable internal +reviews of the manuscript. Shen was supported by National Science Foundation EAR-2221880 and Office +of Science, US Department of Energy under award DE-SC0016605. Gentine acknowledges funding from +the National Science Foundation Science and Technology Center, Learning the Earth with Artificial +intelligence and Physics (LEAP), award #2019625 and Understanding and Modeling the Earth System with +Machine Learning (USMILE) European Research Council grant. Marty Wernimont at the U.S. Geological + +(a) +(b) +1.0 +Differentiable process- +Parameter regionalization +based model - HBV* +0.8 +Precipitation/Temperature +snowfall +Rainfall +0.6 +Sp +Attributes +CDF +soil, land cover, +Static θ +0.4 +geology, others... ++E. +β/β: +or +OptionalNN +Forcing +ireplacement +0.2 +P, T, Ep +9A(A,x) +Dynamic θ +LSTM unit +Q1 +0.0 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +NSE +Q2 +zIs. +MPR+mHM +dPL+evolved HBV (On) +B=492 NSE50: 0.528 +B=671 NSE50: 0.692 +dPL+HBV (61) +dPL+evolved HBV with DP (On(βt, yt) +B=492 NSE50: 0.618 +B=671 NSE50: 0.711 +* Not all parameters and detailed processes of +dPL+HBV (1) +LSTM +HBV are sketched here for the sake of simplicity. +B=671 NSE50:0.628 +B=671 NSE50: 0.719 +(c) +(d) +(e) +Q98 +Q10 +ET Correlation +Annual mean +1.0 +LSTM R2=0.72 +R2=0.55 +R2=0.26 +0.8 +RMSE=0.368 +RMSE=2.873 +0.8 +RMSE=0.071 +0 +0.9 +50 +000 +111 +0.5 +0.6 +0.6 +5 - +25 +0.8 + Simulated trend +0.0 +0 +0.4 +0.4 +0.5 +0.7 +0 +0 +So +0.5 +0.5 +6(βt, yt) R2=0.88 +R2=0.78 +R2=0.28 +0.2 +0.2 +RMSE=0.287 +RMSE=2.732 +RMSE=0.087 +0.6 +1.0 +0.0 +ao +50 +0.0 +5 - +0.5 +25 +0.5 +98 +-0.2 +0 +0.0 +0- +-0.2 +0.4 +In-sample +PUB +PUR +NSE +KGE +0.5 +0 +0 +50 +6 +6(βt) +(βt, yt) +ILSTM +16 +1(βt) +6(βt, yt) +Observed trend16 + +Survey (USGS) greatly improved the presentation of Figures 1 and 2; Wernimont and Appling were +supported by the USGS Water Mission Area, Water Availability and Use Science Program. 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In hydrology, deep networks +such as long short-term memory (LSTM) networks70, and convolutional neural networks (CNNs)71,72 have +shown strong ability with regard to prediction of soil moisture58–60, water supply155, streamflow61–64, +evapotranspiration65–67, groundwater levels68, snow69, and other aspects of the water cycle57. In water +quality studies, LSTMs and CNNs have shown promise in simulating water temperature49–52, dissolved +oxygen53, phosphorous54, and nitrogen55,56, among others47,48. In agriculture, ML approaches have been +widely applied for crop production prediction156–158. In regional climate studies, CNN-based schemes or +generative algorithms have been found to improve the forecasting of precipitation fields44,45 and +prediction of clouds (deep clouds)46. Often the studies have reported state-of-the-art performance when +compared with conventional approaches. Typically, such high-quality predictions can be made even when +a good understanding of the underlying processes is not available. We made an effort to collect a list of +somewhat comparable studies with metrics for both traditional and ML models (Figure S3 and Table S1). +Previous models have been highly useful in advancing science, but these results imply that they were not +fully exploiting the information available in the data28, and they can benefit from leveraging the strength +of ML. + + +35 + +Table S1. ML vs. traditional model performances for a number of scientific applications with data from many sites. The metrics were computed +based on simulations and observations. The lower the values, the better for RMSE, while higher is better for Pearson’s correlation (COR), R2, and +Nash-Sutcliffe model efficiency coefficient (NSE). This is presented with many caveats, such as the ML model is optimized to match observations +while traditional models have many other constraints; a selection bias – where ML did not outperform did not get published (nevertheless, one +could also argue studies where PBM outperformed were not easily found). The point of this table was not to show that ML was always better, but +to support the argument that ML tends to have advantages in accuracy. Also note the limitations of ML we discussed in the Introduction. +Variable +Metric +Deep networks +Traditional +Reference +Stream +Temperature + +RMSE (°C) +1.91 +4.01 +Chen et al.159 +RMSE (°C) +0.89 +1.80 +Rahmani et al.160 and Daraio et al.161 +Pearson COR +0.99 +0.91 +Rahmani et al.160 and van Vliet et al.162 +R2 +0.942 +0.93 +Rahmani et al.160 +NSE +0.98 +0.93 +Rahmani et al.160 +Evapotranspiration + +R2 +0.67 +0.21 +He et al.163 +RMSE (mm/day) +1.21 +2.56 +NSE +0.65 +0.57 +Talib et al.164 +Soil Moisture +RMSE +0.027 +0.085 +Fang et al.58 +Pearson COR +0.87 +0.72 +RMSE +0.027 +0.035 +Pearson COR +0.87 +0.82 +Pearson COR +0.91 +0.77 +Liu et al.143 +RMSE +0.034 +0.08 + +Streamflow +NSE +0.76 +0.68 +Seibert et al.165 and Kratzert et al.62 +NSE +0.9 /0.68 +- +Mohamoud and Parmar166 +Mean R2 +0.71 +- +Merritt et al.167 +Dissolved oxygen + +NSE +0.78 +- +Zhi et al.53 +Median R2 +- +0.64 +Stefan and Fang168 +CC (correlation Coefficient) +0.972 +- +Heddam169 +Median NSE +0.760 +- +Keshtegar and Heddam170 + +36 + +B. Why can differentiable process-based models achieve state-of-the-art predictive performance? +Purely data-driven ML architectures have set a high bar for accuracy in multiple geoscience domains, +such that one would be tempted to predict a loss in accuracy when adding in less-flexible process-based +components. However, here it is argued that generic ML architectures are not necessary to achieve good +model accuracy. As long as some model components are adaptable and learnable, we can learn from data. +If we view the model as a more strongly constrained ML model (perspective “a” in Figure 2), it is easy to +see that there is a potential to achieve ML-level performance if we enlarge the searchable space of PBM to +include a good approximation of the true function, directed by gradient-based training. The paths we take +to upgrade the models will be expert-dependent (prior-dependent), so one should not expect a unified +approach at present. +Many dynamical systems in Geosciences can be written as ordinary differential equations (ODEs), +e.g., rainfall runoff in a basin, crop growth, or nutrient release. While solving these equations, we run the +numerical model for many steps. This is mathematically similar to recurrent neural networks, and the time +integration operation is similar to the functionality achieved by some neural networks like the Residual +Networks171,172. It should not be surprising that learnable process-based models with some ML components +can perform as well as deep networks. +As we discuss in Section S1, multiple studies have already shown that differentiable, learnable models +can approach the performance of purely data-driven models, or exhibit advantages in some cases where +extrapolation is key. Differentiable model formulations can maintain at least two of the three desirable +features: approximating complex, previously unknown functions, and the ability to assimilate information +from big data. Compared to purely data-driven ML, DG trades genericity for interpretability and the ability +to ask specific questions. Deep networks like CNNs, LSTMs, and transformers will be an ingrained part of +DG. Eventually, deep learning will become part of the repertoire of geoscientists, just like with numerical +methods173. + +C. How is DG related to physics-guided machine learning (PGML) and how are they different? +Many ML-physics integration strategies with a wide variety of complexity have been proposed in the +past in a seemingly scattered manner, such that a clear classification is difficult154. It has not been +sufficiently recognized that some of these algorithms work fundamentally because they leveraged the +differentiable programming tools. The scattered nature of those publications makes the landscape of ML- +physics integration daunting and confusing, while hindering us from making innovations based on first +principles. However, once we treat differentiability as the central tool, it serves as a compass to guide us in +understanding newly proposed methods, i.e., we can ask if a method is fully (end-to-end) differentiable, +how it uses gradients, how much prior information is inserted, what questions are asked, and how it scales +with data. Here we outline some similarities and differences between DG and some existing methods. +DG and physics-guided (or physics-informed, theory-guided, or knowledge-guided) machine learning +(PGML)174–176 both seek to combine physics with ML, but they differ in their approaches, purposes, and +philosophies. Many PGML studies seek to introduce physical constraints, e.g., as regularization or pre- +training, to ML methods to gain better generalizability with less training data. PGML does not in theory +need differentiable programming and partial physics could be enforced. In contrast, DG is more thorough +in that it uses the numerical physical model as the backbone and demands that the entire workflow be +differentiable. In terms of purposes, PGML is tasked to make the ML model more robust, while +differentiable modeling seeks to update our assumptions or discover new knowledge. Relatedly, in terms +of philosophies, when a physical law was introduced in PGML, it was treated as truth (albeit sometimes +with some tolerance level, as in Read et al.52). Often, this includes all the calculations and assumptions to + +37 + +support the law. In DG, we do not presume the physical laws to be correct, and, rather, are constantly +looking for opportunities to update existing knowledge. +There are many not-fully-differentiable methods that could be valuable for various applications but +are outside of the scope of DG for this paper. For one, it is possible to incorporate ML algorithms trained +offline on datasets as part of a physical model, e.g., training a neural network on turbulent heat fluxes and +inserting into a hydrologic model177; training pedotransfer functions to infer soil parameters from soil +hydraulic data178; training an atmospheric parameterization network on short-term cloud-resolving +simulations179; or training ocean-mixing parameterizations on data and physical constraints180. While this +approach has the advantage that the physical meaning of the NN is clear and stands alone, direct training +data are needed for the variable of interest (thus having issues with pure ML as discussed earlier) and the +network can no longer evolve and adapt in an interactive fashion, for instance to further update the model +when exposed to observations. In the future these NNs could be further incorporated into DG models. Some +other offline coupling methods include providing outputs of process-based models as inputs to neural +networks (this helps to integrate over spatiotemporal heterogeneity)181,182, or training ML models to predict +the PBM residuals150,183,184. Readers are referred to Reichstein et al.185 which promoted a number of ways +to connect physics and ML for geosciences, with a brief mention of differentiable programming. + +S2. Details for some examples. +Example 1. Part of the effort in Tsai et al.100, which proposed differentiable parameter learning (dPL), +connected the Variable Infiltration Capacity (VIC) process-based hydrologic model to a neural network (������������) +that estimates physical parameters of VIC (������������) using some widely available attributes (A): ������������ = ������������(������������). In an +“end-to-end” workflow, ������������ is then sent to VIC, whose outputs are compared with observations, effectively +turning the parameter calibration problem into a machine learning problem, trained on all sites +simultaneously using backpropagation and gradient descent (Figure S1a). As a result of this global loss +function, dPL exhibits advantages over traditional calibration on multiple fronts, for three different datasets +(soil moisture, CAMELS streamflow, and global headwater runoff). The parameter sets are spatially +coherent (Figure S1b-c) and extrapolate better in space (Figure S1d-e). dPL is hyper efficient: a job that +normally takes a 100-CPU cluster 2-3 days now takes a single Graphical Processing Unit (GPU) one hour. +dPL allows the combined model to output unobserved variables while addressing the notorious problem of +parameter equifinality119. As summarized earlier, these are the great advantages we expect to harness with +differentiable modeling. + + +38 + + +Example 2. Physics-informed neural networks (PINNs)101,140, while first published in 2017 before the +existence of the term “differentiable geosciences”, could be perceived as a genre of DG as the gradient +information is critically employed. PINNs pose problems in a unique way, seeking to train a neural network +with space-time coordinates as inputs, h(t,x) where x represents spatial coordinates and t is time such that +(i) h(t,x) agrees with known data points at (t,x), and (ii) the derivatives dh/dx, dh/dt, etc. agree with the +governing partial differential equations. Physical parameters could also be part of the inputs to the h +network153. PINNs are a highly innovative approach tested on a large variety of applications in many +domains, and there have been a number of good reviews of this work130,154. PINNs have made enormous +strides, with novel inversion uses such data assimilation140 and learning governing equations, but, as with +other methods, there are also some limitations. Obviously, the function h(t,x) is tied to the initial and +boundary conditions so it needs to be trained separately for different initial/boundary condition pairs, and +the form of the inputs limits the neural network to certain types (multilayer perceptron network) that are +not the easiest to train. However, the learned parameters and constitutive relationships can describe the +system under a wide range of boundary and initial conditions. Furthermore, the fidelity of the trained +network to physical equations must be carefully examined. +In geosciences, a PINN method for learning unknown parameter fields and constitutive +relationships was proposed104 (Figure S2). As an example, steady-state groundwater flow in an aquifer with +an unknown conductivity field and unsaturated flow in the vadose zone with an unknown pressure- +dependent conductivity were considered. In the unsaturated flow application, it was assumed that only +sparse measurements of pressure head were available. The quantities of interest were the unsaturated +conductivity as a function of the pressure head, and the pressure head field. Notably, it was assumed that +no measurements of the unknown parameters were available. In the proposed PINN method, both quantities +of interest were represented with neural networks (NNs) (with unknown parameters). This step created a +differentiable model of the unsaturated flow in the vadose zone. It was also assumed that the pressure head +measurements could be described by the steady-state Richards equation. Substituting the NN +approximations into this equation formed the axillary residual NN, which shared the (unknown) parameters + + + + + + + + + +Figure S1. (From Tsai et al.100, reprint allowed via Creative Commons Attribution 4.0 International +License, http://creativecommons.org/licenses/by/4.0/) (a) Structural diagram of one of the dPL +frameworks called gA; (b & c) The estimated infiltrating curve parameter (INFILT) from dPL vs. the site- +by-site calibrated shuffled complex evolutionary algorithm (SCE-UA);(d & e) dPL better matches the +MODIS satellite product for uncalibrated variable ET than does SCE-UA. +(a) + +(e) +(a) + +(d) +(e) +(c) +(b) + +Ideal Line +1000 +dPLgA +MODIS (mm/yr) +750 +Ensemble : +500 +(mean±std) +Bias =-22.75 ++ -34.30± 10.87 +250 +. Corr = 0.76 +0.73 ±0.02 +. +NSE =0.56 +0.50±0.05 +0 +0 +250 +500 +750 +1000 +dPLgaET (mm/yr)dPL -- 9A +Dynamic +PBM or its +inputs +surrogate +Static +Physical +attributes +Parameters(b) +Ideal Line +1000 +SCE +MODIS (mm/yr) +750 +Ensemble : +500 +(mean± std) +Bias =-58.31 ++59.34±0.84 +250 +Corr = 0.68 ++0.69±0.006 +NSE = 0.36 +0.38±0.01 +0 +0 +250 +500 +750 +1000 +SCE-UA ET (mm/yr)INFILT (-) from dPL (gz ) +(a) +(b) +INFILT (-) from SCE-UA +0.14 +0.40 +0.35 +0.12 +0.30 +0.10 +0.25 +0.20 +0.0839 + +with the primary NNs. For the primary NNs to satisfy the governing equation, the residual NN should be +zero everywhere in the domain – in other words, the exact measurements of the residuals are available +everywhere in the domain. The NNs were trained jointly using the pressure head measurements. Since the +conductivity and residual NNs share the same parameters, estimating parameters in the residual NN also +provides the parameterization of the conductivity NN. Figure S2a shows the reference pressure head field +and the locations of the measurements. Figure S2b shows the point errors in the estimated pressure head +field. The reference and estimated unsaturated conductivity functions are shown in Figure S2c. These +figures demonstrate that the PINN method can learn both the state variable and the constitutive relationship +very accurately. + + +Figure S2. (from Tartakovsky et al.104, reprint permission obtained) (a) The reference pressure head field +and the locations of the measurements. (b) The point errors in the estimated head field. (c) The reference +and estimated conductivities as functions of the pressure head. + + +Disclaimer +Any use of trade, firm, or product names is for descriptive purposes only and does not imply +endorsement by the U.S. Government. + +Supplemental References +… + + + + +0.0025 +-5 +8 +8 +0.0020 +-6 +6 +9 +0.0015 +-7 +4 +4 +0.0010 +-8 +2 +2 +0.0005 +-9 +2 +4 +6 +8 +2 +4 +6 +8 +X1 +X10.35 +O +Exact +Prediction +0.30 +0.25 +n) +0.20 +0.15 +0.10 +-10 +-9 +-8 +-7 +-6 +-5 +-4 +n \ No newline at end of file diff --git a/JtE2T4oBgHgl3EQfpQiY/content/tmp_files/load_file.txt b/JtE2T4oBgHgl3EQfpQiY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6dfc79beffb39d4bb9da6705e96a0264fd22df4d --- /dev/null +++ b/JtE2T4oBgHgl3EQfpQiY/content/tmp_files/load_file.txt @@ -0,0 +1,2095 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf,len=2094 +page_content='1 Differentiable modeling to unify machine learning and physical models and advance Geosciences Chaopeng Shen1*, Alison P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Appling2, Pierre Gentine3, Toshiyuki Bandai4, Hoshin Gupta5, Alexandre Tartakovsky6, Marco Baity-Jesi7, Fabrizio Fenicia7, Daniel Kifer8, Li Li1, Xiaofeng Liu1, Wei Ren9, Yi Zheng10, Ciaran J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Harman11, Martyn Clark12, Matthew Farthing13, Dapeng Feng1, Praveen Kumar6,14, Doaa Aboelyazeed1, Farshid Rahmani1, Hylke E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Beck15, Tadd Bindas1, Dipankar Dwivedi16, Kuai Fang17, Marvin Höge7, Chris Rackauckas18, Tirthankar Roy19, Chonggang Xu20, Kathryn Lawson1 1 Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 2 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Geological Survey, Water Mission Area, Integrated Modeling and Prediction Division, Reston, VA, USA 3 National Science Foundation Science and Technology Center for Learning the Earth with Artificial Intelligence and Physics (LEAP), Columbia University, New York, NY USA 4 Life and Environmental Science Department, University of California, Merced, CA, USA 5 Hydrology and Atmospheric Sciences, The University of Arizona, Tucson, AZ, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 6 Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' University of Illinois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Urbana Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 7 Eawag: Swiss Federal Institute of Aquatic Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Dübendorf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Switzerland 8 Computer Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 9 Department of Natural Resources and the Environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' University of Connecticut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Storrs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 10 Southern University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Guangdong Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' China 11 Department of Environmental Health and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' MD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 12 Global Institute for Water Security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' University of Saskatchewan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Canmore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Alberta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Canada 13 US Army Engineer Research and Development Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Vicksburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' MS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 14 Prairie Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' University of Illinois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Urbana Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 15 Physical Science and Engineering Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' King Abdullah University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Thuwal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Saudi Arabia 16 Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 17 Department of Earth System Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Stanford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 18 Computer Science and Artificial Intelligence Laboratory (CSAIL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Massachusetts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 19 Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' University of Nebraska-Lincoln,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' NE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA 20 Earth and Environmental Divisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' USA Corresponding author,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' email cshen@engr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='edu 2 Abstract Process-Based Modeling (PBM) and Machine Learning (ML) are often perceived as distinct paradigms in the geosciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Here we present differentiable geoscientific modeling as a powerful pathway toward dissolving the perceived barrier between them and ushering in a paradigm shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For decades, PBM offered benefits in interpretability and physical consistency but struggled to efficiently leverage large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ML methods, especially deep networks, presented strong predictive skills yet lacked the ability to answer specific scientific questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' While various methods have been proposed for ML-physics integration, an important underlying theme — differentiable modeling — is not sufficiently recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Here we outline the concepts, applicability, and significance of differentiable geoscientific modeling (DG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' “Differentiable” refers to accurately and efficiently calculating gradients with respect to model variables, critically enabling the learning of high-dimensional unknown relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG refers to a range of methods connecting varying amounts of prior knowledge to neural networks and training them together, capturing a different scope than physics-guided machine learning and emphasizing first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Preliminary evidence suggests DG offers better interpretability and causality than ML, improved generalizability and extrapolation capability, and strong potential for knowledge discovery, while approaching the performance of purely data-driven ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG models require less training data while scaling favorably in performance and efficiency with increasing amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' With DG, geoscientists may be better able to frame and investigate questions, test hypotheses, and discover unrecognized linkages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Introduction Geoscientific models encompass a wide range of domains, with evolving scopes and ever-increasing societal importance, especially in the face of rapid climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For example, hydrologic models help us manage water resources1,2 and plan for extremes such as floods and droughts3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' vegetation models can help predict the fate of carbon and other key biogeochemical cycles on land4 or in the ocean5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' agricultural models estimate crop yields and also their environmental impacts6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' geophysical models aim to predict land surface changes via processes like landslides7, land subsidence8, and earthquakes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' biogeochemical reactive transport models aim to understand and predict surface and subsurface water chemistry and quality9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Combining many such components, Earth System Models11–13 and integrated assessment models14–16 provide crucial guidance for resource managers and policy makers17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The uses of such models go beyond making predictions of the future to also facilitating communication with the stakeholders and aiding in the policy-making process18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Geoscientific models often share some commonalities as they describe the dynamic responses of systems to time-dependent forcings as modulated by semi-static attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Many such problems can be described as systems of nonlinear equations, algebraic differential equations, or ordinary and/or partial differential equations (ODE/PDEs), along with parameterizations (empirical representations) of physical processes with spatially-varying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The overall system can contain multiple processes chained together, some of which are well understood while others are not19,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Further, many of these process representations and parameterizations are subject to considerable uncertainty, some of which is related to scale, and thus has significant room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Here we argue that differentiable implementations of geoscientific models offer a transformative approach to simultaneously advancing process representations, parameter estimation, and predictive accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In particular, differentiable implementations provide an unprecedentedly seamless connection between process-based and machine- learning-based model components, potentially enabling us to realize the value and minimize the limitations of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 3 Value and limitations of process-based geoscientific modeling The traditional process-based modeling (PBM) approach has served the geosciences well in helping to improve our understanding of system functions and behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Due to their physical basis, they can be leveraged in hypothesis testing to assess system responses, and cause-effect relationships (see the Physical Laws row in Table 1), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', the impacts of land use changes on flooding trends21 and future warming on glacial melt22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Further, they can simulate a wide variety of observed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', discharge or leaf area index) and unobserved variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', groundwater recharge or fine-root carbon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Such an ability is critical to both advancing scientific understanding and to providing a narrative when communicating with the public and stakeholders, who are engaged in the decision-making process23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' It is possible to ask and examine specific questions regarding processes within the modelled system, by progressively improving the representations of processes 24–27 and evaluating them using controlled experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Despite these benefits, there remain important challenges with PBMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (1) Process-based models often cannot rapidly evolve with and fully exploit the information in “big data” due to the time needed to develop and test process representations and parameterizations28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Traditionally, the differences between model predictions and observations are first reconciled by parameter calibration, which adds significant uncertainty (more about this later)30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For model errors beyond parameter adjustments, modelers then hypothesize different causes, implement structural changes to the model, and iteratively confront the updated model hypotheses with the data24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This iterative process is highly expensive (in both labor and time) and dauntingly complex, and is dependent on developer intuition and legacy31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Consequently, it is common for the structural representation of a specific process in a geoscientific model to stagnate, with years or decades passing between structural updates32–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (2) Process-based models are limited by knowledge gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Extensive physical, biological, and socioeconomic knowledge is required to achieve adequate representations and updates for processes in a geoscientific model, and any deficiencies can amplify errors and ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Another major challenge is the interactions of processes across disciplinary boundaries36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For instance, vegetation, human management, and socioeconomic systems all interact with each other and affect the water and carbon and other biogeochemical cycles37–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' While the intersections of these domains will continue to stimulate scientific discovery, a new paradigm could enable us to make faster progress despite knowledge gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Potential and limitations of machine-learning-based geoscientific modeling Irrespective of the domain of application, one cannot help but notice the “Cambrian explosion” of purely data-driven machine learning (ML) approaches, especially deep neural networks (NNs), applied to a wide range of scientific applications36,41 (see Discussion A in S1, Supplementary Discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In geosciences, NNs have shown strong accuracy in predicting crop production42,43, precipitation fields44,45 and clouds46, water quality variables47,48 such as water temperature49–52, dissolved oxygen53, phosphorous54, and nitrogen55,56, and the full hydrologic cycle57 including soil moisture58–60, streamflow61–64, evapotranspiration65–67, groundwater levels68, and snow69, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Deep networks like long short-term memory (LSTM) networks70, graph neural networks63, and convolutional neural networks (CNNs)71,72 have become widely known in geosciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Many such studies reported noticeably better performance than conventional approaches, revealing that the latter did not fully exploit the information in the data28 (Table S1 in Supplementary Information S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Nevertheless, there remain important challenges with purely data-driven ML: 4 (1) Deep networks are data hungry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The success of deep networks relies on the availability of "big data”, which can, unfortunately, be sparse for many geoscientific pr oblems56,73, where many variables are measured at dozens, hundreds, or thousands of sites only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For example, water quality data are sparse and inconsistent in temporal, spatial, and chemical coverage74,75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For rare and extreme events such as mega floods, droughts, and earthquakes, available data is even scarcer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (2) ML has difficulties with errors, incompleteness, or bias in the inputs or observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The quality of ML models is limited by the quantity, diversity, and quality of training data52,76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Since a purely data-driven model can, at best, nearly-perfectly replicate the patterns in the training data, it invariably inherits various issues from the training data including implicit or explicit biases, inadequate spatiotemporal resolutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', with satellite-based observations), and the inability to account for non-stationarity in time series due to the short data record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (3) Neural Networks remain challenging to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Although explainable AI methods such as layerwise relevance backpropagation77–79 can be highly helpful in revealing some of the internal workings of a network and should be pursued, they are not designed to flexibly query a model or identify missing physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (4) Purely data-driven ML models cannot predict untrained variables (those not provided as training targets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Due to their very nature, ML-based models are designed to only output the training targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' They cannot provide an account of how events unfolded, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', the ability to state that “the flood occurred because the soil was saturated” in a study where soil moisture is unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This hinders both formation of hypotheses and communication with stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (5) Most geoscientific ML algorithms capture correlations and not causality regarding both attributes and temporal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' There are always confounding covarying factors in data, so that ML models can produce the “right” results for the wrong reasons, potentially making projections less reliable when circumstances are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The root of deep network’s success – Differentiable Programming Considering both the exceptional successes and limitations of ML and especially NNs, one can ask: What are the foundational strengths of NNs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' How can we maximize these strengths while overcoming the limitations associated with data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' How can we extract knowledge in an interpretable form while maintaining ML-level performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In answering these questions, we argue that differentiable programming (explained below) is the computing paradigm that supports the efficient training of NNs which, in turn, can deliver many philosophically and practically transformative outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Traditional modeling has been dealing with optimization problems for decades (see the Similarity block in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' However, it is argued here that only by exploiting the power of parallelized gradient-based optimization have we been able to learn from big data and train the large numbers of weights (parameters) necessary to approximate complex unknown functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The ability of generic NN architectures such as CNNs, LSTMs, and attention mechanisms to approximate unknown functions has achieved desirable outcomes (Figure 1 & Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' First, the cost of learning a few generic architectures is lower than the significant domain expertise required by traditional models, making NNs suitable for widening access to usable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Second, NNs can help in the identification of previously unrecognized linkages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Third, NN training can scale up favorably with the amount of data (in terms of accuracy, generalizability, and efficiency)76,80, in contrast with traditional modeling where the learning may saturate after some limited calibration of parameters or functions52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 5 All of these features are possible only because we can now train NNs with a large number of weights, providing a large learnable function space81,82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The number of weights easily exceeds the optimization capabilities of conventional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The most recent computer vision model contains two billion weights83 and LSTM models widely employed in hydrology can contain ~500,000 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In contrast, traditional evolutionary84–86, or genetic87 or particle swarm optimization methods88 can hardly handle more than a few dozen independent parameters (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The computing paradigm that enables efficient training of so many parameters is Differentiable Programming89,90 (Figure 1), where accurate derivatives of the model outputs with respect to inputs and/or intermediate variables can be efficiently computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Without getting into details, this paradigm is often (but not always) enabled by ML platforms, which support reverse- or forward-mode automatic differentiation (AD)89–91 using various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Models written on these platforms can, often without much effort, be programmatically differentiable – even where certain operations are mathematically indifferentiable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', thresholding or if statements), the fact that they are piecewise differentiable enables gradient computations to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The chain rule can be applied to efficiently accumulate the derivatives in a process called “backpropagation”92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Note that differentiability is normally only needed for training, not when running the model in forward mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Here we expand the scope and use the term differentiable modeling to include any method that can produce the gradients rapidly and accurately at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' A non-AD example is that of adjoint methods, which solve accompanying equations (called adjoint equations)93–95 for the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' AD differentiates through the code in an automated manner and is independent of the problem, while adjoint methods differentiate through the mathematical model equations and thus require manual derivations of adjoint equations for each problem96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Many alternative gradient estimation methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', finite differences, are intractable for any reasonably-sized NNs (10,000 weights would require 10,001 forward model evaluations) and can be challenged by stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Cheaply obtained gradients allow for parameter updates via various first-order gradient-descent methods97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Second-order methods, such as Newton Raphson, have not gained popularity for the training of NNs due to the cost of computing the Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The vast majority of NNs are implemented on platforms supporting differentiable programming, while most existing PBMs are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Historical differences in the training of geoscientists vs ML practitioners (Education row in Table 1) may give the impression that ML and process-based modeling are fundamentally unrelated, but the perceived divide is more of a legacy issue now that differentiable modeling is broadly accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In reality, both ML and parametric physical models can be expressed in nearly identical mathematical forms (Mathematical form row in Table 1), and the code forms also converge when both process-based and ML components are implemented within the differentiable programming paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This leads us to the conclusion of this section: differentiable programming is the core distinguishing feature of neural networks, and differentiable modeling can serve as the basis for unifying NN and process-based geoscientific modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' As we will discuss in the following sections, this unification requires only minor modifications to our conceptual modeling and implementation strategies, but it opens new doors to scientific discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Similarities and differences between purely data-driven neural networks and purely process- based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' [Pro] annotates the comparative strengths, also shown in green text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In the equations, W stands for weights of the neural network ������������;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ������������ stands for the physical parameters of the process-based model f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' x, u and A are dynamic forcings, state variables, and semi-static attributes, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' and L represents the loss function which quantifies the difference between simulation outputs and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Purely data-driven neural networks Purely process-based models Similarities 6 Mathematical form ������������ = ������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ������������) ������������ = ������������������������������������������������������������������������(������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ������������∗)) ������������ = ������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ������������) ������������ = ������������������������������������������������������������������������(������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ������������∗)) Programmatically differentiable Yes Traditionally no,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' but could be reimplemented on differentiable platforms or supported by new libraries Differences Ease of use [Pro] Generic model architecture – Easy to develop even without domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Specialized domain knowledge Architecture [Pro] Generic structure with a large number of weights that allow the model to approximate a wide range of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Specific structural priors representing human understanding of physics, with a small number of parameters Data [Pro] Capable of efficiently learning from and obtaining scaling benefits from big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Typically calibrated at a few sites, or a few parameters are calibrated in a regionalization equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Learning saturates at a small data quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' [Pro] The potential to overcome data limitations in accuracy, resolution, and availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Training/ Calibration [Pro] Trained using gradient descent, supported by differentiable programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Calibrated using various small-scale algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Normally code does not support differentiable programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Unknown processes [Pro] Data can be used to make up for processes we are not certain about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This also means we can learn unrecognized connections and expand knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' We must specify the processes to be used in the model, even if they are only assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Outputs Output trained variables only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' [Pro] Output many intermediate variables that facilitate providing an interpretable full narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Physical laws May not fully respect physical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' [Pro] Respect physical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Help us to assess cause-effect relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Interpretation Difficult to interpret [Pro] Elucidate physical processes, allowing us to ask specific science questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Education Taught in computer science or data science curricula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Taught in engineering or science curricula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 7 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (a) ML (blue boxes) gives us great results with easy-to-use models, resulting from the complexity of neural networks (many parameters) and the technologies that make it feasible to train such complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The most fundamental of these technologies is differentiable programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (b) In the DG paradigm, which incorporates differentiable non-ML model components (physically based structural priors), we can now obtain additional great features (plum boxes) while retaining and augmenting the old ones (blue and blue-plum boxes, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Differentiable Geosciences: Absorbing the core power of scientific ML into geoscientific domains What is differentiable modeling in the geosciences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Here we advocate for a new modeling paradigm: “Differentiable Geoscientific modeling”, or simply “Differentiable Geosciences” (DG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG refers to the use of models intermingling process-based descriptions and NNs to simulate geoscientific processes, update our physical process representations, learn physically meaningful parameters, quantify uncertainty, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG allows us to replace poorly-understood or low-accuracy process-based model components with ML components that may be more accurate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' while ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='a) Machine learning (ML) paradigm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Technology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Scientific Benefits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='KnowledgeOutcomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='ay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Differentiable programming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='TechnicalBenefits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Represent complex& ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='ap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Highlyaccurate predictions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='unknownprocesses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Efficient convergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='despite many parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Gain accuracy & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Progress despite knowledge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Data-driven training methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='generalizability with big data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='gaps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='b) Differentiable geosciences (DG) paradigm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Scientific Benefits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Represent complex& ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='unknownprocesses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Technology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='Gain accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='efficiency& KnowledgeOutcomes generalizability with big data ay Differentiableprogramming TechnicalBenefits Highlyaccuratepredictions ap Efficient convergence Interpret &narrate model despite many parameters behavior Progress despiteknowledge Data-driventrainingmethods gaps 亚 Narrowersearchspace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='still 目目 Learn causalityfrom precise including true function physical questions Physicallybased structural Reduction of knowledge priors gaps Rapidexperimentation with Predict meaningful yet process representations untrained variables Overcome data quality& quantity limitations8 retaining those process-based model components that we already trust or want to improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG may also exploit gradients for other purposes such as sensitivity analysis or trajectory optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' A distinct feature of DG is its full programmatical differentiability – that is, the whole model needs to support gradient calculation from the start to the end of the workflow – to ensure that we can incorporate neural network units that can adapt to and evolve from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The process-based descriptions retained in the model can be called the structural priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG seeks to marry the core of NN models – their optimizing and learning capabilities – to geoscientific process descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG can be considered a branch of scientific machine learning98,99 that emphasizes improving process representations and understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' With DG, we trade the model genericity for physical interpretability, with minimal compromises to accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG reduces the cost (in terms of data) of finding good solutions because the structural priors serve to constrain the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Meanwhile it also scales well with data quantity and can reap the benefits of big data, just as does purely data-driven ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' There are two perspectives from which we can view DG models (Figure 2): (a) they are ML models constrained to a smaller searchable space by the structural priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (b) they are PBMs augmented with learnable and adaptable components (and thus an expanded searchable space) provided by NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In DG, NNs can be commissioned in a wide variety of ways, ranging from learning parameters100 to updating assumptions used in the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', process representations)76, and from estimating time- dependent forcing terms to describing the whole space-time solution101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The next section provides some forms of use cases, and examples are provided in Classes of DG methods with examples section below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG is different from previous concepts of physics-guided machine learning (PGML) or not-fully-differentiable models in the methodology (must be fully differentiable), mission (to advance process understanding), and philosophy (whether treating physical law as truth or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Please see Supplementary Information S1, Discussion C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' From technical breakthrough to philosophical change – why will DG be transformative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' While efficient gradient calculation may appear to be merely a technical change, it is likely to transform our modeling philosophy and scientific objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' First, the ability to approximate complex, unknown functions greatly broadens the type of questions we can ask, by enabling us to treat trusted components as priors and focus on improving uncertain model components, one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' To explain this idea in concise mathematical terms, let us consider a physics-based model y=g(u, x, θ) where u, x, θ represent state variables, dynamic forcings, and physical parameters, respectively (This representation encompasses differential equations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', ∂u/∂t= g(u, x, θ), but is more generic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Traditional inversion algorithms only estimate the parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', asking “θ =?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=') while requiring that the functional form g be assumed a priori (except for some rigid methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', nonparametric regression, which require complicated derivations and specialized training algorithms, and thus have not gained popularity) [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' However, differentiable models allow us to ask questions about the functional form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', “������������ =?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', by training a neural network (NN) (or parameterized functions) to replace ������������: y = NNW(u, x, θ) where W is the high-dimensional weights (see examples later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Hence, with DG, we now can place our question mark precisely in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The functions to estimate could be (i) a parameterization scheme, as done in differentiable parameter learning100: ������������ = ������������(������������, ������������, ������������ = ������������������������������������(������������));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (ii) a module in a model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', where we can replace ������������3 in ������������ = ������������(������������1 , ������������2, ������������3(������������, ������������, ������������)) with NN: ������������ = ������������(������������1 , ������������2, ������������������������������������(������������, ������������, ������������)), as Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='102 optionally replaced the runoff function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' or 9 (iii) a part of a governing equation or constitutive laws, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', we can estimate ������������������������������������ in ������������������������/������������������������ = ������������(������������1 , ������������2, ������������������������������������(������������, ������������, ������������))103,104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In the above equations, physical process equations provide a backbone for the overall model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', in (i) the physical backbone is ������������;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' in (ii) and (iii) the physical backbone is ������������, ������������1, ������������2 and ������������3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The unchanged parts (structural priors), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', ������������, ������������1, ������������2 in (ii) and (iii), critically serve as physical constraints, allowing us to isolate and focus our attention (and data) on the most unknown model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' We may gain insights by simply visualizing the relationships learned by NNW 63,105 or applying knowledge distillation methods106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' We are also able to evolve better process representations for some model components like ������������1 or ������������2 mentioned above, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', the relation between soil moisture and effective rainfall in conceptual hydrologic models, without needing a full understanding of all the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This precision of questioning is opposed to some popular off-the-shelf interpretive AI approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', layerwise relevance propagation77,107, Shapley additive explanations108, or local surrogate methods109, that are limited to only asking a few fixed questions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', which parts of the inputs caused this result?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Moreover, in geoscientific modeling, directly interpreting the trained sensitivities may be risky – with only limited measurement sites, the trained relationship related to the spatial attributes tends to be overfitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG provides a framework for combining deductive reasoning and inductive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Purely data- driven models are inductive and seek to derive almost all relationships from data, whereas process-based models first posit hypotheses and then test those hypothesis using data, albeit facing many challenges in doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The DG paradigm posits a user-defined number of structural priors, and then identifies many other parts of the model from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This design follows the traditional scientific approach that identifies parsimonious models to reflect the general properties of the phenomenon, along with a quantification of the predictable aspects that are not yet understood110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Moreover, differentiable, learnable models can and have obtained state-of-the-art performance that can match fully data-driven models (Supplementary Information S1, Discussion B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Differentiable models can be viewed as (A) machine learning models guided into a smaller searchable space by structural priors or (B) process-based models with expanded search space supported by learnable units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The background fill colors indicate model optimality, related to the cost function if we had infinite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' X Process-Based optimal B Differentiable Geosciences Machine Learning searchablefunctionspace10 Why is differentiable modeling particularly valuable for the geosciences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' First of all, geoscientific data are strongly imbalanced in spatial extent, temporal coverage, and in terms of variables of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' While satellites can measure leaf area index111 or coarse-resolution surface soil moisture112,113 all over the world, there are a limited number of sites measuring photosynthesis rates114 or streamflow, especially in Africa and Asia115;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' and there is very limited knowledge of subsurface properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Purely data-driven ML may be biased or stymied by these data limitations, which may be overcome by the inclusion of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Indeed, preliminary analysis shows that differentiable models with a physical model as the backbone can outperform LSTM in regional extrapolation116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The second major motivation is system nonstationarity induced by climate change, which could drive many systems out of the previously observed range of variability117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Data-driven methods are tailored to the training data and may not maintain accuracy in the face of strongly changing conditions (this is a nuanced statement as models like LSTM may have highly competitive scores even in long-term projection tests62,116, but nonetheless experience large declines in accuracy when faced with nonstationary processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Careful testing suggests adding stronger priors may lead to better future projections116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' As DG models can also output any diagnostic (latent) variable available from the process-based equations within the DG model, we can perform model conditioning and/or data assimilation operations with sparse and scattered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' By conditioning, we mean constraining the model using an observable to learn more realistic parameters or processes so that the overall model dynamics are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For example, satellite-based soil moisture data can condition a hydrologic model to better predict vegetation water use100 or primary productivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' streamflow can constrain a model to better simulate snow water equivalent118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For data assimilation, the model can use recent observations of B to improve the short-term forecast of A, as B can also help to update our model state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Physical parameters play key roles in geoscientific models in modulating the behaviors of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Parameter estimation transfers information from either (i) raw observable physiographical variables or (ii) fine-scale dynamics to parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Quite often, we have no ground truth information for the parameters and they require inversion using observations or high-resolution simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Parameter estimation has, for decades, been fraught with uncertainty, ambiguity, and frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Due to different parameters producing very similar output and their sensitivity to spatiotemporal resolutions, calibration at a geographic location can often lead to nonunique inference (sometimes referred to as “equifinality”)119–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Extending parameters to unmonitored locations requires “regionalization”, which also introduces uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Because of increasing geospatial data availability, parameter estimation is an area where machine learning is well- poised to make significant progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' A novel aspect is that, as with purely data-driven ML, DG methods provide favorable scaling relationships – more training data leads to improved performance, efficiency, and generalizability100 (discussed in Supplementary Information Text S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' What are the promises of differentiable modeling in geosciences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' We hope to evolve differentiable models so that we can gain process knowledge while improving the model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Success can be claimed if we obtain models with the following features: (i) Predictive accuracy and transferability equal to or superseding purely data-driven models for extensively measured variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (ii) Models capable of structural evolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', we can improve the parameterization and formulation of the processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (iii) Accurate generalizability to data-sparse regions or into long-term future;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (iv) Conservation of mass/energy/momentum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 11 (v) Consistency of internal physical fluxes and states that can provide a full narrative of the events and full support to downstream processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (vi) Permits efficient isolation of one uncertain model component at a time to learn physics with less ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This wish list is ambitious and yet partial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' However, as shown below, some examples already demonstrated the plausibility of these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Motivating questions for DG With differentiable geosciences models, we hope to ask and answer the following types of questions: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' What is the relationship between variable x and variable y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' What is the missing physics as part of the differential equation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' What should have been the assumption or function here?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' How does factor A influence parameter β?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Which process is causing phenomenon P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' What will happen in new environmental conditions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' What is the information content of datasets, either input or target data for training?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Most domains in geosciences could benefit from DG (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' To provide more concrete motivating examples, we now list one example question that DG is primed to answer from each of the domains below (ordered alphabetically): Agriculture: Can we predict crop phenology dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', planting, shooting, flowering, harvesting) and assess potential production risk under future climate change (type f), which involves interconnected biotic, abiotic, and human influences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG can optimize model representations of more and less understood components of this interconnected system for accuracy even in climate extrapolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Climate: Can we predict cloud processes and ocean eddies and their impact on climate sensitivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' PBM- NN hybrids implemented with DG can help to improve cloud representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Ecosystem: Should we parameterize ecosystem models regarding carbon and nutrient cycles on the plant functional type level or the trait level (type c)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Testing the configurations of differentiable parameter learning schemes could answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Coastal: Can we better leverage emerging sensing platforms while improving our model representations of sediment transport and nonlinear wave-wave interactions in order to infer nearshore bathymetry at large scales (type g)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Cryosphere: Can we leverage both physics and data to create more accurate models for ice dynamics within the cryosphere and better constrain its fate under climate change(type f)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For example, the plumbing system for melted water and its influence on ice-basal bed rock friction are two of the key components for ice mass movement121,122, with increasingly available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 12 Coastal: Can we better leverage emerging sensing platforms while improving our model representations of sediment transport and nonlinear wave-wave interactions in order to infer nearshore bathymetry at large scales (type g)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Geohazards: Can we use space-based observations of geohazards, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', landslides122, to quantify subsurface properties (type d) so we can better predict future events (type f)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Space-based observations and combined with differentiable parameter learning provides an opportunity to inversely estimate properties like soil cohesion and friction angle which are challenging and expensive to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Hydraulics: How do we estimate floodplain hydraulic parameter values efficiently at large scales using new sensing data (type a, d)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Estimation and inversion are most difficult problems facing the hydraulics research community, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', Manning’s n for flow resistance and sediment transport rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Another example is bathymetry which is required to run any hydraulics model but hard to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Hydrology: How does global groundwater-dominated baseflow respond to climate change (type a)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' What is a proper, scale-appropriate way to parameterize groundwater storage and flow at the global scale (type c)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For this question, we cannot answer it using a purely data-driven method, but could leverage differentiable models for the diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Soil science: Can we find functional forms to express soil hydraulic properties (water retention and hydraulic conductivity function) that describes non-equilibrium flow (type c or b)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Water quality: How and to what extent do denitrification rates vary across gradients of climate, vegetation, land use, and geology conditions (type d) and thus how do they change under different climates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Nitrate is one of the most widespread and persistent contaminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Denitrification removes nitrate from water but the rates and extent of denitrification however depend on an array of entangled environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 13 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Differentiable Geosciences can help almost all geoscientific domains in knowledge discovery and improving simulation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Green and blue highlighting is used to show how there can be multiple uses for neural networks within a single model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Classes of DG methods with examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG is a young modeling paradigm that could benefit from wider participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This section briefly describes early explorations of DG, categorized by how gradients are computed and employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This section also gives examples, which are by no means exhaustive, to explain the concepts and to inspire more innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Directly differentiating through numerical models and connecting them to NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Among the several options, directly differentiating numerical models is the most straightforward method and is most similar to traditional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Utilizing the AD functionality provided by modern ML platforms, one can reimplement an existing model to obtain a differentiable model version (and ensure reproducibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Then the differentiable model is connected to NNs as discussed in the section “Why will DG be transformative”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Because the model being trained is the same one for the forward simulation, the physics is clearly enforced, and the user can apply the forward simulator for any initial, boundary and forcing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' They can also migrate the learned relationships to existing implementations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', the national water model, to immediately support operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' However, reimplementing a model does incur non-trivial initial development cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Mathematical changes may be required to adapt previously non- differentiable mathematical operations to be mathematically differentiable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', by replacing indexing with convolutions, and to improve parallel efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' While DG models may not always have to run on Graphical Process Units (GPUs), enabling GPUs will improve the computational efficiency by orders of magnitude, notwithstanding some current challenges (described in the Challenges to address for DG Water Soil science Cryosphere quality Hydraulics Hydrology Coastal Ecosystem Differentiable process-based mode Geohazards Climate f(u, x, o, NNw(u, x, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=') = y Agriculture Other replacinga moduleina model Forcings parameterization Observations Attributes14 section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Our position is that in most cases, the cost is well justified due to the potential to interrogate into the model, make changes, and learn physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The reimplementation may provide a “reset” opportunity to reexamine many of the habitually-made assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' As an example, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='102 implemented the conceptual hydrologic model HBV (a system of ODEs) on PyTorch and used coupled NNs for parameterization and optionally replaced processes with NNs (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Strikingly, they approached the performance level of LSTM, giving a median Nash Sutcliffe model Efficiency coefficient (NSE) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='732 for the CAMELS streamflow benchmark, compared to LSTM’s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='748 for the same dataset, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='715 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='722 for another forcing dataset (Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' They also output untrained variables such as evapotranspiration and baseflow, which agreed well with alternative estimates (Figure 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Moreover, in spatial extrapolation test cases, the differentiable model outperformed LSTM with respect to daily metrics and decadal trends116 (Figure 4 c-d) due to the structural constraints, demonstrating its potential for global hydrologic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Similarly, Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='118 encoded the hydrologic model EXP- HYDRO as a recurrent NN architecture and coupled it with fully connected NNs which served as the parameterization pipeline as well as postprocessor to improve runoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' They showed that a symbiotic integration between NN and physics led to robust transferability and that snow water equivalent was well captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In the Biogeosciences or ecosystem modeling, differentiable models found improved parameters for photosynthesis123 at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Apart from models similar to ODEs, direct differentiation can also be applied to models operating on graphs representing the natural systems such as river networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Bindas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='124 created a differentiable river routing model that was trained on daily discharge at a gauge downstream of a river network (with pretrained LSTM producing runoff as inputs to the graph) to learn a parameterization scheme for Manning’s roughness coefficient (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' They obtained a power-law-like curve between n and catchment area that was consistent with the expected n behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Similarly, Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='125 implemented an advective dispersion equation on the river graph to simulate stream water temperature and found that the model performed better in data-sparse situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 15 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (From Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='102,116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Reprint permission obtained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (a) Sketch of a differentiable hydrologic model using process-based hydrologic model HBV as a backbone (b) For temporal test using NLDAS forcings, δ models can approach the performance of LSTM and greatly outperform traditional approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (c) For prediction in ungauged regions (PUR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' train in some regions and test in another large ungauged region), δ models can surpass the performance of LSTM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (d) For the PUR test, δ models can predict long-term trends of annual flow percentiles more reliably than LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (d) δ models can predict high-quality evapotranspiration estimate (not used in training) compared to a satellite product for both in-sample and spatial generalization tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ADDITIONAL CLASSES, CHALLENGES TO DG, AND CONCLUDING REMARKS REDACTED BEFORE PAPER ACCEPTANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Acknowledgements We attribute many ideas of the paper to a discussion in the HydroML symposium, University Park, PA, May 2022, https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='ly/3g3DQNX, sponsored by National Science Foundation EAR #2015680 and Penn State Institute for Computational and Data Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Content related to this paper was also presented in some presentations, including Artificial Intelligence for Earth System Processes (AI4ESP) talk online https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='ly/3etm5aI in Nov 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' We thank Jordan Read and James McCreight for valuable internal reviews of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Shen was supported by National Science Foundation EAR-2221880 and Office of Science, US Department of Energy under award DE-SC0016605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Gentine acknowledges funding from the National Science Foundation Science and Technology Center, Learning the Earth with Artificial intelligence and Physics (LEAP), award #2019625 and Understanding and Modeling the Earth System with Machine Learning (USMILE) European Research Council grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Marty Wernimont at the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Geological (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0 Differentiable process- Parameter regionalization based model - HBV* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='8 Precipitation/Temperature snowfall Rainfall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='6 Sp Attributes CDF soil, land cover, Static θ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='4 geology, others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' +E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' β/β: or OptionalNN Forcing ireplacement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='2 P, T, Ep 9A(A,x) Dynamic θ LSTM unit Q1 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='4 In-sample PUB PUR NSE KGE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='5 0 0 50 6 6(βt) (βt, yt) ILSTM 16 1(βt) 6(βt, yt) Observed trend16 Survey (USGS) greatly improved the presentation of Figures 1 and 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Wernimont and Appling were supported by the USGS Water Mission Area, Water Availability and Use Science Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Competing Interests KL and CS have financial interests in HydroSapient, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', a company which could potentially benefit from the results of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This interest has been reviewed by the University in accordance with its Individual Conflict of Interest policy, for the purpose of maintaining the objectivity and the integrity of research at The Pennsylvania State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 17 Main Bibliography 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Ajami, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', Gupta, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', Wagener, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' & Sorooshian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Calibration of a semi-distributed hydrologic model for streamflow estimation along a river system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Journal of Hydrology 298, 112–135 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' van Griensven, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' & Meixner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' A global and efficient multi-objective auto-calibration and uncertainty estimation method for water quality catchment models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Journal of Hydroinformatics 9, 277–291 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Barendrecht, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The value of empirical data for estimating the parameters of a sociohydrological flood risk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Water Resour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 55, 1312–1336 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Post, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', Vrugt, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Application of deep learning neural networks for nitrate prediction in the Klokot River, Bosnia and Herzegovina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' in 2021 7th International Conference on Control, Instrumentation and Automation (ICCIA) 1–6 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='1109/ICCIA52082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='2021.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Editorial: Broadening the use of machine learning in hydrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Water 3, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 22 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Fang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', Shen, C.' metadata={'source': 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Hydrology and Earth System Sciences Discussions (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='5194/hess-2022-111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Reichstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Deep learning and process understanding for data-driven Earth system science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Nature 566, 195–204 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 34 Supplementary Information S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Supplementary Discussion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Recent progress in geoscientific domains with purely data-driven machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Machine learning (ML) has gradually but pervasively permeated the vast majority of scientific disciplines, and it is transforming those sciences at an unprecedented pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In hydrology, deep networks such as long short-term memory (LSTM) networks70, and convolutional neural networks (CNNs)71,72 have shown strong ability with regard to prediction of soil moisture58–60, water supply155, streamflow61–64, evapotranspiration65–67, groundwater levels68, snow69, and other aspects of the water cycle57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In water quality studies, LSTMs and CNNs have shown promise in simulating water temperature49–52, dissolved oxygen53, phosphorous54, and nitrogen55,56, among others47,48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In agriculture, ML approaches have been widely applied for crop production prediction156–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In regional climate studies, CNN-based schemes or generative algorithms have been found to improve the forecasting of precipitation fields44,45 and prediction of clouds (deep clouds)46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Often the studies have reported state-of-the-art performance when compared with conventional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Typically, such high-quality predictions can be made even when a good understanding of the underlying processes is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' We made an effort to collect a list of somewhat comparable studies with metrics for both traditional and ML models (Figure S3 and Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Previous models have been highly useful in advancing science, but these results imply that they were not fully exploiting the information available in the data28, and they can benefit from leveraging the strength of ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 35 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' ML vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' traditional model performances for a number of scientific applications with data from many sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The metrics were computed based on simulations and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The lower the values, the better for RMSE, while higher is better for Pearson’s correlation (COR), R2, and Nash-Sutcliffe model efficiency coefficient (NSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This is presented with many caveats, such as the ML model is optimized to match observations while traditional models have many other constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' a selection bias – where ML did not outperform did not get published (nevertheless, one could also argue studies where PBM outperformed were not easily found).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The point of this table was not to show that ML was always better, but to support the argument that ML tends to have advantages in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Also note the limitations of ML we discussed in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Variable Metric Deep networks Traditional Reference Stream Temperature RMSE (°C) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='01 Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='159 RMSE (°C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='80 Rahmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='160 and Daraio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='161 Pearson COR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='91 Rahmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='160 and van Vliet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='162 R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='93 Rahmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='160 NSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='93 Rahmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='160 Evapotranspiration R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='21 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='163 RMSE (mm/day) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='56 NSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='57 Talib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='164 Soil Moisture RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='085 Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='58 Pearson COR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='72 RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='035 Pearson COR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='82 Pearson COR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='77 Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='143 RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='08 Streamflow NSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='68 Seibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='165 and Kratzert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='62 NSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='9 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='68 Mohamoud and Parmar166 Mean R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='71 Merritt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='167 Dissolved oxygen NSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='78 Zhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='53 Median R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='64 Stefan and Fang168 CC (correlation Coefficient) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='972 Heddam169 Median NSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='760 Keshtegar and Heddam170 36 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Why can differentiable process-based models achieve state-of-the-art predictive performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Purely data-driven ML architectures have set a high bar for accuracy in multiple geoscience domains, such that one would be tempted to predict a loss in accuracy when adding in less-flexible process-based components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' However, here it is argued that generic ML architectures are not necessary to achieve good model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' As long as some model components are adaptable and learnable, we can learn from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' If we view the model as a more strongly constrained ML model (perspective “a” in Figure 2), it is easy to see that there is a potential to achieve ML-level performance if we enlarge the searchable space of PBM to include a good approximation of the true function, directed by gradient-based training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The paths we take to upgrade the models will be expert-dependent (prior-dependent), so one should not expect a unified approach at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Many dynamical systems in Geosciences can be written as ordinary differential equations (ODEs), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', rainfall runoff in a basin, crop growth, or nutrient release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' While solving these equations, we run the numerical model for many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This is mathematically similar to recurrent neural networks, and the time integration operation is similar to the functionality achieved by some neural networks like the Residual Networks171,172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' It should not be surprising that learnable process-based models with some ML components can perform as well as deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' As we discuss in Section S1, multiple studies have already shown that differentiable, learnable models can approach the performance of purely data-driven models, or exhibit advantages in some cases where extrapolation is key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Differentiable model formulations can maintain at least two of the three desirable features: approximating complex, previously unknown functions, and the ability to assimilate information from big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Compared to purely data-driven ML, DG trades genericity for interpretability and the ability to ask specific questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Deep networks like CNNs, LSTMs, and transformers will be an ingrained part of DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Eventually, deep learning will become part of the repertoire of geoscientists, just like with numerical methods173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' How is DG related to physics-guided machine learning (PGML) and how are they different?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Many ML-physics integration strategies with a wide variety of complexity have been proposed in the past in a seemingly scattered manner, such that a clear classification is difficult154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' It has not been sufficiently recognized that some of these algorithms work fundamentally because they leveraged the differentiable programming tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The scattered nature of those publications makes the landscape of ML- physics integration daunting and confusing, while hindering us from making innovations based on first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' However, once we treat differentiability as the central tool, it serves as a compass to guide us in understanding newly proposed methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', we can ask if a method is fully (end-to-end) differentiable, how it uses gradients, how much prior information is inserted, what questions are asked, and how it scales with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Here we outline some similarities and differences between DG and some existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' DG and physics-guided (or physics-informed, theory-guided, or knowledge-guided) machine learning (PGML)174–176 both seek to combine physics with ML, but they differ in their approaches, purposes, and philosophies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Many PGML studies seek to introduce physical constraints, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', as regularization or pre- training, to ML methods to gain better generalizability with less training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' PGML does not in theory need differentiable programming and partial physics could be enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In contrast, DG is more thorough in that it uses the numerical physical model as the backbone and demands that the entire workflow be differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In terms of purposes, PGML is tasked to make the ML model more robust, while differentiable modeling seeks to update our assumptions or discover new knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Relatedly, in terms of philosophies, when a physical law was introduced in PGML, it was treated as truth (albeit sometimes with some tolerance level, as in Read et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Often, this includes all the calculations and assumptions to 37 support the law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In DG, we do not presume the physical laws to be correct, and, rather, are constantly looking for opportunities to update existing knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' There are many not-fully-differentiable methods that could be valuable for various applications but are outside of the scope of DG for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For one, it is possible to incorporate ML algorithms trained offline on datasets as part of a physical model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=', training a neural network on turbulent heat fluxes and inserting into a hydrologic model177;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' training pedotransfer functions to infer soil parameters from soil hydraulic data178;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' training an atmospheric parameterization network on short-term cloud-resolving simulations179;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' or training ocean-mixing parameterizations on data and physical constraints180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' While this approach has the advantage that the physical meaning of the NN is clear and stands alone, direct training data are needed for the variable of interest (thus having issues with pure ML as discussed earlier) and the network can no longer evolve and adapt in an interactive fashion, for instance to further update the model when exposed to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In the future these NNs could be further incorporated into DG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Some other offline coupling methods include providing outputs of process-based models as inputs to neural networks (this helps to integrate over spatiotemporal heterogeneity)181,182, or training ML models to predict the PBM residuals150,183,184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Readers are referred to Reichstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='185 which promoted a number of ways to connect physics and ML for geosciences, with a brief mention of differentiable programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Details for some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Part of the effort in Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='100, which proposed differentiable parameter learning (dPL), connected the Variable Infiltration Capacity (VIC) process-based hydrologic model to a neural network (������������) that estimates physical parameters of VIC (������������) using some widely available attributes (A): ������������ = ������������(������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In an “end-to-end” workflow, ������������ is then sent to VIC, whose outputs are compared with observations, effectively turning the parameter calibration problem into a machine learning problem, trained on all sites simultaneously using backpropagation and gradient descent (Figure S1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' As a result of this global loss function, dPL exhibits advantages over traditional calibration on multiple fronts, for three different datasets (soil moisture, CAMELS streamflow, and global headwater runoff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The parameter sets are spatially coherent (Figure S1b-c) and extrapolate better in space (Figure S1d-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' dPL is hyper efficient: a job that normally takes a 100-CPU cluster 2-3 days now takes a single Graphical Processing Unit (GPU) one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' dPL allows the combined model to output unobserved variables while addressing the notorious problem of parameter equifinality119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' As summarized earlier, these are the great advantages we expect to harness with differentiable modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' 38 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Physics-informed neural networks (PINNs)101,140, while first published in 2017 before the existence of the term “differentiable geosciences”, could be perceived as a genre of DG as the gradient information is critically employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' PINNs pose problems in a unique way, seeking to train a neural network with space-time coordinates as inputs, h(t,x) where x represents spatial coordinates and t is time such that (i) h(t,x) agrees with known data points at (t,x), and (ii) the derivatives dh/dx, dh/dt, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' agree with the governing partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Physical parameters could also be part of the inputs to the h network153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' PINNs are a highly innovative approach tested on a large variety of applications in many domains, and there have been a number of good reviews of this work130,154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' PINNs have made enormous strides, with novel inversion uses such data assimilation140 and learning governing equations, but, as with other methods, there are also some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Obviously, the function h(t,x) is tied to the initial and boundary conditions so it needs to be trained separately for different initial/boundary condition pairs, and the form of the inputs limits the neural network to certain types (multilayer perceptron network) that are not the easiest to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' However, the learned parameters and constitutive relationships can describe the system under a wide range of boundary and initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Furthermore, the fidelity of the trained network to physical equations must be carefully examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In geosciences, a PINN method for learning unknown parameter fields and constitutive relationships was proposed104 (Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' As an example, steady-state groundwater flow in an aquifer with an unknown conductivity field and unsaturated flow in the vadose zone with an unknown pressure- dependent conductivity were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In the unsaturated flow application, it was assumed that only sparse measurements of pressure head were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The quantities of interest were the unsaturated conductivity as a function of the pressure head, and the pressure head field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Notably, it was assumed that no measurements of the unknown parameters were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' In the proposed PINN method, both quantities of interest were represented with neural networks (NNs) (with unknown parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' This step created a differentiable model of the unsaturated flow in the vadose zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' It was also assumed that the pressure head measurements could be described by the steady-state Richards equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Substituting the NN approximations into this equation formed the axillary residual NN, which shared the (unknown) parameters Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (From Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='100, reprint allowed via Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0 International License, http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0/) (a) Structural diagram of one of the dPL frameworks called gA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (b & c) The estimated infiltrating curve parameter (INFILT) from dPL vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' the site- by-site calibrated shuffled complex evolutionary algorithm (SCE-UA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='(d & e) dPL better matches the MODIS satellite product for uncalibrated variable ET than does SCE-UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (a) (e) (a) (d) (e) (c) (b) Ideal Line 1000 dPLgA MODIS (mm/yr) 750 Ensemble : 500 (mean±std) Bias =-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='75 + -34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='30± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='87 250 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Corr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='73 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' NSE =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='05 0 0 250 500 750 1000 dPLgaET (mm/yr)dPL -- 9A Dynamic PBM or its inputs surrogate Static Physical attributes Parameters(b) Ideal Line 1000 SCE MODIS (mm/yr) 750 Ensemble : 500 (mean± std) Bias =-58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='31 +59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='84 250 Corr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='68 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='006 NSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='01 0 0 250 500 750 1000 SCE-UA ET (mm/yr)INFILT (-) from dPL (gz ) (a) (b) INFILT (-) from SCE-UA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0839 with the primary NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' For the primary NNs to satisfy the governing equation, the residual NN should be zero everywhere in the domain – in other words, the exact measurements of the residuals are available everywhere in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The NNs were trained jointly using the pressure head measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Since the conductivity and residual NNs share the same parameters, estimating parameters in the residual NN also provides the parameterization of the conductivity NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Figure S2a shows the reference pressure head field and the locations of the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Figure S2b shows the point errors in the estimated pressure head field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' The reference and estimated unsaturated conductivity functions are shown in Figure S2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' These figures demonstrate that the PINN method can learn both the state variable and the constitutive relationship very accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (from Tartakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='104, reprint permission obtained) (a) The reference pressure head field and the locations of the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (b) The point errors in the estimated head field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' (c) The reference and estimated conductivities as functions of the pressure head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Disclaimer Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content=' Supplemental References … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0025 5 8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0020 6 6 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0015 7 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0010 8 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='0005 9 2 4 6 8 2 4 6 8 X1 X10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='35 O Exact Prediction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='25 n) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} +page_content='10 10 9 8 7 6 5 4 n' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfpQiY/content/2301.04027v1.pdf'} diff --git a/MdFPT4oBgHgl3EQfkzU3/content/tmp_files/2301.13119v1.pdf.txt b/MdFPT4oBgHgl3EQfkzU3/content/tmp_files/2301.13119v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..58190016433f0d3b393c25211bdefea8f34df7d8 --- /dev/null +++ b/MdFPT4oBgHgl3EQfkzU3/content/tmp_files/2301.13119v1.pdf.txt @@ -0,0 +1,2202 @@ +Floquet Exceptional Topological Insulator +Gaurab Kumar Dash, Subhajyoti Bid, and Manisha Thakurathi +Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India 110016 +(Dated: January 31, 2023) +We propose a novel way of modulating exceptional topology by implementing Floquet engineering +in non-hermitian (NH) systems. +We introduce Floquet exceptional topological insulator which +results from shining light on a conventional three-dimensional NH topological insulator. +Light- +matter interaction facilitates the quantum phases of matter to exhibit a novel phenomenon, where, +the point gaps in the bulk host surface states. These distinct surface states either fill the point gap +in the complex eigenspectrum or exhibit exceptional points in the presence of a magnetic field. We +also highlight the existence of a quantum anomaly generated by photo-induced modulation. The +existence of the Floquet biorthogonal Chern number and spectral winding number show that the +momentum slices exhibit NH skin effect, even though the system as a whole does not. We also +employ wave-dynamics evolution to illustrate the NH surface skin effect. +Introduction.— The non-hermitian (NH) topological +phases have been the center of attraction for theoretical +as well as experimental studies [1, 2]. Many experimen- +tally realizable models of NH systems have been proposed +in condensed matter physics hosting exceptional points +(EPs), lines, rings, and nodal planes [3, 4]. +Recently, +NH generalization of topological insulators (TI) known +as Exceptional TIs has been studied where the surface +hosts either a 2D band structure with a single EP (a +point where both eigenenergies and eigenvectors coalesce) +or a single band, which represents a vortex [5]. This is +obtained either by varying the imbalance between the g- +factors of the orbitals or the magnetic field strength while +keeping it isotropic along [111] direction. On the other +hand, the Floquet engineering of the topological phases +in Hermitian [6–17], as well as NH systems, has defined +exotic ramifications in stroboscopic limits which are oth- +erwise not realizable in their static counterparts[18, 19]. +Therefore, Floquet engineering of exceptional topology +has captured the desirable attention [20]. +In this letter, we interlink these two ideas and in- +troduce a 3D Floquet exceptional topological insulator +(FETI). We also aim to answer the following questions. +Is it possible to achieve the ETI phase without an exter- +nal magnetic field, convert a hugely defective point into +non-defective points, and modulate exceptional topology +using Floquet theory? +In order to answer these ques- +tions, we start with conventional 3D NHTI and shine +circularly polarised light (CPL) on it [21]. The system +hosts a central point gap in the bulk which can be tran- +sitioned into a central line gap by tuning the amplitude +of the vector potential of CPL. The point gaps, which +are characterized by the quantized value of topological +invariant in the bulk, host an infernal point [5] lead- +ing to the defectiveness of the system in the static part. +However, in the stroboscopic phase, the system experi- +ences a photo-induced pseudo-magnetic field and hosts +robust non-defective doubly degenerate surface states lo- +calized at opposite surfaces. The doubly degenerate sur- +face states fill the point gap and exhibit different disper- +FIG. 1: Schematic picture of a cubic lattice with s and p +orbital at each site where a CPL light is irradiated along ⃗n +direction with θ and φ being the angles of polarization. +sion relations. The surface state appears as a single sheet +which represents a vortex without an anti-vortex partner +giving rise to the quantum anomaly in 3D NHTI systems. +This sheet can be shown to establish homotopy with a +torus-shaped 2D Brillouin zone (BZ). +However, when the system is subjected to a constant +and isotropic magnetic field, it hosts a single second- +order EP on the surface of the 3D NHTI models. +In +the presence of CPL, it also experiences a photo-induced +pseudo-magnetic field which couples with the external +magnetic field [22] to modulate the EP on the surface. +Thus, a constant magnetic field in the static phase gains +dynamics in the stroboscopic picture thereby dramati- +cally photo-modulating the Land´e-g-factor. The system +possesses NH surface skin effect (NHSSE) [23, 24] in both, +static as well as dynamic phases. Here we explain this +surface quenching phenomenon with the help of wave +dynamics[25, 26] without analyzing the photovoltaic and +chiral transport phenomenons associated with it. +Static model— We start with the quantum mechanical +model of spin-full 3D NHTI, given by following Hamilto- +arXiv:2301.13119v1 [cond-mat.mes-hall] 30 Jan 2023 + +2 +FIG. 2: (a) Bulk complex eigenspectrum for point gap region +satisfying |M + A2 +2 − 3| ≤ δ, cyan (pink) color highlights the +value of W3D = 1(−2). (b) represents the central point gap +region of complex eigenspectrum for OBC along the z axis. +The degenerate surface states fill the point gap completely +and appear as a single sheet structure. (c) Plot of Abs(E) as +a function of kx for ky = 0, hosting a surface Dirac node at +kx = 0. Notably, when ky is non-zero the surface is gapped +and the states fill the Dirac cone completely as appears in the +contour plot (d). System parameters are M = 2.5, λ = 1, +δ = 1, A2 +0/ω = 1/5 and momentum resolution ∆k = π/400. +nian, +H0(k) = +� +j=x,y,z +� +(cos kj − M) τzσ0 + λ sin kjτxσj +� ++ i δ τxσ0. +(1) +In the tight-binding model, the system has s and p or- +bital at each lattice site [5, 25] (implying four degrees of +freedom at each site) see Fig. 1(a), σµ and τµ denotes +the Pauli matrices acting independently of spin and or- +bit degrees of freedom respectively where µ = 0, 1, 2, 3. +Here, M accounts for the band inversion for the s and +p orbital, λ controls the intrinsic spin-orbit coupling, δ +denotes the NH contribution either from electron-phonon +scattering or via short-lived f-electron coupling to the s +and p orbitals [5]. We note that the Hermitian counter- +part of the above model mentioned in Eq. 1 resembles +a four-band model of 3D TI hosting a Dirac node. The +static Hamiltonian hosts two kinds of gaps in the bulk +due to its NH property, point gap for |M − 3| ≤ δ as +shown in Fig. 1(b) and line gap for |M − 3| ≥ δ. +Floquet exceptional topological insulator (FETI).— +We +irradiate +CPL +[27] +along +the +direction +⃗n += +(sin θ cos φ, sin θ sin φ, cos θ), where, θ and φ are polar +and azimuthal angles respectively in conventional spher- +ical polar coordinates(see Fig. 1). The vector potential +has the following form, +⃗A = A0[cos (ωt) ⃗e1 + η sin (ωt) ⃗e2], +(2) +where, A0 and ω represent the amplitude of the vector +potential and frequency of the CPL, respectively. The +parameter η regulates the orientation of the polarization +(η = +1(−1) for left (right) CPL). All the three unit +vectors ⃗e1, ⃗e2, and ⃗n must be orthogonal to each other. +Thus, we choose unit vectors ⃗e1 = (cos θ cos φ, cos θ sin φ,- +sin θ) and ⃗e2 = (sin φ, − cos φ, 0). We derive an effective +stroboscopic Hamiltonian invoking the high-frequency +Floquet formalism (see [28]), given by following form, +HF (k) = +� +j=x,y,z +� +(cos kj − M − A2/2)τzσ0 + λ sin kjτxσj +� ++ τ0 (⃗n.⃗σ) + i δ τxσ0, +(3) +where, the vector ⃗n is given by, +⃗n = ηλ2A2 +ω +(sin θ cos φ, sin θ sin φ, cos θ). +(4) +Here, the term coupled to the unit direction ⃗n can be +defined as the photo-induced magnetic field generated +by the light-matter interaction as a result of the Floquet +driving [29]. This pseudo-magnetic field is analogous to +artificial gauge fields realized in ultra-cold atoms but dif- +fers in the sense that, the pseudo-magnetic field compo- +nents are the functions of parameters of irradiated light +into the system which is generally anisotropic. The mag- +nitude of the pseudo-magnetic field scales inversely to the +frequency of the drive and its direction can be altered by +changing the orientation of the polarization. +Thus, the stroboscopic phase represents a FETI re- +sulting from the periodic driving of static 3DNHTI in +the high-frequency limit. As the time-reversal symmetry +of the Hamiltonian is broken due to the pseudo-magnetic +field, the model hosts a pair of chiral Weyl nodes along +any desired axis with a suitable choice of θ and φ and +gets connected over the imaginary axis due to NH term, +thus giving rise to the point gap. Hence, for FETI, the +system hosts a point gap for |M + A2 +2 − 3| ≤ δ [see Fig. +2(a)] and a line gap for |M + A2 +2 − 3| ≥ δ, see [28]. The +Floquet Hamiltonian has A2/2 onsite term in the diago- +nal, thereby modulating M, which is responsible for the +band inversion. Therefore, the FETI phase can be real- +ized in the system even if its static counterpart is in the +trivial phase. +Surface States.— We define the critical angles of po- +larization as, θc = tan−1( +√ +2) and φc = π/4. +This +makes the pseudo-magnetic field isotropic along [111] di- +rection. At these critical values of the angle, in the com- +plex energy spectrum, the point gap is filled isotropically +with the doubly degenerate surface states having a sin- +gle Fermi point, shown in Fig. 2(b). We also consider +another unconventional model where two Fermi points + +1 +a +0.5 +Im(E) +0 +-0.5 +-1 +-6 +-4 +-2 +0 +2 +4 +6 +Re(E)0.5 +0.5 +IPR +Im( +0 +0.3 +-0.5 +0.1 +Re(E)5 +c +4 +Abs(E) +3 +2 +1 +0 +-1 +-0.5 +0 +0.5 +1 +kc3.6 +d) +0.4 +0.2 +0- +-0.2 +-0.4 +-3 +-2 +kr1 +2 +33 +FIG. 3: For one axis OBC along the z axis, (a) depicts a signature of second order EP in the static phase for M = 3, λ = 1, +δ = 1, B0 = 0.2 by an octagon shape. (b)-(d) represents the dynamics of the EP via Floquet driving the 3DNHTI in the +presence of the magnetic field. We periodically vary +A2 +0 +ω sin χ and magnetic field as B0 cos χ for χ = 0.6, 0.65, 0.7 in (b), (c), +and (d) respectively. As χ varies, the two second-order EPs collide with each other and form a third-order EP which further +splits into two second-order EPs. Other parameters are the same as (a) along with M = 2.5 and A2 +0/ω = 1/5. +can appear, see [28]. +Therefore, it exhibits a single- +sheet structure. The degenerate partners of surface states +are localized to the opposite edges of the truncated lat- +tice. The surface state has a dispersion relation kx +iky, +whereas, its degenerate partner has the dispersion rela- +tion ky + ikx. Thus, we infer that the single sheet es- +tablishes homotopy with 2D torus-shaped Brillouin zone +(BZ), see [28] for more details. +Additionally, the sur- +face states appear as a Dirac cone in the kx −ky plane as +shown in Figs. 2(c) and (d). However, the photo-induced +homotopy does not remain futile from the infinitesimal +perturbation from the critical angle. We also note that +the Hamiltonian hosts a hugely defective infernal point at +kx = ky = 0 in the complex eigenspectrum which leads to +numerical instability in the static model, see [28]. How- +ever, for the dynamic case due to the pseudo-magnetic +field, this infernal point converts into doubly degenerate +non-defective points. +Effect of external magnetic field— We consider the +system in the presence of an external magnetic field, +by adding the term τz( ⃗B0 · ⃗σ) to the Hamiltonian writ- +ten in Eq.(1) and Floquet Hamiltonian also changes to +HF + τz( ⃗B0 · ⃗σ). Thus, a constant magnetic field in the +static phase gains dynamics in the stroboscopic limit. +The resultant magnetic field modulates the imbalance be- +tween lande-g-factors between the s and p orbital. Thus +we can photo-modulate the Land´e-g-factor by parameter- +izing Floquet term as λ2A2 +0 +ω +sin χ and the magnetic field +B0 cos χ with +χ = tan−1 +� λA2 +0 +ωB0 +� +. +(5) +The bulk Hamiltonian has a point gap however, the +finite system along the z axis, possesses a single second- +order EP on the surface for the non-zero magnetic field +accounting for a quantum anomaly on the surface, see +Fig. 3(a). We note that if there is nth order EP, then +one has to take 4n turns around it to come back to the +original state in the square momentum grid [5]. The dy- +namics and the order of the EP residing on the surface +can be photo-tuned by Floquet driving which can never +be realized in its static counterpart with a constant unidi- +rectional magnetic field. By adiabatically varying χ, two +second-order EPs come closer and collide with each other, +thereby creating a third-order EP. This third-order EP +can then be further transformed into two second-order by +moving them away from each other (see Figs. 3(b)-(d)). +We then apply open boundary condition (OBC) along +y and z, while retaining periodic BC along the x axis. +In the absence of the magnetic field, the surface states +in the point gaps are localized in opposite corners. This +remarkable phenomenon is the well-known NH surface +skin effect (NHSSE) and can be related to the higher- +order skin effect [30, 31]. However, as the magnetic field +is turned on, the surface states get confined in one of the +corners (refer Figs. 4(a),(b)). To demonstrate this novel +effect, we prepare a trial wavefunction localized at a finite +y − z sheet of dimension L2. The trial wavefunction is +given by: +|ψ0⟩ = N0e− (y−y0)2 +α2 +e− (z−z0)2 +β2 +eikxax|ζ0⟩ +(6) +where, N0 is the normalization factor of the wave func- +tion, y0 and z0 are constants that determine the local- +ization of the wave function in the finite sheet, α and +β control its Gaussian width, and |ζ0⟩ is a spinor. We +perform the time evolution of the trial wavefunction as +e−iHt|Ψ0⟩ with respect to the Hamiltonian obtained by +truncating y and z axes. The skin modes are not com- +pletely localized at the corners and they have noticeable +overlap with the bulk modes (Figs. 4(a)-(b)). So this +effect allows the wavepacket which was initially localized +at one of the edges, to travel into the opposite edge by +permeating into the bulk (Figs. 4(c)-(f)). However, there +is a significant dynamical quenching of the time-evolved +probability of the trial wavefunction at the edges where +surface states are localized. + +a +0.5 +-0.5 +-0.5 +0 +0.5 +1 +1.5 +2 +Re(E)b +0.5 +(α)ul +-0.5 +-0.5 +0 +0.5 +1.5 +2 +Re(E)0.5 +Im(E) +-0.5 +-0.5 +0 +0.5 +1.5 +2 +Re(E)0.6 +0.5 +1 +0.4 +IPR +0.2 +-0.5 +-0.5 +0 +0.5 +1.5 +2 +Re(E)4 +FIG. 4: Probability amplitudes demonstrate the NHSEE for +(a) B0 = 0 and (b) B0 = 0.2. (c)-(f) demonstrates the dy- +namical evolution of the trial wavefunction from time t = 0 +to t = 60. The constants α and β are set to be 2 and 6 re- +spectively. +We choose the spinor |ζ0⟩ as [1, i, 0, 0], and the +momentum kx is evaluated numerically considering those val- +ues which experience NHSSE. Other system parameters are +the same as Fig. 3 +. +Topological Invariants.— We compute three topolog- +ical invariants starting with the spectral winding num- +ber along a particular momentum axis say kz while +fixing the other two momenta values at kx0 and ky0. +Thereafter, along the kz axis, the Hamiltonian be- +comes one-dimensional and can be defined as HF +1D(kz) = +HF (kx0, ky0, kz). We calculate the spectral winding num- +ber as, +νkx0,ky0(Ep) = +1 +2πi +� π +−π +dkzTr[Q1D(kz)], +(7) +where Q1D(kz) = [HF +1D(kz) − Ep]−1∂kz[HF +1D(kz) − Ep] +with Ep as a reference energy inside the corresponding +spectral region. +Thus, for a suitable choice of com- +plex reference energy in the eigenspectrum, the quan- +tized value of the spectral winding number implies bro- +ken bulk-boundary correspondence (BBC) resulting an +NH skin effect for those particular kx0 and ky0 along kz +direction. As shown in Fig. 5(a), there are four bands +with different values of ν along with the zero-valued re- +gion that can be visualized as the outcome of adding two +ν’s in the overlapping complex eigenspectrum. +Furthermore, along with three spectral winding num- +bers in three directions [24], system has three 1D winding +numbers W1D,l defined as, +W1D,l = −i +� +d3k +(2π)3 Tr[Ql(k)]. +(8) +where Ql = [HF (k)−E]−1∂kl[HF (k)−E] with E being +the reference energy in the point gap. For our system, +W1D,l = 0 for all values of l, which also necessitates the +absence of NHSE [24, 32]. Thus, the uniqueness of the +model is evident from the fact that the system as a whole +does not exhibit the collapse of the BBC although the +momentum slice in the Hamiltonian experience NHSE. +There is also presence of another 3D topological invariant +W3D[33–36] defined in the bulk spectrum which is given +by, +W3D = −1 +24π2 +� +d3kϵijkTr[Qi(k)Qj(k)Qk(k)]. +(9) +The topological invariant defined in the bulk deter- +mines the fate of the surface states. The quantization +of W3D, however, does not require any symmetry for +its stabilization. +Finally, we also employ a biorthog- +onal approach to compute the Floquet open-boundary +Chen number for each kx value [37]. +This unique ap- +proach, however, re-establishes the BBC. The Floquet +open-boundary Chern number is given by; +Cα = 2πi +l′yl′z +Tr′ � +ˆPα[[ˆry, ˆPα], [ˆrz, ˆPα]] +� +. +(10) +Here ˆry (ˆrz) is the coordinate operator along y (z) di- +rection and defined as ˆry(z)mn = ry(z)δmn with rx, ry ≤ l, +l × l is the size of the system and l′ +y = l′ +z = l − 2l0 +where l0 is a boundary layer that has been removed +from lx/y, see [28]. The bulk band projection operator, +Pβ = � +n∈β |nR⟩⟨nL|, where β denotes all the unoccu- +pied bands with |nL⟩ (|nR⟩) being the left (right) eigen- +states of Floquet Hamiltonian. The Floquet biorthogo- +nal Chern number gives the quantized value of one where +there are surface states in the Hamiltonian [see Figs. 2(c) +and 4(b)] and can be modulated by varying A2 +0 +ω . +FIG. 5: (a) The quantized spectral winding number in the +complex eigenspectrum plane for kx0 = −π and ky0 = −π. +(b) represents the Floquet biorthogonal Chen number depict- +ing the quantized value of one in the region where there are +surface states. +System parameters are the same as in the +previous figure with B0 = 0. +In conclusion, we have studied 3D NHTI in the pres- +ence of CPL. The system hosts a novel phase of quantum +matter, namely, FETI in the stroboscopic limit with no +static counterpart. FETI has a point gap that is filled +by either a single band at the surface or a 2D band with +EPs when 3D NHTI is subjected to an external mag- +netic field. Due to the NH fermion doubling theorem, an +odd number of Fermi points or EPs are impossible in a +2D model. That’s why the surface states in FETI are +anomalous. The exceptional topology i.e. the number + +0 +0.3 +Y +[亚|2 +0.15 +10 +20 +10Z +200 +0.7 +亚|2 +(b) +0.35 +10 +20 +10 Z +200 +0=↑ +0.25 +Y +0.15 +10 +(c) +20 +0 +10 Z +200 +t= 10 +0.02 +Y +10 +(d) +0.01 +20 +10 +)Z +200 +t2 +20 +0.02 +Y +(e) +10 +0.01 +20 +0 +10 +Z +200 +t = 30 +0.03 +Y +0.02 +10 +(f) +20 +10 +Z +200 +t = 40 +0.02 +Y +(g) +10 +0.01 +20 +0 +10Z +200 +09 = ↑ +0.012 +Y +[ 亚|2 +10 +(h) +0.006 +20 +0 +10Z4 +(a) +0 +2 +00 +R +-2 +0 +-4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +Im(E)C 0.5 +-0.5 +-1 +0 +0.5 +1 +kα/T5 +of EPs can also be modulated using CPL. We also es- +tablished using different topological invariants, that the +NH skin effect does not exist in the entire system, but +it is present in momentum slices. Finally, the NHSSE +exhibited by FETI has been explained by the dynamical +quenching of the wave-function. 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Wang, X.-M. Zhao, L. Zhuang, and W.-M. Liu, +arXiv preprint arXiv:2105.10980 (2021). +[39] J. Avron, P. Exner, and Y. Last, Physical review letters +72, 896 (1994). +SUPPLEMENTARY MATERIAL: Floquet Exceptional Topological Insulator +Gaurab Kumar Dash, Subhajyoti Bid, Manisha Thakurathi +Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India 110016 +3D NON-HERMITIAN TOPOLOGICAL INSULATOR +The Hermitian counterpart of the Hamiltonian H0(k) written in the main text is a four-band model of 3DTI hosting +a Dirac node. For 1 ≤ |M| ≤ 3, the system exhibits a trivial phase. The phase transition from trivial to topological +phase occurs at |M| = 1 and |M| = 3. Due to the NH property, the Hamiltonian is accompanied by two kinds of +gaps in the complex eigenspectrum[1, 5, 25]. For |M − 3| ≤ δ and |M − 3| ≥ δ, it showcases a central point gap and +a central line gap respectively [refer Fig. S1(a)]. +The Hamiltonian is hugely defective in the surface for kx = ky = 0 and leads to numerical instability (refer Fig. +S1(b)-(e)). Thus, the model exhibits an infernal point in the thermodynamic limit at kx = ky = 0, which accounts +for the states to be localized at one of the edges. An analytical derivation of the dispersion relation of the infernal +point is presented in [5] +FIG. S1: (a) shows a line gap for M = 2.3, λ = 1, δ = 0.5, B0 = 0. (b)-(d) shows the complex eigenspectrum of the Hamiltonian +defined in Eq. S1 for lattice site N = 10, 30, and 70 respectively accounting for the numerical instability of the system. +LATTICE REALIZATION OF 3DNHTI +The Hamiltonian in the main text can be realized in a cubic lattice with an electron with spin up and down in s +and p orbital respectively. Thereafter, the tight binding Hamiltonian is given by, +H = +� +r,γ +C† +r,γH(k)Cr,γ, +(S1) +where, r = x, y, z denotes the position of lattice and γ = 0(1) notifies the s(p) orbitals. We try to expand each term +on the basis of the lattice mentioned in the main text. The constant part of the onsite term in the second quantization +notation is written as, + +a +0.4 +0.2 +Im( +0 +-0.2 +-0.4 +-4 +-2 +2 +4 +Re(E)n +0.5 +国 +Im( +-0.5 +Re(E)0.5 +Im( +-0.5 +2 +2 +Re() +E0.3 +IPR +0.5 +Im(E) +0.2 +-0.1 +-0.5 +-1 +Re(E)2 +� +r,γ +C† +r,γ(−M)τzσ0Cr,γ += +� +r,γ +� +C† +r,s,↑ C† +r,s,↓ C† +r,p,↑ C† +r,p,↓ +� +� +� +� +� +−M +0 +0 +0 +0 +−M +0 +0 +0 +0 +M +0 +0 +0 +0 +M +� +� +� +� +� +� +� +� +Cr,s,↑ +Cr,s,↓ +Cr,p,↑ +Cr,p,↓ +� +� +� +� += +� +r +[−MC† +r,s,↑Cr,s,↑ − MC† +r,s,↓Cr,s,↓ − MC† +r,p,↑Cr,p,↑ − MC† +r,p,↓Cr,,↓] += −M +� +r,γ +(−1)γC† +r,γσ0Cr,γ +(S2) +where, C† +r,γ = +� +C† +r,γ,↑ C† +r,p,γ,↑ +� +. +The k dependent onsite terms are of the following form, +� +r,γ +C† +r,γ(cos kx)τzσ0Cr,γ = +� +r +� +C† +r,s,↑ C† +r,s,↓ C† +r,p,↑ C† +r,p,↓ +� +� +� +� +� +cos kx +0 +0 +0 +0 +cos kx +0 +0 +0 +0 +− cos kx +0 +0 +0 +0 +− cos kx +� +� +� +� +� +� +� +� +Cr,s,↑ +Cr,s,↓ +Cr,p,↑ +Cr,p,↓ +� +� +� +� += +� +r +[cos kxC† +r,s,↑Cr,s,↑ − cos kxC† +r,s,↓Cr,s,↓ − cos kxC† +r,p,↑Cr,p,↑ − cos kxC† +r,p,↓Cr,p,↓] += 1 +2 +� +r +(C† +r+ex,s,↑Cr,s,↑ + C† +r+ex,s,↓Cr,s,↓ − C† +r+ex,p,↑Cr,p,↑ − C† +r+ex,p,↓Cr,p,↓) += 1 +2 +� +r,γ +(−1)γC† +r+ex,γσ0Cr,γ + H.C. +(S3) +Hence, the collective term cos kx + cos ky + cos kz can be evaluated as: +� +r,γ +C† +r,γ(cos kx + cos ky + cos kz)τzσ0Cr,γ = 1 +2 +� +r,γ +� +i=x,y,z +(−1)γC† +r+ei,γσ0Cr,γ + H.C. +(S4) +Similarly, following the same steps of the calculation, the rest of the terms can be converted as: +� +r,γ +� +i=x,y,z +C† +r,γ(sin ki)τxσiCr,γ = λ +2i +� +r,γ +� +i=x,y,z +(−1)γC† +r+ei,γ+1σiCr,γ + H.C, +(S5) +� +r,γ +� +i=x,y,z +C† +r,γ(Bτzσi)Cr,γ = B +� +r,γ +� +i=x,y,z +(−1)γC† +r,γσiCr,γ + H.C, +(S6) +and +� +r,γ +� +i=x,y,z +C† +r,γiδτxσ0Cr,γ = iδ +� +r,γ +(−1)γC† +r,γ+1σ0Cr,γ. +(S7) +After collecting the terms the lattice Hamiltonian takes the following form, +H = +−M +� +r,γ +(−1)γC† +r,γσ0Cr,γ + [1 +2 +� +r,γ +(−1)γC† +r+ex,γσ0Cr,γ + H.C] ++ λ +2i +� +r,γ +� +i=x,y,z +(−1)γC† +r+ei,γ+1σiCr,γ + H.C. ++B +� +r,γ +� +i=x,y,z +(−1)γC† +r,γσiCr,γ + H.C. + iδ +� +r,γ +(−1)γC† +r,γ+1σ0Cr,γ, +(S8) +where, the NH part in the Hamiltonian can be realized as the electron-phonon interaction between s and p orbital. + +3 +FORMALISM FOR FLOQUET THEORY +We define a non-unitary time evolution operator U(t, t +′) which evolves the system from time t to t′ with periodicity +τ = 2π +ω , then the Floquet theorem states that, +U(t + nτ, t0) = U(t, t0)[U(t0 + τ, t0)]n +(S9) +and we define the non-unitary Floquet Hamiltonian as: +U(t0 + τ, t0) = exp(iHF τ +ℏ +) +(S10) +where, HF is the Floquet NH Hamiltonian. For the stroboscopic analysis, we can always set t0 = 0, without the +loss of generality, as the original time period of the periodically driven system dominates any time scale that may +be acquired by the unitary operator. The eigenstates corresponding to the Floquet operators defined above for time +t0 = 0 are called the Floquet modes and are given by ψα(0). Thus, the Floquet operator can be rewritten as[22], +U(τ, 0) = +� +α +e−iϵατ/ℏ |ψα(0)⟩ ⟨ψα(0)| , +(S11) +where, ϵα are the complex eigenvalues corresponding to the Floquet NH Hamiltonian[38]. The Floquet modes also +satisfy the periodic relation as, ψα(τ) = ψα(0). The time evolution of the Floquet modes is given by: +Ψα(t) = e−iϵαt/ℏψα(0). +(S12) +Substituting the time-dependent ansatz into the time-dependent Schrodinger’s equation we get, +[H(t) − iℏ ∂ +∂t]Ψα(t) = ϵαΨα(t), +(S13) +where, K(t) = H(t) − iℏ ∂ +∂t can be termed as Floquet extended Hamiltonian. +Floquet-space-time representation of periodically driven systems +We can further expand the Floquet modes in the time-periodic Fourier series as: +Ψα(t) = +∞ +� +j=−∞ +|φα,j⟩ eijωt. +(S14) +Plugging this into the above equation yields: +∞ +� +j=−∞ +Hj−j′ φα,j + jℏωφα,j = ϵαφα,j, +(S15) +where, +Hj−j′ = Hn = 1 +τ +� τ/2 +−τ/2 +H(k, t)einωt. +(S16) +Thus, the above equation can be written in matrix formulation as: +Hslab = +� +����������� +H0 +H−1 +0 +. . . +0 +0 +0 +H1 H0 − ℏω +H−1 +. . . +0 +0 +0 +0 +H1 +H0 − 2ℏω . . . +0 +0 +0 +... +... +... +... +... +... +... +0 +0 +0 +. . . ... H−1 +0 +0 +0 +0 +. . . H1 +... +H−1 +0 +0 +0 +. . . +0 +H1 +H0 − jℏω +� +����������� +. +(S17) + +4 +FIG. S2: The Floquet variable onsite terms as the slabs and temporal hoppings are shown by red arrows +Thus, the harmonic indices j and j +′ represent the fictitious temporal direction such that a d-dimensional Hamiltonian +can always be visualized in the d + 1 dimensional space-time representation(refer Fig. S2). The term jℏω represents +the variable onsite term (similar to a 1D chain under stark electric field) and H +′ +j−j represents the hopping between j +and j +′ temporal site. In addition to this, quasi-energy also satisfies the periodic relation of ϵα = ϵα + jℏω. Thus the +space-time picture can also equally be mapped into the Wannier-Stark ladder[39]. +considering the periodicity of the drive to be very high, we can write the Hamiltonian as : +H = H0 + H1eiωt + H−1e−iωt. +(S18) +In the infinite frequency domain, the hopping along the temporal direction becomes completely ineffective breaking +a d+1-dimensional system (4-dimensional system in our case) to the isolated d-dimensional systems. However, in the +low energy approximation, the perturbation theory yields unique results in the second order. If ϵ is the energy which +is associated with the zeroth mode level(H0), then going from n = 0 and n = 1 and coming back is described by the +term H−1 +1 +(ϵ+ℏω)−ϵH† +1 and going from n = 0 to n = −1 and coming back is described by the term H1 +1 +(ϵ−ℏω)−ϵH† +−1. +Thus, the whole process described above can be mathematically written as: +Heff = H0 +F + +inf +� +n=1 +1 +ω [H−n +F , Hn +F ]. +(S19) +FLOQUET DRIVING OF 3DNHTI +We use the above-developed perturbation theory in the conventional 3DNHTI and try to develop FETI. We use +the vector potential(mentioned in the main text) as: +A = A0[cos ωt⃗e1 + η sin ωt⃗e2], +(S20) +where, A0 is the amplitude and ω is the frequency of the driven system. η = ±1 signifies the right circularly and +left circularly polarized light respectively. We choose e1 = (cos θ cos φ, cos θ sin φ, sinθ) and e2 = (sin φ, − cos φ, 0). +Thus, the vector potential is given by: +A = A0 (cos θ cos φ cos ωt + η sin φ sin ωt, cos θ sin φ cos ωt − η cos φ sin ωt, − sin θ cos ωt) , +(S21) +with minimal coupling the time-dependent Hamiltonian becomes, +H(k) = � +j(cos kj − M − A2 +0 +4 (1 + η2))τzσ0 + λ � +j[sin kx − A0(cos θ cos φ cos ωt + η sin φ sin ωt)τxσx ++ sin ky − A0(cos θ sin φ cos ωt − η cos φ sin ωt)τxσy + sin kz − A0(− sin θ cos ωt)τxσz] + iδτxσ0. +(S22) + +Ho + hw +Hi1 +H_1 +Ho +Hi +H_1 +Ho- hw5 +Thus, the various Floquet terms can be recovered as: +H0 = +� +j +(cos kj − M − A2 +0 +4 (1 + η2))τzσ0 + λ +� +j +sin kjτxσj + iδτxσ0, +(S23) +H1 = −λA0 +2 ([cos θ cos φ + iη sin φ]τxσx + [cos θ sin φ − iη cos φ]τxσy + sin θτxσz), +(S24) +and +H−1 = −λA0 +2 ([cos θ cos φ − iη sin φ]τxσx + [cos θ sin φ + iη cos φ]τxσy + sin θτxσz). +(S25) +Thus, in the Fourier space-time representation, each onsite term represents a 3DNHTI with onsite loss and gain +term whereas the hopping between two connected 3DNHTI can be modulated by the angle of polarization. Thus the +whole system can be modulated by A0, θ, φ to realize exotic quantum phases of matter. +We neglect the periodic fluctuating terms with coefficient A2 +0 +ω . Thus, by using equation 13 the effective Hamiltonian +can be calculated as: +H(k) = +� +j +(cos kj − M − A2 +4 (1 + η2))τzσ0 + λ +� +j +sin kjτxσj + τ0(n.σ) + iδτxσ0, +(S26) +where, the vector n is given by : +n = λ2A2η +ω +(sin θ cos φ, sin θ sin φ, cos θ). +(S27) +However, in the presence of the extrinsic magnetic field, the Hamiltonian in the stroboscopic phase is given by: +H(k) = +� +j +(cos kj − M − A2 +4 (1 + η2))τzσ0 + λ +� +j +sin kjτxσj + (nτ0 + Bτz).σ + iδτxσ0 . +(S28) +LATTICE REALIZATION OF FETI +The FETI can also be calculated in the cubic lattice as: +H = −(M + A2 +0 +2 ) � +r,γ(−1)γC† +r,γσ0Cr,γ + [ 1 +2 +� +r,γ(−1)γC† +r+ex,γσ0Cr,γ + H.C] ++[ λ +2i +� +r,γ +� +i=x,y,z(−1)γC† +r+ei,γ+1σiCr,γ + H.C] +� +r,γ +� +i=x,y,z(ni + B(−1)γ)C† +r,γσiCr,γ + iδ � +r,γ C† +r,γ+1σ0Cr,γ. +(S29) +Thus, irradiating light in a 3DNHTI generates Onsite excitation due to the coupling of light-matter interaction +which is analogous to a photo-induced magnetic field. The uniqueness of such a fictitious magnetic field reveals its +true nature from the fact that it is generally anisotropic(except at the critical angle defined in the main text). The +amplitude of such a field is dependent on the frequency and handedness of the light used. +PHOTO-INDUCED HOMOTOPY IN FETI +We truncate x-axis while retaining PBC along y and z axes. For simplicity, we use the angle of polarization as +θ = 0 and φ = φ′ where 0 ≤ φ′ ≤ 2π. Then the truncated hamiltonian can be written as: +H = 1 +2[� +x C† +x+1(τzσ0 + iλτxσx)Cx + � +x C† +x−1(τzσ0 − iλτxσx)Cx] ++C† +x[(2 − M − A2 +0 +2 )]τzσ0 + λ2A2 +0 +ω +τ0σz + iδτxσ0]Cx. +(S30) + +6 +We use the trial wavefunction as: +|Ψ¯100⟩ = +� +x +αx |x⟩ |ζ0⟩ +(S31) +We write the Harper’s equation (Γ0 = τzσ0) as[25], +Γ0 +�1 − λτyσx +2 +α−1 + 1 + λτyσx +2 +α + [(2 − M − A2 +0 +2 ) + λ2A2 +0 +ω +τzσz − δτyσ0] +� +|ζ0⟩ = 0 +(S32) +The terms τzσz and τyσ0 commutes with τyσx.Thus the eigenstates of τyσx are given by: +|ψ1⟩ = (−i, 0, 0, 1)T +√ +2 +(S33) +|ψ2⟩ = (0, −i, 1, 0)T +√ +2 +(S34) +|ψ−1⟩ = (i, 0, 0, 1)T +√ +2 +(S35) +|ψ−2⟩ = (0, i, 1, 0)T +√ +2 +(S36) +Thus we write the spinor |ζ0⟩ as: +|ζ0⟩ = p1 |ψ1⟩ + p2 |ψ2⟩ +(S37) +We set p1 = cos θ′ and p2 = sin θ′eiφ′. Thus, after normalization, the Harper equation reduces to: +� +α − M − A2 +0 +2 + λ2A2 +0 +ω +� +cos θ′ − δ sin θ′eiφ′ = 0 +(S38) +� +α − M − A2 +0 +2 + λ2A2 +0 +ω +� +sin θ′eiφ′ − δ cos θ′ = 0 +(S39) +Thus the value of the constants are given by +α = +� +(δ2 + (λ2A2 +0 +ω +)2) + (M + A2 +0 +2 ) − 2 +(S40) +θ′ = tan−1(α − M − A2 +0 +2 + 2 + λ2A2 +0 +ω +δ +) +(S41) +φ′ = 0 +(S42) +Following the similar steps, we can write the trial wavefunction for (100) surface as : +|Ψ¯100⟩ = +� +x +αx |x⟩ |ζ′ +0⟩ +(S43) +Then , after normalisation, the surface states are given by: +|Ψ¯100⟩ = +� +x +αL−x |x⟩ [cos θ′ |ψ−1⟩ − sin θ′ |ψ−2⟩] +(S44) + +7 +|Ψ100⟩ = +� +x +αx |x⟩ [cos θ′ |ψ−1⟩ + sin θ′ |ψ−2⟩] +(S45) +For the system to establish homotopy with the torus shaped BZ and exhibits a single sheet in the surface, the top +and buttom surfaces must couple to each other demanding the surface state in the complex eigenspectrum to be a +superposition of |Ψ100⟩ and |Ψ¯100⟩.Thus we assume the surface state to have the following form. +|ψ⟩ = (|Ψ100⟩) ± (|Ψ¯100⟩) +√ +2 +. +(S46) +We then consider the perturbative correction of ky and kz respectively. Since ⟨ψ1| τxσy |ψ1⟩ = − ⟨ψ2| τxσy |ψ2⟩ = +⟨ψ−1| τxσy |ψ−1⟩ = ⟨ψ−2| τxσy |ψ−2⟩ = 1.Hence ⟨Ψ¯100| τxσy |Ψ¯100⟩ = − ⟨Ψ100| τxσy |Ψ100⟩ = cos 2θ′, thus ky terms +results in the energy splitting of ± cos 2θ′ky for small value of ky which would cage the surface states to localise +at one surface. +Following the similar calculations for the kz term, we have, ⟨ψ1| τxσy |ψ2⟩ = − ⟨ψ2| τxσy |ψ1⟩ = +− ⟨ψ−1| τxσy |ψ−2⟩ = ⟨ψ−2| τxσy |ψ−1⟩ = i. But kz dependent terms are not diagonal unlike the ky dependent terms. +So, |Ψ100⟩ and |Ψ¯100⟩ are not the good basis of the perturbed Hamiltonian when kz dependent perturbations are +included in the system. To Tackle this difficulty, we re-solve the Harper’s equation again without the NH terms. +Γ0 +�1 − λτyσx +2 +α−1 + 1 + λτyσx +2 +α + [(2 − M − A2 +0 +2 ) + λ2A2 +0 +ω +τzσz − δτyσ0 + i sin kzτyσz] +� +|ζ0⟩ = 0 +(S47) +As [τyσz, τxσy] = 0,the kz dependent term inter-twins and mixes the Hilbert space of the unperturbed Hamiltonian +since τyσz |ψ1⟩ = |ψ−2⟩, τyσz |ψ2⟩ = |ψ−1⟩, τyσz |ψ−2⟩ = |ψ1⟩, and τyσz |ψ−1⟩ = − |ψ2⟩.Thus for (¯100) surface we +redefine the spinor as +|ζ⟩ = cos θ1 |ψ1⟩ + sin θ1eiφ1 |ψ−2⟩ +(S48) +Re-solving the Harper’s equation we get +α′ = β1 + β2 +β3 +(S49) +β1 = −1 + (λ2A2 +0 +ω +)2 − (sin kz)2 − (M + A2 +0 +2 ) +(S50) +β2 = − +� +(1 − (λ2A2 +0 +ω +)2 + (sin kz)2) + 4((λ2A2 +0 +ω +)2 − (M + A2 +0 +2 )2) + (M + A2 +0 +2 )2 +(S51) +β3 = 2(M + A2 +0 +2 − λ2A2 +0 +ω +) +(S52) +θ1 = − tan−1( +sin kz +α′ + M + A2 +0 +2 + λ2A2 +0 +ω +) +(S53) +φ1 = −π +2 +(S54) +similarly for (100) surface, we can write +|ζ1⟩ = cos θ1 |ψ1⟩ − i sin θ1 |ψ2⟩ +(S55) +|ζ2⟩ = cos θ1 |ψ−1⟩ + i sin θ1 |ψ−2⟩ +(S56) +Now including the NH electron-phonon interaction in the Hilbert space of |ζ1,2⟩: ⟨ζ1| iτxσ0 |ζ1⟩ = ⟨ζ2| iτxσ0 |ζ2⟩ = +i sin 2θ1. Thus the NH interaction splits the energy level equally. For small kz, θ1 is directly proportional to kz and +energy splitting in the NH complex energy is directly proportional to ikz. Thus, the surface states have zero energy +for finite δ since the NH term does not alter the surface states for kz = 0 as θ1 = 0 when kz = 0 thereby forming a +single sheet in the complex eigenspectrum. + +8 +PHASE SPACE OF FETI +We neglect A2 +ω terms in the Hamiltonian to invoke the hidden sub-lattice symmetry of it by rotating the basis as, +τx ⇒ τy ⇒ τz ⇒ τx. Then the Hamiltonian can be converted into the off-block diagonal form[5], +H(k, M, λ, δ, A) = � +j(cos kj − M − A2 +4 (1 + η2))τxσ0 + λ � +j sin kjτyσj + iδτyσ0 +⇒ H(k, M, λ, δ, A) = +� +0 +h† +UR +hLL +0 +� +, +(S57) +where, h† +UR and hLL denotes the upper right and lower left matrices respectively and given by, +h† +UR = +�f(k, M, A) − iλ sin kz + δ +−iλ sin kx − λ sin ky +−iλ sin kx + λ sin ky +f(k, M, A) + iλ sin kz + δ +� +(S58) +and +hLL = +�f(k, M, A) + iλ sin kz − δ +iλ sin kx + λ sin ky +iλ sin kx − λ sin ky +f(k, M, A) − iλ sin kz − δ +� +. +(S59) +Here, f(k, M, A) = � +j(cos kj−M− A2 +4 (1+η2) we can then define Qi = H−1∂kjH = +�h−1 +LL∂kihLL +0 +0 +(h† +UR)−1∂kihLL +� +. +The topological invariant W3D can then be expressed as, +W3D = W LL +3D − W UR +3D . +(S60) +We can further link both of the off diagonal matrices as considering, +hLL(k, M, A, λ, δ) = h0(k, M + A2 +4 (1 + η2) + δ, λ), +(S61) +h† +UR(k, M, A, λ, δ) = h0(k, M + A2 +4 (1 + η2) − δ, λ) +(S62) +and the Hamiltonian h0 is written in the form, +h0 = dµσµ +(S63) +where, dµ = (cos kj −M +′, iλ sin kx, iλ sin ky, iλ sin kz) and σµ = (σ0, σ1, σ2, σ3). Thus it represents a two-band model +of 3DTI with NH spin-orbit-coupling. For vanishing point gap the system demands that h0(k, M +′, λ) = 0, implies, +cos kj = M +′. This yields, M +′ = ±1, ±3 which decides the phase space of the block matrices. Thus, the winding +number can be expressed as, +W3D = W 0 +3D(k, M + A2 +4 (1 + η2) + δ, λ) − W 0 +3D(k, M + A2 +4 (1 + η2) − δ, λ), +(S64) +and, M +′ = ±3 implies | M + A2 +4 (1 + η2) |= 3. Thus, | M + A2 +4 (1 + η2) − 3 |≤ δ corresponds to a central point-gap +whereas | M + A2 +4 (1 + η2) − 3 |≥ δ corresponds to a central line-gap. Let us consider the model to be in static +topological phase boundary for which it demands the value of M = 3. Then by shining light on such a quantum +phases of matter in the topological phase boundary, a phase transition from point gap to the line gap can be achieved. +A2 +4 (1 + η2) ≤ δ demands a central point gap and A2 +4 (1 + η2) ≥ δ demands a central line gap. + +9 +FIG. S3: (a)-(h) corresponds to the dynamic evolution of the probabilities of the trial wavefunction defined in the main text +for M = 3,δ = 1,λ = 1,B0 = 0.2,from time t = 0 to t = 70 in the interval of 10. |ζ0⟩ = [1, 0, i, 0] and the value of α and β +remains same as in the main text. +SLAB CALCULATION AND WAVE DYNAMICS +We prepare a trial wave function, defined in the main text, and evolve it with respect to the slab Hamiltonian in +which two of the axes are truncated. In the tight-binding representation, it is given by, +H = +� +i,j +H0C† +i,jCi,j + +� +tyC† +i,jCi+1,j + tzC† +i,jCi,j+1 + H.C +� +(S65) +where, H0 is given by, +H0 = +� +j +� +cos kj − M − A2 +2 +� +τzσ0 + λ sin kyτxσy + τ0 (n.σ) + τz (B.σ) + iδτxσ0. +(S66) +The hoppings takes the following form, +ty = 1 +2 (τzσ0 − iλτxσx) , +(S67) +t† +y = 1 +2 (τzσ0 + iλτxσx) , +(S68) +tz = 1 +2 (τzσ0 − iλτxσz) , +(S69) +and +t† +z = 1 +2 (τzσ0 + iλτxσz) . +(S70) +We have mentioned wave-dynamics evolution for critical angles in the main text. In this section, we demonstrate +the wave-dynamics evolution of the wave function for the static case in the presence of an external magnetic field. +The wave-packet initially localized in one of the corners of the 2D sheet permeates into the bulk to travel to the +opposite corner and returns to the same corner when the system evolved with respect to time. This evolution of an +arbitrary Gaussian wave-packet account for the mimic phenomena of NHSSE exhibited by the model in the static +phase with an external magnetic field where part of the wave-function probabilities are always quenched dynamically +along the corners (see Fig.S3(a)-(h)). Thus, although the time evolution makes the wave-function to get absorbed +into the bulk, it immediately escapes the bulk by penetrating to the other corner since the duration for which the it +remains in the bulk is negligible compared to the duration for which it remains in the corners. + +0 +0 = ↑ +0.5 +Y +10 +(a) +0.3 +20 +10 Z +0 +200 +t = 10 +0.04 +Y +10 +(b) +0.02 +20 +0 +200+ +t = 20 +0.025 +Y +(C) +10 +0.01 +20 +10 +Z +200 +t = 30 +0.05 +5 +Y +(d) +0.03 +10 +15 +0.01 +20 +5 +10 +Z 15 +200 +t = 40 +0.08 +Y +(e) +10 +0.04 +20 +10 +Z +200- +t = 50 +0.014 +Y +(f) +0.007 +10 +20 +10z +(0 +200 +09 = +0.020 +Y +(g) +10 +0.010 +20 +20 +0 +10 +ZFO +02 = 4 +0.0175 +亚|2 +Y +(h) +10 +0.0075 +20 +10 +Z +2010 +UNCONVENTIONAL 3DNHTI +We define an unconventional 3DNHTI as a cubic lattice of 3DNHTI (mentioned in the main text) with intrinsic +spin orbit coupling (SOC) preserving the spin alignment controlled by the parameter ∆. The rest of the terms has +usual meaning as discussed for the previous model. The tight binding Hamiltonian is given by, +H(k) = +� +j +(cos kj − M)τzσ0 + λ +� +j +sin kjτxσj + ∆ +� +j +sin kjτyσ0 + iδτxσj . +(S71) +The model exhibits similar complex eigenspectrum as that of 3DNHTI. For |M − 3| ≤ δ and |M − 3| ≥ δ, it +showcases a central point gap and a central line gap respectively (refer Fig. S4(a)-(b)).It is, however, a band strained +version of 3DNHTI near the BZ which accounts for the flatness of the bands at BZ boundary. However, the phase +diagram of this unconventional 3DNHTI remains same as the 3DNHTI discussed above. +FIG. S4: (a) and (b) depicts the central point gap and line gap for M = 3,λ = 1,δ = 1,∆ = 1 and M = 2.3,λ = 1,δ = 0.5,∆ = 1 +respectively.(c) shows a single sheet structure in the complex eigenspectrum for M = 2.5,λ = 1,δ = 1,∆ = 1,A0 = 1,ω = 5 +when the model is truncated along z-direction. +LATTICE REALIZATION OF UNCONVENTIONAL 3DNHTI +We express its tight binding Hamiltonian in the cubic lattice as, +H = −M � +r,γ(−1)γC† +r,γσ0Cr,γ + [ 1 +2 +� +r,γ(−1)γC† +r+ex,γσ0Cr,γ + H.C] ++[ λ +2i +� +r,γ +� +i=x,y,z(−1)γC† +r+ei,γ+1σiCr,γ + H.C] ++[ ∆ +2 +� +r,γ C† +r+ex,γ+1σ0Cr,γ + H.C] + +iδ � +r,γ(−1)γC† +r,γ+1σ0Cr,γ. +(S72) +FLOQUET DRIVING OF UNCONVENTIONAL 3DNHTI +With the vector potential defined in equation S21, minimal coupling yields the time-dependent Hamiltonian, +H(k) = � +j(cos kj − M − A2 +0 +4 (1 + η2))τzσ0 + λ[sin kx − A0(cos θ cos φ cos ωt + η sin φ sin ωt)τxσx ++ sin ky − A0(cos θ sin φ cos ωt − η cos φ sin ωt)τxσy + sin kz − A0(− sin θ cos ωt)τxσz] +∆((sin kx − A0(cos θ cos φ cos ωt + η sin φ sin ωt)) + (sin ky − A0(cos θ sin φ cos ωt − η cos φ sin ωt)) ++(sin kz − A0(− sin θ cos ωt)))τyσ0 + iδτxσ0. +(S73) +The various Floquet modes are extracted as, +H0 = +� +j +(cos kj − M − A2 +0 +4 (1 + η2))τzσ0 + λ +� +j +sin kjτxσj + ∆ +� +j +sin kjτyσ0 + iδτxσ0, +(S74) + +1 +(a) +0.5 +Im(E) +0 +-0.5 +-1 +-6 +-4 +-2 +0 +2 +4 +6 +Re(E)(b) +0.4 +0.2 +Im( +0 +-0.2 +-0.4 +-2 +2 +4 +Re(E)0.5 +IPR +0.5 +Im(E) +0.3 +0 +-0.5 +0.1 +Re(E)11 +H1 = − λA0 +2 ([cos θ cos φ + iη sin φ]τxσx + [cos θ sin φ − iη cos φ]τxσy ++ sin θτxσz) − − ∆A0 +2 ([cos θ cos φ + iη sin φ] ++[cos θ sin φ − iη cos φ] + sin θ)τyσ0, +(S75) +and +H−1 = − λA0 +2 ([cos θ cos φ − iη sin φ]τxσx + [cos θ sin φ + iη cos φ]τxσy ++ sin θτxσz) − − ∆A0 +2 ([cos θ cos φ − iη sin φ] ++[cos θ sin φ + iη cos φ] + sin θ)τyσ0. +(S76) +Thus, expanding the time dependent Hamiltonian in the stroboscopic phase yields: +H(k) = +� +j +(cos kj − M − A2 +2 )τzσ0 + λ +� +j +sin kjτxσj + ∆ +� +j +sin kjτyσ0 − τ0(n.σ) − τz(n′.σ) + iδτxσ0 , +(S77) +where, n is the photodressed vector obtained above and n′ vector is given by, +n′ = (cos θ − sin θ sin φ, cos θ − sin θ cos φ, sin θ(cos φ − sin φ)) . +(S78) +FIG. S5: (a) denotes an Abs(E) contour plot for M = 2.5,λ = 1,δ = 1,∆ = 1,A0 = 1,ω = 5,θ = tan−1√ +2 and φ = π +4 .(b) and +(c) show the sliding of two of the sheets M = 2.5,λ = 1,δ = 1,A0 = 1,ω = 5,θ = π +2 and φ = π +4 by changing ∆ as -0.6 and -0.9 +respectively. +Thus, the Floquet driving generates two kinds of photo-dressed vectors that behave like the magnetic field. For +the critical angle, the vectors become isotropic and exhibit a single sheet hosting two Fermi point(unlike the single +sheet of FETI hosting a single Fermi point) and establish homotopy with the BZ (see Fig. S4(c) and S5(a)), but for +the choice of the suitable angle of polarization, the single sheet evolves into a double sheet by sliding over each other +hosting a non-degenerate edge state at each of the surfaces (see Fig. S5(b)-(c)). Therefore, the angle of polarization +helps one to switch between both cases of the band spectrum on the surface. Hence ∆ modulates the photo-dressed +Land´e-g-factor, in the sense, deciding the alignment of s and p orbital along the photo-dressed magnetic field even +when the system does not experience any external magnetic field. The interaction which gives rise to additional +spin-orbit coupling in the static phase and transforms a conventional 3DNHTI into an unconventional 3DNHTI also +gives rise to the photo-dressed lande-g-factor for the dynamic case. + +0.4 +a +0.2 +-0.2 +-0.4 +-3 +-1 +0 +1 +2 +30.4 +0.2 +ky O +-0.2 +-0.4 +3 +-2 +-1 +0 +1 +2 +3c +0.4- +1.4 +0.2 +1.2亩 +ky0 +bs( +9 +0.8A +-0.2 +0.4 +-0.4 +0 +-2 +0 +2 +312 +LATTICE REALIZATION OF FLOQUET UNCONVENTIONAL 3DNHTI +The FETI can be realized in the cubic lattice as, +H = −(M + A2 +0 +2 ) � +r,γ(−1)γC† +r,γσ0Cr,γ + [ 1 +2 +� +r,γ(−1)γC† +r+ex,γσ0Cr,γ + H.C] ++[ λ +2i +� +r,γ +� +i=x,y,z(−1)γC† +r+ei,γ+1σiCr,γ + H.C] ++[ ∆ +2 +� +r,γ C† +r+ex,γ+1σ0Cr,γ + H.C] + � +r,γ niC† +r,γσiCr,γ ++ � +r,γ(−1)γn +′ +iC† +r,γσiCr,γ + iδ � +r,γ C† +r,γ+1σ0Cr,γ. +(S79) +SLAB HAMILTONIAN AND WAVE DYNAMICS EVOLUTION +As mentioned in Eq. 26, we determine the slab Hamiltonian and calculate the onsite and hopping part of the +Hamiltonian. These are given by, +H0 = +� +j +� +cos kj − M − A2 +2 +� +τzσ0 + λ sin kyτxσy + τ0 (n.σ) + τz +� +n +′.σ +� ++ iδτxσ0, +(S80) +ty = 1 +2 (τzσ0 − iλτxσy − i∆τyσ0) , +(S81) +t† +y = 1 +2 (τzσ0 + iλτxσy + i∆τyσ0) , +(S82) +tz = 1 +2 (τzσ0 − iλτxσz − i∆τyσ0) , +(S83) +and +t† +z = 1 +2 (τzσ0 + iλτxσz + i∆τyσ0) +(S84) +respectively. For case (1), the surface states are localized in one of the edges and it has a significant amount of +overlap integral with the bulk (refer Fig. S6(a)-(h)). So, the wavefunction initially localized in the middle of the +Z-axis also has a significant amount of non-vanishing amplitude in the bulk which makes the localized wavefunction +penetrate into the bulk and allow a part of it to travel along the edges. The wavepacket stays on the edge for the +longer time where the surface states are caged. +For the second case, the wavefunction initially localized in one of the corners travels to the other corners and gets +localized with a quasi-static phase in the dynamic evolution (see Fig. S7(a)-(h)). The localization of the wavefunction +increases at the corner where the majority of them are trapped. So, the wave dynamic evolution of the localized +Gaussian wavefunction is guided by the NHSSE entrailed within the Hamiltonian which can be modulated by varying +the angle of polarization of the CPL. + +13 +FIG. S6: (a)-(h) corresponds to the dynamic evolution of the probabilities of the trial wavefunction initially localized at the +middle of one of the edges for M = 2.5,δ = 1,λ = 1,∆ = 1 = 0.2,A0 = 1,ω = 5,θ = tan−1√ +2 and φ = π +4 from time t = 0 to +t = 70 in the interval of 10.|ζ0⟩ = [1, i, i, 0] and the value of α and β remains same as in the main text. +FIG. S7: (a)-(h) corresponds to the dynamic evolution of the probabilities of the trial wavefunction initially localized at one +of the corners for M = 2.5,δ = 1,λ = 1,∆ = 1 = 0.2,A0 = 1,ω = 5,θ = tan−1√ +2 and φ = π +4 from time t = 0 to t = 70 in the +interval of 10.|ζ0⟩ = [1, i, i, 0] and the value of α and β remains same as in the main text. + +0 +0=↑ +0.25 +Y +0.15 +10 +(a) +0.05 +20 +10 +Z +200 +t = 10 +0.014 +10 +(b) +0.007 +20 +0 +0 +10 +Z. +200 +t = 20 +0.025 +Y +0.015 +10 +(c) +20 +10 +Z +200- +t = 20 +0.03 +Y +0.02 +10 +(c) +0.01 +20 +0 +10 +Z +200 +t =40 +0.01 +Y +10 +(e) +0.006 +0.002 +20 +10 +Z +200 +t = 50 +0.012 +Y +(f) +10 +0.006 +20 +0 +10 +Z +20F0 +=60 +0.0175 +Y +(g) +10 +0.0075 +20 +20 +0 +10 +Z0- +0.035 +/亚|2 +Y +(h) +10 +0.015 +20 +10 +Z. +200 +0=↑ +0.4 +Y +10 +(a) +0.2 +20 +0 +0 +10 +Z +200 +t = 10 +0.014 +10 +0.007 +(b) +20 +0 +10 +Z +200 +t = 20 +0.016 +Y +10 +(c) +0.008 +20 +0 +10 +Z +200 +t = 30 +0.03 +Y +0.02 +10 +(d) +0.01 +20 +10 +Z +200 +t = 40 +0.03 +Y +0.02 +10 +(d) +0.01 +20 +10 +z +200 +t = 50 +0.020 +Y +10 +(e) +0.010 +20 +20 +10 +Z0 +0.02 +10 +(g) +0.01 +20 +10 +Z +200 +70 +0.012 +[亚|2 +Y +10 +(h) +0.006 +20 +10 +Z +20 \ No newline at end of file diff --git a/MdFPT4oBgHgl3EQfkzU3/content/tmp_files/load_file.txt b/MdFPT4oBgHgl3EQfkzU3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ba7420f8233ef5277936b00914e70c3f7e59447 --- /dev/null +++ b/MdFPT4oBgHgl3EQfkzU3/content/tmp_files/load_file.txt @@ -0,0 +1,1059 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf,len=1058 +page_content='Floquet Exceptional Topological Insulator Gaurab Kumar Dash, Subhajyoti Bid, and Manisha Thakurathi Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India 110016 (Dated: January 31, 2023) We propose a novel way of modulating exceptional topology by implementing Floquet engineering in non-hermitian (NH) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We introduce Floquet exceptional topological insulator which results from shining light on a conventional three-dimensional NH topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Light- matter interaction facilitates the quantum phases of matter to exhibit a novel phenomenon, where, the point gaps in the bulk host surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' These distinct surface states either fill the point gap in the complex eigenspectrum or exhibit exceptional points in the presence of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We also highlight the existence of a quantum anomaly generated by photo-induced modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The existence of the Floquet biorthogonal Chern number and spectral winding number show that the momentum slices exhibit NH skin effect, even though the system as a whole does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We also employ wave-dynamics evolution to illustrate the NH surface skin effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='— The non-hermitian (NH) topological phases have been the center of attraction for theoretical as well as experimental studies [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Many experimen- tally realizable models of NH systems have been proposed in condensed matter physics hosting exceptional points (EPs), lines, rings, and nodal planes [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Recently, NH generalization of topological insulators (TI) known as Exceptional TIs has been studied where the surface hosts either a 2D band structure with a single EP (a point where both eigenenergies and eigenvectors coalesce) or a single band, which represents a vortex [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This is obtained either by varying the imbalance between the g- factors of the orbitals or the magnetic field strength while keeping it isotropic along [111] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' On the other hand, the Floquet engineering of the topological phases in Hermitian [6–17], as well as NH systems, has defined exotic ramifications in stroboscopic limits which are oth- erwise not realizable in their static counterparts[18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Therefore, Floquet engineering of exceptional topology has captured the desirable attention [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' In this letter, we interlink these two ideas and in- troduce a 3D Floquet exceptional topological insulator (FETI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We also aim to answer the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Is it possible to achieve the ETI phase without an exter- nal magnetic field, convert a hugely defective point into non-defective points, and modulate exceptional topology using Floquet theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' In order to answer these ques- tions, we start with conventional 3D NHTI and shine circularly polarised light (CPL) on it [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The system hosts a central point gap in the bulk which can be tran- sitioned into a central line gap by tuning the amplitude of the vector potential of CPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The point gaps, which are characterized by the quantized value of topological invariant in the bulk, host an infernal point [5] lead- ing to the defectiveness of the system in the static part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' However, in the stroboscopic phase, the system experi- ences a photo-induced pseudo-magnetic field and hosts robust non-defective doubly degenerate surface states lo- calized at opposite surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The doubly degenerate sur- face states fill the point gap and exhibit different disper- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1: Schematic picture of a cubic lattice with s and p orbital at each site where a CPL light is irradiated along ⃗n direction with θ and φ being the angles of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' sion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The surface state appears as a single sheet which represents a vortex without an anti-vortex partner giving rise to the quantum anomaly in 3D NHTI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This sheet can be shown to establish homotopy with a torus-shaped 2D Brillouin zone (BZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' However, when the system is subjected to a constant and isotropic magnetic field, it hosts a single second- order EP on the surface of the 3D NHTI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' In the presence of CPL, it also experiences a photo-induced pseudo-magnetic field which couples with the external magnetic field [22] to modulate the EP on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, a constant magnetic field in the static phase gains dynamics in the stroboscopic picture thereby dramati- cally photo-modulating the Land´e-g-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The system possesses NH surface skin effect (NHSSE) [23, 24] in both, static as well as dynamic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Here we explain this surface quenching phenomenon with the help of wave dynamics[25, 26] without analyzing the photovoltaic and chiral transport phenomenons associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Static model— We start with the quantum mechanical model of spin-full 3D NHTI, given by following Hamilto- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='13119v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='mes-hall] 30 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 2: (a) Bulk complex eigenspectrum for point gap region satisfying |M + A2 2 − 3| ≤ δ, cyan (pink) color highlights the value of W3D = 1(−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (b) represents the central point gap region of complex eigenspectrum for OBC along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The degenerate surface states fill the point gap completely and appear as a single sheet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (c) Plot of Abs(E) as a function of kx for ky = 0, hosting a surface Dirac node at kx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Notably, when ky is non-zero the surface is gapped and the states fill the Dirac cone completely as appears in the contour plot (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' System parameters are M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5, λ = 1, δ = 1, A2 0/ω = 1/5 and momentum resolution ∆k = π/400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' nian, H0(k) = � j=x,y,z � (cos kj − M) τzσ0 + λ sin kjτxσj � + i δ τxσ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (1) In the tight-binding model, the system has s and p or- bital at each lattice site [5, 25] (implying four degrees of freedom at each site) see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1(a), σµ and τµ denotes the Pauli matrices acting independently of spin and or- bit degrees of freedom respectively where µ = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Here, M accounts for the band inversion for the s and p orbital, λ controls the intrinsic spin-orbit coupling, δ denotes the NH contribution either from electron-phonon scattering or via short-lived f-electron coupling to the s and p orbitals [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We note that the Hermitian counter- part of the above model mentioned in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1 resembles a four-band model of 3D TI hosting a Dirac node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The static Hamiltonian hosts two kinds of gaps in the bulk due to its NH property, point gap for |M − 3| ≤ δ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1(b) and line gap for |M − 3| ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Floquet exceptional topological insulator (FETI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='— We irradiate CPL [27] along the direction ⃗n = (sin θ cos φ, sin θ sin φ, cos θ), where, θ and φ are polar and azimuthal angles respectively in conventional spher- ical polar coordinates(see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The vector potential has the following form, ⃗A = A0[cos (ωt) ⃗e1 + η sin (ωt) ⃗e2], (2) where, A0 and ω represent the amplitude of the vector potential and frequency of the CPL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The parameter η regulates the orientation of the polarization (η = +1(−1) for left (right) CPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' All the three unit vectors ⃗e1, ⃗e2, and ⃗n must be orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, we choose unit vectors ⃗e1 = (cos θ cos φ, cos θ sin φ,- sin θ) and ⃗e2 = (sin φ, − cos φ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We derive an effective stroboscopic Hamiltonian invoking the high-frequency Floquet formalism (see [28]), given by following form, HF (k) = � j=x,y,z � (cos kj − M − A2/2)τzσ0 + λ sin kjτxσj � + τ0 (⃗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='⃗σ) + i δ τxσ0, (3) where, the vector ⃗n is given by, ⃗n = ηλ2A2 ω (sin θ cos φ, sin θ sin φ, cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (4) Here, the term coupled to the unit direction ⃗n can be defined as the photo-induced magnetic field generated by the light-matter interaction as a result of the Floquet driving [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This pseudo-magnetic field is analogous to artificial gauge fields realized in ultra-cold atoms but dif- fers in the sense that, the pseudo-magnetic field compo- nents are the functions of parameters of irradiated light into the system which is generally anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The mag- nitude of the pseudo-magnetic field scales inversely to the frequency of the drive and its direction can be altered by changing the orientation of the polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the stroboscopic phase represents a FETI re- sulting from the periodic driving of static 3DNHTI in the high-frequency limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' As the time-reversal symmetry of the Hamiltonian is broken due to the pseudo-magnetic field, the model hosts a pair of chiral Weyl nodes along any desired axis with a suitable choice of θ and φ and gets connected over the imaginary axis due to NH term, thus giving rise to the point gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Hence, for FETI, the system hosts a point gap for |M + A2 2 − 3| ≤ δ [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 2(a)] and a line gap for |M + A2 2 − 3| ≥ δ, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The Floquet Hamiltonian has A2/2 onsite term in the diago- nal, thereby modulating M, which is responsible for the band inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Therefore, the FETI phase can be real- ized in the system even if its static counterpart is in the trivial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Surface States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='— We define the critical angles of po- larization as, θc = tan−1( √ 2) and φc = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This makes the pseudo-magnetic field isotropic along [111] di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' At these critical values of the angle, in the com- plex energy spectrum, the point gap is filled isotropically with the doubly degenerate surface states having a sin- gle Fermi point, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We also consider another unconventional model where two Fermi points 1 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 Im(E) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1 6 4 2 0 2 4 6 Re(E)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 IPR Im( 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1 Re(E)5 c 4 Abs(E) 3 2 1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1 kc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='6 d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 3 2 kr1 2 33 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 3: For one axis OBC along the z axis, (a) depicts a signature of second order EP in the static phase for M = 3, λ = 1, δ = 1, B0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 by an octagon shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (b)-(d) represents the dynamics of the EP via Floquet driving the 3DNHTI in the presence of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We periodically vary A2 0 ω sin χ and magnetic field as B0 cos χ for χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='65, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='7 in (b), (c), and (d) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' As χ varies, the two second-order EPs collide with each other and form a third-order EP which further splits into two second-order EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Other parameters are the same as (a) along with M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 and A2 0/ω = 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' can appear, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Therefore, it exhibits a single- sheet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The degenerate partners of surface states are localized to the opposite edges of the truncated lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The surface state has a dispersion relation kx +iky, whereas, its degenerate partner has the dispersion rela- tion ky + ikx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, we infer that the single sheet es- tablishes homotopy with 2D torus-shaped Brillouin zone (BZ), see [28] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Additionally, the sur- face states appear as a Dirac cone in the kx −ky plane as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 2(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' However, the photo-induced homotopy does not remain futile from the infinitesimal perturbation from the critical angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We also note that the Hamiltonian hosts a hugely defective infernal point at kx = ky = 0 in the complex eigenspectrum which leads to numerical instability in the static model, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' How- ever, for the dynamic case due to the pseudo-magnetic field, this infernal point converts into doubly degenerate non-defective points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Effect of external magnetic field— We consider the system in the presence of an external magnetic field, by adding the term τz( ⃗B0 · ⃗σ) to the Hamiltonian writ- ten in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (1) and Floquet Hamiltonian also changes to HF + τz( ⃗B0 · ⃗σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, a constant magnetic field in the static phase gains dynamics in the stroboscopic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The resultant magnetic field modulates the imbalance be- tween lande-g-factors between the s and p orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus we can photo-modulate the Land´e-g-factor by parameter- izing Floquet term as λ2A2 0 ω sin χ and the magnetic field B0 cos χ with χ = tan−1 � λA2 0 ωB0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (5) The bulk Hamiltonian has a point gap however, the finite system along the z axis, possesses a single second- order EP on the surface for the non-zero magnetic field accounting for a quantum anomaly on the surface, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We note that if there is nth order EP, then one has to take 4n turns around it to come back to the original state in the square momentum grid [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The dy- namics and the order of the EP residing on the surface can be photo-tuned by Floquet driving which can never be realized in its static counterpart with a constant unidi- rectional magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' By adiabatically varying χ, two second-order EPs come closer and collide with each other, thereby creating a third-order EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This third-order EP can then be further transformed into two second-order by moving them away from each other (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 3(b)-(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We then apply open boundary condition (OBC) along y and z, while retaining periodic BC along the x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' In the absence of the magnetic field, the surface states in the point gaps are localized in opposite corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This remarkable phenomenon is the well-known NH surface skin effect (NHSSE) and can be related to the higher- order skin effect [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' However, as the magnetic field is turned on, the surface states get confined in one of the corners (refer Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 4(a),(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' To demonstrate this novel effect, we prepare a trial wavefunction localized at a finite y − z sheet of dimension L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The trial wavefunction is given by: |ψ0⟩ = N0e− (y−y0)2 α2 e− (z−z0)2 β2 eikxax|ζ0⟩ (6) where, N0 is the normalization factor of the wave func- tion, y0 and z0 are constants that determine the local- ization of the wave function in the finite sheet, α and β control its Gaussian width, and |ζ0⟩ is a spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We perform the time evolution of the trial wavefunction as e−iHt|Ψ0⟩ with respect to the Hamiltonian obtained by truncating y and z axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The skin modes are not com- pletely localized at the corners and they have noticeable overlap with the bulk modes (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 4(a)-(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' So this effect allows the wavepacket which was initially localized at one of the edges, to travel into the opposite edge by permeating into the bulk (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 4(c)-(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' However, there is a significant dynamical quenching of the time-evolved probability of the trial wavefunction at the edges where surface states are localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 2 Re(E)b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 (α)ul 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 2 Re(E)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 Im(E) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 2 Re(E)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 IPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 2 Re(E)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 4: Probability amplitudes demonstrate the NHSEE for (a) B0 = 0 and (b) B0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (c)-(f) demonstrates the dy- namical evolution of the trial wavefunction from time t = 0 to t = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The constants α and β are set to be 2 and 6 re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We choose the spinor |ζ0⟩ as [1, i, 0, 0], and the momentum kx is evaluated numerically considering those val- ues which experience NHSSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Other system parameters are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Topological Invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='— We compute three topolog- ical invariants starting with the spectral winding num- ber along a particular momentum axis say kz while fixing the other two momenta values at kx0 and ky0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thereafter, along the kz axis, the Hamiltonian be- comes one-dimensional and can be defined as HF 1D(kz) = HF (kx0, ky0, kz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We calculate the spectral winding num- ber as, νkx0,ky0(Ep) = 1 2πi � π −π dkzTr[Q1D(kz)], (7) where Q1D(kz) = [HF 1D(kz) − Ep]−1∂kz[HF 1D(kz) − Ep] with Ep as a reference energy inside the corresponding spectral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, for a suitable choice of com- plex reference energy in the eigenspectrum, the quan- tized value of the spectral winding number implies bro- ken bulk-boundary correspondence (BBC) resulting an NH skin effect for those particular kx0 and ky0 along kz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 5(a), there are four bands with different values of ν along with the zero-valued re- gion that can be visualized as the outcome of adding two ν’s in the overlapping complex eigenspectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Furthermore, along with three spectral winding num- bers in three directions [24], system has three 1D winding numbers W1D,l defined as, W1D,l = −i � d3k (2π)3 Tr[Ql(k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (8) where Ql = [HF (k)−E]−1∂kl[HF (k)−E] with E being the reference energy in the point gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For our system, W1D,l = 0 for all values of l, which also necessitates the absence of NHSE [24, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the uniqueness of the model is evident from the fact that the system as a whole does not exhibit the collapse of the BBC although the momentum slice in the Hamiltonian experience NHSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' There is also presence of another 3D topological invariant W3D[33–36] defined in the bulk spectrum which is given by, W3D = −1 24π2 � d3kϵijkTr[Qi(k)Qj(k)Qk(k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (9) The topological invariant defined in the bulk deter- mines the fate of the surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The quantization of W3D, however, does not require any symmetry for its stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Finally, we also employ a biorthog- onal approach to compute the Floquet open-boundary Chen number for each kx value [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This unique ap- proach, however, re-establishes the BBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The Floquet open-boundary Chern number is given by;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Cα = 2πi l′yl′z Tr′ � ˆPα[[ˆry, ˆPα], [ˆrz, ˆPα]] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (10) Here ˆry (ˆrz) is the coordinate operator along y (z) di- rection and defined as ˆry(z)mn = ry(z)δmn with rx, ry ≤ l, l × l is the size of the system and l′ y = l′ z = l − 2l0 where l0 is a boundary layer that has been removed from lx/y, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The bulk band projection operator, Pβ = � n∈β |nR⟩⟨nL|, where β denotes all the unoccu- pied bands with |nL⟩ (|nR⟩) being the left (right) eigen- states of Floquet Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The Floquet biorthogo- nal Chern number gives the quantized value of one where there are surface states in the Hamiltonian [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 2(c) and 4(b)] and can be modulated by varying A2 0 ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 5: (a) The quantized spectral winding number in the complex eigenspectrum plane for kx0 = −π and ky0 = −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (b) represents the Floquet biorthogonal Chen number depict- ing the quantized value of one in the region where there are surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' System parameters are the same as in the previous figure with B0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' In conclusion, we have studied 3D NHTI in the pres- ence of CPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The system hosts a novel phase of quantum matter, namely, FETI in the stroboscopic limit with no static counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' FETI has a point gap that is filled by either a single band at the surface or a 2D band with EPs when 3D NHTI is subjected to an external mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Due to the NH fermion doubling theorem, an odd number of Fermi points or EPs are impossible in a 2D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' That’s why the surface states in FETI are anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The exceptional topology i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' the number 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3 Y [亚|2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='15 10 20 10Z 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='7 亚|2 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='35 10 20 10 Z 200 0=↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='25 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='15 10 (c) 20 0 10 Z 200 t= 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='02 Y 10 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='01 20 10 )Z 200 t2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='02 Y (e) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='01 20 0 10 Z 200 t = 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='03 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='02 10 (f) 20 10 Z 200 t = 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='02 Y (g) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='01 20 0 10Z 200 09 = ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='012 Y [ 亚|2 10 (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='006 20 0 10Z4 (a) 0 2 00 R 2 0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3 Im(E)C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1 kα/T5 of EPs can also be modulated using CPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We also es- tablished using different topological invariants, that the NH skin effect does not exist in the entire system, but it is present in momentum slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Finally, the NHSSE exhibited by FETI has been explained by the dynamical quenching of the wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, a photo-induced modulation of the transport and quantum anomaly can also be realized in such a modeled NH system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Acknowledgments— For financial support, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' thanks CSIR, India and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' thanks Science and Engineer- ing Research Board (India) grant SRG/2022/001408 and Young Faculty Incentive Fellowship from IIT Delhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The authors would like to thank Ravi Gilani for computa- tional resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Bergholtz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Budich, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Kunst, Rev.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='045701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Denner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Skurativska, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='L121113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Mondal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Ghosh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Nag, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Saha, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' B 107, 035427 (2023), URL https://link.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Loss, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Klinovaja, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' B 95, 155407 (2017), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/ 10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1, 032013 (2019), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1103/PhysRevResearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='032013.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='235403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thakurathi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Sengupta, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Sen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' B 89, 235434 (2014), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Narayan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' B 102, 205423 (2020), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1103/PhysRevB.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Deng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Duan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Fu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Wang, L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='206601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [28] See Supplemental Material for the details on infernal point, formalism of Floquet theory, photo-induced ho- motopy in FETI, lattice realization of 3D NHTI, FETI, and unconventional 3D NHTI of the main text.' metadata={'source': 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+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1103/PhysRevX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='031079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Ghatak and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Das, Journal of Physics: Condensed Matter 31, 263001 (2019), URL https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1088/1361-648X/ab11b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1 [36] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Kitagawa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Berg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Rudner, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Demler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' B 82, 235114 (2010), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/ doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='235114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [37] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Yao, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 123, 246801 (2019), URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='246801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Zhuang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Liu, arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='10980 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Avron, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Exner, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Last, Physical review letters 72, 896 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL: Floquet Exceptional Topological Insulator Gaurab Kumar Dash, Subhajyoti Bid, Manisha Thakurathi Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India 110016 3D NON-HERMITIAN TOPOLOGICAL INSULATOR The Hermitian counterpart of the Hamiltonian H0(k) written in the main text is a four-band model of 3DTI hosting a Dirac node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For 1 ≤ |M| ≤ 3, the system exhibits a trivial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The phase transition from trivial to topological phase occurs at |M| = 1 and |M| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Due to the NH property, the Hamiltonian is accompanied by two kinds of gaps in the complex eigenspectrum[1, 5, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For |M − 3| ≤ δ and |M − 3| ≥ δ, it showcases a central point gap and a central line gap respectively [refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The Hamiltonian is hugely defective in the surface for kx = ky = 0 and leads to numerical instability (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S1(b)-(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the model exhibits an infernal point in the thermodynamic limit at kx = ky = 0, which accounts for the states to be localized at one of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' An analytical derivation of the dispersion relation of the infernal point is presented in [5] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S1: (a) shows a line gap for M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3, λ = 1, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5, B0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (b)-(d) shows the complex eigenspectrum of the Hamiltonian defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S1 for lattice site N = 10, 30, and 70 respectively accounting for the numerical instability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' LATTICE REALIZATION OF 3DNHTI The Hamiltonian in the main text can be realized in a cubic lattice with an electron with spin up and down in s and p orbital respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thereafter, the tight binding Hamiltonian is given by, H = � r,γ C† r,γH(k)Cr,γ, (S1) where, r = x, y, z denotes the position of lattice and γ = 0(1) notifies the s(p) orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We try to expand each term on the basis of the lattice mentioned in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The constant part of the onsite term in the second quantization notation is written as, a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 Im( 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 4 2 2 4 Re(E)n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 国 Im( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 Re(E)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 Im( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 2 2 Re() E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3 IPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 Im(E) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1 Re(E)2 � r,γ C† r,γ(−M)τzσ0Cr,γ = � r,γ � C† r,s,↑ C† r,s,↓ C† r,p,↑ C† r,p,↓ � � � � � −M 0 0 0 0 −M 0 0 0 0 M 0 0 0 0 M � � � � � � � � Cr,s,↑ Cr,s,↓ Cr,p,↑ Cr,p,↓ � � � � = � r [−MC† r,s,↑Cr,s,↑ − MC† r,s,↓Cr,s,↓ − MC† r,p,↑Cr,p,↑ − MC† r,p,↓Cr,,↓] = −M � r,γ (−1)γC† r,γσ0Cr,γ (S2) where, C† r,γ = � C† r,γ,↑ C† r,p,γ,↑ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The k dependent onsite terms are of the following form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='γ C† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='γ(cos kx)τzσ0Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='γ = � r � C† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑ C† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓ C† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑ C† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓ � � � � � cos kx 0 0 0 0 cos kx 0 0 0 0 − cos kx 0 0 0 0 − cos kx � � � � � � � � Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑ Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓ Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑ Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓ � � � � = � r [cos kxC† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑ − cos kxC† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓ − cos kxC† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑ − cos kxC† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓] = 1 2 � r (C† r+ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑ + C† r+ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓ − C† r+ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↑ − C† r+ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='↓) = 1 2 � r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='γ (−1)γC† r+ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='γσ0Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S3) Hence, the collective term cos kx + cos ky + cos kz can be evaluated as: � r,γ C† r,γ(cos kx + cos ky + cos kz)τzσ0Cr,γ = 1 2 � r,γ � i=x,y,z (−1)γC† r+ei,γσ0Cr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S4) Similarly, following the same steps of the calculation, the rest of the terms can be converted as: � r,γ � i=x,y,z C† r,γ(sin ki)τxσiCr,γ = λ 2i � r,γ � i=x,y,z (−1)γC† r+ei,γ+1σiCr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C, (S5) � r,γ � i=x,y,z C† r,γ(Bτzσi)Cr,γ = B � r,γ � i=x,y,z (−1)γC† r,γσiCr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C, (S6) and � r,γ � i=x,y,z C† r,γiδτxσ0Cr,γ = iδ � r,γ (−1)γC† r,γ+1σ0Cr,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S7) After collecting the terms the lattice Hamiltonian takes the following form, H = −M � r,γ (−1)γC† r,γσ0Cr,γ + [1 2 � r,γ (−1)γC† r+ex,γσ0Cr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] + λ 2i � r,γ � i=x,y,z (−1)γC† r+ei,γ+1σiCr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' +B � r,γ � i=x,y,z (−1)γC† r,γσiCr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' + iδ � r,γ (−1)γC† r,γ+1σ0Cr,γ, (S8) where, the NH part in the Hamiltonian can be realized as the electron-phonon interaction between s and p orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 3 FORMALISM FOR FLOQUET THEORY We define a non-unitary time evolution operator U(t, t ′) which evolves the system from time t to t′ with periodicity τ = 2π ω , then the Floquet theorem states that, U(t + nτ, t0) = U(t, t0)[U(t0 + τ, t0)]n (S9) and we define the non-unitary Floquet Hamiltonian as: U(t0 + τ, t0) = exp(iHF τ ℏ ) (S10) where, HF is the Floquet NH Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For the stroboscopic analysis, we can always set t0 = 0, without the loss of generality, as the original time period of the periodically driven system dominates any time scale that may be acquired by the unitary operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The eigenstates corresponding to the Floquet operators defined above for time t0 = 0 are called the Floquet modes and are given by ψα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the Floquet operator can be rewritten as[22], U(τ, 0) = � α e−iϵατ/ℏ |ψα(0)⟩ ⟨ψα(0)| , (S11) where, ϵα are the complex eigenvalues corresponding to the Floquet NH Hamiltonian[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The Floquet modes also satisfy the periodic relation as, ψα(τ) = ψα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The time evolution of the Floquet modes is given by: Ψα(t) = e−iϵαt/ℏψα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S12) Substituting the time-dependent ansatz into the time-dependent Schrodinger’s equation we get, [H(t) − iℏ ∂ ∂t]Ψα(t) = ϵαΨα(t), (S13) where, K(t) = H(t) − iℏ ∂ ∂t can be termed as Floquet extended Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Floquet-space-time representation of periodically driven systems We can further expand the Floquet modes in the time-periodic Fourier series as: Ψα(t) = ∞ � j=−∞ |φα,j⟩ eijωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S14) Plugging this into the above equation yields: ∞ � j=−∞ Hj−j′ φα,j + jℏωφα,j = ϵαφα,j, (S15) where, Hj−j′ = Hn = 1 τ � τ/2 −τ/2 H(k, t)einωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S16) Thus, the above equation can be written in matrix formulation as: Hslab = � ����������� H0 H−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 0 0 0 H1 H0 − ℏω H−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 0 0 0 0 H1 H0 − 2ℏω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' H−1 0 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' H−1 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 0 H1 H0 − jℏω � ����������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S17) 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S2: The Floquet variable onsite terms as the slabs and temporal hoppings are shown by red arrows Thus, the harmonic indices j and j ′ represent the fictitious temporal direction such that a d-dimensional Hamiltonian can always be visualized in the d + 1 dimensional space-time representation(refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The term jℏω represents the variable onsite term (similar to a 1D chain under stark electric field) and H ′ j−j represents the hopping between j and j ′ temporal site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' In addition to this, quasi-energy also satisfies the periodic relation of ϵα = ϵα + jℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus the space-time picture can also equally be mapped into the Wannier-Stark ladder[39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' considering the periodicity of the drive to be very high, we can write the Hamiltonian as : H = H0 + H1eiωt + H−1e−iωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S18) In the infinite frequency domain, the hopping along the temporal direction becomes completely ineffective breaking a d+1-dimensional system (4-dimensional system in our case) to the isolated d-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' However, in the low energy approximation, the perturbation theory yields unique results in the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' If ϵ is the energy which is associated with the zeroth mode level(H0), then going from n = 0 and n = 1 and coming back is described by the term H−1 1 (ϵ+ℏω)−ϵH† 1 and going from n = 0 to n = −1 and coming back is described by the term H1 1 (ϵ−ℏω)−ϵH† −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the whole process described above can be mathematically written as: Heff = H0 F + inf � n=1 1 ω [H−n F , Hn F ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S19) FLOQUET DRIVING OF 3DNHTI We use the above-developed perturbation theory in the conventional 3DNHTI and try to develop FETI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We use the vector potential(mentioned in the main text) as: A = A0[cos ωt⃗e1 + η sin ωt⃗e2], (S20) where, A0 is the amplitude and ω is the frequency of the driven system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' η = ±1 signifies the right circularly and left circularly polarized light respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We choose e1 = (cos θ cos φ, cos θ sin φ, sinθ) and e2 = (sin φ, − cos φ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the vector potential is given by: A = A0 (cos θ cos φ cos ωt + η sin φ sin ωt, cos θ sin φ cos ωt − η cos φ sin ωt, − sin θ cos ωt) , (S21) with minimal coupling the time-dependent Hamiltonian becomes, H(k) = � j(cos kj − M − A2 0 4 (1 + η2))τzσ0 + λ � j[sin kx − A0(cos θ cos φ cos ωt + η sin φ sin ωt)τxσx + sin ky − A0(cos θ sin φ cos ωt − η cos φ sin ωt)τxσy + sin kz − A0(− sin θ cos ωt)τxσz] + iδτxσ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S22) Ho + hw Hi1 H_1 Ho Hi H_1 Ho- hw5 Thus, the various Floquet terms can be recovered as: H0 = � j (cos kj − M − A2 0 4 (1 + η2))τzσ0 + λ � j sin kjτxσj + iδτxσ0, (S23) H1 = −λA0 2 ([cos θ cos φ + iη sin φ]τxσx + [cos θ sin φ − iη cos φ]τxσy + sin θτxσz), (S24) and H−1 = −λA0 2 ([cos θ cos φ − iη sin φ]τxσx + [cos θ sin φ + iη cos φ]τxσy + sin θτxσz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S25) Thus, in the Fourier space-time representation, each onsite term represents a 3DNHTI with onsite loss and gain term whereas the hopping between two connected 3DNHTI can be modulated by the angle of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus the whole system can be modulated by A0, θ, φ to realize exotic quantum phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' We neglect the periodic fluctuating terms with coefficient A2 0 ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, by using equation 13 the effective Hamiltonian can be calculated as: H(k) = � j (cos kj − M − A2 4 (1 + η2))τzσ0 + λ � j sin kjτxσj + τ0(n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='σ) + iδτxσ0, (S26) where, the vector n is given by : n = λ2A2η ω (sin θ cos φ, sin θ sin φ, cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S27) However, in the presence of the extrinsic magnetic field, the Hamiltonian in the stroboscopic phase is given by: H(k) = � j (cos kj − M − A2 4 (1 + η2))τzσ0 + λ � j sin kjτxσj + (nτ0 + Bτz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='σ + iδτxσ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S28) LATTICE REALIZATION OF FETI The FETI can also be calculated in the cubic lattice as: H = −(M + A2 0 2 ) � r,γ(−1)γC† r,γσ0Cr,γ + [ 1 2 � r,γ(−1)γC† r+ex,γσ0Cr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] +[ λ 2i � r,γ � i=x,y,z(−1)γC† r+ei,γ+1σiCr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] � r,γ � i=x,y,z(ni + B(−1)γ)C† r,γσiCr,γ + iδ � r,γ C† r,γ+1σ0Cr,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S29) Thus, irradiating light in a 3DNHTI generates Onsite excitation due to the coupling of light-matter interaction which is analogous to a photo-induced magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The uniqueness of such a fictitious magnetic field reveals its true nature from the fact that it is generally anisotropic(except at the critical angle defined in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The amplitude of such a field is dependent on the frequency and handedness of the light used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' PHOTO-INDUCED HOMOTOPY IN FETI We truncate x-axis while retaining PBC along y and z axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For simplicity, we use the angle of polarization as θ = 0 and φ = φ′ where 0 ≤ φ′ ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Then the truncated hamiltonian can be written as: H = 1 2[� x C† x+1(τzσ0 + iλτxσx)Cx + � x C† x−1(τzσ0 − iλτxσx)Cx] +C† x[(2 − M − A2 0 2 )]τzσ0 + λ2A2 0 ω τ0σz + iδτxσ0]Cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S30) 6 We use the trial wavefunction as: |Ψ¯100⟩ = � x αx |x⟩ |ζ0⟩ (S31) We write the Harper’s equation (Γ0 = τzσ0) as[25], Γ0 �1 − λτyσx 2 α−1 + 1 + λτyσx 2 α + [(2 − M − A2 0 2 ) + λ2A2 0 ω τzσz − δτyσ0] � |ζ0⟩ = 0 (S32) The terms τzσz and τyσ0 commutes with τyσx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='Thus the eigenstates of τyσx are given by: |ψ1⟩ = (−i, 0, 0, 1)T √ 2 (S33) |ψ2⟩ = (0, −i, 1, 0)T √ 2 (S34) |ψ−1⟩ = (i, 0, 0, 1)T √ 2 (S35) |ψ−2⟩ = (0, i, 1, 0)T √ 2 (S36) Thus we write the spinor |ζ0⟩ as: |ζ0⟩ = p1 |ψ1⟩ + p2 |ψ2⟩ (S37) We set p1 = cos θ′ and p2 = sin θ′eiφ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' after normalization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' the Harper equation reduces to: � α − M − A2 0 2 + λ2A2 0 ω � cos θ′ − δ sin θ′eiφ′ = 0 (S38) � α − M − A2 0 2 + λ2A2 0 ω � sin θ′eiφ′ − δ cos θ′ = 0 (S39) Thus the value of the constants are given by α = � (δ2 + (λ2A2 0 ω )2) + (M + A2 0 2 ) − 2 (S40) θ′ = tan−1(α − M − A2 0 2 + 2 + λ2A2 0 ω δ ) (S41) φ′ = 0 (S42) Following the similar steps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' we can write the trial wavefunction for (100) surface as : |Ψ¯100⟩ = � x αx |x⟩ |ζ′ 0⟩ (S43) Then ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' after normalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' the surface states are given by: |Ψ¯100⟩ = � x αL−x |x⟩ [cos θ′ |ψ−1⟩ − sin θ′ |ψ−2⟩] (S44) 7 |Ψ100⟩ = � x αx |x⟩ [cos θ′ |ψ−1⟩ + sin θ′ |ψ−2⟩] (S45) For the system to establish homotopy with the torus shaped BZ and exhibits a single sheet in the surface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' the top and buttom surfaces must couple to each other demanding the surface state in the complex eigenspectrum to be a superposition of |Ψ100⟩ and |Ψ¯100⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='Thus we assume the surface state to have the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' |ψ⟩ = (|Ψ100⟩) ± (|Ψ¯100⟩) √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S46) We then consider the perturbative correction of ky and kz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Since ⟨ψ1| τxσy |ψ1⟩ = − ⟨ψ2| τxσy |ψ2⟩ = ⟨ψ−1| τxσy |ψ−1⟩ = ⟨ψ−2| τxσy |ψ−2⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='Hence ⟨Ψ¯100| τxσy |Ψ¯100⟩ = − ⟨Ψ100| τxσy |Ψ100⟩ = cos 2θ′, thus ky terms results in the energy splitting of ± cos 2θ′ky for small value of ky which would cage the surface states to localise at one surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Following the similar calculations for the kz term, we have, ⟨ψ1| τxσy |ψ2⟩ = − ⟨ψ2| τxσy |ψ1⟩ = − ⟨ψ−1| τxσy |ψ−2⟩ = ⟨ψ−2| τxσy |ψ−1⟩ = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' But kz dependent terms are not diagonal unlike the ky dependent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' So, |Ψ100⟩ and |Ψ¯100⟩ are not the good basis of the perturbed Hamiltonian when kz dependent perturbations are included in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' To Tackle this difficulty, we re-solve the Harper’s equation again without the NH terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Γ0 �1 − λτyσx 2 α−1 + 1 + λτyσx 2 α + [(2 − M − A2 0 2 ) + λ2A2 0 ω τzσz − δτyσ0 + i sin kzτyσz] � |ζ0⟩ = 0 (S47) As [τyσz, τxσy] = 0,the kz dependent term inter-twins and mixes the Hilbert space of the unperturbed Hamiltonian since τyσz |ψ1⟩ = |ψ−2⟩, τyσz |ψ2⟩ = |ψ−1⟩, τyσz |ψ−2⟩ = |ψ1⟩, and τyσz |ψ−1⟩ = − |ψ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='Thus for (¯100) surface we redefine the spinor as |ζ⟩ = cos θ1 |ψ1⟩ + sin θ1eiφ1 |ψ−2⟩ (S48) Re-solving the Harper’s equation we get α′ = β1 + β2 β3 (S49) β1 = −1 + (λ2A2 0 ω )2 − (sin kz)2 − (M + A2 0 2 ) (S50) β2 = − � (1 − (λ2A2 0 ω )2 + (sin kz)2) + 4((λ2A2 0 ω )2 − (M + A2 0 2 )2) + (M + A2 0 2 )2 (S51) β3 = 2(M + A2 0 2 − λ2A2 0 ω ) (S52) θ1 = − tan−1( sin kz α′ + M + A2 0 2 + λ2A2 0 ω ) (S53) φ1 = −π 2 (S54) similarly for (100) surface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' we can write |ζ1⟩ = cos θ1 |ψ1⟩ − i sin θ1 |ψ2⟩ (S55) |ζ2⟩ = cos θ1 |ψ−1⟩ + i sin θ1 |ψ−2⟩ (S56) Now including the NH electron-phonon interaction in the Hilbert space of |ζ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2⟩: ⟨ζ1| iτxσ0 |ζ1⟩ = ⟨ζ2| iτxσ0 |ζ2⟩ = i sin 2θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus the NH interaction splits the energy level equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For small kz, θ1 is directly proportional to kz and energy splitting in the NH complex energy is directly proportional to ikz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the surface states have zero energy for finite δ since the NH term does not alter the surface states for kz = 0 as θ1 = 0 when kz = 0 thereby forming a single sheet in the complex eigenspectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 8 PHASE SPACE OF FETI We neglect A2 ω terms in the Hamiltonian to invoke the hidden sub-lattice symmetry of it by rotating the basis as, τx ⇒ τy ⇒ τz ⇒ τx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Then the Hamiltonian can be converted into the off-block diagonal form[5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' H(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' A) = � j(cos kj − M − A2 4 (1 + η2))τxσ0 + λ � j sin kjτyσj + iδτyσ0 ⇒ H(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' A) = � 0 h† UR hLL 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S57) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' h† UR and hLL denotes the upper right and lower left matrices respectively and given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' h† UR = �f(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' A) − iλ sin kz + δ −iλ sin kx − λ sin ky −iλ sin kx + λ sin ky f(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' A) + iλ sin kz + δ � (S58) and hLL = �f(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' A) + iλ sin kz − δ iλ sin kx + λ sin ky iλ sin kx − λ sin ky f(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' A) − iλ sin kz − δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S59) Here, f(k, M, A) = � j(cos kj−M− A2 4 (1+η2) we can then define Qi = H−1∂kjH = �h−1 LL∂kihLL 0 0 (h† UR)−1∂kihLL � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The topological invariant W3D can then be expressed as, W3D = W LL 3D − W UR 3D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S60) We can further link both of the off diagonal matrices as considering, hLL(k, M, A, λ, δ) = h0(k, M + A2 4 (1 + η2) + δ, λ), (S61) h† UR(k, M, A, λ, δ) = h0(k, M + A2 4 (1 + η2) − δ, λ) (S62) and the Hamiltonian h0 is written in the form, h0 = dµσµ (S63) where, dµ = (cos kj −M ′, iλ sin kx, iλ sin ky, iλ sin kz) and σµ = (σ0, σ1, σ2, σ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus it represents a two-band model of 3DTI with NH spin-orbit-coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For vanishing point gap the system demands that h0(k, M ′, λ) = 0, implies, cos kj = M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This yields, M ′ = ±1, ±3 which decides the phase space of the block matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the winding number can be expressed as, W3D = W 0 3D(k, M + A2 4 (1 + η2) + δ, λ) − W 0 3D(k, M + A2 4 (1 + η2) − δ, λ), (S64) and, M ′ = ±3 implies | M + A2 4 (1 + η2) |= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, | M + A2 4 (1 + η2) − 3 |≤ δ corresponds to a central point-gap whereas | M + A2 4 (1 + η2) − 3 |≥ δ corresponds to a central line-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Let us consider the model to be in static topological phase boundary for which it demands the value of M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Then by shining light on such a quantum phases of matter in the topological phase boundary, a phase transition from point gap to the line gap can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' A2 4 (1 + η2) ≤ δ demands a central point gap and A2 4 (1 + η2) ≥ δ demands a central line gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S3: (a)-(h) corresponds to the dynamic evolution of the probabilities of the trial wavefunction defined in the main text for M = 3,δ = 1,λ = 1,B0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2,from time t = 0 to t = 70 in the interval of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' |ζ0⟩ = [1, 0, i, 0] and the value of α and β remains same as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' SLAB CALCULATION AND WAVE DYNAMICS We prepare a trial wave function, defined in the main text, and evolve it with respect to the slab Hamiltonian in which two of the axes are truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' In the tight-binding representation, it is given by, H = � i,j H0C† i,jCi,j + � tyC† i,jCi+1,j + tzC† i,jCi,j+1 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C � (S65) where, H0 is given by, H0 = � j � cos kj − M − A2 2 � τzσ0 + λ sin kyτxσy + τ0 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='σ) + τz (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='σ) + iδτxσ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S66) The hoppings takes the following form, ty = 1 2 (τzσ0 − iλτxσx) , (S67) t† y = 1 2 (τzσ0 + iλτxσx) , (S68) tz = 1 2 (τzσ0 − iλτxσz) , (S69) and t† z = 1 2 (τzσ0 + iλτxσz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S70) We have mentioned wave-dynamics evolution for critical angles in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' In this section, we demonstrate the wave-dynamics evolution of the wave function for the static case in the presence of an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The wave-packet initially localized in one of the corners of the 2D sheet permeates into the bulk to travel to the opposite corner and returns to the same corner when the system evolved with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' This evolution of an arbitrary Gaussian wave-packet account for the mimic phenomena of NHSSE exhibited by the model in the static phase with an external magnetic field where part of the wave-function probabilities are always quenched dynamically along the corners (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='S3(a)-(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, although the time evolution makes the wave-function to get absorbed into the bulk, it immediately escapes the bulk by penetrating to the other corner since the duration for which the it remains in the bulk is negligible compared to the duration for which it remains in the corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 0 0 = ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 Y 10 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3 20 10 Z 0 200 t = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='04 Y 10 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='02 20 0 200+ t = 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='025 Y (C) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='01 20 10 Z 200 t = 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='05 5 Y (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='03 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='01 20 5 10 Z 15 200 t = 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='08 Y (e) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='04 20 10 Z 200- t = 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='014 Y (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='007 10 20 10z (0 200 09 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='020 Y (g) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='010 20 20 0 10 ZFO 02 = 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='0175 亚|2 Y (h) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='0075 20 10 Z 2010 UNCONVENTIONAL 3DNHTI We define an unconventional 3DNHTI as a cubic lattice of 3DNHTI (mentioned in the main text) with intrinsic spin orbit coupling (SOC) preserving the spin alignment controlled by the parameter ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The rest of the terms has usual meaning as discussed for the previous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The tight binding Hamiltonian is given by, H(k) = � j (cos kj − M)τzσ0 + λ � j sin kjτxσj + ∆ � j sin kjτyσ0 + iδτxσj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S71) The model exhibits similar complex eigenspectrum as that of 3DNHTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For |M − 3| ≤ δ and |M − 3| ≥ δ, it showcases a central point gap and a central line gap respectively (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S4(a)-(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='It is, however, a band strained version of 3DNHTI near the BZ which accounts for the flatness of the bands at BZ boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' However, the phase diagram of this unconventional 3DNHTI remains same as the 3DNHTI discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S4: (a) and (b) depicts the central point gap and line gap for M = 3,λ = 1,δ = 1,∆ = 1 and M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3,λ = 1,δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5,∆ = 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (c) shows a single sheet structure in the complex eigenspectrum for M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5,λ = 1,δ = 1,∆ = 1,A0 = 1,ω = 5 when the model is truncated along z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' LATTICE REALIZATION OF UNCONVENTIONAL 3DNHTI We express its tight binding Hamiltonian in the cubic lattice as, H = −M � r,γ(−1)γC† r,γσ0Cr,γ + [ 1 2 � r,γ(−1)γC† r+ex,γσ0Cr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] +[ λ 2i � r,γ � i=x,y,z(−1)γC† r+ei,γ+1σiCr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] +[ ∆ 2 � r,γ C† r+ex,γ+1σ0Cr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] + +iδ � r,γ(−1)γC† r,γ+1σ0Cr,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S72) FLOQUET DRIVING OF UNCONVENTIONAL 3DNHTI With the vector potential defined in equation S21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' minimal coupling yields the time-dependent Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' H(k) = � j(cos kj − M − A2 0 4 (1 + η2))τzσ0 + λ[sin kx − A0(cos θ cos φ cos ωt + η sin φ sin ωt)τxσx + sin ky − A0(cos θ sin φ cos ωt − η cos φ sin ωt)τxσy + sin kz − A0(− sin θ cos ωt)τxσz] ∆((sin kx − A0(cos θ cos φ cos ωt + η sin φ sin ωt)) + (sin ky − A0(cos θ sin φ cos ωt − η cos φ sin ωt)) +(sin kz − A0(− sin θ cos ωt)))τyσ0 + iδτxσ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S73) The various Floquet modes are extracted as, H0 = � j (cos kj − M − A2 0 4 (1 + η2))τzσ0 + λ � j sin kjτxσj + ∆ � j sin kjτyσ0 + iδτxσ0, (S74) 1 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 Im(E) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 1 6 4 2 0 2 4 6 Re(E)(b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 Im( 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 2 2 4 Re(E)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 IPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 Im(E) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='1 Re(E)11 H1 = − λA0 2 ([cos θ cos φ + iη sin φ]τxσx + [cos θ sin φ − iη cos φ]τxσy + sin θτxσz) − − ∆A0 2 ([cos θ cos φ + iη sin φ] +[cos θ sin φ − iη cos φ] + sin θ)τyσ0, (S75) and H−1 = − λA0 2 ([cos θ cos φ − iη sin φ]τxσx + [cos θ sin φ + iη cos φ]τxσy + sin θτxσz) − − ∆A0 2 ([cos θ cos φ − iη sin φ] +[cos θ sin φ + iη cos φ] + sin θ)τyσ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S76) Thus, expanding the time dependent Hamiltonian in the stroboscopic phase yields: H(k) = � j (cos kj − M − A2 2 )τzσ0 + λ � j sin kjτxσj + ∆ � j sin kjτyσ0 − τ0(n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='σ) − τz(n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='σ) + iδτxσ0 , (S77) where, n is the photodressed vector obtained above and n′ vector is given by, n′ = (cos θ − sin θ sin φ, cos θ − sin θ cos φ, sin θ(cos φ − sin φ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S78) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S5: (a) denotes an Abs(E) contour plot for M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5,λ = 1,δ = 1,∆ = 1,A0 = 1,ω = 5,θ = tan−1√ 2 and φ = π 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (b) and (c) show the sliding of two of the sheets M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5,λ = 1,δ = 1,A0 = 1,ω = 5,θ = π 2 and φ = π 4 by changing ∆ as -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='6 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Thus, the Floquet driving generates two kinds of photo-dressed vectors that behave like the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For the critical angle, the vectors become isotropic and exhibit a single sheet hosting two Fermi point(unlike the single sheet of FETI hosting a single Fermi point) and establish homotopy with the BZ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S4(c) and S5(a)), but for the choice of the suitable angle of polarization, the single sheet evolves into a double sheet by sliding over each other hosting a non-degenerate edge state at each of the surfaces (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S5(b)-(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Therefore, the angle of polarization helps one to switch between both cases of the band spectrum on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' Hence ∆ modulates the photo-dressed Land´e-g-factor, in the sense, deciding the alignment of s and p orbital along the photo-dressed magnetic field even when the system does not experience any external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The interaction which gives rise to additional spin-orbit coupling in the static phase and transforms a conventional 3DNHTI into an unconventional 3DNHTI also gives rise to the photo-dressed lande-g-factor for the dynamic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 3 1 0 1 2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 ky O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 3 2 1 0 1 2 3c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2亩 ky0 bs( 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='8A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='4 0 2 0 2 312 LATTICE REALIZATION OF FLOQUET UNCONVENTIONAL 3DNHTI The FETI can be realized in the cubic lattice as, H = −(M + A2 0 2 ) � r,γ(−1)γC† r,γσ0Cr,γ + [ 1 2 � r,γ(−1)γC† r+ex,γσ0Cr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] +[ λ 2i � r,γ � i=x,y,z(−1)γC† r+ei,γ+1σiCr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] +[ ∆ 2 � r,γ C† r+ex,γ+1σ0Cr,γ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='C] + � r,γ niC† r,γσiCr,γ + � r,γ(−1)γn ′ iC† r,γσiCr,γ + iδ � r,γ C† r,γ+1σ0Cr,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' (S79) SLAB HAMILTONIAN AND WAVE DYNAMICS EVOLUTION As mentioned in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 26, we determine the slab Hamiltonian and calculate the onsite and hopping part of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' These are given by, H0 = � j � cos kj − M − A2 2 � τzσ0 + λ sin kyτxσy + τ0 (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='σ) + τz � n ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='σ � + iδτxσ0, (S80) ty = 1 2 (τzσ0 − iλτxσy − i∆τyσ0) , (S81) t† y = 1 2 (τzσ0 + iλτxσy + i∆τyσ0) , (S82) tz = 1 2 (τzσ0 − iλτxσz − i∆τyσ0) , (S83) and t† z = 1 2 (τzσ0 + iλτxσz + i∆τyσ0) (S84) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For case (1), the surface states are localized in one of the edges and it has a significant amount of overlap integral with the bulk (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S6(a)-(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' So, the wavefunction initially localized in the middle of the Z-axis also has a significant amount of non-vanishing amplitude in the bulk which makes the localized wavefunction penetrate into the bulk and allow a part of it to travel along the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The wavepacket stays on the edge for the longer time where the surface states are caged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' For the second case, the wavefunction initially localized in one of the corners travels to the other corners and gets localized with a quasi-static phase in the dynamic evolution (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S7(a)-(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' The localization of the wavefunction increases at the corner where the majority of them are trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' So, the wave dynamic evolution of the localized Gaussian wavefunction is guided by the NHSSE entrailed within the Hamiltonian which can be modulated by varying the angle of polarization of the CPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S6: (a)-(h) corresponds to the dynamic evolution of the probabilities of the trial wavefunction initially localized at the middle of one of the edges for M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5,δ = 1,λ = 1,∆ = 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2,A0 = 1,ω = 5,θ = tan−1√ 2 and φ = π 4 from time t = 0 to t = 70 in the interval of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='|ζ0⟩ = [1, i, i, 0] and the value of α and β remains same as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' S7: (a)-(h) corresponds to the dynamic evolution of the probabilities of the trial wavefunction initially localized at one of the corners for M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='5,δ = 1,λ = 1,∆ = 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='2,A0 = 1,ω = 5,θ = tan−1√ 2 and φ = π 4 from time t = 0 to t = 70 in the interval of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='|ζ0⟩ = [1, i, i, 0] and the value of α and β remains same as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 0 0=↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='25 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='15 10 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='05 20 10 Z 200 t = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='014 10 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='007 20 0 0 10 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content=' 200 t = 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='025 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='015 10 (c) 20 10 Z 200- t = 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdFPT4oBgHgl3EQfkzU3/content/2301.13119v1.pdf'} +page_content='03 Y 0.' metadata={'source': 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of Euclidean windows of the hadronic vacuum polarization +T. Blum,1 P. A. Boyle,2, 3 M. Bruno,4, 5 D. Giusti,6 V. G¨ulpers,3 R. C. Hill,3 +T. Izubuchi,2, 7 Y.-C. Jang,8, 9 L. Jin,1, 7 C. Jung,2 A. J¨uttner,10, 11 C. Kelly,12 +C. Lehner,6, ∗ N. Matsumoto,7 R. D. Mawhinney,9 A. S. Meyer,13, 14 and J. T. Tsang10, 15 +(RBC and UKQCD Collaborations) +1Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA +2Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA +3School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK +4Dipartimento di Fisica, Universit´a di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy +5INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy +6Fakult¨at f¨ur Physik, Universit¨at Regensburg, Universit¨atsstraße 31, 93040 Regensburg, Germany +7RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA +8Electron-Ion Collider, Brookhaven National Laboratory, Upton, NY 11973, USA +9Physics Department, Columbia University, New York, NY 10027, USA +10CERN, Theoretical Physics Department, Geneva, Switzerland +11School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK +12Computational Science Initiative, Brookhaven National Laboratory, Upton, NY 11973, USA +13University of California, Berkeley, CA 94720, USA +14Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA +15CP3-Origins & Department of Mathematics and Computer Science, +University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark +(Dated: January 23, 2023) +We compute the standard Euclidean window of the hadronic vacuum polarization using multiple +independent blinded analyses. +We improve the continuum and infinite-volume extrapolations of +the dominant quark-connected light-quark isospin-symmetric contribution and address additional +sub-leading systematic effects from sea-charm quarks and residual chiral-symmetry breaking from +first principles. We find aW +µ = 235.56(65)(50) × 10−10, which is in 3.8σ tension with the recently +published dispersive result of aW +µ = 229.4(1.4)×10−10 [1] and in agreement with other recent lattice +determinations. +We also provide a result for the standard short-distance window. +The results +reported here are unchanged compared to our presentation at the Edinburgh workshop of the g-2 +Theory Initiative in 2022 [2]. +PACS numbers: +12.38.Gc +I. +INTRODUCTION +The anomalous magnetic moment of the muon aµ is defined as the relative deviation of the muon’s Land´e factor +gµ from Dirac’s relativistic quantum mechanics result, aµ = gµ/2 − 1. It is one of the most precisely determined +quantities in particle physics and has exhibited a persistent tension between the experimentally measured value and +the Standard Model theory result. +In order to reduce the experimental uncertainties, substantial efforts are currently undertaken at Fermilab (E989) +and planned at J-PARC (E34) [3]. In 2021 the Fermilab experiment released first results [4] confirming the previously +best result obtained by the BNL E821 experiment [5] and reducing the experimental uncertainty from 0.54 ppm to +0.46 ppm. Over the next few years, the Fermilab experiment aims to reduce the uncertainty further to approximately +0.14 ppm [6]. +The Standard Model result provided by the Muon g-2 Theory Initiative [7–27] currently has an uncertainty of +0.37 ppm and is in 4.2σ tension with the experimental value. A further reduction of the theory uncertainty by at least +a factor of two is therefore needed [28] to match the expected experimental progress over the next few years. More +than 90% of the theory uncertainty is due to the leading-order hadronic vacuum polarization (HVP) contribution +such that a reduction of its uncertainty is particularly pressing. +The leading-order HVP contribution aHVP LO +µ +can be related to e+e− decays using a dispersion relation such that, to +the degree that there is no new physics in e+e− decays, it can be used to represent the Standard Model theory result. +The Muon g-2 Theory Initiative result quoted above uses this method to determine the HVP contribution. One can +also relate the HVP contribution to hadronic τ decays, however, this requires precise first-principles knowledge of the +needed isospin rotation. Our collaboration is working on such a calculation [29] and we will report on related progress +in a separate publication. Finally, the HVP contribution can be computed from first principles using systematically +improvable lattice QCD+QED methods. +arXiv:2301.08696v1 [hep-lat] 20 Jan 2023 + +2 +Until recently, lattice QCD+QED methods have not yet been competitive with the precision provided by the +dispersive method. +The BMW collaboration, however, has now produced a lattice QCD+QED result with 0.8% +precision [30], which is close to the current 0.6% precision of the dispersive method. The BMW value taken by itself +only leads to a 1.5σ tension for aµ. At the same time, the BMW value for the HVP contribution is in a 2.1σ tension +with the dispersive result provided by the Muon g-2 Theory Initiative. +In 2018, our collaboration introduced Euclidean window quantities [31], which allow for the separation of the +most challenging short and long time-distance contributions to aHVP LO +µ +. The remaining standard window quantity, +aHVP LO W +µ +, is much easier to compute at high precision in lattice QCD+QED and can also be computed using the +dispersive method [1, 30–32]. The BMW collaboration’s calculation of aHVP LO W +µ +is in fact in 3.7σ tension with the +dispersive result, which has motivated many lattice collaborations to focus on high-precision calculations of aHVP LO W +µ +first in order to clarify the situation. In this work, we provide a significantly improved calculation of aHVP LO W +µ +. We +focus on the the quark-connected light-quark contribution in the isospin-symmetric limit, which accounts for almost +90% of aHVP LO W +µ +. Special attention is given to the continuum limit for which we replace our previous continuum +extrapolation based on a single approach using 2 lattice spacings with one based on 8 distinct approaches using 3 +lattice spacings. We perform this update using a blinding procedure with five independent analysis groups. This +blinding procedure is implemented to avoid bias toward our previous computation of aHVP LO W +µ +in Ref. [31], the +dispersive results, or other lattice results. +This paper is organized as follows. In Sec. II, we describe our methodology before giving computational details +in Sec. III. In Sec. IV, we discuss blinded results and explain convergence to the final prescription to determine +aHVP LO W +µ +. Finally, in Sec. V, we present unblinded results and compare them to other groups’ results, including +data-driven ones, before concluding in Sec. VI. +II. +METHODOLOGY +We first define the time-momentum representation in Sec. II A, which provides the basis for the definition of the +Euclidean windows in Sec. II B. In Sec. II C we define the isospin-symmetric world around which we expand. Special +care is taken such that the isospin-symmetric contribution can be compared directly with other lattice results. In +Sec. II D, we describe our blinding procedure. +A. +Time-momentum representation +Starting from the vector current Jµ(x) = i � +f QfΨf(x)γµΨf(x) with fractional electric charge Qf and sum over +quark flavors f we may write +aHVP LO +µ += +∞ +� +t=0 +wtC(t) +(1) +with correlator +C(t) = 1 +3 +� +⃗x +� +j=0,1,2 +⟨Jj(⃗x, t)Jj(0)⟩ , +(2) +where the weights wt capture the photon and muon part of the HVP diagrams. A complete list of diagrams is given +in Fig. 1. The weights can be expressed as a one-dimensional integral [33] +wt = 8α2 +� ∞ +0 +dQ2 +�cos (Qt) − 1 +Q2 ++ 1 +2 t2 +� +f(Q) +(3) +with +f(Q) = m2 +µQ2Z3(Q)(1 − Q2Z(Q)) +1 + m2µQ2Z2(Q) +, +Z(Q) = +� +Q4 + 4Q2m2µ − Q2 +2m2µQ2 +, +(4) +where mµ is the muon mass. Note that we sum only over non-negative t in Eq. (1), yielding an additional symmetry +factor of two in wt. Using a lattice discretization for the photon momenta, an alternative weight +ˆwt = 8α2 +� ∞ +0 +dQ2 +�cos (Qt) − 1 +(2 sin Q/2)2 + 1 +2 t2 +� +f(Q) +(5) + +3 +Diagrams +Isospin +limit +QED +corrections +Strong +isospin +breaking +Diagrams – Isospin limit +2 +with C(t) = 1 +3 +P +~x +P +j=0,1,2hJj(~x, t)Jj(0)i. With appro- +priate definition of wt, we can therefore write +aµ = +X +t +wtC(t) . +(4) +The correlator C(t) is computed in lattice QCD+QED +with dynamical up, down, and strange quarks and non- +degenerate up and down quark masses. We compute the +missing contributions to aµ from bottom quarks and from +charm sea quarks in perturbative QCD [13] by integrating +the time-like region above 2 GeV and find them to be +smaller than 0.3 ⇥ 10�10. +We tune the bare up, down, and strange quark masses +mup, mdown, and mstrange such that the ⇡0, ⇡+, K0, and +K+ meson masses computed in our calculation agree with +the respective experimental measurements [14]. The lat- +tice spacing is determined by setting the �� mass to +its experimental value. We perform the calculation as a +perturbation around an isospin-symmetric lattice QCD +computation [15, 16] with two degenerate light quarks +with mass mlight and a heavy quark with mass mheavy +tuned to produce a pion mass of 135.0 MeV and a kaon +mass of 495.7 MeV [17]. The correlator is expanded in +the fine-structure constant ↵ as well as �mup, down = +mup, down � mlight, and �mstrange = mstrange � mheavy. +We write +C(t) = C(0)(t) + ↵C(1) +QED(t) + +X +f +�mfC(1) +�mf(t) ++ O(↵2, ↵�m, �m2) , +(5) +where C(0)(t) is obtained in the lattice QCD calculation +at the isospin symmetric point and the expansion terms +define the QED and strong isospin-breaking (SIB) correc- +tions, respectively. We keep only the leading corrections +in ↵ and �mf which is su�cient for the desired precision. +We insert the photon-quark vertices perturbatively +with photons coupled to local lattice vector currents mul- +tiplied by the renormalization factor ZV [17]. +We use +ZA � ZV for the charm [22] and QED corrections. The +SIB correction is computed by inserting scalar operators +in the respective quark lines. +The procedure used for +e�ective masses in such a perturbative expansion is ex- +plained in Ref. [18]. +We use the finite-volume QEDL +prescription [19] and remove the universal 1/L and 1/L2 +corrections to the masses [20] with spatial lattice size L. +The e�ect of 1/L3 corrections is small compared to our +statistical uncertainties. We find �mup = �0.00050(1), +�mdown = 0.00050(1), and �mstrange = �0.0002(2) for +the 48I lattice ensemble described in Ref. [17]. The shift +of the �� mass due to the QED correction is significantly +smaller than the lattice spacing uncertainty and its e�ect +on C(t) is therefore not included separately. +Figure 1 shows the quark-connected and quark- +disconnected contributions to C(0). +Similarly, Fig. 2 +shows the relevant diagrams for the QED correction to +FIG. 1. +Quark-connected (left) and quark-disconnected +(right) diagram for the calculation of aHVP LO +µ +. We do not +draw gluons but consider each diagram to represent all orders +in QCD. + 0 + 0.01 + 0.02 + 0.03 + 0.04 + 0.05 + 0.06 + 0.07 + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 +r +Resulting two-point p(d) from p(r)=(1.5 + r)-5 +Figure 6: Displacement probability for 48c run 1. +(a) V +(b) S +(c) T +(d) D1 +(e) D2 +(f) F +(g) D3 +Figure 7: Mass-splitting and HVP 1-photon diagrams. In the former the dots +are meson operators, in the latter the dots are external photon vertices. Note +that for the HVP some of them (such as F with no gluons between the two +quark loops) are counted as HVP NLO instead of HVP LO QED corrections. +We need to make sure not to double-count those, i.e., we need to include the +appropriate subtractions! Also note that some diagrams are absent for flavor +non-diagonal operators. +8 +FIG. 2. QED-correction diagrams with external pseudo-scalar +or vector operators. +the meson spectrum and the hadronic vacuum polariza- +tion. The external vertices are pseudo-scalar operators +for the former and vector operators for the latter. We +refer to diagrams S and V as the QED-connected and to +diagram F as the QED-disconnected contribution. We +note that only the parts of diagram F with additional +gluons exchanged between the two quark loops contribute +to aHVP LO +µ +as otherwise an internal cut through a single +photon line is possible. For this reason, we subtract the +separate quantum-averages of quark loops in diagram F. +In the current calculation, we neglect diagrams T, D1, +D2, and D3. This approximation is estimated to yield an +O(10%) correction for isospin splittings [21] for which the +neglected diagrams are both SU(3) and 1/Nc suppressed. +For the hadronic vacuum polarization the contribution of +neglected diagrams is still 1/Nc suppressed and we adopt +a corresponding 30% uncertainty. +In Fig. 3, we show the SIB diagrams. In the calcu- +x +x +x +(a) M +x +x +x +(b) R +x +x +x +(c) O +Figure 8: Mass-counterterm diagrams for mass-splitting and HVP 1-photon +diagrams. Diagram M gives the valence, diagram R the sea quark mass shift +e�ects to the meson masses. Diagram O would yield a correction to the HVP +disconnected contribution (that likely is very small). +9 +FIG. 3. +Strong isospin-breaking correction diagrams. +The +crosses denote the insertion of a scalar operator. +Diagrams – QED corrections +and fit d�. +red For the finite-volume errors, the two-pion states in d are identical to the +I = 1 contributions of c and can be calculated using the GSL estimate which +we use for c. For the omega-related finite-volume errors, I will take the fitted +d� and E� and use this as the full result at finite-volume and compare it to +a GS model with omega mass from the fitted E� and width from the PDG +in infinite-volume. I should also compare this to R-ratio results for the I = 0 +channel. +Do this entire exercise for 24ID and 32ID to estimate discretization errors. +4 +QED and SIB diagrams +We will perform a full first-principles calculation of all O(↵) and O(mu � md) +corrections. The corresponding list of diagrams is given in Figs. 1 and 2. +(a) V +(b) S +(c) T +(d) Td +(e) D1 +(f) D1d +(g) D2 +(h) D2d +(i) F +(j) D3 +Figure 1: QED corrections +x +x +x +(a) M +x +x +x +(b) R +x +(c) Rd +x +x +x +(d) O +Figure 2: SIB corrections +4 +Diagrams – Strong isospin breaking +8 / 20 +and fit d�. +red For the finite-volume errors, the two-pion states in d are identical to the +I = 1 contributions of c and can be calculated using the GSL estimate which +we use for c. For the omega-related finite-volume errors, I will take the fitted +d� and E� and use this as the full result at finite-volume and compare it to +a GS model with omega mass from the fitted E� and width from the PDG +in infinite-volume. I should also compare this to R-ratio results for the I = 0 +channel. +Do this entire exercise for 24ID and 32ID to estimate discretization errors. +4 +QED and SIB diagrams +We will perform a full first-principles calculation of all O(↵) and O(mu � md) +corrections. The corresponding list of diagrams is given in Figs. 1 and 2. +(a) V +(b) S +(c) T +(d) Td +(e) D1 +(f) D1d +(g) D2 +(h) D2d +(i) F +(j) D3 +Figure 1: QED corrections +x +x +x +(a) M +x +x +x +(b) R +x +(c) Rd +x +x +x +(d) O +Figure 2: SIB corrections +4 +and fit d�. +red For the finite-volume errors, the two-pion states in d are identical to the +I = 1 contributions of c and can be calculated using the GSL estimate which +we use for c. For the omega-related finite-volume errors, I will take the fitted +d� and E� and use this as the full result at finite-volume and compare it to +a GS model with omega mass from the fitted E� and width from the PDG +in infinite-volume. I should also compare this to R-ratio results for the I = 0 +channel. +Do this entire exercise for 24ID and 32ID to estimate discretization errors. +4 +QED and SIB diagrams +We will perform a full first-principles calculation of all O(↵) and O(mu � md) +corrections. The corresponding list of diagrams is given in Figs. 1 and 2. +(a) V +(b) S +(c) T +(d) Td +(e) D1 +(f) D1d +(g) D2 +(h) D2d +(i) F +(j) D3 +Figure 1: QED corrections +x +x +x +(a) M +x +x +x +(b) R +x +(c) Rd +x +x +x +(d) O +Figure 2: SIB corrections +4 +3 / 25 +FIG. 1. +The diagrams of a complete calculation of aHVP LO +µ +when formulated as an expansion around an isospin-symmetric +limit. In the isospin-symmetric limit, there is a quark-connected (left) and quark-disconnected contribution (right). For the +QED- and strong-isospin-breaking (SIB) corrections, we indicate the photon vertices that connect to the muon with filled dots +and only show the respective sub-diagrams. For the QED corrections, one has to enforce the exchange of gluons between the +quark loops in diagram F to avoid double-counting of higher-order HVP contributions. For the SIB corrections, the crosses +denote scalar operator insertions to allow for a linear correction in the respective quark masses. +can be defined, which gives the same value of aHVP LO +µ +in the continuum limit. We use both versions to scrutinize the +continuum extrapolation. +The correlator C(t) is computed in lattice QCD+QED at physical pion mass with non-degenerate up- and down- +quark masses including up-, down-, strange-, and charm-quark contributions. The missing bottom-quark contributions +are estimated using perturbative QCD. +B. +Euclidean windows +In the following, we suppress the leading-order HVP LO label for brevity. Following [31], we define Euclidean +windows that partition the contributions of time-slices t in Eq. (1) into short-distance (SD), window (W), and long- +distance (LD) contributions. To make the quantities well-defined at non-zero lattice spacing, we introduce smearing +kernels with width ∆. We write +aµ = aSD +µ ++ aW +µ + aLD +µ +, +(6) + +4 +where +aSD +µ (t0, ∆) = +∞ +� +t=0 +C(t)wt[1 − Θ(t, t0, ∆)] , +(7) +aW +µ (t0, t1, ∆) = +∞ +� +t=0 +C(t)wt[Θ(t, t0, ∆) − Θ(t, t1, ∆)] , +(8) +aLD +µ (t1, ∆) = +∞ +� +t=0 +C(t)wtΘ(t, t1, ∆) , +(9) +Θ(t, t′, ∆) = [1 + tanh [(t − t′)/∆]] /2 . +(10) +All contributions are well-defined individually and can be computed using lattice methods as well as dispersive methods +by relating the correlator +C(t) = +1 +12π2 +� ∞ +0 +d(√s)R(s)se−√st +(11) +to the R-ratio +R(s) = +3s +4πα2 σ(s, e+e− → had). +(12) +Within a lattice calculation, discretization effects are most severe for the SD contribution, while statistical noise and +finite-volume effects are most pronounced in the LD contribution. The window quantity aW +µ has small statistical and +systematic errors. +As recently argued in Ref. [1], the systematic study of window quantities aW +µ (t0, t1, ∆) as a function of t0 and t1 is +useful to constrain energy regions within the R-ratio contributing to a possible tension between lattice and dispersive +results. First lattice results with a high resolution in t0 and t1 are already available [34]. Windows with larger values of +t0 and t1 are more sensitive to low-energy states and are useful for checking effective field theory as argued in Ref. [35]. +A systematic study of the short-distance window aSD +µ (t0, ∆) as a function of t0 is also useful as argued in Ref. [36], +where the aSD +µ (t0, ∆) defined as above are called one-sided windows since 1−Θ(t, t0, ∆) = [1 − tanh [(t − t0)/∆]] /2 = +Θ(t0, t, ∆). In the current work, we focus on the short-distance and window contributions for the standard values of +t0 = 0.4 fm, t1 = 1.0 fm, and ∆ = 0.15 fm [31]. +C. +Isospin-symmetric world +It is convenient to perform the calculation as an expansion around an isospin-symmetric point [31, 37–39]. We +therefore compute the diagrams of Fig. 1 individually. The exact choice of the expansion point is inconsequential +for the total aµ, however, care is needed if one attempts to compare isospin-symmetric results provided by different +groups [40]. +In this work, we present results for two choices of the isospin-symmetric world. The first choice is the RBC/UKQCD18 +world defined by +mπ = 0.135 GeV , +mK = 0.4957 GeV , +mΩ = 1.67225 GeV , +(13) +consistent with our previous work [31]. In this update, we also consider the effects from dynamical sea-charm quarks +from first principles and therefore extend this choice by +mDs = 1.96847 GeV . +(14) +Since one of the main goals of this work is to scrutinize the result of Ref. [30], we also consider a second choice +mπ = 0.13497 GeV , +mss∗ = 0.6898 GeV , +w0 = 0.17236 fm , +(15) +which we label as the BMW20 world. The quantity mss∗ is obtained from the ground-state energy of the quark- +connected pseudoscalar ¯ss meson two-point function. +This choice is consistent with the isospin-symmetric world +defined in Ref. [30]. For the sea-charm study, we adopt Eq. (14) also in this case. + +5 +ID +a−1/GeV +Nf +L3 × T × Ls/a4 +b + c +amres × 104 +mπ/MeV +mK/MeV +mDs/GeV +mπL +48I +1.7312(28) +2+1 +483 × 96 × 24 +2 +6.1 +139.32(30) +499.44(88) +– +3.9 +64I +2.3549(49) +2+1 +643 × 128 × 12 +2 +3.1 +138.98(43) +507.5(1.5) +– +3.8 +96I +2.6920(67) +2+1 +963 × 192 × 12 +2 +2.3 +131.29(66) +484.5(2.3) +– +4.7 +1 +1.7310(35) +2+1 +323 × 64 × 24 +2 +6.3 +208.1(1.1) +514.0(1.8) +– +3.8 +2 +1.7257(74) +2+1 +243 × 48 × 32 +2 +4.6 +285.4(2.9) +537.8(4.6) +– +4.0 +3 +1.7306(46) +2+1 +323 × 64 × 24 +2 +6.5 +211.3(2.3) +603.8(6.1) +– +3.9 +4 +1.7400(73) +2+1 +243 × 48 × 24 +2 +6.2 +274.8(2.5) +530.1(3.1) +– +3.8 +5 +1.7498(73) +2+1+1 +243 × 48 × 24 +2 +6.7 +279.8(3.5) +539.1(5.3) +1.9902(69) +3.8 +7 +1.7566(81) +2+1+1 +243 × 48 × 24 +2 +7.9 +272.5(5.9) +523(10) +1.3882(57) +3.7 +A +1.7556(83) +2+1 +243 × 48 × 8 +2 +42 +307.4(3.5) +557.3(5.7) +– +4.2 +24ID +1.0230(20) +2+1 +243 × 64 × 24 +4 +23 +142.96(30) +515.7(1.0) +– +3.4 +32ID +1.0230(20) +2+1 +323 × 64 × 24 +4 +23 +142.96(30) +515.7(1.0) +– +4.5 +TABLE I. List of ensembles with parameters determined in the RBC/UKQCD18 isospin symmetric world. Unless specified +otherwise, the ensembles have Iwasaki gauge action and M¨obius [42] domain-wall [43, 44] fermion sea quarks with b − c = 1. +The parameters b and c are defined in Ref. [41]. For the Nf = 2 + 1 + 1 ensembles, the charm quarks couple to three-times +ρ = 0.1 stout smeared gauge fields as in Refs. [45, 46]. The scripts generating the new ensembles are publicly available [47]. The +24ID and 32ID ensembles have an additional DSDR term [41] in the gauge action. The 24ID and 32ID ensemble parameters +are taken from Ref. [48]. +We define these parameters to the exact values given above without additional uncertainty. This avoids an unnec- +essary inflation of uncertainties when comparing isospin-symmetric lattice results. The experimental uncertainties of +the physical hadron spectrum are then taken into account when applying the isospin-breaking corrections. +To support the careful tuning of the isospin-symmetric world, we generated additional near-physical-pion-mass +ensembles allowing for the explicit calculation of light and strange quark-mass derivatives. Our choice of discretisation +and simulation parameters is summarised in Tab. I. We also generated ensembles with dynamical charm quarks and +ensembles with varying extent of the fifth dimension of our domain-wall fermions, Ls, to control for residual chiral- +symmetry-breaking effects. Finally, we include results at physical pion mass and a finer lattice spacing of a−1 ≈ 2.7 +GeV. +We determined the ensemble parameters in two ways. First, we used the new ensembles to obtain the quark-mass +dependence of the quantities defined in Eqs. (13) and (15). We then tune the dimensionless mπ/mΩ and mK/mΩ +for the RBC/UKQCD18 world and w0mπ and w0mss∗ for the BMW20 world to the values provided in Eqs. (13) and +(15). Any of the three dimensionful values can then equivalently be used to determine the lattice spacing a for a +given ensemble. For the Nf = 2 + 1 + 1 ensembles, we also tune mDs/mΩ for the RBC/UKQCD18 world and w0mDs +for the BMW20 world to the value provided in Eq. (14). We provide the results for the RBC/UKQCD18 world in +Tab. I. In addition, we also performed an update of our global fit [41] for which we found consistent results. A detailed +discussion of the updated global fit will be published separately. The two determinations of ensemble parameters were +performed by disjoint sub-groups of authors. +D. +Blinding procedure +Since we provide an update of a previous result [31] compared to which a lower value would mean agreement with +the dispersive method and a higher value would mean agreement with the lattice result of Ref. [30], two values that +are in 3.7σ tension with each other, we believe it is crucial to perform this update in a blinded manner. +We implement the blinding by creating modified correlators Cb(t) from the unaltered correlators C0(t). For each +lattice ensemble, we use +Cb(t) = (b0 + b1a2 + b2a4)C0(t) +(16) +with respective lattice spacing a and random coefficients b0, b1, and b2 that are common for each ensemble but different +for each analysis group. The parameter b0 is drawn from a Gaussian distribution with mean µ = 1.0 and standard +deviation σ = 0.2. The dimensionful parameters b1 and b2 are drawn from a flat distribution with maximum values +of |b1a2| = 0.05 and |b2a4| = 0.0025 for our coarsest lattice cutoff a−1 = 1.73 GeV. This procedure based on three +random numbers per analysis group prevents the possibility of complete unblinding based on previously shared data +on the coarser two ensembles [31]. The blinding factors were generated and directly applied to C0 by author CL. This + +6 +process took a given seed for the random number generator as input such that only this seed and not the blinding +factors were directly accessible to CL. +For the current update, we established five analysis groups (called A–E in the following), composed of non- +overlapping sub-groups of authors. The different analysis groups were provided with the ensemble parameters and the +respectively blinded correlator data. They then separately decided on their respective analysis procedures without +interacting with other groups. The chosen methods are described in Sec. IV A. After the groups completed their +analyses, we started a relative unblinding procedure during which two groups would jointly discuss and scrutinize +their approaches. In this process some important findings emerged, as described in Sec. IV C. Based on these discus- +sions, the collaboration then converged on a preferred prescription that is described in Sec. IV D. At this point the +prescription was frozen and a complete unblinding performed. The results are discussed in Sec. V. +III. +COMPUTATIONAL DETAILS +In the following, we describe in detail the computational methods used in this work. We explain aspects of data +generation as well as crucial components of the various aµ analyses. +A. +Overview of improvements +Compared to our previous calculation of Ref. [31], we have made several substantial improvements. With regard to +the statistical uncertainty, we increased the statistical sample size for the correlators on ensembles 48I and 64I by a +factor of four. Improvements reducing systematic uncertainties are described in the following. +To improve the continuum extrapolation, we add a finer lattice spacing at physical pion mass with a−1 = 2.7 +GeV. We also consider an additional discretization for the vector current by studying both local-conserved as well +as local-local correlators. This can be done in a cost-efficient manner as described in Sec. III B. In addition, we use +two different renormalization procedures for the local vector current. The first procedure, which we label ZV , follows +Ref. [41] and uses that the expectation value of the charge operator in a pion state equals one. The second procedure, +which we label Z⋆ +V , uses the ratio of local-conserved to local-local correlators interpolated to fixed Euclidean time t⋆ +to define the current normalization. The particular choice of t⋆ is described in Sec. IV A. Finally, we use two different +weight functions wt and ˆwt, see Eqs. (3) and (5), at a given lattice spacing. This gives a total of 3×2×2×2 = 24 data +points to study the continuum extrapolation, which improves our previous extrapolation based on two data points. +To reduce parametric uncertainties, we generated new near-physical pion- and kaon-mass ensembles to calculate +parametric derivatives with respect to quark masses. In Sec. III E, we also show how to obtain parametric derivatives +inspired by master-field methodology [49]. +We previously estimated the missing sea-charm effects using perturbative QCD [41]. For this update, we have +generated new ensembles with dynamical charm quarks, which we match to our Nf = 2 + 1 ensembles as described +in Sec. III C. +Domain-wall fermions exhibit only small chiral symmetry breaking which is commonly quantified using the residual +mass mres [44, 50]. For this reason, a very small linear discretization error is allowed. We previously neglected such +effects but have now generated new ensembles with different extents of the fifth dimension Ls to quantify them from +first principles. +Since we only have a small number of configurations for the new 96I ensemble, we also investigate a new five- +dimensional master-field statistical error estimate in Sec. III D to considerably reduce the uncertainty on our estimate +of statistical variance concerning this ensemble. +B. +Local- and conserved-current correlators +In addition to the local lattice vector current Jµ, which we denote in the following as Jl +µ, we consider the conserved +lattice vector current Jc +µ as defined in Ref. [41]. We consider the correlators +Cab(t) = 1 +3 +� +⃗x +� +j=0,1,2 +⟨Jb +j (⃗x, t)Ja +j (0)⟩ +(17) +in the local-local (Cll) and local-conserved (Clc) versions. After performing the fermionic Wick contraction, the source +is always local and the sink varies between local and conserved. The contraction code is publicly available [51]. It +uses an all-mode-averaging procedure [52–55] combined with additional averaging of the low-low component of the + +7 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 0 + 5 + 10 + 15 + 20 +t / a +Clc(t) / Cll(t) +FIG. 2. Ratio Clc(t)/Cll(t) as a function of Euclidean time t on the 96I ensemble. +correlator [31]. Our approach again relies on approximating the low-mode space on a coarse grid as introduced in +Ref. [56]. For the 96I ensemble, this yields a reduction of data volume by a factor of 30. This is crucial not just for +data storage but also for the computational performance of low-mode estimates due to the reduced memory-transfer +requirements. +For a given point source, the local-local and local-conserved correlators are highly correlated. We therefore compute +the ratio Clc/Cll using only a few correlated source positions and multiply this ratio with our full-statistics estimator +of Cll to obtain our estimator for Clc. In Fig. 2, we plot the ratio for the 96I ensemble. +For the 96I ensemble an additional improvement was made. For this ensemble, we generate a data set in which two +source positions at time-slice t and t + 96 are combined with a Z2 number. For short and intermediate distances, this +effectively doubles our statistics at the same cost. A second lower-statistics single time-slice data set is provided to +account for the effects of the backwards propagation of the additional time slice. +Finally, all correlators are provided with identical valence- and sea-quark masses. In this manner, we can perform a +purely unitary data analysis. For the 64I ensemble, however, for historical reasons the eigenvectors were generated for +a partially-quenched mass am = 0.0006203 instead of the unitary mass am = 0.0006780 [41]. For this reason, a small +additional correlated data set was generated at am = 0.001774 such that the unitary correlators can be obtained by +aµ(0.0006780) = aµ(0.0006203) + (0.0006780 − 0.0006203)aµ(0.001774) − aµ(0.0006203) +0.001774 − 0.0006203 +≈ aµ(0.0006203) + aµ(0.001774) − aµ(0.0006203) +20 +. +(18) +Non-linear effects in the small quark-mass shift are negligible for the precision goals of the present calculation. +C. +Sea-charm effects +In this work, we estimate the effects of sea-charm quarks from first principles. Most of our ensembles have Nf = 2+1 +sea quarks with an isospin-symmetric up- and down-quark pair and an additional strange quark. To study the sea- +charm effects from first principles, we have generated additional Nf = 2+1+1 ensembles with different charm masses +to separate the physical effects from a modification of discretization errors. We list the ensemble parameters in Tab. I. +We match the Nf = 2+1 and Nf = 2+1+1 ensembles to the same pion and kaon masses and the Wilson-flowed [57] +energy density at long-distance. In Fig. 3, we show tfE(tf) with flow-time tf and Wilson-flowed energy density E(tf) +for the nominal ensemble 4, 5, and 7 of Tab. I. At shorter distances, we observe a clear signal of charm effects in +the energy density. For the lighter charm mass, this effect extends to longer distances. We plot tfE(tf) instead of +the dimensionless t2 +fE(tf) since all plotted ensembles share the same lattice spacing and the interesting features are +better highlighted in this way. + +8 + 0.16 + 0.18 + 0.2 + 0.22 + 0.24 + 0.26 + 0.28 + 0.3 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 +a2 tf E(tf) +tf / a2 +Nf=2+1 +Nf=2+1+1, mDs=2.0 GeV +Nf=2+1+1, mDs=1.4 GeV +FIG. 3. +Wilson-flowed energy density E(tf) multiplied with the flow-time tf for Nf = 2 + 1 and Nf = 2 + 1 + 1 ensembles. +The small statistical uncertainties for each line are shown as an error band. +We use these matched ensembles to measure the sea-charm contributions to the HVP. We do this in particular for +the short-distance window, where most of the effect should appear. The exact approach used by the different analysis +groups is explained in Sec. IV A. +D. +Five-dimensional master-field statistical errors +For the 96I ensemble, we currently only have 33 gauge field configurations in contrast to the 64I and 48I ensembles +for which we have 238 and 386 gauge field configurations, respectively. In order to obtain a reliable statistical-error +estimate on the 96I ensemble, we have performed a slightly modified master-field error analysis [49]. In our approach, +we improve the covariance estimate by considering a five-dimensional master field with Markov time as an additional +fifth dimension. We expect exponential locality in the fifth dimension governed by the eigen-modes of the Markov +transition matrix and in the four space-time dimensions governed by the eigen-modes of the QCD Hamiltonian. +For an observable Oτ,x with Markov time τ and space-time coordinate x, we consider the statistical average +O = +1 +|V| +� +(τ,x)∈V +Oτ,x +(19) +with set V that contains all tuples (τ, x) for which the observable was determined. Note that we explicitly allow for +sparse sampling in space-time as well as Markov time. The covariance of two such observables O and O′ is then given +by +Covτc,xc(O, O′) ≡ +1 +|V||V′| +� +(τ,x)∈V,(τ′,x′)∈V′, +|x−x′|≤xc,|τ−τ ′|≤τc +� +⟨Oτ,xO′ +τ ′,x′⟩ − ⟨Oτ,x⟩⟨O′ +τ ′,x′⟩ +� +(20) +and studying Covτc,xc(O, O′) as a function of τc and xc to identify a plateau for large τc and xc. +In practice, +we estimate Covτc,xc(O, O′) based on a given set of gauge configurations, which adds an error suppressed by the +inverse square root of the number of sampled five-dimensional points. In comparison, the Jackknife estimator has an +uncertainty suppressed by the inverse square root of the number of gauge configurations, such that its uncertainty +is generally much larger. The distance |x − x′| takes the field boundary conditions into account, i.e., for periodic +boundary conditions, we consider the shortest distance between mirror images. +For arbitrarily sparse V, the various Oτ,x are effectively all statistically independent such that we expect a plateau +already for very small τc and xc. In general, just before reaching the gauge noise limit, the plateaus still start early +in xc. Conversely, a rising behavior in xc signals that our sample points are significantly correlated. We tune the +sampling of our vector correlators to be such that we almost reach the gauge noise limit and therefore plateaus + +9 + 0 + 0.5 + 1 + 1.5 + 2 + 0 + 5 + 10 + 15 + 20 + 25 +c Covτc,xc(C(t),C(t))1/2 +xc / a +t/a=8 +t/a=10 +t/a=12 +t/a=16 +t/a=20 +FIG. 4. +The statistical uncertainty of C(t) determined by Eq. (20) multiplied with a blinding factor c determined by the +five-dimensional master-field approach (individual data points) compared to the Jacknife estimate (solid lines). +For these +estimates, we use randomly selected 660 point sources per 33 configurations on the 96I ensemble. Due to the sparseness of +our measurement setup, we observe a plateau in xc starting essentially from the smallest value. The plot is made after having +established a plateau in τc. +are reached for modest values of xc. In Fig. 4, we compare the statistical uncertainty of C(t) on the 96I ensemble +determined by the five-dimensional master-field approach to the Jackknife estimate. +E. +Master-field parametric derivatives +In order to tune the Nf = 2 + 1 + 1 ensembles described in Sec. III C, we found the master-field formalism useful to +get initial estimates of parametric derivatives with respect to the gauge-action parameter β as well as the sea-charm +mass. To simplify the discussion, we set a = 1 in this sub-section. +Consider a general gauge action +S = −β Nd(Nd − 1) +2 +� +x +Ax +(21) +with space-time dimension Nd and field of Wilson loops Ax anchored at a point x. It is not crucial how we exactly +identify the location x as long as the coordinate behaves properly under translations of the system. One can then +show that for a general observable O in Nd = 4 without explicit β dependence, +∂β⟨O⟩ = 6 lim +xc→∞ Cov0,xc(O, A) , +(22) +with Cov0,xc defined in Eq. (20). Setting O to the Wilson-flowed energy density E(tf), e.g., allows us to determine +the β-derivative of the Wilson-flow scales t0 and w0 [57, 58]. +We can also show that +∂m⟨O⟩ = lim +xc→∞ Cov0,xc(O, Tr[ ˜D−1 +ov (m)]) , +(23) +for sea-quark mass m and +˜D−1 +ov (m) = +1 +1 − m +� +D−1 +ov (m) − 1 +� +, +(24) +with overlap operator Dov [42, 59]. We find that the traces of ˜D−1 +ov (m) can be efficiently estimated using our tadpole +field approach of Ref. [60]. For domain-wall fermions, an additional flavor enters the path integral as the determinant +ratio +det(D(m)D−1(1)) +(25) + +10 + 0 + 0.0005 + 0.001 + 0.0015 + 0.002 + 0.0025 + 0 + 5 + 10 + 15 + 20 + 25 + 30 +xc / a +Cov0,xc(E(2.01),Tr[D~-1 +ov(0.8)]) +-0.035 +-0.03 +-0.025 +-0.02 +-0.015 +-0.01 +-0.005 + 0 + 0 + 5 + 10 + 15 + 20 + 25 + 30 +xc / a +Cov0,xc(E(2.01),A) +FIG. 5. +We plot for the 96I ensemble Cov0,xc(E(tf), Tr[ ˜D−1 +ov (0.8)]) on the left and Cov0,xc(E(tf), A) on the right for +tf = 2.01 ≈ t0/2. The Wilson-loop field A is defined in Eq. (21). +with five-dimensional Dirac operator D(m). For m = 1 this factor is trivial and we can view including an additional +flavor as changing the sea-quark mass down from m = 1 to the target value. In this way integrating the parametric +derivative with respect to m allows us to determine the effects of introducing an additional sea-charm quark. Setting +O to the Wilson-flowed energy density, allows us to determine the effect of the additional sea-charm quark to the +Wilson-flow scales t0 and w0. In Fig. 5, we show the convergence as a function of xc for the β derivative as well as +the charm-quark mass derivative at m = 0.8 of E(tf) with tf = 2.01 ≈ t0/2 on the 96I ensemble. The lower scale +t0/2 allows for a statistically more precise estimate of the dependence of the lattice spacing on β and the charm-quark +mass. +F. +Finite-volume effects +In order to determine the finite-volume effects on C(t), the analysis groups explored two methods: a direct fit to +the 24ID and 32ID data as well as the Hansen-Patella approach [61, 62]. Details of the former approach are given in +Sec. IV A. For the latter approach, we use a monopole ansatz of the electromagnetic pion form factor +F(k2) = +1 +1 − k2/m2ρ +(26) +and study the dependence on mρ. For this ansatz Ref. [62] gives an expression for the finite-volume corrections for +C(t) in terms of a simple integral +CL(t) − C∞(t) = +� +⃗n̸=⃗0 +1 +6π|⃗n|L +� +Im +� +R+iµ +dk3 +2π +eik3|x0|(4m2 +π + k2 +3)m4 +ρ +(m2ρ + k2 +3)2 +e−|⃗n|L +� +m2π+ +k2 +3 +4 +4k3 ++ +� dp3 +2π e−|⃗n|L√ +m2π+p2 +3 d +dz +� +e−z|x0|(z2 − 4m2 +π)m4 +ρ +(z + mρ)2(z2 + 4p2 +3) +� +z=mρ +� +, +(27) +where CL is the correlator at finite spatial volume L3 and C∞ is the infinite-volume version. The equation depends +on the pion mass mπ and the monopole-mass parameter mρ. The complex shift iµ of the integration contour has to +be chosen in the range 0 < µ < 2mπ, however, the integral does not depend on the exact choice. Equation (27) only +considers the pole contribution to the Compton amplitude and neglects terms of order e−√ +2+ +√ +3mπL as well as effects +of finite Euclidean time. This is well justified for our current precision goal. The effects of the regular contribution +to the Compton amplitude and effects of the finite Euclidean time extent are known [61, 62] and may be considered +in future work. +Note that the finite-volume corrections for the quark-connected diagram are 10 +9 of the total as is easily seen from the +following argument. Consider a theory with quark charges Qu = 1 +2 = −Qd instead of the physical Qu = 2 +3 = −2Qd. +The QED charges of mesons made of up and down quarks are identical in both cases, however, in the Qu = 1 +2 = −Qd + +11 +theory the quark-disconnected diagram does not contribute, while the quark-connected diagram contributes with a +Q2 +u + Q2 +d = 1 +2 factor instead of the physical Q2 +u + Q2 +d = 5 +9. We therefore find that 1 +2 +9 +5 = +9 +10 of the quark-connected +contribution is equal to the total contribution and equivalently that the total correction needs to be multiplied by 10 +9 +to obtain the correction for the quark-connected piece. This simple argument is consistent with partially quenched +Chiral Perturbation Theory studies [32, 34, 63]. +IV. +RELATIVE UNBLINDING +In the following, we summarize the different approaches of the five analysis groups and show the result of our +relative unblinding process. We highlight important findings and explain the prescription that all five groups agreed +to be used for the full unblinding. +A. +Distinct methods of the five analysis groups +Each analysis group received the blinded correlator data as described in Sec. II D. The separate analysis groups +then discussed the data and agreed on the respective analysis methods within each group. The confinement of these +discussions to the separate groups lead to a diverse set of approaches to the data analysis. In the following sub-sections, +we briefly describe the approaches of each group, focusing on the differences. +1. +Group A +Analysis group A provides results for aW +µ +as well as aSD +µ . Statistical errors are obtained from a super-jackknife +procedure [64, 65] for most ensembles combined with a binning study and using the master-field error estimates of +Sec. III D on ensemble 96I. The continuum extrapolations are performed based on the 24 data points over three lattice +spacings described in Sec. III A, where small linear corrections to shift the individual points to the lines of constant +physics (LCP) are applied first. Finite-volume corrections are also applied before the continuum extrapolation. To +this end, the Hansen-Patella Eq. (27) is used for finite-volume corrections with nominal parameters mρ = 727 MeV +and errors estimated from the variation to mρ = 770 MeV. An additional ad-hoc 20% uncertainty is added to the +finite-volume corrections to account for the limitations discussed in Sec. III F. Combinations of the fit ansaetze +f2(a2) = c0 + c1a2 , +(28) +f2,4(a2) = c0 + c1a2 + c2a4 , +(29) +f2α(a2) = c0 + c1a2αs(µ = 1/a) , +(30) +f2α,4(a2) = c0 + c1a2αs(µ = 1/a) + c2a4 +(31) +are then considered with four-loop running coupling αs in the MS scheme [66]. +For aW +µ , the central value is chosen as the average of the f2 fits to the (ωt, Clc, Z⋆ +V ), (ωt, Cll, Z⋆ +V ), (ωt, Clc, ZV ) +trajectories with t⋆ = 1 fm. These trajectories had the smallest a4 contributions. For aW +µ , the effect of ωt compared +to ˆωt is negligible. The continuum extrapolation error is estimated by varying f2 to f2α and by considering the spread +of the mean to the individual (ωt, Clc, ZV ) and (ωt, Cll, Z⋆ +V ) fits. +For aSD +µ , the fit form f2,4 is used for all trajectories and the average of (ˆωt, Clc, ZV ) and (ˆωt, Clc, Z⋆ +V ) is used for the +central value since they exhibit the smallest a4 coefficients. The variation from f2,4 to f2α,4 as well as the maximal +variation to (ˆωt, Cll, ZV ), (ωt, Clc, ZV ), (ˆωt, Clc, ZV ), (ˆωt, Cll, Z⋆ +V ), (ωt, Clc, Z⋆ +V ), and (ˆωt, Clc, Z⋆ +V ) is then used for the +continuum extrapolation error. +The effects of the residual mass and the sea-charm quark are studied separately and found to be small compared +to the quoted uncertainties. +2. +Group B +Analysis group B provides results for aW +µ +as well as aSD +µ . +The strategy is to employ a global fit to all of the +measurements on the ensembles listed in Sec. I. Statistical errors for each measurement, including lattice spacings, +pion masses, and so on, are incorporated through a super-jackknife method. + +12 +Several terms comprise the global fit function for the intermediate window. A second-order polynomial in a2 is used +to extrapolate non-zero lattice spacing to the continuum limit. Finite-volume effects are treated explicitly through +a term exponential in mπL and are mainly constrained by the two Iwasaki-DSDR ensembles in Tab. I. Small light- +quark-mass mistunings are treated linearly in the appropriate meson-mass squared and a simple linear ansatz for the +residual mass is applied. Charm-quark mistunings are corrected with inverse mass-squared of the Ds meson. All +together, the fit function takes the form +aµ(...) = aµ +� +1 + c1a2 + c2a4� � +1 + c3e−mπL� � +1 + c4(m2 +π − m2 +π,phys) +� � +1 + c5(m2 +K − m2 +K,phys) +� +× (1 + c6amres) +� +1 + c7 +� +1 +m2 +Ds +− +1 +m2 +Ds,phys +�� +. +(32) +The coefficients c1 and c2 take on different values for the Iwasaki-DSDR ensembles, and the residual mass term is +treated as an O(a) artifact. +To fit the data to Eq. (32), the (log of) C(t) is first cubically interpolated between time-slices and then integrated +with the continuum form of the one-loop QED kernel, Eq. (3). The central value of the procedure is determined from +the average of conserved-local and local-local correlation functions for the HVP. The main part of the systematic error +arises from the difference of these two results in the continuum limit. +For the short-distance window, the procedure is similar except that the discrete version of the one-loop kernel ˆωt is +also used (approximated as wt(1−a2/t2)) and an a2 log a2 term is considered. The systematic error is computed from +differences between pairwise combinations of a2, a4 and a2 log a2 terms, using both wt and ˆwt weights, all added in +quadrature. The central value is taken as the wt version with the conserved-local correlation function since empirically +it has the smallest a4 contamination. +3. +Group C +Analysis group C provides results for aW +µ . The strategy is divided in a few steps. First, using the ensembles listed in +Tab. I the derivatives of the intermediate window with respect to the quark masses are calculated. Additional cutoff +or finite-volume effects on the derivatives are neglected. The derivatives are then used to shift the three reference +ensembles, 48I, 64I and 96I, to the LCP. Additionally, all windows are shifted to mπL = 4 using Chiral Perturbation +Theory and additional systematic effects are not considered since they are well below the statistical uncertainty. +After multiplying by the normalization factors ZV or Z⋆ +V , the intermediate windows from the 3 ensembles and 2 +discretizations (Cll and Clc) are extrapolated to the continuum limit with a constrained fit. Note that also a2/t0 +used in the extrapolation is shifted to the proper LCP. The following three types of fits are considered: linear and +quadratic in a2 with all 6 data points and linear in a2 with the finest 4 data points (96I, 64I). A systematic error from +the spread of the central values of the fitted continuum windows is included in the error budget. Both correlated and +uncorrelated fits are used, and for the latter their quality is assessed using the method developed in Ref [67]. The 3 +fits described above are performed separately using ZV and a variant of Z⋆ +V . For the former it is observed that the +linear fit in a2 is not acceptable, and that a quadratic term is necessary to describe the data. Hence, the preferred +strategy is based on Z⋆ +V and the preferred fit is the constrained linear fit to all 6 data points. For the variant of +Z⋆ +V , a slight modification of the definition provided in Sec. III A is considered, i.e., the ratio of Clc over Cll is used +individually integrated using the smearing function Θ(t, t⋆ − ∆/2, ∆)Θ(t⋆ + ∆/2, t, ∆) with ∆ = 0.15 fm. Several +values of t⋆ are explored and for the final analysis t⋆ = 1 fm is adopted. No particular difference is observed with +respect to the interpolation described in Sec. III A, as one can easily infer from the long plateau in Fig. 2. +The statistical analysis is carried out by propagating all fluctuations of observables using both the Jackknife method +and the Γ-method [68]. No large autocorrelations in the extrapolated continuum window are observed. Finite-volume +effects to correct from mπL = 4 to ∞ are obtained from an independent implementation of Eq. (27). Final shifts for +residual mass effects and dynamical charm effects are applied in the same manner as also done by group B. +4. +Group D +Analysis group D provides results for aW +µ +from the physical pion-mass ensembles 48I, 64I, and 96I, which are +computed with a binned super-jackknife analysis with weight function wt and vector current normalizations ZV and +Z⋆ +V . In addition, a version of ZV is used, where the pion state is replaced by a kaon state. The mass extrapolation +to the physical point is done by assuming linear dependence on the quark masses taken from ensemble 1 with 4 and +ensemble 1 with 3, respectively. Finite-Ls effects are corrected by assuming linearity in mres using ensembles 1, 2, + +13 +4, and A. The values of aW +µ on the 48I and 64I ensembles are corrected by an exponential dependence to the lattice +extent, exp(−mπL), whose coefficient is taken from the 24ID and 32ID ensembles, to match for the volume of 96I. A +50% systematic uncertainty for these finite-volume corrections is added. It is noted that within the statistical noise +of the 24ID and 32ID ensembles, their difference is reproduced by the Hansen-Patella finite-volume formula as well +as the Meyer-Lellouch-L¨uscher-Gounaris-Sakurai [69–71] approach. +After these corrections for 18 data points from three ensembles, two vector currents Cll and Clc, and three vector +current normalizations, the continuum extrapolation is performed by combinations of the fit formulae f2(a2), f2,4(a2), +f2α(a2), and f2α,4(a2) by requiring a universal continuum limit for all 18 data points. f2(a2) poorly fits Cll(t) with +the coarsest ensemble 48I, and it is decided to drop this combination from the final results. In analysis group D, +the central value for the continuum extrapolation is chosen from fit f2(a2) to Cll(t) and f2α(a2) to Clc(t). The error +of the continuum extrapolation is determined to cover all central values of the considered fit forms. The continuum +extrapolation for each of the 6 individual combination of currents and normalizations is also performed. The results are +consistent with that of the universal fit except, again, the f2(a2) fit for Cll(t). Finally, a small volume correction from +the 96I volume to infinity is carried out using the Meyer-Lellouch-L¨uscher-Gounaris-Sakurai approach. For each of the +isospin-symmetric worlds, RBC/UKQCD18 and BMW20, the lattice spacing is determined in two different scaling +trajectories (either keeping w0 or mΩ fixed). +The fit results are consistent between the two scaling trajectories, +providing an additional check for the continuum extrapolation of aW +µ . +5. +Group E +Analysis group E provides results for aW +µ . +The strategy is entirely data driven. +Statistical uncertainties are +determined from a bootstrap analysis with measurements within 20 MD units binned into an effective measurement. +The input uncertainties are propagated via re-sampling (Gaussian error propagation). Both ωt and ˆωt kernels are +used. In addition to ZV a variant of Z⋆ +V is used that for a given window is defined as +ZC +V = alc,bare +µ +all,bare +µ +, +(33) +where aab,bare +µ +is obtained without vector-current normalization factors from the bare correlators Cab. When referring +to aZ,K +µ +below, all,bare +µ +is normalized using two powers of ZV or two powers of ZC +V . +The chiral, strange-quark, +discretization, and finite-volume effects are fitted to all ensembles for a given choice of renormalization procedure and +kernel to the ansatz +aZ,K +µ += aphys +µ +× +� +1 + Cχ +(m2 +π − (m2 +π) +phys) +(m2π)phys +� +× +� +1 + Cs +(Xs − Xphys +s +) +Xphys +s +� +(34) +× +� +1 + CV e−mπL� +× +� +1 + CZ,K +CL,0(aΛ)2 + CZ,K +CL,1(aΛ)4� +× +� +1 + CZ,K +5 +amres +� +. +(35) +In this formula Xs stands for mK for the RBC/UKQCD18 world and for mss⋆ for the BMW20 world. The ratios +RZ,K +Z′,K′ on the three physical point Iwasaki ensembles are simultaneously fitted to the model fR, +RZ,K +Z′,K′ ≡ aZ,K +µ +aZ′,K′ +µ +, +fR ≡ +1 + CZ,K +CL,0(aΛ)2 + CZ,K +CL,1(aΛ)4 +1 + CZ′,K′ +CL,0 (aΛ)2 + CZ′,K′ +CL,1 (aΛ)4 +(36) +and the ratio RID +V +for the ensembles 32ID and 24ID to the model gV , +RID +V ≡ a32ID +µ +a24ID +µ +, +gV ≡ 1 + CV e−(mπL)32ID +1 + CV e−(mπL)24ID . +(37) +All correlations between data points on the same ensembles are included in this fit. Systematic uncertainties are +estimated by variations on the data that enters the fit and/or the terms included in the model(s). +B. +Comparison of results +After the analysis groups had individually converged on their respective methodology described above, we started the +process of relative unblinding. The relative unblinding of groups X and Y was conducted by sharing the individually + +14 + 0.985 + 0.99 + 0.995 + 1 + 1.005 + 1.01 + 1.015 +A +B +C +D +E +aµ +W(t0 = 0.4 fm, t1 = 1.0 fm, ∆ = 0.15 fm) + 0.96 + 0.97 + 0.98 + 0.99 + 1 + 1.01 + 1.02 + 1.03 + 1.04 +A +B +aµ +SD(t0 = 0.4 fm, ∆ = 0.15 fm) +FIG. 6. +Result of the relative unblinding procedure for aW +µ (left) and aSD +µ +(right). The results are normalized to the preferred +prescription described in Sec. IV D. The inner error bars show the statistical uncertainty, the outer error bars show the statistical +and systematic uncertainties added in quadrature. + 0 + 0.02 + 0.04 + 0.06 + 0.08 + 0.1 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +t / fm +t3 C(t) in O(α4) massless perturbative QCD +t3 Clc(t) on 48I with a-1=1.73 GeV +t3 Clc(t) on 64I with a-1=2.35 GeV +t3 Clc(t) on 96I with a-1=2.68 GeV + 0 + 0.02 + 0.04 + 0.06 + 0.08 + 0.1 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +t / fm +t3 C(t) in O(α4) massless perturbative QCD +t3 Cll(t) on 48I with a-1=1.73 GeV +t3 Cll(t) on 64I with a-1=2.35 GeV +t3 Cll(t) on 96I with a-1=2.68 GeV +FIG. 7. The dimensionless correlation function combinations t3Clc(t) (left) and t3Cll(t) (right) as well as the perturbative +result obtained from Ref. [72]. +blinded data sets of group X with group Y and vice versa. One of the groups then re-ran their analysis without +modifications on the other data set. This allowed for a direct comparison of groups X to Y while still keeping the +absolute blinding intact. +In Fig. 6, we show the final result of the relative unblinding procedure for aW +µ , for which all five groups participated. +The inner error bars give the statistical uncertainty, the outer error bars give statistical and systematic uncertainties +added in quadrature. We first note that the statistical uncertainties quoted by the separate analysis groups are consis- +tent. In addition, the different systematic approaches described in Sec. IV A yield different systematic uncertainties, +however, all results are consistent within total uncertainties. +The blinding procedure described in Sec. II D allows the a4 term to affect the comparison at the level of ±0.0025 if +the a4 terms are not included in the fits. This effect is small compared to the quoted uncertainties and is completely +eliminated in Sec. V, where we show the results of all groups after they repeated their unmodified analysis with the +fully unblinded data sets. +C. +Important findings +After the relative unblinding process, the analysis groups exchanged their most important findings for our data +sets. We discuss these findings in this sub-section. They form the basis, determined entirely on blinded data, of +formulating the preferred prescription to produce the combined collaboration result described in Sec. IV D. + +15 + 0.985 + 0.99 + 0.995 + 1 + 1.005 + 1.01 + 1.015 +A +B +C +D +E +RBC/UKQCD 23 +aµ +W(t0 = 0.4 fm, t1 = 1.0 fm, ∆ = 0.15 fm) +FIG. 8. +Result of the relative unblinding procedure for aW +µ inlcuding the preferred prescription RBC/UKQCD 23 described in +Sec. IV D. The data is normalized to the RBC/UKQCD 23 prescription. The inner error bars show the statistical uncertainty, +the outer error bars show the statistical and systematic uncertainties added in quadrature. +Finding 1: The correlator Cll has significantly larger a2/t2 and a4/t4 errors compared to Clc. These errors also +noticeably affect aW +µ . In Fig. 7, we plot the dimensionless t3C(t) to highlight this effect. +Finding 2: Mean-field improved lattice perturbation theory finds the discretization errors of Cll to be approximately +double the discretization errors of Clc. +Finding 3: When analyzing aSD +µ , where both a2 and a4 coefficients were determined, the size of the a4 coefficient is +substantially larger for Cll compared to Clc. +Finding 4: The continuum extrapolation is sensitive to how finite-volume corrections are applied to the individual +ensembles. This is an important effect in our analyses since the new finest 96I ensemble has a larger physical +volume compared to the 64I and 48I ensembles. +D. +Preferred prescription +Based on the findings outlined in Sec. IV C, the collaboration decided on the following principles for the combined +analysis that will be used for the full unblinding. First, when using Cll, we always add a a4 term to the fits. Second, +we use the Hansen-Patella finite-volume corrections instead of the data-driven fits to e−mπL since we expect the +Hansen-Patella formalism to more precisely map out the volume dependence. +These principles are then implemented in the following prescription for aW +µ . For the vector current renormalization +factor, we use ZV as well as Z⋆ +V with t⋆ = 1 fm. For the weight functions we use ˆwt as well as wt. For the continuum +extrapolation, we perform a simultaneous fit to the Cll and Clc data sets using +fll(a2) = c0 + c1a2 + c2a4 , +(38) +flc(a2) = c0 + c3a2 +(39) +as well as +fll,α(a2) = c0 + c1a2αs(µ = 1/a) + c2a4 , +(40) +flc,α(a2) = c0 + c3a2αs(µ = 1/a) . +(41) +We therefore perform 8 fits in total. We take the average of the minimum and maximum result as the central value for +our prediction. We take the difference of the central value to the maximum as our systematic error for the continuum +extrapolation. In Fig. 8, we show the final result of the relative unblinding for each group as well as the preferred +prescription, labelled RBC/UKQCD 23. For aSD +µ +the results of groups A and B were close to identical and we adopt +the prescription of group A as the preferred result. + +16 +RBC/UKQCD 2023 +ETMC 2022 +Mainz 2022 +ChiQCD 2022 OV/HISQ +ChiQCD 2022 OV/DWF +Aubin et al. 2022 +LM 2020 +BMW 2020 +ETMC 2021 +Aubin et al. 2019 +RBC/UKQCD 2018 + 195 + 200 + 205 + 210 + 215 +aµ +W,iso,conn,ud(0.4 fm, 1.0 fm, 0.15 fm) × 1010 +FIG. 9. +Comparison of the up and down quark, connected, isospin-symmetric contribution to the intermediate window. For +historical completeness, we also show results that are superseded by newer results of the same collaboration at the top in gray. +The inner error bars show the statistical uncertainty, the outer error bars show the statistical and systematic uncertainties +added in quadrature. RBC/UKQCD 2018 [31], Aubin et al. 2019 [32], ETMC 2021 [73], BMW 2020 [30], LM 2020 [34], Aubin +et al. 2022 [35], χQCD 2022 [74], Mainz 2022 [75], ETMC 2022 [76]. +V. +ABSOLUTE UNBLINDING +After the collaboration converged on the preferred prescription described in Sec. IV D, the analysis was frozen and +the absolute unblinding was performed. To this end, the unblinded data sets were distributed to the analysis groups, +who then re-ran their analysis without modifications. The results were presented by our collaboration already at the +Edinburgh workshop of the g-2 Theory Initiative [2] in 2022 and are stated without modifications in the following. +A. +Intermediate-distance window aW +µ +For the intermediate-distance window aW +µ in the isospin-symmetric limit with t0 = 0.4 fm, t1 = 1.0 fm, and ∆ = 0.15 +fm, we find the up and down quark-connected contribution to be +aW,iso,conn,ud +µ += 206.36(44)S(42)C(01)FV(00)mπ FV(08)∂m C(00)WF order(03)mres × 10−10 +(42) +in the BMW20 world and +aW,iso,conn,ud +µ += 206.46(53)S(43)C(01)FV(01)mπ FV(09)∂m C(00)WF order(03)mres × 10−10 +(43) +in the RBC/UKQCD18 world. We separately quote the statistical uncertainties (S), the continuum limit uncertainties +(C), the finite-volume uncertainties for the vector correlators (FV), the finite-volume uncertainties of the measured +pion masses (mπ FV), the uncertainties associated with the linear corrections to the line of constant physics (∂m C), +the uncertainties from the discretization of the Wilson flow equation (WF order), as well as the uncertainties due to +the non-zero chiral symmetry breaking (mres). The uncertainties from the ensemble-parameter and renormalization- +factor determinations are fully propagated in the quoted uncertainties. In Fig. 9, we compare Eq. (42) with previously +published results. In this work, we consistently use the BMW20 world for comparison plots of isospin-symmetric +contributions. +Compared to our earlier result presented in Ref. [31], where aW +µ was defined and computed for the first time, we +increase the basis for our continuum extrapolation from 2 data points over two lattice spacings to 24 data points over +three lattice spacings. If we were to repeat the continuum extrapolation through the 2 data points already available +in Ref. [31] with lower statistical precision, we obtain a result consistent with the earlier work of aW,iso,conn,ud +µ += +202.9(1.4) × 10−10. This is shown in Fig. 10. The approximate 2σ upward shift compared to Ref. [31] can therefore +dominantly be attributed to our improved continuum extrapolation. +In Ref. [31], we also computed the QED, strong-isospin-breaking, strange, charm, and quark-disconnected contribu- +tions to the intermediate window quantity. These contributions are much smaller in magnitude and their uncertainties +due to the continuum extrapolation are much smaller in absolute terms compared to aW,iso,conn,ud +µ +. By combining +these contributions with our improved light quark-connected, isospin-symmetric result of Eq. (43), we obtain our + +17 + 200 + 205 + 210 + 215 + 220 + 225 + 230 + 0 + 0.002 0.004 0.006 0.008 + 0.01 + 0.012 0.014 +aµ +W,iso,conn,ud x 1010 +a2 / fm2 +ZV, ˆω, Cll +ZV, ˆω, Clc +ZV, ω, Cll +ZV, ω, Clc +ZV*, ˆω, Cll +ZV*, ˆω, Clc +ZV*, ω, Cll +ZV*, ω, Clc + 200 + 205 + 210 + 215 + 220 + 225 + 230 + 0 + 0.002 0.004 0.006 0.008 + 0.01 + 0.012 0.014 +aµ +W,iso,conn,ud x 1010 +a2 / fm2 +ZV, ˆω, Cll +ZV, ˆω, Clc +ZV, ω, Cll +ZV, ω, Clc +ZV*, ˆω, Cll +ZV*, ˆω, Clc +ZV*, ω, Cll +ZV*, ω, Clc +FIG. 10. +Continuum extrapolation of aW,iso,conn,ud +µ +× 1010. On the left, we show the 8 fits of our preferred prescription. On +the right, we show the fit through the two data points already available in Ref. [31] with lower statistical precision. +Colangelo et al. 2022 +BMW 2020/KNT +Aubin et al. 2019/CL/KNT +RBC/UKQCD 2018/FJ +RBC/UKQCD 2023 +ETMC 2022 +Mainz 2022 +BMW 2020 +ETMC 2021 +RBC/UKQCD 2018 + 224 226 228 230 232 234 236 238 240 +aµ +W(0.4 fm, 1.0 fm, 0.15 fm) × 1010 +FIG. 11. +Comparison of the total intermediate window contribution. For historical completeness, we also show results that +are superseded by newer results of the same collaboration at the top in gray. Dispersive resuls are shown in purple, lattice +results are shown in green. The inner error bars show the statistical uncertainty, the outer error bars show the statistical and +systematic uncertainties added in quadrature. RBC/UKQCD 2018 [31], ETMC 2021 [73], BMW 2020 [30], Mainz 2022 [75], +ETMC 2022 [76], RBC/UKQCD 2018/FJ [77], Aubin et al. 2019/CL/KNT [78], BMW 2020/KNT [79], Colangelo et al. 2022 +[1]. +prediction for the total intermediate window contribution +aW +µ = 235.56(65)(50) × 10−10 +(44) +with statistical (left) and systematic (right) errors given separately. This can be compared with other lattice results +as well as results based on the R-ratio, see Fig. 11. Our result is in 3.8σ tension with the recently published dispersive +result of aW +µ = 229.4(1.4) × 10−10 [1] and in agreement with recent lattice results [30, 75, 76]. +B. +Short-distance window aSD +µ +For the short-distance window aSD +µ +in the isospin-symmetric limit with t0 = 0.4 fm and ∆ = 0.15 fm, we find the +up and down quark-connected contribution to be +aSD,iso,conn,ud +µ += 48.7(0.5)(1.6) × 10−10 +(45) +in the BMW20 world and +aSD,iso,conn,ud +µ += 49.0(0.6)(1.4) × 10−10 +(46) + +18 + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 +-0.1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 +tp / fm +aµ +SD,pQCD(tp,∆) x 1010 (massless) +aµ +W(tp,t0,∆) x 1010 (massive) +aµ +W(tp,t0,∆) x 1010 (massive minus massless) +(aµ +SD,pQCD(tp,∆) + aµ +W(tp,t0,∆)) x 1010 +FIG. 12. +Stability plot of Eq. (48) for t0 = 0.4 fm and ∆ = 0.15 fm. The massless perturbative QCD result is taken from +Ref. [72]. The correction from zero quark mass to non-zero quark mass is obtained from a linear extrapolation in the quark +mass using ensembles 48I, 1, and 4. The horizontal lines give the result of lattice QCD without combination with perturbative +QCD. Only the quark-connected isospin-symmetric up and down quark contribution is shown. +in the RBC/UKQCD18 world. We can substantially improve this result by replacing the very shortest distances +with perturbative QCD. Such a hybrid result of perturbative and non-perturbative QCD is still a first-principles +determination but may combine the strength of both approaches. In addition, the study of the consistency of lattice +QCD and perturbative QCD at short distances may play an important role in understanding the origin of the tension +for aW +µ described in Sec. V A. +To establish a hybrid method, we use the additive property of the windows, i.e., +aSD +µ (t0, ∆) = aSD +µ (tp, ∆) + aW +µ (tp, t0, ∆) . +(47) +We can then evaluate the first term in perturbative QCD at O(α4) [72] and the second term in lattice QCD, i.e., we +write +aSD +µ (t0, ∆) = aSD,pQCD +µ +(tp, ∆) + aW +µ (tp, t0, ∆) . +(48) +In Fig. 12, we study this separation as a function of tp. To the degree that perturbative QCD agrees with lattice QCD +at distance tp, the plot should exhibit a plateau. We find that lattice QCD and perturbative QCD are consistent +within 1.5 × 10−10 up to 0.4 fm. For a related study of matching perturbative QCD to short-distance vector current +correlators, see Ref. [80]. If we choose tp = 0.1 fm, we find +aSD,iso,conn,ud +µ += 48.51(43)(53) × 10−10 +(49) +in the BMW20 world and +aSD,iso,conn,ud +µ += 48.70(52)(59) × 10−10 +(50) +in the RBC/UKQCD18 world. This is our preferred prescription for aSD,iso,conn,ud +µ +. We compare Eq. (49) to previous +results in Fig. 13. +The hybrid method reduces the large discretization errors for the short-distance window and +specifically also reduces the logarithmic discretization errors described in Refs. [81] and [82]. +Finally, we note that the short-distance correlator is insensitive to the quark mass, see Fig. 14. This motivates a +new approach to study the continuum limit of the HVP. Since discretization errors largely cancel in the difference +between vector currents evaluated at different quark masses, we proposed a mass-splitting approach in Ref. [83]. In +this approach, we generate pairs of ensembles with mπ and Mπ with Mπ ≫ mπ to compute +aµ(mπ) = aµ(mπ) − aµ(Mπ) +� +�� +� +≡δaµ ++aµ(Mπ) . +(51) + +19 +RBC/UKQCD 2023 +ETMC 2022 +ETMC 2021 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 +aµ +SD,iso,conn,ud(0.4 fm, 0.15 fm) × 1010 +FIG. 13. +Comparison of our preferred result with previous determinations. For historical completeness, we also show results +that are superseded by newer results of the same collaboration at the top in gray. The inner error bars show the statistical +uncertainty, the outer error bars show the statistical and systematic uncertainties added in quadrature. ETMC 2021 [73], +ETMC 2022 [76]. + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0 + 5 + 10 + 15 + 20 + 25 +t / a +C(t,mπ = 140 MeV) t3 +C(t,mπ = 280 MeV) t3 +(C(t,mπ = 140 MeV) - C(t,mπ = 280 MeV)) t3 +FIG. 14. +Mass dependence of the vector correlator on a lattice with a−1 = 1.73 GeV. At very short distances, the vector +correlator is effectively independent of the quark mass. +This allows us to consider the continuum limit of δaµ and aµ(Mπ) separately. The costly term δaµ can then be +calculated at coarser lattice spacings compared to aµ(Mπ). This method will be used in upcoming improvements to +the present calculation. +C. +Isospin-symmetric scheme dependence +For comparisons of quantities defined in an isospin-symmetric world, it is crucial to precisely match the definitions +of the isospin-symmetric point. In Sec. II C, we defined two hadronic schemes to define the isospin-symmetric world +that match results previously presented by the RBC/UKQCD and BMW collaborations. In previous sections, we +presented our results separately for both schemes. In this section, we provide results for the correlated difference of +the BMW20 minus the RBC/UKQCD18 world. For the intermediate window we find +∆aW,iso,conn,ud +µ += −0.10(24)(07) × 10−10 +(52) +and for the short-distance window we find +∆aSD,iso,conn,ud +µ += −0.33(36)(36) × 10−10 +(53) +using the lattice results of Eqs. (45) and (46). We can therefore not yet resolve the difference in isospin-symmetric +schemes and they can be viewed as compatible at the current precision. +D. +Retrospective discussion of the blinding procedure +In the current paper, we performed a blinded analysis as described in Sec. II D. The goal of this procedure was +to eliminate psychological bias that may have influenced systematic decisions of the analysis groups to favor either + +20 + 0.985 + 0.99 + 0.995 + 1 + 1.005 + 1.01 + 1.015 +A +B +C +D +E +RBC/UKQCD 23 +aµ +W(t0 = 0.4 fm, t1 = 1.0 fm, ∆ = 0.15 fm) + 0.985 + 0.99 + 0.995 + 1 + 1.005 + 1.01 + 1.015 +A +B +C +D +E +RBC/UKQCD 23 +aµ +W(t0 = 0.4 fm, t1 = 1.0 fm, ∆ = 0.15 fm) +FIG. 15. +We show the result of the relative unblinding for aW +µ including the preferred prescription. On the left side, each +group used its own blinded data set including the a2 and a4 terms added in Eq. (16). On the right side, each group re-ran +their unmodified analysis after the absolute unblinding on the unblinded dataset. As anticipated, the artificial discretization +errors in the blinded data can change central values and error estimates at the ±0.0025 level. The data is normalized to the +RBC/UKQCD 23 prescription. The inner error bars show the statistical uncertainty, the outer error bars show the statistical +and systematic uncertainties added in quadrature. +a larger value for aW +µ , confirming the lattice QCD result of the BMW collaboration for this window quantity, or a +smaller value, confirming the result based on the R-ratio. To this end, we added artificial discretization errors using +both a2 and a4 terms such that it is impossible for those who had access to our previous results for the coarser two +lattice spacings of Ref. [31] to completely unblind themselves by comparing the new blinded correlators with the +previously shared data. This is the reason for the three parameters of Eq. (16) exceeding the number of previously +available lattice spacings. +Nevertheless, the possibility of an analysis group computing unblinded correlators based on the used gauge fields +always remains. Given the reduced statistical noise of short-distance time-slices of C(t), even our chosen blinding +procedure can in principle be circumvented with sufficient effort. It therefore remains an important task to evaluate +the balance between the threshold preventing such unblinding and the possible drawbacks introduced by the blinding +procedure. We suggest that a reasonable balance is found when everybody acting in good faith is protected from +psychological bias. +For the current calculation, we believe the chosen blinding procedure to be successful in that regard. However, it +came at the cost of a ±0.0025 level uncertainty, limiting the optimization of our preferred procedure. This uncertainty +is introduced by the a4 terms in Eq. (16) that are not always eliminated by the continuum extrapolation. The analysis +groups, however, had to make decisions and freeze their analyses based on the blinded data set. In Fig. 15, we highlight +this effect by contrasting the relative unblinding as performed on the blinded data sets compared to the case, where +we re-run the unmodified analyses on the unblinded data sets. +In future studies, we will have to reconsider our exact approach since adding even higher-order terms (such as a6) +with sufficiently small coefficients to account for additional finer data sets would have a diminishing effect. We may +therefore decide to use only lattice-spacing-independent blinding factors in the future. +VI. +CONCLUSIONS AND OUTLOOK +In this work we compute the standard Euclidean window of the hadronic vacuum polarization. +We employ a +blinded setup to avoid a possible bias towards reproducing previously published results. We focus on the dominant +quark-connected light-quark isospin-symmetric contribution and significantly improve its continuum extrapolation +and address additional sub-leading systematic effects from sea-charm quarks and residual chiral-symmetry breaking +from first principles. Our result for the total intermediate window aW +µ is in 3.8σ tension with the recently published +dispersive result of Ref. [1] and in agreement with other lattice results [30, 75, 76]. +For the isospin-symmetric +connected up and down quark contribution aW,iso,conn,ud +µ +more lattice results are available [30, 34, 35, 74–76] that are +all in agreement with the result presented in this work. +The tension for the intermediate window between lattice QCD and the dispersive result needs to be addressed in +future work and a systematic study of additional windows may provide further insights. As it stands, this tension +may be interpreted as a yet to be understood new physics contribution to hadronic e+e− decays. In the context of + +21 +the 4.2σ tension for aµ [4], +aµ(EXP) − aµ(SM) = 25.1(5.9) × 10−10 , +(54) +we note that the difference of the dispersive and lattice results for aW +µ (SM) is only 6 × 10−10. +In addition, we provide a result for the short-distance window for which our result is compatible with the recently +published result of the ETMC collaboration [76]. At short distances, we contrast lattice QCD and perturbative QCD +and find agreement up to 0.4 fm at the level of 1.5 × 10−10. We also provide results for a hybrid method in which +we use perturbative QCD below 0.1 fm and lattice QCD at longer distances. The effective mass-independence of the +vector correlators at short distances finally motivates us to define a mass-splitting procedure to further improve the +continuum extrapolation of the HVP. +We are currently generating additional ensembles with lattice spacings at a−1 = 3.5 GeV and 4.7 GeV that will +support a five-lattice spacing continuum extrapolation using the mass-splitting method. +Finally, we are also preparing an update for the long-distance window using the improved bounding method [84] +and an update of our QED and strong-isospin-breaking corrections re-using data from our hadronic light-by-light +program [26, 85–87]. Upon completion of our HVP program, we expect to be able to match the FNAL E989 target +precision. +VII. +ACKNOWLEDGMENTS +We thank our colleagues of the RBC and UKQCD collaborations for many valuable discussions and joint efforts +over the years. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) +for funding this project by providing computing time on the GCS Supercomputer JUWELS at J¨ulich Supercomputing +Centre (JSC). An award of computer time was provided by the ASCR Leadership Computing Challenge (ALCC) and +Innovative and Novel Computational Impact on Theory and Experiment (INCITE) programs. This research used +resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported +under contract DE-AC02-06CH11357. This research also used resources of the Oak Ridge Leadership Computing +Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. This research +used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy +Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE- +AC02-05CH11231 using NERSC award NESAP m1759 for 2020. This work used the DiRAC Blue Gene Q Shared +Petaflop system at the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre on behalf of the +STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital +grant ST/K000411/1, STFC capital grant ST/H008845/1, and STFC DiRAC Operations grants ST/K005804/1 and +ST/K005790/1. DiRAC is part of the National E-Infrastructure. We gratefully acknowledge disk and tape storage +provided by USQCD and by the University of Regensburg with support from the DFG. The lattice data analyzed +in this project was generated using GPT [88], Grid [89], and CPS [90] and analyzed, in part, using pyobs [91]. +TB is supported by the US DOE under grant DE-SC0010339. PB, TI, CJ, and CL were supported in part by US +DOE Contract DESC0012704(BNL), and PB, TI, and CJ were supported in part by the Scientific Discovery through +Advanced Computing (SciDAC) program LAB 22-2580. The research of MB is funded through the MUR program +for young researchers “Rita Levi Montalcini”. This project has received funding from Marie Sk�lodowska-Curie grant +894103 (EU Horizon 2020). VG and RH are supported by UK STFC Grant No. ST/P000630/1. NM is supported +by the Special Postdoctoral Researchers Program of RIKEN. TI is also supported by the Department of Energy, +Laboratory Directed Research and Development (LDRD No. 23-051) of BNL and RIKEN BNL Research Center. +LJ acknowledges the support of DOE Office of Science Early Career Award DE-SC0021147 and DOE grant DE- +SC0010339. RM is supported in part by the US DOE under grant DE-SC0011941. The work of ASM was supported +by the Department of Energy, Office of Nuclear Physics, under Contract No. DE-SC00046548. +∗ Corresponding author; christoph.lehner@ur.de +[1] G. Colangelo, A. X. El-Khadra, M. Hoferichter, A. Keshavarzi, C. Lehner, P. Stoffer, and T. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' NY 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' USA 13University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' CA 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' USA 14Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' CA 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' USA 15CP3-Origins & Department of Mathematics and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' University of Southern Denmark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Campusvej 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 5230 Odense M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Denmark (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2023) We compute the standard Euclidean window of the hadronic vacuum polarization using multiple independent blinded analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We improve the continuum and infinite-volume extrapolations of the dominant quark-connected light-quark isospin-symmetric contribution and address additional sub-leading systematic effects from sea-charm quarks and residual chiral-symmetry breaking from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We find aW µ = 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='56(65)(50) × 10−10, which is in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8σ tension with the recently published dispersive result of aW µ = 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4)×10−10 [1] and in agreement with other recent lattice determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We also provide a result for the standard short-distance window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The results reported here are unchanged compared to our presentation at the Edinburgh workshop of the g-2 Theory Initiative in 2022 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' PACS numbers: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='Gc I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' INTRODUCTION The anomalous magnetic moment of the muon aµ is defined as the relative deviation of the muon’s Land´e factor gµ from Dirac’s relativistic quantum mechanics result, aµ = gµ/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' It is one of the most precisely determined quantities in particle physics and has exhibited a persistent tension between the experimentally measured value and the Standard Model theory result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In order to reduce the experimental uncertainties, substantial efforts are currently undertaken at Fermilab (E989) and planned at J-PARC (E34) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In 2021 the Fermilab experiment released first results [4] confirming the previously best result obtained by the BNL E821 experiment [5] and reducing the experimental uncertainty from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='54 ppm to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='46 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Over the next few years, the Fermilab experiment aims to reduce the uncertainty further to approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='14 ppm [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The Standard Model result provided by the Muon g-2 Theory Initiative [7–27] currently has an uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='37 ppm and is in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2σ tension with the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A further reduction of the theory uncertainty by at least a factor of two is therefore needed [28] to match the expected experimental progress over the next few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' More than 90% of the theory uncertainty is due to the leading-order hadronic vacuum polarization (HVP) contribution such that a reduction of its uncertainty is particularly pressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The leading-order HVP contribution aHVP LO µ can be related to e+e− decays using a dispersion relation such that, to the degree that there is no new physics in e+e− decays, it can be used to represent the Standard Model theory result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The Muon g-2 Theory Initiative result quoted above uses this method to determine the HVP contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' One can also relate the HVP contribution to hadronic τ decays, however, this requires precise first-principles knowledge of the needed isospin rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Our collaboration is working on such a calculation [29] and we will report on related progress in a separate publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finally, the HVP contribution can be computed from first principles using systematically improvable lattice QCD+QED methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='08696v1 [hep-lat] 20 Jan 2023 2 Until recently, lattice QCD+QED methods have not yet been competitive with the precision provided by the dispersive method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The BMW collaboration, however, has now produced a lattice QCD+QED result with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8% precision [30], which is close to the current 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6% precision of the dispersive method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The BMW value taken by itself only leads to a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5σ tension for aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' At the same time, the BMW value for the HVP contribution is in a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1σ tension with the dispersive result provided by the Muon g-2 Theory Initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In 2018, our collaboration introduced Euclidean window quantities [31], which allow for the separation of the most challenging short and long time-distance contributions to aHVP LO µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The remaining standard window quantity, aHVP LO W µ , is much easier to compute at high precision in lattice QCD+QED and can also be computed using the dispersive method [1, 30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The BMW collaboration’s calculation of aHVP LO W µ is in fact in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7σ tension with the dispersive result, which has motivated many lattice collaborations to focus on high-precision calculations of aHVP LO W µ first in order to clarify the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this work, we provide a significantly improved calculation of aHVP LO W µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We focus on the the quark-connected light-quark contribution in the isospin-symmetric limit, which accounts for almost 90% of aHVP LO W µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Special attention is given to the continuum limit for which we replace our previous continuum extrapolation based on a single approach using 2 lattice spacings with one based on 8 distinct approaches using 3 lattice spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We perform this update using a blinding procedure with five independent analysis groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This blinding procedure is implemented to avoid bias toward our previous computation of aHVP LO W µ in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31], the dispersive results, or other lattice results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II, we describe our methodology before giving computational details in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV, we discuss blinded results and explain convergence to the final prescription to determine aHVP LO W µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' V, we present unblinded results and compare them to other groups’ results, including data-driven ones, before concluding in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' METHODOLOGY We first define the time-momentum representation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II A, which provides the basis for the definition of the Euclidean windows in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II C we define the isospin-symmetric world around which we expand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Special care is taken such that the isospin-symmetric contribution can be compared directly with other lattice results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II D, we describe our blinding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Time-momentum representation Starting from the vector current Jµ(x) = i � f QfΨf(x)γµΨf(x) with fractional electric charge Qf and sum over quark flavors f we may write aHVP LO µ = ∞ � t=0 wtC(t) (1) with correlator C(t) = 1 3 � ⃗x � j=0,1,2 ⟨Jj(⃗x, t)Jj(0)⟩ , (2) where the weights wt capture the photon and muon part of the HVP diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A complete list of diagrams is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The weights can be expressed as a one-dimensional integral [33] wt = 8α2 � ∞ 0 dQ2 �cos (Qt) − 1 Q2 + 1 2 t2 � f(Q) (3) with f(Q) = m2 µQ2Z3(Q)(1 − Q2Z(Q)) 1 + m2µQ2Z2(Q) , Z(Q) = � Q4 + 4Q2m2µ − Q2 2m2µQ2 , (4) where mµ is the muon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Note that we sum only over non-negative t in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (1), yielding an additional symmetry factor of two in wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Using a lattice discretization for the photon momenta, an alternative weight ˆwt = 8α2 � ∞ 0 dQ2 �cos (Qt) − 1 (2 sin Q/2)2 + 1 2 t2 � f(Q) (5) 3 Diagrams Isospin limit QED corrections Strong isospin breaking Diagrams – Isospin limit 2 with C(t) = 1 3 P ~x P j=0,1,2hJj(~x, t)Jj(0)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' With appro- priate definition of wt, we can therefore write aµ = X t wtC(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (4) The correlator C(t) is computed in lattice QCD+QED with dynamical up, down, and strange quarks and non- degenerate up and down quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We compute the missing contributions to aµ from bottom quarks and from charm sea quarks in perturbative QCD [13] by integrating the time-like region above 2 GeV and find them to be smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3 ⇥ 10�10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We tune the bare up, down, and strange quark masses mup, mdown, and mstrange such that the ⇡0, ⇡+, K0, and K+ meson masses computed in our calculation agree with the respective experimental measurements [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The lat- tice spacing is determined by setting the �� mass to its experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We perform the calculation as a perturbation around an isospin-symmetric lattice QCD computation [15, 16] with two degenerate light quarks with mass mlight and a heavy quark with mass mheavy tuned to produce a pion mass of 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 MeV and a kaon mass of 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7 MeV [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The correlator is expanded in the fine-structure constant ↵ as well as �mup, down = mup, down � mlight, and �mstrange = mstrange � mheavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We write C(t) = C(0)(t) + ↵C(1) QED(t) + X f �mfC(1) �mf(t) + O(↵2, ↵�m, �m2) , (5) where C(0)(t) is obtained in the lattice QCD calculation at the isospin symmetric point and the expansion terms define the QED and strong isospin-breaking (SIB) correc- tions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We keep only the leading corrections in ↵ and �mf which is su�cient for the desired precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We insert the photon-quark vertices perturbatively with photons coupled to local lattice vector currents mul- tiplied by the renormalization factor ZV [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We use ZA � ZV for the charm [22] and QED corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The SIB correction is computed by inserting scalar operators in the respective quark lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The procedure used for e�ective masses in such a perturbative expansion is ex- plained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We use the finite-volume QEDL prescription [19] and remove the universal 1/L and 1/L2 corrections to the masses [20] with spatial lattice size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The e�ect of 1/L3 corrections is small compared to our statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We find �mup = �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='00050(1), �mdown = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='00050(1), and �mstrange = �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0002(2) for the 48I lattice ensemble described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The shift of the �� mass due to the QED correction is significantly smaller than the lattice spacing uncertainty and its e�ect on C(t) is therefore not included separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Figure 1 shows the quark-connected and quark- disconnected contributions to C(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Similarly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2 shows the relevant diagrams for the QED correction to FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Quark-connected (left) and quark-disconnected (right) diagram for the calculation of aHVP LO µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We do not draw gluons but consider each diagram to represent all orders in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='07 0 10 20 30 40 50 60 70 r Resulting two-point p(d) from p(r)=(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 + r)-5 Figure 6: Displacement probability for 48c run 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (a) V (b) S (c) T (d) D1 (e) D2 (f) F (g) D3 Figure 7: Mass-splitting and HVP 1-photon diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In the former the dots are meson operators, in the latter the dots are external photon vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Note that for the HVP some of them (such as F with no gluons between the two quark loops) are counted as HVP NLO instead of HVP LO QED corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We need to make sure not to double-count those, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=', we need to include the appropriate subtractions!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Also note that some diagrams are absent for flavor non-diagonal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' QED-correction diagrams with external pseudo-scalar or vector operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' the meson spectrum and the hadronic vacuum polariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The external vertices are pseudo-scalar operators for the former and vector operators for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We refer to diagrams S and V as the QED-connected and to diagram F as the QED-disconnected contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We note that only the parts of diagram F with additional gluons exchanged between the two quark loops contribute to aHVP LO µ as otherwise an internal cut through a single photon line is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For this reason, we subtract the separate quantum-averages of quark loops in diagram F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In the current calculation, we neglect diagrams T, D1, D2, and D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This approximation is estimated to yield an O(10%) correction for isospin splittings [21] for which the neglected diagrams are both SU(3) and 1/Nc suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the hadronic vacuum polarization the contribution of neglected diagrams is still 1/Nc suppressed and we adopt a corresponding 30% uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 3, we show the SIB diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In the calcu- x x x (a) M x x x (b) R x x x (c) O Figure 8: Mass-counterterm diagrams for mass-splitting and HVP 1-photon diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Diagram M gives the valence, diagram R the sea quark mass shift e�ects to the meson masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Diagram O would yield a correction to the HVP disconnected contribution (that likely is very small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Strong isospin-breaking correction diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The crosses denote the insertion of a scalar operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Diagrams – QED corrections and fit d�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' red For the finite-volume errors, the two-pion states in d are identical to the I = 1 contributions of c and can be calculated using the GSL estimate which we use for c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the omega-related finite-volume errors, I will take the fitted d� and E� and use this as the full result at finite-volume and compare it to a GS model with omega mass from the fitted E� and width from the PDG in infinite-volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I should also compare this to R-ratio results for the I = 0 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Do this entire exercise for 24ID and 32ID to estimate discretization errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 4 QED and SIB diagrams We will perform a full first-principles calculation of all O(↵) and O(mu � md) corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The corresponding list of diagrams is given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (a) V (b) S (c) T (d) Td (e) D1 (f) D1d (g) D2 (h) D2d (i) F (j) D3 Figure 1: QED corrections x x x (a) M x x x (b) R x (c) Rd x x x (d) O Figure 2: SIB corrections 4 Diagrams – Strong isospin breaking 8 / 20 and fit d�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' red For the finite-volume errors, the two-pion states in d are identical to the I = 1 contributions of c and can be calculated using the GSL estimate which we use for c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the omega-related finite-volume errors, I will take the fitted d� and E� and use this as the full result at finite-volume and compare it to a GS model with omega mass from the fitted E� and width from the PDG in infinite-volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I should also compare this to R-ratio results for the I = 0 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Do this entire exercise for 24ID and 32ID to estimate discretization errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 4 QED and SIB diagrams We will perform a full first-principles calculation of all O(↵) and O(mu � md) corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The corresponding list of diagrams is given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (a) V (b) S (c) T (d) Td (e) D1 (f) D1d (g) D2 (h) D2d (i) F (j) D3 Figure 1: QED corrections x x x (a) M x x x (b) R x (c) Rd x x x (d) O Figure 2: SIB corrections 4 and fit d�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' red For the finite-volume errors, the two-pion states in d are identical to the I = 1 contributions of c and can be calculated using the GSL estimate which we use for c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the omega-related finite-volume errors, I will take the fitted d� and E� and use this as the full result at finite-volume and compare it to a GS model with omega mass from the fitted E� and width from the PDG in infinite-volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I should also compare this to R-ratio results for the I = 0 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Do this entire exercise for 24ID and 32ID to estimate discretization errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 4 QED and SIB diagrams We will perform a full first-principles calculation of all O(↵) and O(mu � md) corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The corresponding list of diagrams is given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (a) V (b) S (c) T (d) Td (e) D1 (f) D1d (g) D2 (h) D2d (i) F (j) D3 Figure 1: QED corrections x x x (a) M x x x (b) R x (c) Rd x x x (d) O Figure 2: SIB corrections 4 3 / 25 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The diagrams of a complete calculation of aHVP LO µ when formulated as an expansion around an isospin-symmetric limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In the isospin-symmetric limit, there is a quark-connected (left) and quark-disconnected contribution (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the QED- and strong-isospin-breaking (SIB) corrections, we indicate the photon vertices that connect to the muon with filled dots and only show the respective sub-diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the QED corrections, one has to enforce the exchange of gluons between the quark loops in diagram F to avoid double-counting of higher-order HVP contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the SIB corrections, the crosses denote scalar operator insertions to allow for a linear correction in the respective quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' can be defined, which gives the same value of aHVP LO µ in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We use both versions to scrutinize the continuum extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The correlator C(t) is computed in lattice QCD+QED at physical pion mass with non-degenerate up- and down- quark masses including up-, down-, strange-, and charm-quark contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The missing bottom-quark contributions are estimated using perturbative QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Euclidean windows In the following, we suppress the leading-order HVP LO label for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Following [31], we define Euclidean windows that partition the contributions of time-slices t in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (1) into short-distance (SD), window (W), and long- distance (LD) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To make the quantities well-defined at non-zero lattice spacing, we introduce smearing kernels with width ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We write aµ = aSD µ + aW µ + aLD µ , (6) 4 where aSD µ (t0, ∆) = ∞ � t=0 C(t)wt[1 − Θ(t, t0, ∆)] , (7) aW µ (t0, t1, ∆) = ∞ � t=0 C(t)wt[Θ(t, t0, ∆) − Θ(t, t1, ∆)] , (8) aLD µ (t1, ∆) = ∞ � t=0 C(t)wtΘ(t, t1, ∆) , (9) Θ(t, t′, ∆) = [1 + tanh [(t − t′)/∆]] /2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (10) All contributions are well-defined individually and can be computed using lattice methods as well as dispersive methods by relating the correlator C(t) = 1 12π2 � ∞ 0 d(√s)R(s)se−√st (11) to the R-ratio R(s) = 3s 4πα2 σ(s, e+e− → had).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (12) Within a lattice calculation, discretization effects are most severe for the SD contribution, while statistical noise and finite-volume effects are most pronounced in the LD contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The window quantity aW µ has small statistical and systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' As recently argued in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [1], the systematic study of window quantities aW µ (t0, t1, ∆) as a function of t0 and t1 is useful to constrain energy regions within the R-ratio contributing to a possible tension between lattice and dispersive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' First lattice results with a high resolution in t0 and t1 are already available [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Windows with larger values of t0 and t1 are more sensitive to low-energy states and are useful for checking effective field theory as argued in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A systematic study of the short-distance window aSD µ (t0, ∆) as a function of t0 is also useful as argued in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [36], where the aSD µ (t0, ∆) defined as above are called one-sided windows since 1−Θ(t, t0, ∆) = [1 − tanh [(t − t0)/∆]] /2 = Θ(t0, t, ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In the current work, we focus on the short-distance and window contributions for the standard values of t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, t1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 fm, and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Isospin-symmetric world It is convenient to perform the calculation as an expansion around an isospin-symmetric point [31, 37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We therefore compute the diagrams of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 1 individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The exact choice of the expansion point is inconsequential for the total aµ, however, care is needed if one attempts to compare isospin-symmetric results provided by different groups [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this work, we present results for two choices of the isospin-symmetric world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The first choice is the RBC/UKQCD18 world defined by mπ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='135 GeV , mK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4957 GeV , mΩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='67225 GeV , (13) consistent with our previous work [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this update, we also consider the effects from dynamical sea-charm quarks from first principles and therefore extend this choice by mDs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='96847 GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (14) Since one of the main goals of this work is to scrutinize the result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [30], we also consider a second choice mπ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='13497 GeV , mss∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6898 GeV , w0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='17236 fm , (15) which we label as the BMW20 world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The quantity mss∗ is obtained from the ground-state energy of the quark- connected pseudoscalar ¯ss meson two-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This choice is consistent with the isospin-symmetric world defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the sea-charm study, we adopt Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (14) also in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 5 ID a−1/GeV Nf L3 × T × Ls/a4 b + c amres × 104 mπ/MeV mK/MeV mDs/GeV mπL 48I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7312(28) 2+1 483 × 96 × 24 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='32(30) 499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='44(88) – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='9 64I 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3549(49) 2+1 643 × 128 × 12 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='98(43) 507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5) – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8 96I 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6920(67) 2+1 963 × 192 × 12 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='29(66) 484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3) – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7310(35) 2+1 323 × 64 × 24 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1) 514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8) – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7257(74) 2+1 243 × 48 × 32 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='9) 537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6) – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7306(46) 2+1 323 × 64 × 24 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3) 603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1) – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='9 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7400(73) 2+1 243 × 48 × 24 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5) 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1) – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7498(73) 2+1+1 243 × 48 × 24 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5) 539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='9902(69) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7566(81) 2+1+1 243 × 48 × 24 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='9 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='9) 523(10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3882(57) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7556(83) 2+1 243 × 48 × 8 2 42 307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5) 557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7) – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 24ID 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0230(20) 2+1 243 × 64 × 24 4 23 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='96(30) 515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0) – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 32ID 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0230(20) 2+1 323 × 64 × 24 4 23 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='96(30) 515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0) – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' List of ensembles with parameters determined in the RBC/UKQCD18 isospin symmetric world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Unless specified otherwise, the ensembles have Iwasaki gauge action and M¨obius [42] domain-wall [43, 44] fermion sea quarks with b − c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The parameters b and c are defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the Nf = 2 + 1 + 1 ensembles, the charm quarks couple to three-times ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 stout smeared gauge fields as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The scripts generating the new ensembles are publicly available [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The 24ID and 32ID ensembles have an additional DSDR term [41] in the gauge action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The 24ID and 32ID ensemble parameters are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We define these parameters to the exact values given above without additional uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This avoids an unnec- essary inflation of uncertainties when comparing isospin-symmetric lattice results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The experimental uncertainties of the physical hadron spectrum are then taken into account when applying the isospin-breaking corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To support the careful tuning of the isospin-symmetric world, we generated additional near-physical-pion-mass ensembles allowing for the explicit calculation of light and strange quark-mass derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Our choice of discretisation and simulation parameters is summarised in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We also generated ensembles with dynamical charm quarks and ensembles with varying extent of the fifth dimension of our domain-wall fermions, Ls, to control for residual chiral- symmetry-breaking effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finally, we include results at physical pion mass and a finer lattice spacing of a−1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We determined the ensemble parameters in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' First, we used the new ensembles to obtain the quark-mass dependence of the quantities defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (13) and (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We then tune the dimensionless mπ/mΩ and mK/mΩ for the RBC/UKQCD18 world and w0mπ and w0mss∗ for the BMW20 world to the values provided in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (13) and (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Any of the three dimensionful values can then equivalently be used to determine the lattice spacing a for a given ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the Nf = 2 + 1 + 1 ensembles, we also tune mDs/mΩ for the RBC/UKQCD18 world and w0mDs for the BMW20 world to the value provided in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We provide the results for the RBC/UKQCD18 world in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In addition, we also performed an update of our global fit [41] for which we found consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A detailed discussion of the updated global fit will be published separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The two determinations of ensemble parameters were performed by disjoint sub-groups of authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Blinding procedure Since we provide an update of a previous result [31] compared to which a lower value would mean agreement with the dispersive method and a higher value would mean agreement with the lattice result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [30], two values that are in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7σ tension with each other, we believe it is crucial to perform this update in a blinded manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We implement the blinding by creating modified correlators Cb(t) from the unaltered correlators C0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For each lattice ensemble, we use Cb(t) = (b0 + b1a2 + b2a4)C0(t) (16) with respective lattice spacing a and random coefficients b0, b1, and b2 that are common for each ensemble but different for each analysis group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The parameter b0 is drawn from a Gaussian distribution with mean µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 and standard deviation σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The dimensionful parameters b1 and b2 are drawn from a flat distribution with maximum values of |b1a2| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='05 and |b2a4| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0025 for our coarsest lattice cutoff a−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='73 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This procedure based on three random numbers per analysis group prevents the possibility of complete unblinding based on previously shared data on the coarser two ensembles [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The blinding factors were generated and directly applied to C0 by author CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This 6 process took a given seed for the random number generator as input such that only this seed and not the blinding factors were directly accessible to CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the current update, we established five analysis groups (called A–E in the following), composed of non- overlapping sub-groups of authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The different analysis groups were provided with the ensemble parameters and the respectively blinded correlator data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' They then separately decided on their respective analysis procedures without interacting with other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The chosen methods are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' After the groups completed their analyses, we started a relative unblinding procedure during which two groups would jointly discuss and scrutinize their approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this process some important findings emerged, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Based on these discus- sions, the collaboration then converged on a preferred prescription that is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' At this point the prescription was frozen and a complete unblinding performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The results are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' COMPUTATIONAL DETAILS In the following, we describe in detail the computational methods used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We explain aspects of data generation as well as crucial components of the various aµ analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Overview of improvements Compared to our previous calculation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31], we have made several substantial improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' With regard to the statistical uncertainty, we increased the statistical sample size for the correlators on ensembles 48I and 64I by a factor of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Improvements reducing systematic uncertainties are described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To improve the continuum extrapolation, we add a finer lattice spacing at physical pion mass with a−1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We also consider an additional discretization for the vector current by studying both local-conserved as well as local-local correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This can be done in a cost-efficient manner as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In addition, we use two different renormalization procedures for the local vector current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The first procedure, which we label ZV , follows Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [41] and uses that the expectation value of the charge operator in a pion state equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The second procedure, which we label Z⋆ V , uses the ratio of local-conserved to local-local correlators interpolated to fixed Euclidean time t⋆ to define the current normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The particular choice of t⋆ is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finally, we use two different weight functions wt and ˆwt, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (3) and (5), at a given lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This gives a total of 3×2×2×2 = 24 data points to study the continuum extrapolation, which improves our previous extrapolation based on two data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To reduce parametric uncertainties, we generated new near-physical pion- and kaon-mass ensembles to calculate parametric derivatives with respect to quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III E, we also show how to obtain parametric derivatives inspired by master-field methodology [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We previously estimated the missing sea-charm effects using perturbative QCD [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For this update, we have generated new ensembles with dynamical charm quarks, which we match to our Nf = 2 + 1 ensembles as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Domain-wall fermions exhibit only small chiral symmetry breaking which is commonly quantified using the residual mass mres [44, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For this reason, a very small linear discretization error is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We previously neglected such effects but have now generated new ensembles with different extents of the fifth dimension Ls to quantify them from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Since we only have a small number of configurations for the new 96I ensemble, we also investigate a new five- dimensional master-field statistical error estimate in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III D to considerably reduce the uncertainty on our estimate of statistical variance concerning this ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Local- and conserved-current correlators In addition to the local lattice vector current Jµ, which we denote in the following as Jl µ, we consider the conserved lattice vector current Jc µ as defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We consider the correlators Cab(t) = 1 3 � ⃗x � j=0,1,2 ⟨Jb j (⃗x, t)Ja j (0)⟩ (17) in the local-local (Cll) and local-conserved (Clc) versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' After performing the fermionic Wick contraction, the source is always local and the sink varies between local and conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The contraction code is publicly available [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' It uses an all-mode-averaging procedure [52–55] combined with additional averaging of the low-low component of the 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8 1 0 5 10 15 20 t / a Clc(t) / Cll(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Ratio Clc(t)/Cll(t) as a function of Euclidean time t on the 96I ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' correlator [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Our approach again relies on approximating the low-mode space on a coarse grid as introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the 96I ensemble, this yields a reduction of data volume by a factor of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This is crucial not just for data storage but also for the computational performance of low-mode estimates due to the reduced memory-transfer requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For a given point source, the local-local and local-conserved correlators are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We therefore compute the ratio Clc/Cll using only a few correlated source positions and multiply this ratio with our full-statistics estimator of Cll to obtain our estimator for Clc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2, we plot the ratio for the 96I ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the 96I ensemble an additional improvement was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For this ensemble, we generate a data set in which two source positions at time-slice t and t + 96 are combined with a Z2 number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For short and intermediate distances, this effectively doubles our statistics at the same cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A second lower-statistics single time-slice data set is provided to account for the effects of the backwards propagation of the additional time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finally, all correlators are provided with identical valence- and sea-quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this manner, we can perform a purely unitary data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the 64I ensemble, however, for historical reasons the eigenvectors were generated for a partially-quenched mass am = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006203 instead of the unitary mass am = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006780 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For this reason, a small additional correlated data set was generated at am = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='001774 such that the unitary correlators can be obtained by aµ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006780) = aµ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006203) + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006780 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006203)aµ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='001774) − aµ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006203) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='001774 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006203 ≈ aµ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006203) + aµ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='001774) − aµ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0006203) 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (18) Non-linear effects in the small quark-mass shift are negligible for the precision goals of the present calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Sea-charm effects In this work, we estimate the effects of sea-charm quarks from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Most of our ensembles have Nf = 2+1 sea quarks with an isospin-symmetric up- and down-quark pair and an additional strange quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To study the sea- charm effects from first principles, we have generated additional Nf = 2+1+1 ensembles with different charm masses to separate the physical effects from a modification of discretization errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We list the ensemble parameters in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We match the Nf = 2+1 and Nf = 2+1+1 ensembles to the same pion and kaon masses and the Wilson-flowed [57] energy density at long-distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 3, we show tfE(tf) with flow-time tf and Wilson-flowed energy density E(tf) for the nominal ensemble 4, 5, and 7 of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' At shorter distances, we observe a clear signal of charm effects in the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the lighter charm mass, this effect extends to longer distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We plot tfE(tf) instead of the dimensionless t2 fE(tf) since all plotted ensembles share the same lattice spacing and the interesting features are better highlighted in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 3 a2 tf E(tf) tf / a2 Nf=2+1 Nf=2+1+1, mDs=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 GeV Nf=2+1+1, mDs=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Wilson-flowed energy density E(tf) multiplied with the flow-time tf for Nf = 2 + 1 and Nf = 2 + 1 + 1 ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The small statistical uncertainties for each line are shown as an error band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We use these matched ensembles to measure the sea-charm contributions to the HVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We do this in particular for the short-distance window, where most of the effect should appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The exact approach used by the different analysis groups is explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Five-dimensional master-field statistical errors For the 96I ensemble, we currently only have 33 gauge field configurations in contrast to the 64I and 48I ensembles for which we have 238 and 386 gauge field configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In order to obtain a reliable statistical-error estimate on the 96I ensemble, we have performed a slightly modified master-field error analysis [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In our approach, we improve the covariance estimate by considering a five-dimensional master field with Markov time as an additional fifth dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We expect exponential locality in the fifth dimension governed by the eigen-modes of the Markov transition matrix and in the four space-time dimensions governed by the eigen-modes of the QCD Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For an observable Oτ,x with Markov time τ and space-time coordinate x, we consider the statistical average O = 1 |V| � (τ,x)∈V Oτ,x (19) with set V that contains all tuples (τ, x) for which the observable was determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Note that we explicitly allow for sparse sampling in space-time as well as Markov time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The covariance of two such observables O and O′ is then given by Covτc,xc(O, O′) ≡ 1 |V||V′| � (τ,x)∈V,(τ′,x′)∈V′, |x−x′|≤xc,|τ−τ ′|≤τc � ⟨Oτ,xO′ τ ′,x′⟩ − ⟨Oτ,x⟩⟨O′ τ ′,x′⟩ � (20) and studying Covτc,xc(O, O′) as a function of τc and xc to identify a plateau for large τc and xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In practice, we estimate Covτc,xc(O, O′) based on a given set of gauge configurations, which adds an error suppressed by the inverse square root of the number of sampled five-dimensional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In comparison, the Jackknife estimator has an uncertainty suppressed by the inverse square root of the number of gauge configurations, such that its uncertainty is generally much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The distance |x − x′| takes the field boundary conditions into account, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=', for periodic boundary conditions, we consider the shortest distance between mirror images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For arbitrarily sparse V, the various Oτ,x are effectively all statistically independent such that we expect a plateau already for very small τc and xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In general, just before reaching the gauge noise limit, the plateaus still start early in xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Conversely, a rising behavior in xc signals that our sample points are significantly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We tune the sampling of our vector correlators to be such that we almost reach the gauge noise limit and therefore plateaus 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 2 0 5 10 15 20 25 c Covτc,xc(C(t),C(t))1/2 xc / a t/a=8 t/a=10 t/a=12 t/a=16 t/a=20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The statistical uncertainty of C(t) determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (20) multiplied with a blinding factor c determined by the five-dimensional master-field approach (individual data points) compared to the Jacknife estimate (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For these estimates, we use randomly selected 660 point sources per 33 configurations on the 96I ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Due to the sparseness of our measurement setup, we observe a plateau in xc starting essentially from the smallest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The plot is made after having established a plateau in τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' are reached for modest values of xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 4, we compare the statistical uncertainty of C(t) on the 96I ensemble determined by the five-dimensional master-field approach to the Jackknife estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Master-field parametric derivatives In order to tune the Nf = 2 + 1 + 1 ensembles described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III C, we found the master-field formalism useful to get initial estimates of parametric derivatives with respect to the gauge-action parameter β as well as the sea-charm mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To simplify the discussion, we set a = 1 in this sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Consider a general gauge action S = −β Nd(Nd − 1) 2 � x Ax (21) with space-time dimension Nd and field of Wilson loops Ax anchored at a point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' It is not crucial how we exactly identify the location x as long as the coordinate behaves properly under translations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' One can then show that for a general observable O in Nd = 4 without explicit β dependence, ∂β⟨O⟩ = 6 lim xc→∞ Cov0,xc(O, A) , (22) with Cov0,xc defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Setting O to the Wilson-flowed energy density E(tf), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=', allows us to determine the β-derivative of the Wilson-flow scales t0 and w0 [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We can also show that ∂m⟨O⟩ = lim xc→∞ Cov0,xc(O, Tr[ ˜D−1 ov (m)]) , (23) for sea-quark mass m and ˜D−1 ov (m) = 1 1 − m � D−1 ov (m) − 1 � , (24) with overlap operator Dov [42, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We find that the traces of ˜D−1 ov (m) can be efficiently estimated using our tadpole field approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For domain-wall fermions, an additional flavor enters the path integral as the determinant ratio det(D(m)D−1(1)) (25) 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0025 0 5 10 15 20 25 30 xc / a Cov0,xc(E(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01),Tr[D~-1 ov(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8)]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='005 0 0 5 10 15 20 25 30 xc / a Cov0,xc(E(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01),A) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We plot for the 96I ensemble Cov0,xc(E(tf), Tr[ ˜D−1 ov (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8)]) on the left and Cov0,xc(E(tf), A) on the right for tf = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 ≈ t0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The Wilson-loop field A is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' with five-dimensional Dirac operator D(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For m = 1 this factor is trivial and we can view including an additional flavor as changing the sea-quark mass down from m = 1 to the target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this way integrating the parametric derivative with respect to m allows us to determine the effects of introducing an additional sea-charm quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Setting O to the Wilson-flowed energy density, allows us to determine the effect of the additional sea-charm quark to the Wilson-flow scales t0 and w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 5, we show the convergence as a function of xc for the β derivative as well as the charm-quark mass derivative at m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8 of E(tf) with tf = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 ≈ t0/2 on the 96I ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The lower scale t0/2 allows for a statistically more precise estimate of the dependence of the lattice spacing on β and the charm-quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finite-volume effects In order to determine the finite-volume effects on C(t), the analysis groups explored two methods: a direct fit to the 24ID and 32ID data as well as the Hansen-Patella approach [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Details of the former approach are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the latter approach, we use a monopole ansatz of the electromagnetic pion form factor F(k2) = 1 1 − k2/m2ρ (26) and study the dependence on mρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For this ansatz Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [62] gives an expression for the finite-volume corrections for C(t) in terms of a simple integral CL(t) − C∞(t) = � ⃗n̸=⃗0 1 6π|⃗n|L � Im � R+iµ dk3 2π eik3|x0|(4m2 π + k2 3)m4 ρ (m2ρ + k2 3)2 e−|⃗n|L � m2π+ k2 3 4 4k3 + � dp3 2π e−|⃗n|L√ m2π+p2 3 d dz � e−z|x0|(z2 − 4m2 π)m4 ρ (z + mρ)2(z2 + 4p2 3) � z=mρ � , (27) where CL is the correlator at finite spatial volume L3 and C∞ is the infinite-volume version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The equation depends on the pion mass mπ and the monopole-mass parameter mρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The complex shift iµ of the integration contour has to be chosen in the range 0 < µ < 2mπ, however, the integral does not depend on the exact choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Equation (27) only considers the pole contribution to the Compton amplitude and neglects terms of order e−√ 2+ √ 3mπL as well as effects of finite Euclidean time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This is well justified for our current precision goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The effects of the regular contribution to the Compton amplitude and effects of the finite Euclidean time extent are known [61, 62] and may be considered in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Note that the finite-volume corrections for the quark-connected diagram are 10 9 of the total as is easily seen from the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Consider a theory with quark charges Qu = 1 2 = −Qd instead of the physical Qu = 2 3 = −2Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The QED charges of mesons made of up and down quarks are identical in both cases, however, in the Qu = 1 2 = −Qd 11 theory the quark-disconnected diagram does not contribute, while the quark-connected diagram contributes with a Q2 u + Q2 d = 1 2 factor instead of the physical Q2 u + Q2 d = 5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We therefore find that 1 2 9 5 = 9 10 of the quark-connected contribution is equal to the total contribution and equivalently that the total correction needs to be multiplied by 10 9 to obtain the correction for the quark-connected piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This simple argument is consistent with partially quenched Chiral Perturbation Theory studies [32, 34, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' RELATIVE UNBLINDING In the following, we summarize the different approaches of the five analysis groups and show the result of our relative unblinding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We highlight important findings and explain the prescription that all five groups agreed to be used for the full unblinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Distinct methods of the five analysis groups Each analysis group received the blinded correlator data as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The separate analysis groups then discussed the data and agreed on the respective analysis methods within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The confinement of these discussions to the separate groups lead to a diverse set of approaches to the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In the following sub-sections, we briefly describe the approaches of each group, focusing on the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Group A Analysis group A provides results for aW µ as well as aSD µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Statistical errors are obtained from a super-jackknife procedure [64, 65] for most ensembles combined with a binning study and using the master-field error estimates of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III D on ensemble 96I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The continuum extrapolations are performed based on the 24 data points over three lattice spacings described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III A, where small linear corrections to shift the individual points to the lines of constant physics (LCP) are applied first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finite-volume corrections are also applied before the continuum extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To this end, the Hansen-Patella Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (27) is used for finite-volume corrections with nominal parameters mρ = 727 MeV and errors estimated from the variation to mρ = 770 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' An additional ad-hoc 20% uncertainty is added to the finite-volume corrections to account for the limitations discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Combinations of the fit ansaetze f2(a2) = c0 + c1a2 , (28) f2,4(a2) = c0 + c1a2 + c2a4 , (29) f2α(a2) = c0 + c1a2αs(µ = 1/a) , (30) f2α,4(a2) = c0 + c1a2αs(µ = 1/a) + c2a4 (31) are then considered with four-loop running coupling αs in the MS scheme [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For aW µ , the central value is chosen as the average of the f2 fits to the (ωt, Clc, Z⋆ V ), (ωt, Cll, Z⋆ V ), (ωt, Clc, ZV ) trajectories with t⋆ = 1 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' These trajectories had the smallest a4 contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For aW µ , the effect of ωt compared to ˆωt is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The continuum extrapolation error is estimated by varying f2 to f2α and by considering the spread of the mean to the individual (ωt, Clc, ZV ) and (ωt, Cll, Z⋆ V ) fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For aSD µ , the fit form f2,4 is used for all trajectories and the average of (ˆωt, Clc, ZV ) and (ˆωt, Clc, Z⋆ V ) is used for the central value since they exhibit the smallest a4 coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The variation from f2,4 to f2α,4 as well as the maximal variation to (ˆωt, Cll, ZV ), (ωt, Clc, ZV ), (ˆωt, Clc, ZV ), (ˆωt, Cll, Z⋆ V ), (ωt, Clc, Z⋆ V ), and (ˆωt, Clc, Z⋆ V ) is then used for the continuum extrapolation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The effects of the residual mass and the sea-charm quark are studied separately and found to be small compared to the quoted uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Group B Analysis group B provides results for aW µ as well as aSD µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The strategy is to employ a global fit to all of the measurements on the ensembles listed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Statistical errors for each measurement, including lattice spacings, pion masses, and so on, are incorporated through a super-jackknife method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 12 Several terms comprise the global fit function for the intermediate window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A second-order polynomial in a2 is used to extrapolate non-zero lattice spacing to the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finite-volume effects are treated explicitly through a term exponential in mπL and are mainly constrained by the two Iwasaki-DSDR ensembles in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Small light- quark-mass mistunings are treated linearly in the appropriate meson-mass squared and a simple linear ansatz for the residual mass is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Charm-quark mistunings are corrected with inverse mass-squared of the Ds meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' All together, the fit function takes the form aµ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=') = aµ � 1 + c1a2 + c2a4� � 1 + c3e−mπL� � 1 + c4(m2 π − m2 π,phys) � � 1 + c5(m2 K − m2 K,phys) � × (1 + c6amres) � 1 + c7 � 1 m2 Ds − 1 m2 Ds,phys �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (32) The coefficients c1 and c2 take on different values for the Iwasaki-DSDR ensembles, and the residual mass term is treated as an O(a) artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To fit the data to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (32), the (log of) C(t) is first cubically interpolated between time-slices and then integrated with the continuum form of the one-loop QED kernel, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The central value of the procedure is determined from the average of conserved-local and local-local correlation functions for the HVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The main part of the systematic error arises from the difference of these two results in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the short-distance window, the procedure is similar except that the discrete version of the one-loop kernel ˆωt is also used (approximated as wt(1−a2/t2)) and an a2 log a2 term is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The systematic error is computed from differences between pairwise combinations of a2, a4 and a2 log a2 terms, using both wt and ˆwt weights, all added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The central value is taken as the wt version with the conserved-local correlation function since empirically it has the smallest a4 contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Group C Analysis group C provides results for aW µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The strategy is divided in a few steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' First, using the ensembles listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' I the derivatives of the intermediate window with respect to the quark masses are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Additional cutoff or finite-volume effects on the derivatives are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The derivatives are then used to shift the three reference ensembles, 48I, 64I and 96I, to the LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Additionally, all windows are shifted to mπL = 4 using Chiral Perturbation Theory and additional systematic effects are not considered since they are well below the statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' After multiplying by the normalization factors ZV or Z⋆ V , the intermediate windows from the 3 ensembles and 2 discretizations (Cll and Clc) are extrapolated to the continuum limit with a constrained fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Note that also a2/t0 used in the extrapolation is shifted to the proper LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The following three types of fits are considered: linear and quadratic in a2 with all 6 data points and linear in a2 with the finest 4 data points (96I, 64I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A systematic error from the spread of the central values of the fitted continuum windows is included in the error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Both correlated and uncorrelated fits are used, and for the latter their quality is assessed using the method developed in Ref [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The 3 fits described above are performed separately using ZV and a variant of Z⋆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the former it is observed that the linear fit in a2 is not acceptable, and that a quadratic term is necessary to describe the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Hence, the preferred strategy is based on Z⋆ V and the preferred fit is the constrained linear fit to all 6 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the variant of Z⋆ V , a slight modification of the definition provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III A is considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=', the ratio of Clc over Cll is used individually integrated using the smearing function Θ(t, t⋆ − ∆/2, ∆)Θ(t⋆ + ∆/2, t, ∆) with ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Several values of t⋆ are explored and for the final analysis t⋆ = 1 fm is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' No particular difference is observed with respect to the interpolation described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' III A, as one can easily infer from the long plateau in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The statistical analysis is carried out by propagating all fluctuations of observables using both the Jackknife method and the Γ-method [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' No large autocorrelations in the extrapolated continuum window are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finite-volume effects to correct from mπL = 4 to ∞ are obtained from an independent implementation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Final shifts for residual mass effects and dynamical charm effects are applied in the same manner as also done by group B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Group D Analysis group D provides results for aW µ from the physical pion-mass ensembles 48I, 64I, and 96I, which are computed with a binned super-jackknife analysis with weight function wt and vector current normalizations ZV and Z⋆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In addition, a version of ZV is used, where the pion state is replaced by a kaon state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The mass extrapolation to the physical point is done by assuming linear dependence on the quark masses taken from ensemble 1 with 4 and ensemble 1 with 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finite-Ls effects are corrected by assuming linearity in mres using ensembles 1, 2, 13 4, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The values of aW µ on the 48I and 64I ensembles are corrected by an exponential dependence to the lattice extent, exp(−mπL), whose coefficient is taken from the 24ID and 32ID ensembles, to match for the volume of 96I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A 50% systematic uncertainty for these finite-volume corrections is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' It is noted that within the statistical noise of the 24ID and 32ID ensembles, their difference is reproduced by the Hansen-Patella finite-volume formula as well as the Meyer-Lellouch-L¨uscher-Gounaris-Sakurai [69–71] approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' After these corrections for 18 data points from three ensembles, two vector currents Cll and Clc, and three vector current normalizations, the continuum extrapolation is performed by combinations of the fit formulae f2(a2), f2,4(a2), f2α(a2), and f2α,4(a2) by requiring a universal continuum limit for all 18 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' f2(a2) poorly fits Cll(t) with the coarsest ensemble 48I, and it is decided to drop this combination from the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In analysis group D, the central value for the continuum extrapolation is chosen from fit f2(a2) to Cll(t) and f2α(a2) to Clc(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The error of the continuum extrapolation is determined to cover all central values of the considered fit forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The continuum extrapolation for each of the 6 individual combination of currents and normalizations is also performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The results are consistent with that of the universal fit except, again, the f2(a2) fit for Cll(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finally, a small volume correction from the 96I volume to infinity is carried out using the Meyer-Lellouch-L¨uscher-Gounaris-Sakurai approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For each of the isospin-symmetric worlds, RBC/UKQCD18 and BMW20, the lattice spacing is determined in two different scaling trajectories (either keeping w0 or mΩ fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The fit results are consistent between the two scaling trajectories, providing an additional check for the continuum extrapolation of aW µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Group E Analysis group E provides results for aW µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The strategy is entirely data driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Statistical uncertainties are determined from a bootstrap analysis with measurements within 20 MD units binned into an effective measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The input uncertainties are propagated via re-sampling (Gaussian error propagation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Both ωt and ˆωt kernels are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In addition to ZV a variant of Z⋆ V is used that for a given window is defined as ZC V = alc,bare µ all,bare µ , (33) where aab,bare µ is obtained without vector-current normalization factors from the bare correlators Cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' When referring to aZ,K µ below, all,bare µ is normalized using two powers of ZV or two powers of ZC V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The chiral, strange-quark, discretization, and finite-volume effects are fitted to all ensembles for a given choice of renormalization procedure and kernel to the ansatz aZ,K µ = aphys µ × � 1 + Cχ (m2 π − (m2 π) phys) (m2π)phys � × � 1 + Cs (Xs − Xphys s ) Xphys s � (34) × � 1 + CV e−mπL� × � 1 + CZ,K CL,0(aΛ)2 + CZ,K CL,1(aΛ)4� × � 1 + CZ,K 5 amres � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (35) In this formula Xs stands for mK for the RBC/UKQCD18 world and for mss⋆ for the BMW20 world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The ratios RZ,K Z′,K′ on the three physical point Iwasaki ensembles are simultaneously fitted to the model fR, RZ,K Z′,K′ ≡ aZ,K µ aZ′,K′ µ , fR ≡ 1 + CZ,K CL,0(aΛ)2 + CZ,K CL,1(aΛ)4 1 + CZ′,K′ CL,0 (aΛ)2 + CZ′,K′ CL,1 (aΛ)4 (36) and the ratio RID V for the ensembles 32ID and 24ID to the model gV , RID V ≡ a32ID µ a24ID µ , gV ≡ 1 + CV e−(mπL)32ID 1 + CV e−(mπL)24ID .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (37) All correlations between data points on the same ensembles are included in this fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Systematic uncertainties are estimated by variations on the data that enters the fit and/or the terms included in the model(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Comparison of results After the analysis groups had individually converged on their respective methodology described above, we started the process of relative unblinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The relative unblinding of groups X and Y was conducted by sharing the individually 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='995 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='015 A B C D E aµ W(t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, t1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 fm, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='04 A B aµ SD(t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Result of the relative unblinding procedure for aW µ (left) and aSD µ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The results are normalized to the preferred prescription described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The inner error bars show the statistical uncertainty, the outer error bars show the statistical and systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 t / fm t3 C(t) in O(α4) massless perturbative QCD t3 Clc(t) on 48I with a-1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='73 GeV t3 Clc(t) on 64I with a-1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='35 GeV t3 Clc(t) on 96I with a-1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='68 GeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 t / fm t3 C(t) in O(α4) massless perturbative QCD t3 Cll(t) on 48I with a-1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='73 GeV t3 Cll(t) on 64I with a-1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='35 GeV t3 Cll(t) on 96I with a-1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='68 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The dimensionless correlation function combinations t3Clc(t) (left) and t3Cll(t) (right) as well as the perturbative result obtained from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' blinded data sets of group X with group Y and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' One of the groups then re-ran their analysis without modifications on the other data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This allowed for a direct comparison of groups X to Y while still keeping the absolute blinding intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 6, we show the final result of the relative unblinding procedure for aW µ , for which all five groups participated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The inner error bars give the statistical uncertainty, the outer error bars give statistical and systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We first note that the statistical uncertainties quoted by the separate analysis groups are consis- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In addition, the different systematic approaches described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV A yield different systematic uncertainties, however, all results are consistent within total uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The blinding procedure described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II D allows the a4 term to affect the comparison at the level of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0025 if the a4 terms are not included in the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This effect is small compared to the quoted uncertainties and is completely eliminated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' V, where we show the results of all groups after they repeated their unmodified analysis with the fully unblinded data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Important findings After the relative unblinding process, the analysis groups exchanged their most important findings for our data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We discuss these findings in this sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' They form the basis, determined entirely on blinded data, of formulating the preferred prescription to produce the combined collaboration result described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='995 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='015 A B C D E RBC/UKQCD 23 aµ W(t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, t1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 fm, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Result of the relative unblinding procedure for aW µ inlcuding the preferred prescription RBC/UKQCD 23 described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The data is normalized to the RBC/UKQCD 23 prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The inner error bars show the statistical uncertainty, the outer error bars show the statistical and systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finding 1: The correlator Cll has significantly larger a2/t2 and a4/t4 errors compared to Clc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' These errors also noticeably affect aW µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 7, we plot the dimensionless t3C(t) to highlight this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finding 2: Mean-field improved lattice perturbation theory finds the discretization errors of Cll to be approximately double the discretization errors of Clc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finding 3: When analyzing aSD µ , where both a2 and a4 coefficients were determined, the size of the a4 coefficient is substantially larger for Cll compared to Clc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finding 4: The continuum extrapolation is sensitive to how finite-volume corrections are applied to the individual ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This is an important effect in our analyses since the new finest 96I ensemble has a larger physical volume compared to the 64I and 48I ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Preferred prescription Based on the findings outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV C, the collaboration decided on the following principles for the combined analysis that will be used for the full unblinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' First, when using Cll, we always add a a4 term to the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Second, we use the Hansen-Patella finite-volume corrections instead of the data-driven fits to e−mπL since we expect the Hansen-Patella formalism to more precisely map out the volume dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' These principles are then implemented in the following prescription for aW µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the vector current renormalization factor, we use ZV as well as Z⋆ V with t⋆ = 1 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the weight functions we use ˆwt as well as wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the continuum extrapolation, we perform a simultaneous fit to the Cll and Clc data sets using fll(a2) = c0 + c1a2 + c2a4 , (38) flc(a2) = c0 + c3a2 (39) as well as fll,α(a2) = c0 + c1a2αs(µ = 1/a) + c2a4 , (40) flc,α(a2) = c0 + c3a2αs(µ = 1/a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (41) We therefore perform 8 fits in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We take the average of the minimum and maximum result as the central value for our prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We take the difference of the central value to the maximum as our systematic error for the continuum extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 8, we show the final result of the relative unblinding for each group as well as the preferred prescription, labelled RBC/UKQCD 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For aSD µ the results of groups A and B were close to identical and we adopt the prescription of group A as the preferred result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 16 RBC/UKQCD 2023 ETMC 2022 Mainz 2022 ChiQCD 2022 OV/HISQ ChiQCD 2022 OV/DWF Aubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2022 LM 2020 BMW 2020 ETMC 2021 Aubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2019 RBC/UKQCD 2018 195 200 205 210 215 aµ W,iso,conn,ud(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 fm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm) × 1010 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Comparison of the up and down quark, connected, isospin-symmetric contribution to the intermediate window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For historical completeness, we also show results that are superseded by newer results of the same collaboration at the top in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The inner error bars show the statistical uncertainty, the outer error bars show the statistical and systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' RBC/UKQCD 2018 [31], Aubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2019 [32], ETMC 2021 [73], BMW 2020 [30], LM 2020 [34], Aubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2022 [35], χQCD 2022 [74], Mainz 2022 [75], ETMC 2022 [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' ABSOLUTE UNBLINDING After the collaboration converged on the preferred prescription described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' IV D, the analysis was frozen and the absolute unblinding was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To this end, the unblinded data sets were distributed to the analysis groups, who then re-ran their analysis without modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The results were presented by our collaboration already at the Edinburgh workshop of the g-2 Theory Initiative [2] in 2022 and are stated without modifications in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Intermediate-distance window aW µ For the intermediate-distance window aW µ in the isospin-symmetric limit with t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, t1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 fm, and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm, we find the up and down quark-connected contribution to be aW,iso,conn,ud µ = 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='36(44)S(42)C(01)FV(00)mπ FV(08)∂m C(00)WF order(03)mres × 10−10 (42) in the BMW20 world and aW,iso,conn,ud µ = 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='46(53)S(43)C(01)FV(01)mπ FV(09)∂m C(00)WF order(03)mres × 10−10 (43) in the RBC/UKQCD18 world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We separately quote the statistical uncertainties (S), the continuum limit uncertainties (C), the finite-volume uncertainties for the vector correlators (FV), the finite-volume uncertainties of the measured pion masses (mπ FV), the uncertainties associated with the linear corrections to the line of constant physics (∂m C), the uncertainties from the discretization of the Wilson flow equation (WF order), as well as the uncertainties due to the non-zero chiral symmetry breaking (mres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The uncertainties from the ensemble-parameter and renormalization- factor determinations are fully propagated in the quoted uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 9, we compare Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (42) with previously published results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this work, we consistently use the BMW20 world for comparison plots of isospin-symmetric contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Compared to our earlier result presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31], where aW µ was defined and computed for the first time, we increase the basis for our continuum extrapolation from 2 data points over two lattice spacings to 24 data points over three lattice spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' If we were to repeat the continuum extrapolation through the 2 data points already available in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31] with lower statistical precision, we obtain a result consistent with the earlier work of aW,iso,conn,ud µ = 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='9(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4) × 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The approximate 2σ upward shift compared to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31] can therefore dominantly be attributed to our improved continuum extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31], we also computed the QED, strong-isospin-breaking, strange, charm, and quark-disconnected contribu- tions to the intermediate window quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' These contributions are much smaller in magnitude and their uncertainties due to the continuum extrapolation are much smaller in absolute terms compared to aW,iso,conn,ud µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' By combining these contributions with our improved light quark-connected, isospin-symmetric result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (43), we obtain our 17 200 205 210 215 220 225 230 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='014 aµ W,iso,conn,ud x 1010 a2 / fm2 ZV, ˆω, Cll ZV, ˆω, Clc ZV, ω, Cll ZV, ω, Clc ZV*, ˆω, Cll ZV*, ˆω, Clc ZV*, ω, Cll ZV*, ω, Clc 200 205 210 215 220 225 230 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='014 aµ W,iso,conn,ud x 1010 a2 / fm2 ZV, ˆω, Cll ZV, ˆω, Clc ZV, ω, Cll ZV, ω, Clc ZV*, ˆω, Cll ZV*, ˆω, Clc ZV*, ω, Cll ZV*, ω, Clc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Continuum extrapolation of aW,iso,conn,ud µ × 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' On the left, we show the 8 fits of our preferred prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' On the right, we show the fit through the two data points already available in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31] with lower statistical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Colangelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2022 BMW 2020/KNT Aubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2019/CL/KNT RBC/UKQCD 2018/FJ RBC/UKQCD 2023 ETMC 2022 Mainz 2022 BMW 2020 ETMC 2021 RBC/UKQCD 2018 224 226 228 230 232 234 236 238 240 aµ W(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 fm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm) × 1010 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Comparison of the total intermediate window contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For historical completeness, we also show results that are superseded by newer results of the same collaboration at the top in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Dispersive resuls are shown in purple, lattice results are shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The inner error bars show the statistical uncertainty, the outer error bars show the statistical and systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' RBC/UKQCD 2018 [31], ETMC 2021 [73], BMW 2020 [30], Mainz 2022 [75], ETMC 2022 [76], RBC/UKQCD 2018/FJ [77], Aubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2019/CL/KNT [78], BMW 2020/KNT [79], Colangelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 2022 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' prediction for the total intermediate window contribution aW µ = 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='56(65)(50) × 10−10 (44) with statistical (left) and systematic (right) errors given separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This can be compared with other lattice results as well as results based on the R-ratio, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Our result is in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8σ tension with the recently published dispersive result of aW µ = 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4) × 10−10 [1] and in agreement with recent lattice results [30, 75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Short-distance window aSD µ For the short-distance window aSD µ in the isospin-symmetric limit with t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm, we find the up and down quark-connected contribution to be aSD,iso,conn,ud µ = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5)(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6) × 10−10 (45) in the BMW20 world and aSD,iso,conn,ud µ = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='6)(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4) × 10−10 (46) 18 0 10 20 30 40 50 60 70 80 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 tp / fm aµ SD,pQCD(tp,∆) x 1010 (massless) aµ W(tp,t0,∆) x 1010 (massive) aµ W(tp,t0,∆) x 1010 (massive minus massless) (aµ SD,pQCD(tp,∆) + aµ W(tp,t0,∆)) x 1010 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Stability plot of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (48) for t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The massless perturbative QCD result is taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The correction from zero quark mass to non-zero quark mass is obtained from a linear extrapolation in the quark mass using ensembles 48I, 1, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The horizontal lines give the result of lattice QCD without combination with perturbative QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Only the quark-connected isospin-symmetric up and down quark contribution is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' in the RBC/UKQCD18 world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We can substantially improve this result by replacing the very shortest distances with perturbative QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Such a hybrid result of perturbative and non-perturbative QCD is still a first-principles determination but may combine the strength of both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In addition, the study of the consistency of lattice QCD and perturbative QCD at short distances may play an important role in understanding the origin of the tension for aW µ described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' V A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To establish a hybrid method, we use the additive property of the windows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=', aSD µ (t0, ∆) = aSD µ (tp, ∆) + aW µ (tp, t0, ∆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (47) We can then evaluate the first term in perturbative QCD at O(α4) [72] and the second term in lattice QCD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=', we write aSD µ (t0, ∆) = aSD,pQCD µ (tp, ∆) + aW µ (tp, t0, ∆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (48) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 12, we study this separation as a function of tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To the degree that perturbative QCD agrees with lattice QCD at distance tp, the plot should exhibit a plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We find that lattice QCD and perturbative QCD are consistent within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 × 10−10 up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For a related study of matching perturbative QCD to short-distance vector current correlators, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' If we choose tp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 fm, we find aSD,iso,conn,ud µ = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='51(43)(53) × 10−10 (49) in the BMW20 world and aSD,iso,conn,ud µ = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='70(52)(59) × 10−10 (50) in the RBC/UKQCD18 world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This is our preferred prescription for aSD,iso,conn,ud µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We compare Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (49) to previous results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The hybrid method reduces the large discretization errors for the short-distance window and specifically also reduces the logarithmic discretization errors described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [81] and [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finally, we note that the short-distance correlator is insensitive to the quark mass, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This motivates a new approach to study the continuum limit of the HVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Since discretization errors largely cancel in the difference between vector currents evaluated at different quark masses, we proposed a mass-splitting approach in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this approach, we generate pairs of ensembles with mπ and Mπ with Mπ ≫ mπ to compute aµ(mπ) = aµ(mπ) − aµ(Mπ) � �� � ≡δaµ +aµ(Mπ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (51) 19 RBC/UKQCD 2023 ETMC 2022 ETMC 2021 45 46 47 48 49 50 51 52 53 aµ SD,iso,conn,ud(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm) × 1010 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Comparison of our preferred result with previous determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For historical completeness, we also show results that are superseded by newer results of the same collaboration at the top in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The inner error bars show the statistical uncertainty, the outer error bars show the statistical and systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' ETMC 2021 [73], ETMC 2022 [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2 0 5 10 15 20 25 t / a C(t,mπ = 140 MeV) t3 C(t,mπ = 280 MeV) t3 (C(t,mπ = 140 MeV) - C(t,mπ = 280 MeV)) t3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Mass dependence of the vector correlator on a lattice with a−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='73 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' At very short distances, the vector correlator is effectively independent of the quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This allows us to consider the continuum limit of δaµ and aµ(Mπ) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The costly term δaµ can then be calculated at coarser lattice spacings compared to aµ(Mπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This method will be used in upcoming improvements to the present calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Isospin-symmetric scheme dependence For comparisons of quantities defined in an isospin-symmetric world, it is crucial to precisely match the definitions of the isospin-symmetric point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II C, we defined two hadronic schemes to define the isospin-symmetric world that match results previously presented by the RBC/UKQCD and BMW collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In previous sections, we presented our results separately for both schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In this section, we provide results for the correlated difference of the BMW20 minus the RBC/UKQCD18 world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the intermediate window we find ∆aW,iso,conn,ud µ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='10(24)(07) × 10−10 (52) and for the short-distance window we find ∆aSD,iso,conn,ud µ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='33(36)(36) × 10−10 (53) using the lattice results of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (45) and (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We can therefore not yet resolve the difference in isospin-symmetric schemes and they can be viewed as compatible at the current precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Retrospective discussion of the blinding procedure In the current paper, we performed a blinded analysis as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The goal of this procedure was to eliminate psychological bias that may have influenced systematic decisions of the analysis groups to favor either 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='995 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='015 A B C D E RBC/UKQCD 23 aµ W(t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, t1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 fm, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='995 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='015 A B C D E RBC/UKQCD 23 aµ W(t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm, t1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0 fm, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15 fm) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We show the result of the relative unblinding for aW µ including the preferred prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' On the left side, each group used its own blinded data set including the a2 and a4 terms added in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' On the right side, each group re-ran their unmodified analysis after the absolute unblinding on the unblinded dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' As anticipated, the artificial discretization errors in the blinded data can change central values and error estimates at the ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0025 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The data is normalized to the RBC/UKQCD 23 prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The inner error bars show the statistical uncertainty, the outer error bars show the statistical and systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' a larger value for aW µ , confirming the lattice QCD result of the BMW collaboration for this window quantity, or a smaller value, confirming the result based on the R-ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' To this end, we added artificial discretization errors using both a2 and a4 terms such that it is impossible for those who had access to our previous results for the coarser two lattice spacings of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31] to completely unblind themselves by comparing the new blinded correlators with the previously shared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This is the reason for the three parameters of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (16) exceeding the number of previously available lattice spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Nevertheless, the possibility of an analysis group computing unblinded correlators based on the used gauge fields always remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Given the reduced statistical noise of short-distance time-slices of C(t), even our chosen blinding procedure can in principle be circumvented with sufficient effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' It therefore remains an important task to evaluate the balance between the threshold preventing such unblinding and the possible drawbacks introduced by the blinding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We suggest that a reasonable balance is found when everybody acting in good faith is protected from psychological bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the current calculation, we believe the chosen blinding procedure to be successful in that regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' However, it came at the cost of a ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='0025 level uncertainty, limiting the optimization of our preferred procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This uncertainty is introduced by the a4 terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (16) that are not always eliminated by the continuum extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The analysis groups, however, had to make decisions and freeze their analyses based on the blinded data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 15, we highlight this effect by contrasting the relative unblinding as performed on the blinded data sets compared to the case, where we re-run the unmodified analyses on the unblinded data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In future studies, we will have to reconsider our exact approach since adding even higher-order terms (such as a6) with sufficiently small coefficients to account for additional finer data sets would have a diminishing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We may therefore decide to use only lattice-spacing-independent blinding factors in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' CONCLUSIONS AND OUTLOOK In this work we compute the standard Euclidean window of the hadronic vacuum polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We employ a blinded setup to avoid a possible bias towards reproducing previously published results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We focus on the dominant quark-connected light-quark isospin-symmetric contribution and significantly improve its continuum extrapolation and address additional sub-leading systematic effects from sea-charm quarks and residual chiral-symmetry breaking from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Our result for the total intermediate window aW µ is in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='8σ tension with the recently published dispersive result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [1] and in agreement with other lattice results [30, 75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' For the isospin-symmetric connected up and down quark contribution aW,iso,conn,ud µ more lattice results are available [30, 34, 35, 74–76] that are all in agreement with the result presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The tension for the intermediate window between lattice QCD and the dispersive result needs to be addressed in future work and a systematic study of additional windows may provide further insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' As it stands, this tension may be interpreted as a yet to be understood new physics contribution to hadronic e+e− decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In the context of 21 the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='2σ tension for aµ [4], aµ(EXP) − aµ(SM) = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='9) × 10−10 , (54) we note that the difference of the dispersive and lattice results for aW µ (SM) is only 6 × 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' In addition, we provide a result for the short-distance window for which our result is compatible with the recently published result of the ETMC collaboration [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' At short distances, we contrast lattice QCD and perturbative QCD and find agreement up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='4 fm at the level of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 × 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We also provide results for a hybrid method in which we use perturbative QCD below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='1 fm and lattice QCD at longer distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The effective mass-independence of the vector correlators at short distances finally motivates us to define a mass-splitting procedure to further improve the continuum extrapolation of the HVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We are currently generating additional ensembles with lattice spacings at a−1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='5 GeV and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='7 GeV that will support a five-lattice spacing continuum extrapolation using the mass-splitting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Finally, we are also preparing an update for the long-distance window using the improved bounding method [84] and an update of our QED and strong-isospin-breaking corrections re-using data from our hadronic light-by-light program [26, 85–87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Upon completion of our HVP program, we expect to be able to match the FNAL E989 target precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank our colleagues of the RBC and UKQCD collaborations for many valuable discussions and joint efforts over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The authors gratefully acknowledge the Gauss Centre for Supercomputing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='gauss-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='eu) for funding this project by providing computing time on the GCS Supercomputer JUWELS at J¨ulich Supercomputing Centre (JSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' An award of computer time was provided by the ASCR Leadership Computing Challenge (ALCC) and Innovative and Novel Computational Impact on Theory and Experiment (INCITE) programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under contract DE-AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This research also used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' DE- AC02-05CH11231 using NERSC award NESAP m1759 for 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This work used the DiRAC Blue Gene Q Shared Petaflop system at the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre on behalf of the STFC DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This equipment was funded by BIS National E-infrastructure capital grant ST/K000411/1, STFC capital grant ST/H008845/1, and STFC DiRAC Operations grants ST/K005804/1 and ST/K005790/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' DiRAC is part of the National E-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' We gratefully acknowledge disk and tape storage provided by USQCD and by the University of Regensburg with support from the DFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The lattice data analyzed in this project was generated using GPT [88], Grid [89], and CPS [90] and analyzed, in part, using pyobs [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' TB is supported by the US DOE under grant DE-SC0010339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' PB, TI, CJ, and CL were supported in part by US DOE Contract DESC0012704(BNL), and PB, TI, and CJ were supported in part by the Scientific Discovery through Advanced Computing (SciDAC) program LAB 22-2580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The research of MB is funded through the MUR program for young researchers “Rita Levi Montalcini”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' This project has received funding from Marie Sk�lodowska-Curie grant 894103 (EU Horizon 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' VG and RH are supported by UK STFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' ST/P000630/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' NM is supported by the Special Postdoctoral Researchers Program of RIKEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' TI is also supported by the Department of Energy, Laboratory Directed Research and Development (LDRD No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' 23-051) of BNL and RIKEN BNL Research Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' LJ acknowledges the support of DOE Office of Science Early Career Award DE-SC0021147 and DOE grant DE- SC0010339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' RM is supported in part by the US DOE under grant DE-SC0011941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' The work of ASM was supported by the Department of Energy, Office of Nuclear Physics, under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' DE-SC00046548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' ∗ Corresponding author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' christoph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='lehner@ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='de [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Colangelo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' X.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Yang (chiQCD), Muon g-2 with overlap valence fermion, (2022), arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='01280 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [75] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' C`e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=', Window observable for the hadronic vacuum polarization contribution to the muon g − 2 from lattice QCD, (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='06582 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [76] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' Alexandrou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=', Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization con- tributions to the muon magnetic moment using twisted-mass fermions, (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content='15084 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [77] This result was produced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [31] using data provided by Fred Jegerlehner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [78] This result was produced by Christoph Lehner for Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [32] using data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} +page_content=' [79] This result was produced in Ref.' 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[hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FAT4oBgHgl3EQfzB6w/content/2301.08696v1.pdf'} diff --git a/ONFPT4oBgHgl3EQfmjW_/content/2301.13126v1.pdf b/ONFPT4oBgHgl3EQfmjW_/content/2301.13126v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e10b1096b3e151de15d759f5c1e15902888d5e73 --- /dev/null +++ b/ONFPT4oBgHgl3EQfmjW_/content/2301.13126v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:011fd66d7f8bba58348d061446e276ce88c54f45dc34dbb5552235a82c8eced7 +size 2150272 diff --git a/ONFPT4oBgHgl3EQfmjW_/vector_store/index.pkl b/ONFPT4oBgHgl3EQfmjW_/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..78ce5cfd35e552827e70b2ba6cec30f8bd66dfe1 --- /dev/null +++ b/ONFPT4oBgHgl3EQfmjW_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cde195b15ecaaa27a4213867db57625f24effd7676b26fbbebca5d6d453e9273 +size 252836 diff --git a/OdE4T4oBgHgl3EQf9w5t/content/tmp_files/2301.05358v1.pdf.txt b/OdE4T4oBgHgl3EQf9w5t/content/tmp_files/2301.05358v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e9e15210d4eaf3c02ff61a9ba9707edeede5a6b --- /dev/null +++ b/OdE4T4oBgHgl3EQf9w5t/content/tmp_files/2301.05358v1.pdf.txt @@ -0,0 +1,1753 @@ +1 +Experimental System Identification and Disturbance +Observer-based Control for a Monolithic Zθxθy +Precision Positioning System +Mohammadali Ghafarian1,2, Bijan Shirinzadeh1, Ammar Al-Jodah1,3, Tilok Kumar Das1, Tianyao Shen1 +Abstract—A compliant parallel micromanipulator is a mech- +anism in which the moving platform is connected to the base +through a number of flexural components. Utilizing parallel- +kinematics configurations and flexure joints, the monolithic +micromanipulators can achieve extremely high motion resolution +and accuracy. In this work, the focus was towards the experimen- +tal evaluation of a 3-DOF (Zθxθy) monolithic flexure-based piezo- +driven micromanipulator for precise out-of-plane micro/nano +positioning applications. The monolithic structure avoids the +deficiencies of non-monolithic designs such as backlash, wear, +friction, and improves the performance of micromanipulator +in terms of high resolution, accuracy, and repeatability. A +computational study was conducted to investigate and obtain +the inverse kinematics of the proposed micromanipulator. As a +result of computational analysis, the developed prototype of the +micromanipulator is capable of executing large motion range +of ±238.5µm × ±4830.5µrad × ±5486.2µrad. Finally, a slid- +ing mode control strategy with nonlinear disturbance observer +(SMC-NDO) was designed and implemented on the proposed mi- +cromanipulator to obtain system behaviours during experiments. +The obtained results from different experimental tests validated +the fine micromanipulator’s positioning ability and the efficiency +of the control methodology for precise micro/nano manipulation +applications. The proposed micromanipulator achieved very fine +spatial and rotational resolutions of ±4nm, ±250nrad, and +±230nrad throughout its workspace. +Note to Practitioners—Piezo-actuated precision positioning +systems play an increasingly important role in the fields of +micro/nano manipulation robots. They have the advantages of +fine resolution, high accuracy, fast response speed, and large +output displacement. However, such systems inherently exhibit +vibration, hysteresis behaviors, and are affected by external +disturbances that could cause oscillations and positioning errors. +This study presents a robust control methodology implemented +on a 3-DOF positioning system (Zθxθy), which is among the most +prone system to be affected by existing disturbances. This control +methodology is designed to improve the tracking performance in +the presence of hysteresis nonlinearity, disturbances, and model- +ing errors. The effectiveness of the proposed control methodology +is demonstrated by conducting a series of experiments. Due to +the ease of implementation, the developed control methodology +can be applied to other positioning systems as well. +Index Terms—Sliding mode control, Nonlinear disturbance +observer, Precision positioning, Monolithic parallel manipulator, +Amplification mechanism +I. INTRODUCTION +1Robotics and Mechatronics Research Laboratory (RMRL), Department of +Mechanical and Aerospace Engineering, Monash University, Clayton, VIC +3800, Australia. 2Institute for Intelligent Systems Research and Innovation +(IISRI), Deakin University, Geelong Waurn Ponds, VIC 3216, Australia. 3The +University of Western Australia, Perth, WA 6009, Australia. +Corresponding author: m.ghafarian@deakin.edu.au +F +Lexure joints have dominant superiority over traditional +mechanical joints in precision engineering including mi- +cromanipulation mechanisms. Flexure-based parallel micro- +manipulators benefit from the advantages of both flexure +joints and parallel-kinematics configurations, and addition- +ally utilize the important characteristics of micromanipulators +such as frictionless motion, absence of mechanical play and +backlash, and no need for lubrication. These features are +important and effective for various micro/nano positioning +and nano-alignment applications, and it is not surprising that +flexure-based parallel micro/nano manipulation systems stand +out among others as the key element in the ultra-precision +technologies [1]–[10]. Wei and Xu [11] proposed a force- +sensing cell microinjector based on a single-axis compliant +small-stiffness mechanism. Yong et al. [12] presented single- +and dual-stage vertical positioners for high-speed piezoelectric +nanopositioning applications. Wang et al. [13] presented a +decoupled piezo-driven XY micro/nano positioning system +with a travel range of ±27.7 × ±26.6µm2. Tian et al. [14] +presented a custom made atomic force microscopy (AFM) +which was integrated with a 3-DOF XYZ parallel-kinematics +piezo-driven micromanipulator for high-speed imaging. Cai +et al. [15] presented the design and analysis of two par- +allel compliant piezo-driven 3-DOF micro/nano positioning +system. Meanwhile, the inverse Bouc-Wen (BW) model was +applied as a feedforward hysteresis compensator in the feed- +forward/feedback hybrid control method to compensate for +the hysteresis of piezoelectric actuators (PEAs). The proposed +micromanipulator exhibited small translational and rotational +workspaces of ±4.1×±5.2×±6.5µm3 and ±112×±52.5× +±48.5µrad3, respectively. Qin et al. [16] proposed the design +of a 3-PRR (prismatic-revolute-revolute) XYθ micromanipula- +tor. The Scott-Russell (SR) mechanism was utilized in the pro- +posed design to magnify the displacement of the PEA. Dong et +al. [17] presented a 6-DOF piezo-driven micromanipulator. As +a result of having no amplification mechanism, the workspace +of the micromanipulator was very limited. Yang et al. [18] +demonstrated the design, modeling, and experimental analysis +of a piezo-driven XY micromanipulator. The experimental +results illustrated that the XY micromanipulator had a working +range of ±75 × ±73.5µm2 with the resolution of ±0.128µm +and ±0.143µm in the X- and Y-directions, respectively. +Among the flexure-based parallel micromanipulators, the com- +pliant Zθxθy type micro/nano positioning systems have be- +come the research focus due to their important advantages +for in out-of-plane positioning tasks [19]–[22]. Qu et al. [23] +arXiv:2301.05358v1 [cs.RO] 13 Jan 2023 + +2 +Fig. 1: Monolithic Zθxθy parallel micromanipulator +presented the design, modeling and test of a piezo-driven +θxθy flexure-based micro/nano positioning system. The exper- +imental results indicated that the developed micromanipulator +could achieve a workspace of ±515µrad × ±460µrad about +its two working axes with a resolution of ±0.5µrad. Chen +et al. [24] introduced the mechanical design, modeling and +experimental tests of a large-angle Zθxθy macromanipulator +driven by four small air-gap voice coil actuators. The proposed +system could achieve rotational-motion ranges of ±41.59mrad +and ±41.13mrad in the working axes, for which the Z- +mode frequency was 49.6Hz and the rotational ones were +55.45Hz and 56.09Hz, respectively. The motion resolution +of the macromanipulator was ±6.67µrad. Kim et al. [25] +presented an active vibration control system which was con- +structed based on a non-monolithic 3-DOF Zθxθy micro/nano +manipulation system with an in-plane dimension of 160mm +(diameter) and an out-of-plane height of 60mm. Cao and Chen +[26] demonstrated the development of a system identifica- +tion model for a commercially-available 3-DOF piezo-driven +Zθxθy micromanipulator (P-558.TCD, Physik Instrumente). +The system was driven by four PEAs and had a motion range +of ±25µm×±250µrad×±250µrad. The quasi-static analysis +of a non-monolithic compliant tripod system for micro/nano +positioning applications was presented by Wei et al. [27]. The +proposed micromanipulator had an overall positioning range +of ±41µm × ±330µrad × ±385µrad. Considering the above- +mentioned studies, a compliant monolithic 3-DOF piezo- +driven micromanipulator was introduced by the authors with +a larger workspace and fine resolution capable of executing +three out-of-plane motions, one translation and two rotations. +The structure of the proposed monolithic micromanipulator +was optimized completely to have a maximum working range +and bandwidth frequency higher than 100Hz [28]. +Regardless of the types of micromanipulators used to perform +micro/nano manipulation tasks, an effective motion tracking +control strategy can improve the tracking performances of the +system significantly. In addition, disturbances such as cross- +coupling, parametric uncertainties, etc. can practically affect +and degrade the performance of a precision positioning system. +TABLE I: Mechanical and physical properties of ABS +Symbol +Quantity +Value +ν +Poisson’s ratio +0.35 +ρ +Density +0.9087(g/cm3) +E +Young’s modulus +2200(MPa) +σyield +Tensile yield strength +31(MPa) +σultimate +Tensile ultimate strength +55(MPa) +Therefore, designing and utilizing a disturbance observer- +based control methodology to be able to estimate and com- +pensate the effect of disturbances for achieving high precision +applications is very beneficial [29]–[31]. Chen et al. [32] +introduced a nonlinear disturbance observer (NDO) for robotic +manipulators for various purposes such as friction compensa- +tion, independent joint control, sensorless torque control, and +fault diagnosis. Furthermore, the global exponential stability +of the proposed NDO was guaranteed. Lau et al. [33] pre- +sented an enhanced adaptive robust disturbance observer-based +motion tracking control methodology for tracking a desired +motion trajectory in the presence of unknown or uncertain +system’s parameters, nonlinearities including hysteresis, and +disturbances in the motion system. The proposed control +methodology was applied in a semi-automated hand-held ear +surgical device for the treatment of Otitis Media with Effusion +(OME). +Motivated by the previous work [28], an experimental study +of a large range piezo-driven spatial compliant monolithic +parallel Zθxθy micro/nano manipulation system with a fine res- +olution is presented in this paper. Monolithically manufactured +designs are very important for micro/nano applications and +they are preferable in comparison with the assembled manip- +ulation structures because of elimination of the unwanted fea- +tures that affect a smooth and accurate nano-resolution manip- +ulation. Other advantages of the proposed micromanipulator +are low manufacturing and material cost. Nano-meter/radian +resolution, large amplification ratio, repeatability, and stability +are ensured due to the characteristics of the proposed mono- +lithic micromanipulator. To investigate the motion range and +decouple the micromanipulator’s motions, the inverse kinemat- +ics is obtained using FEA. The performances of the developed +micromanipulator are investigated in the real-time experiments +via three feedback control methodologies, i.e. Proportional- +Integral-Derivative control (PID), sliding mode control (SMC), +and nonlinear disturbance observer-based sliding mode control +(SMC-NDO). The role of the developed NDO is to compensate +for the uncertain disturbances in the real-time experiments to +achieve high precision manipulation tasks. In the end, the ex- +perimental results, including frequency, resolution, and several +complex trajectory motion tracking analyses are presented, +and precise manipulations can be guaranteed by the developed +monolithic micromanipulator and control methodologies. +II. MECHANICAL DESIGN +As presented in Figure 1, the structure of the micromanip- +ulator consists of a fixed base platform, six leaf-flexure-based +parallelogram mechanisms, three flexure-based Scott-Russell +amplification mechanisms, three flexure-based spherical joint + +Input Force +3-DOF Monolithic Micromanipulator +MovingPlatform +Input Force +Input Force +Flexure-Based +Spherical Joint +FixedBasePlatform +Flexure-Based Scott-Russell +AmplificationMechanism +Leaf-Flexure-Based +Parallelogram Mechanism3 +Fig. 2: Reachable workspace of the developed micromanipulator +modules, and a moving platform. Two sets of leaf parallel- +ograms are incorporated into the input and output points of +the Scott-Russell amplification mechanism as prismatic joints. +This linearizes the motion of the Scott-Russell mechanism +and increases the micromanipulator’s stiffness. Because PEAs +produce a very small displacement as a proportion of their +length, mechanical displacement amplification is inevitably +required for large displacement applications. The Scott-Russell +amplification mechanism is a well established mechanical +amplifier [16], [28]. Additionally, it has the advantage of +transforming into a horizontal input to a vertical output +which is ideal for the compactness of the designed Zθxθy +micromanipulator. The platform needs to generate rotational +motions along two different axes. A spherical joint, which is +capable of rotating in three different axes is adopted for the +connection between the platform and the manipulation system. +In comparison with previous designs, here the PEAs are placed +outside of the micromanipulator not inside. Thus, different +sizes of PEAs can be used for the proposed monolithic design +without affecting the geometry and overall dimensions of the +structural design. The overall dimensions of the proposed de- +sign are 201mm, 180mm, 75mm. Developments in 3D printing +enable complex geometries to be easily and economically +manufactured in multiple materials. Therefore, the proposed +design is fabricated using high density and high accuracy +3D-printing from Acrylonitrile Butadiene Styrene (ABS). The +mechanical and physical properties of ABS are presented in +Table I. +III. WORKSPACE ANALYSIS +The usable workspace is limited by material stresses. In +order to maintain the stability, repeatability, and capabilities +of the micromanipulator in precise manipulation, it is very +important that the applied stress on the micromanipulator +due to the load remains less than its tensile yield strength. +Using the stress and safety factor analyses in FEA software +(ANSYS), the maximum Von-Mises stress and the minimum +safety factor occur when the input displacement of 90µm is +applied to the micromanipulator as the input of the three PEAs +simultaneously. The values of maximum Von-Mises stress and +minimum safety factor corresponding to the 90µm inputs were +17.585MPa and 1.71, respectively. It is worth noting that +the obtained minimum safety factor was 1.71, as the safety +factor must always be greater than 1. Therefore, more input +force/displacement could have been applied to obtain a larger +workspace. However, the value of 1.71 is considered as a lower +limit for the safety factor of the micromanipulator to avoid +some deficiencies including creep, fatigue, and mechanical +failure of the micromanipulator in the experiment. +Three PEAs (Physik Instrumente P-843.60) with a maxi- +mum displacement of 90µm were assumed to be used for +analysing the workspace based on the FEA. The inverse kine- +matics of the micromanipulator was computed by evaluating +the motion of the system across 80 inputs, spanning the full +input range, and performing a regression on the resulting +outputs. The best fit inverse kinematics is computed as: +J−1 = +� +� +−0.1909 +0.0001 +0.0110 +−0.1877 +−0.0095 +−0.0053 +−0.1875 +0.0093 +−0.0056 +� +� +(1) +The resultant workspace of computational analysis is pre- +sented in Figure 2. The values of motions along Z, θx, +and θy axes are ±238.5µm, ±4830.5µrad, and ±5486.2µrad, +respectively. Since the determinant of the Jacobian matrix is +non-zero, therefore there is no singularity in the workspace of +the micromanipulator. +IV. CONTROLLER DESIGN +The accuracy of a manipulation task is very dependent +on eliminating undesired disturbances affecting the system. +Therefore, in this section, establishment of control method- +ologies will be presented for the utilization in the experi- +mental study to overcome this challenging issue for precise +micro/nano manipulation applications. +A. Sliding Mode Control (SMC) +In general, flexure-based micromanipulators are structures +comprising solid links and flexure hinges. The monolithic +Fig. 3: Block diagram of the proposed SMC-NDO control +methodology + +5000 +6000 +6000 +4000 +4000 +4000 +3000 +2000 +2000 +2000 +1000 +Rx (μrad) +Ry (μrad) +Ry (μrad) +0 +-1000 +-2000 +-2000 +-2000 +-3000 +4000 +4000 +-4000 +-5000 +-6000 +0009 +-250 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +250 +-250-200 +-150-100 +-50 +50 +100 +150 +200 +250 +-5000 -4000 -3000 -2000 -1000 +0 +10002000300040005000 +z(μm) +z (μm) +Rx (μrad)[9] +Plant +Trajectory +Inverse +[9a] +Sliding Mode +u +Piezo +Capacitive +Generator +Kinematics +Amplifier +Sensors ++q +Control +d. +Nonlinear +Disturbance +Observer4 +construction reduces assembly errors and guarantees high +accuracy. A lumped parameter dynamic model for a flexure- +based micromanipulator can be given by: +mq ¨q + cq ˙q + kqq = dq + uq +(2) +where q represents the three output motions of the microma- +nipulator (Z, θx, θy), and mq, cq, and kq are the equivalent +mass, damping and stiffness in the corresponding axis. dq and +uq are the lumped disturbance and control action in a given +axis, respectively. A state-space representation of Eq. (2) can +be obtained by considering x1 = q and x2 = ˙q. The tracking +error between the actual and desired motions can be introduced +as follows: +e = x1 − x1des +(3) +There are two steps in the design of an SMC. The first step +is designing a sliding surface so that the plant restricted to +the sliding surface has a desired system response. This means +Fig. 4: Schematic diagram of sensing, control, and experimen- +tal research facility +a +b +c +Fig. 5: Modal analysis of the micromanipulator: (a) Z-resonant +frequency (b) θx-resonant frequency (c) θy-resonant frequency +the state variables of the plant dynamics are constrained to +satisfy another set of equations which define the so-called +switching surface. The second step is constructing a switched +feedback gains necessary to drive the plant’s state trajectory to +the sliding surface. These strategies are built on the generalized +Lyapunov stability theory. In this study, a PID sliding surface +was selected to reduce the steady-state error and improve the +transient response speed. The selected sliding surface is given +by: +s = λpe + λi +� t +0 +edt + λd ˙e +(4) +where λp, λi, and λd are positive proportional, integral, and +derivative parameters to be selected, respectively. The SMC + +Physik Instrumente +AmplifierE-509.C3A +h52 +255 +255 +3-AxisPiezo Controller +Physik Instrumente E-500 +500 +500 +540 +Elliot ScientificMDE263 +3-axis micropositioners +(PhysikInstrumente) +Capacitive Sensor D-050.00 +(Physik Instrumente) +78.8 +MicromanipulatorElliotScientificMIDE263 +-axismicropositioners +Micromanipulator +P-843.60PEAs +(Physik Instrumente) +Capacitive Sensor D-050.00 +(Physik Instrumente) +Base Platform mounted on a vibration-isolated optical table20 +10 +Magnitude (dB) +-10 +- Experimental +-Estimated +-20 +101 +102 +Frequency (Hz)20 +10 +Magnitude (dB) +-10 +-Experimental +-Estimated +-20 +101 +102 +Frequency (Hz)30 +Experimental +Estimated +25 +Magnitude (dB) +20 +15 +10 +5 +0 +10 1 +102 +Frequency (Hz)5 +Fig. 6: System response to the applied staircase input +law is intended to force the tracking error e to approach the +sliding surface and then proceed to the origin along the sliding +surface. Hence, the sliding surface needs to be stable, and +therefore yields to the fact that ˙s = 0. Additionally, the control +action of the SMC approach consists of two parts; equivalent +control and switching control as follows: +u = ueq + usw +(5) +Thus, it can be realized that to be able to force the actual +motions to converge to the desired ones, the equivalent control +and switching control need to be as follows: +ueq = cx2 + kx1 + m¨x1des − m λi +λd e − m λp +λd ˙e − ˆd +usw = −a1s − a2sgn(s) +(6) +where a1 and a2 are positive constants. The term ˆd is the +estimation of the disturbance which will be derived later. +The deviation between the actual and estimated disturbance +is defined by, ˜d, and can be given as follows: +˜d = d − ˆd +(7) +Lyapunov stability function is used to verify the design +performance of the SMC. In order to have a stable controller, +it is required that V must be bounded, which means that it +Fig. 7: Star trajectory tracking results +is important to prove ˙V ≤ 0. Therefore, the derivative of +Lyapunov function is derived as follows: +˙V = s ˙s = s{λd +m usw + λd +m d − λd +m +ˆd} = s{−k1s +−k2sgn(s) + λd +m +˜d} = −k1s2 − k2|s| ++λd +m +˜ds ≤ −k1s2 − |s|(k2 − λd +m +˜d) +(8) +where k1 and k2 are equal to +λda1 +m +and +λda2 +m , respectively. +The stability of the designed controller is guaranteed, if +the inequality k2 ≥ +λd +m ˜d is established. Furthermore, the +discontinuity of the signum function in the SMC law can +cause chattering. Therefore, in order to avoid this problem, +the discontinuous signum function is replaced by the contin- +uous hyperbolic tangent function (tanh). In other words, the +function is used as an approximator of the signum function. +B. Nonlinear Disturbance Observer (NDO) +An NDO is designed to estimate the undesired disturbance +dq, introduced in the dynamic model, Eq. (2). dq represents +all disturbances that may be caused by unmodeled dynamics. +The objective is to design an observer such that the estima- +tion ˆd yielded by the observer exponentially approaches the +disturbance d under any q and ˙q. +The dynamic model of the micromanipulator, Eq. (2), can be +rewritten in the general description of a form as follows: +˙x = f(x) + g1(x)u + g2(x)d +(9) +where x is a representation of the state variables x1 and x2. +Besides, functions f, g1, and g2 can also be found to be as +follows: +g1(x) = g2(x) = +1 +mq +f(x) = − cq +mq x2 − kq +mq x1 +(10) +The NDO is designed according to the following formula- +tions [34], [35]: +ˆd = z + p(x) +˙z = −l(x)g2(x)z − l(x){g2(x)p(x) + f(x) + g1(x)u} +(11) +where ˆd and z are the estimate of the undesired disturbance +and the internal state of the nonlinear observer, respectively, + +5 +PID +SMC +4 +SMC-NDO +Desired +3 +(μrad) +2 +1 +JLI +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)6 +PID +SMC +5 +SMC-NDO +Desired +4 +(μrad) +3 +2 +1 +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)600 +PID +SMC +400 +SMC-NDO +- Desired +200 +(μrad) +0 +-200 +-400 +-600 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +Z (μm)0.12 +PID +SMC +0.1 +SMC-NDO +Desired +0.08 +uwh +0.06 +(ur) +N +0.04 +15141 +0.02 +0 +-0.02 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)6 +Fig. 8: Experimental control tracking errors in the star trajectory +and p(x) is a nonlinear function to be designed. The nonlinear +observer gain l(x) is defined as follows: +l(x) = ∂p(x) +∂x +(12) +It can be shown that ˆd approaches d exponentially if p(x) is +chosen such that: +˙˜d + ∂p(x) +∂x g2(x) ˜d = 0 +(13) +Therefore, if p(x) is chosen as lx2, it can be realized from +Eq. (13) that the designed NDO yields to following form in +the time domain: +˜d(t) = ˜d(0)exp(− l +mt) +(14) +According to Eq. (14), if l is chosen to be a positive constant, +when time goes to infinity the disturbance estimation error, ˜d, +converges to zero. Therefore, the designed NDO is globally +exponentially stable. +C. Robustness Analysis of SMC-NDO +The robustness of the proposed SMC-NDO is proven in this +section. The feedback control of the system described by Eq. +Fig. 9: RMRL logo trajectory tracking results +(2) is robust, if control law (5) is applied and it satisfies the +conditions of: +lim +s→0+ ˙s < 0 +, +lim +s→0− ˙s > 0 +(15) +The above-mentioned conditions illustrate that if the phase +trajectory around s = 0 points to the s hyper-plane, then it +will enter the sliding mode when the hyper-plane s = 0. +By substituting control law (5) into Eq. (2), the equation of +motion of the closed-loop system is obtained as: +¨q = +1 +mq +� +˜dq − a1s − a2tanh(s) +� ++ +� +¨qdes − λi +λd +e − λp +λd +˙e +� +(16) +Moreover, by substituting Eq. (16) into the differentiation of +Eq. (4), the dynamic sliding mode equation is obtained as: +˙s = λp ˙e + λie + λd¨e += λp ˙e + λie + λd(¨q − ¨qdes) +≈ ˜dq − a1s − a2tanh(s) +(17) +Utilizing the two conditions (15), and the stability condition +that was obtained in Section III.A, (a2 ≥ ˜d), it is trivial that +the robustness conditions are satifisfied as following: +lim +s→0+ ˙s = ˜dq − a2tanh(s) < 0 +lim +s→0− ˙s = ˜dq − a2tanh(s) > 0 +(18) +Thus, the phase trajectory around s = 0 enters the sliding +mode in the hyper-plane and s = 0, so the PID sliding surface +(4) is: +s = (λp + λis−1 + λds)e = 0 +(19) +Using Eq. (19) and having the knowledge of ˙e = ˙x2 − ˙x2des, +Eq. (9) is rewritten as following: +x1 = x1des + e +x2 = x2des − λp +λd e − λi +λd s−1e +(20) +Thus, Eq. (20) becomes the state equation when the system +enters the sliding mode. We know that the second-order system +Eq. (2) can be expressed by the first-order state Eq. (20) and +that the dynamic characteristics of the system are independent +of d(t), therefore, SMC-NDO is highly robust when the system +enters the sliding mode, thereby ensuring that the system +exhibits a good dynamic response and stability. + +0.25 +PID +SMC +SMC-NDO +0.2 +(wn) +0.15 +0.1 +0.05 +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)9 +PID +8 +SMC +SMC-NDO +7 +6 +5 +Error e. +4 +3 +2 +1 +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)1.4 +PID +SMC +1.2 +SMC-NDO +,-axis (μrad) +0.8 +0.6 +Error +0.4 +0.2 +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)600 +PID +SMC +SMC-NDO +400 +Desired +200 +(μrad) +0 +-200 +-400 +-600 +009- +-400 +-200 +0 +200 +400 +600 +a, (μrad)7 +Fig. 10: Experimental control tracking errors in the RMRL logo trajectory +D. Time Convergence Stability Analysis of SMC-NDO +The time convergence of the sliding mode equation is +investigated in this section when the disturbance observer and +chattering reduction methods are used. +Lemma. [36] Assume that a continuous definite function +V (t) ≥ 0 satisfies the subsequent differential inequality ˙V (t)+ +β1V α(t) + β2V (t) ≤ 0, where β1, β2, and α are the positive +constants, and the range of α is between 0 to 1. Based on +which the function V (t) converges to zero in finite time given +as: +ts = +1 +β2(1 − α) ln β2V (1−α)(0) + β1 +β1 +(21) +Theorem. Consider system described by Eq. (2). If the +surface manifold and control law are chosen according to Eqs. +(4) and (6), then sliding mode enforcement along the sliding +surface, Eq. (4), can be validated. As a result, the system states +will converge to the desired reference trajectories and tracking +errors will converge to zero in finite time. +Proof of Theorem. Consider the Lyapunov candidate func- +tions of the form as: +V = 1 +2s2 +(22) +Taking the time derivative of Eq. (22) and utilizing Eq. (17) +yield to: +˙V = s{ ˜d − a1s − a2tanh(s)} ≤ −a1s2 − a2|s| +(23) +Fig. 11: 3D-Archimedean spiral trajectory tracking results +Let β1 = +√ +2a2, β2 = 2a1, and α = 1 +2, then Eq. (23) can be +rewritten as: +˙V (t) + β1V α(t) + β2V (t) ≤ 0 +(24) +Then, according to Lemma, the differential inequality (24) +yields the finite-settling time as represented by Eq. (21), and +that completes the proof. +Finally, Figure 3 provides the information on how the SMC- +NDO was implemented in the real-time experiment to achieve +remarkable performances from the micromanipulator. +V. EXPERIMENTS AND RESULTS +Experimental tests were carried out to investigate the me- +chanical and dynamic performances of the developed flexure- +based micromanipulator, as well as the efficiency of the +proposed control methodologies. The schematic diagram of +the experimental set-up is presented in Figure 4. The ba- +sic operation of the flexure-based micromanipulator was to +change the drive voltage from the voltage control unit (am- +plifier module from Physik Instrumente) to the three PEAs +from Physik Instrumente (model P-843.60) to further drive +the micromanipulator. During installation of the PEAs, pre- +compression forces were applied to keep the actuators’ tip and +the micromanipulator in contact. These PEAs are multilayer +PZT stacked ceramic translators capable of 90µm displace- +ment corresponding to a range of operating voltage from 0 to +100V. The position of the moving platform was measured by +three capacitive sensors (Physik Instrumente D-050.00). The +TABLE II: Control parameters +Description +Parameter +Value +PID +PID constants +kp +50 +ki +17.5e6 +kd +0 +SMC +PID sliding surface constants +λp +50 +λi +13e6 +λd +100 +Switching control constants +a1 +2.5 +a2 +0.6 +SMC-NDO +Nonlinear observer gain +l +100 + +0.09 +PID +0.08 +SMC +SMC-NDO +0.07 +(wr) +0.06 + z-axis +0.05 +0.04 +Error +0.03 +0.02 +0.01 +0 +0 +2 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t(s)8 +PID +SMC +SMC-NDO +6 +-axis (μrad) +5 +3 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)25 +PID +SMC +SMC-NDO +20 +-axis (μrad) +15 +Error +10 +5 +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)450~ +400~ +350~ +300~ +250~ +(μrad) +200~ +150~ +PID +100 +(pejr) +168 +SMC +166 +SMC-NDO +164 +50 +=Desired +106 +3.72 +105 +3.74, +*3.76, +104 +-50 +3.78 +Z (μm) +ox (μrad) +400 +200 +15 +10 +0 +0 +-200 +-5 +-10 +, (μrad) +-400 +-15 +Z (μm)8 +Fig. 12: Experimental control tracking errors in the 3D-Archimedean spiral trajectory +capacitive sensors had a circular active area with a 4 mm radius +surrounded by an annular guard ring, with a specified working +range of 50µm. The electronic interface (Physik Instrumente +E-509.C3A) to the capacitive sensors provided a distance +measurement as an analog voltage, which was recorded at +the control computer with a 16-bit analog-to-digital converter +(ADC) and at the rate of 10kHz. In order to reduce the external +vibration disturbances, all experimental tests were performed +on a vibration-isolated optical table. +A. System Identification +System identification of the developed micromanipulator +was conducted to investigate the dynamic characteristics. +Hence, a sinusoidal sweep signal with the frequency linearly +varying from 1Hz to 250Hz was applied to the PEAs input, +one at a time, in order to excite the modes of vibration of +the platform. The frequency responses to these inputs are +presented in Figure 5. Resonance was observed in the response +along the Z, θx, and θy axes occured at 113.4Hz, 173.1Hz, and +184.2Hz, respectively. Finally, three continuous-time transfer +functions were identified from the frequency response along +each motion axis and formulated as follows: +TFz = +3.774e05 +s2 + 58.44s + 5.08e05 +(25) +TFθx = +5.792e05 +s2 + 59.16s + 1.186e06 +(26) +TFθy = +1.454e06 +s2 + 76.13s + 1.346e06 +(27) +Considering the transfer functions (15)-(17), mq, cq, and kq +can easily be calculated for implementing into to the SMC and +SMC-NDO. +In the next stage, i.e. closed-loop experiment, the control +parameters were found by empirical rules based on observation +of the response to reference and disturbance changes. These +selected control parameters for the PID, SMC, and SMC-NDO +control schemes are presented in Table II. +B. Resolution characterization of the system +To test the closed-loop system positioning resolution, the +staircase response of the positioner was measured along each +working axis. The magnitude of each step was chosen to be +20nm for the translational axis and 900nrad for the rotational +axes. The obtained results from the three control techniques are +presented in Figure 6. The best resolution result was captured +utilizing the SMC-NDO control technique. The minimum +resolution was found to be 8nm, 500nrad, and 460nrad +along the Z, θx, and θy axes, respectively. The positioning +resolution is limited by the measurement noise in motion axes, +quantization in the measurement of rotation, the resolution +of capacitive sensors, and the output resolution of the data +acquisition card. +C. Motion tracking characterization of the system +To further illustrate the motion abilities of the developed +micromanipulator alongside with the designed control ap- +proaches, three complex curves tracking experiments were +conducted. In which, a star, Robotics and Mechatronics Re- +search Laboratory (RMRL) logo, and 3D-Archimedean spiral +were selected and designed as the target tracking trajectories +due to their high level of complexities. The tracking results +are presented in Figures 7 to 12. +The results of the trajectory tracking and tracking errors of +the star trajectory implementing the three proposed control +schemes are presented in Figures 7 and 8. The best tracking +error was obtained by implementing SMC-NDO and the +maximum values of error were 14nm for the Z-axis, 700nrad +and 494nrad for the θx and θy axes, respectively. Figure 7 +illustrates the projection of the tracked trajectory on the ZY- +plane. +The RMRL logo was designed as a symbol of the most +complex trajectory to verify the performance of the microma- +nipulator. The results of the trajectory tracking and tracking +errors are presented in Figures 9 and 10. The projection of the +designed trajectory on the XY-plane is presented in Figure 9. +According to the results, the micromanipulator was capable of +tracking the designed logo with high precision and accuracy. It +must be noted that by using sensors with a higher resolution, +more accurate motion tracking can be achieved even with the +same control methodologies proposed in this work. +The tracking results of the 3D-Archimedean spiral are pre- +sented in Figures 11 and 12. These two figures provide the +difference between the desired and actual trajectories while +implementing different control methodologies. Same as the +previous trajectories tracking, the SMC-NDO control scheme +was observed to outperform the SMC and PID controllers. + +0.2 +PID +0.18 +SMC +SMC-NDO +0.16 +0.14 +(wr) +0.12 +Z-axis ( +0.1 +rror +0.08 +0.06 +0.04 +0.02 +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t(s)PID +SMC +6 +SMC-NDO +5 +-axis (μrad) +4 +3 +Error +2 +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)1.8 +PID +1.6 +SMC +SMC-NDO +1.4 +v-axis (μrad) +1.2 +A +0.8 +rror +0.6 +0.4 +0.2 +0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +t (s)9 +Fig. 13: Hysteresis cancellation of different control methodologies +TABLE III: Summary of the numerical results of trajectory tracking and hysteresis compensation +Controller +Z-axis +θx-axis +θy-axis +Trajectory tracking +RMSE (µm) +%Improvement (SMC-NDO/PID) +RMSE (µrad) +%Improvement (SMC-NDO/PID) +RMSE (µrad) +%Improvement (SMC-NDO/PID) +Star +PID +0.1094 +91.04 +3.9298 +90.80 +0.3903 +66.92 +SMC +0.1012 +3.6336 +0.3596 +SMC-NDO +0.0098 +0.3615 +0.1291 +RMRL logo +PID +0.0239 +74.06 +2.9472 +87.61 +8.8958 +91.07 +SMC +0.0184 +2.2886 +6.9058 +SMC-NDO +0.0062 +0.3652 +0.7942 +3D-Archimedean spiral +PID +0.0819 +96.83 +2.7101 +95.44 +0.5296 +76.25 +SMC +0.0660 +2.1850 +0.4345 +SMC-NDO +0.0026 +0.1237 +0.1258 +Hysteresis compensation +%Hysteresis max. +%Improvement (SMC-NDO/PID & SMC) +%Hysteresis max. +%Improvement (SMC-NDO/PID & SMC) +%Hysteresis max. +%Improvement (SMC-NDO/PID & SMC) +PID +0.49 +100 +0.44 +100 +0.54 +100 +SMC +0.36 +0.38 +0.44 +SMC-NDO +0 +0 +0 +Also, the SMC provided a better tracking performance than the +PID control scheme. Considering SMC-NDO, the maximum +tracking error along the Z, θx, and θy axes were observed to +be 11nm, 644nrad, and 650nrad, respectively. On the other +hand, with respect to the output displacement and rotations, +there were within error of %0.045 along the Z-axis, %0.088 +along the θx-axis and %0.148 along the θy-axis, respectively. +D. Hysteresis Compensation +Hysteresis is an undesired effect for precise manipulation +using a piezo-actuated micromanipulator. Hence, a series of +sinusoidal inputs were generated and applied to the developed +micromanipulator to investigate the capability of the proposed +control methodologies for hysteresis compensation. The results +are presented in Figure 13. It can be observed that the hystere- +sis effect was compensated/eliminated utilizing the SMC-NDO +control methodology. +Finally, the performances of micromanipulator and control +methodologies regarding very complex motion tracking and +hysteresis compensation are summarized in Table III. Accord- +ing to this table, the maximum and minimum improvements of +%100 and %67 were achieved, respectively, by implementing +the SMC-NDO control methodology. The minimum bound of +the improvement will increase if a smooth time-transitioned +trajectory is considered to track, which is the case in many +practical applications. +VI. CONCLUSION +A monolithic parallel Zθxθy micromanipulator with large +range, high resolution, and high bandwidth frequency was +presented. This positioning system utilized flexure-based com- +ponents to produce desired motions in three out-of-plane +directions. The linearized displacement relation between the +actuation space and the output Cartesian space of the mi- +cromanipulator was established utilizing FEA. Hence, the +micromanipulator’s workspace was obtained implementing the +inverse kinematics. Additionally, stress level in the flexure +hinges was found to be below the yield point of the ma- +terial (ABS) under the maximum input of the PEAs, thus +verifying the repeatability and stability of the developed mi- +cromanipulator for the precise manipulation tasks. An SMC +control methodology with nonlinear disturbance observer was +introduced and utilized for the experimental evaluation of the +micromanipulator. This control methodology was established +to track very complex desired motion trajectories with high +accuracy. It was also designed to accommodate system para- +metric uncertainties, cross-axis coupling, and nonlinearities +including unknown disturbances and the hysteresis effect in +the Zθxθy micromanipulator. The stability and robustness of +the proposed closed-loop control technique (SMC-NDO) were +proved, and the convergence of the position tracking errors to +zero was guaranteed by the established system. Furthermore, +an experimental facility was established to investigate the +effectiveness of control methodologies and micromanipulator’s +high precision positioning capabilities. Based on the recorded +experimental results, the positioning system showed very fine +position and orientation resolutions of ±4nm, ±250nrad, and +±230nrad throughout the micromanipulator’s motion range of +±238.5µm × ±4830.5µrad × ±5486.2µrad. Finally, based +on the root mean square error (RMSE) analysis of trajectory + +20 +15 +0.4 +10 +0.3 +5 +(un) +0.2 +desired +0 +0.2 +0.3 +0.4 +XLabel +N +-5 +-10 +PID +-15 +SMC +SMC-NDO +-20 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +Z +(μm) +actual600 +400 +200 +(μrad) +2 +cesired +0 +2 +4 +6 +8 ++ +-200 +-400 +PID +SMC +SMC-NDO +-600 +-600 +-400 +-200 +0 +200 +400 +600 +6. +(μrad) +4 +actual600 +400 +N +0 +200 +(μrad) +2 +cesired +4 +-2 +0 +2 +-200 +-400 +PID +SMC +SMC-NDO +600 +-600 +-400 +-200 +0 +200 +400 +600 +(μrad) +y10 +motion tracking and maximum hysteresis investigation, high +resolution, high accuracy, hysteresis elimination, and conse- +quently the effectiveness of the SMC-NDO control methodol- +ogy were established. +ACKNOWLEDGMENT +This work was funded by the Australian Research Council +(ARC) Discovery Project (DP) grant, and the Australian +Research Council (ARC) Linkage Infrastructure, Equipment +and Facilities (LIEF) grant. +CONFLICT OF INTEREST +The authors declare that they have no conflict of interest. +REFERENCES +[1] M. 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Available: http: +//dx.doi.org/10.1007/s11071-015-2180-4 + diff --git a/OdE4T4oBgHgl3EQf9w5t/content/tmp_files/load_file.txt b/OdE4T4oBgHgl3EQf9w5t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2763ad50bc6f1862566e6dd0a86c6db53807c424 --- /dev/null +++ b/OdE4T4oBgHgl3EQf9w5t/content/tmp_files/load_file.txt @@ -0,0 +1,1003 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf,len=1002 +page_content='1 Experimental System Identification and Disturbance Observer-based Control for a Monolithic Zθxθy Precision Positioning System Mohammadali Ghafarian1,2, Bijan Shirinzadeh1, Ammar Al-Jodah1,3, Tilok Kumar Das1, Tianyao Shen1 Abstract—A compliant parallel micromanipulator is a mech- anism in which the moving platform is connected to the base through a number of flexural components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Utilizing parallel- kinematics configurations and flexure joints, the monolithic micromanipulators can achieve extremely high motion resolution and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In this work, the focus was towards the experimen- tal evaluation of a 3-DOF (Zθxθy) monolithic flexure-based piezo- driven micromanipulator for precise out-of-plane micro/nano positioning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The monolithic structure avoids the deficiencies of non-monolithic designs such as backlash, wear, friction, and improves the performance of micromanipulator in terms of high resolution, accuracy, and repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' A computational study was conducted to investigate and obtain the inverse kinematics of the proposed micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' As a result of computational analysis, the developed prototype of the micromanipulator is capable of executing large motion range of ±238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µm × ±4830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µrad × ±5486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2µrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Finally, a slid- ing mode control strategy with nonlinear disturbance observer (SMC-NDO) was designed and implemented on the proposed mi- cromanipulator to obtain system behaviours during experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The obtained results from different experimental tests validated the fine micromanipulator’s positioning ability and the efficiency of the control methodology for precise micro/nano manipulation applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The proposed micromanipulator achieved very fine spatial and rotational resolutions of ±4nm, ±250nrad, and ±230nrad throughout its workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Note to Practitioners—Piezo-actuated precision positioning systems play an increasingly important role in the fields of micro/nano manipulation robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' They have the advantages of fine resolution, high accuracy, fast response speed, and large output displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' However, such systems inherently exhibit vibration, hysteresis behaviors, and are affected by external disturbances that could cause oscillations and positioning errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' This study presents a robust control methodology implemented on a 3-DOF positioning system (Zθxθy), which is among the most prone system to be affected by existing disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' This control methodology is designed to improve the tracking performance in the presence of hysteresis nonlinearity, disturbances, and model- ing errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The effectiveness of the proposed control methodology is demonstrated by conducting a series of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Due to the ease of implementation, the developed control methodology can be applied to other positioning systems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Index Terms—Sliding mode control, Nonlinear disturbance observer, Precision positioning, Monolithic parallel manipulator, Amplification mechanism I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' INTRODUCTION 1Robotics and Mechatronics Research Laboratory (RMRL), Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 2Institute for Intelligent Systems Research and Innovation (IISRI), Deakin University, Geelong Waurn Ponds, VIC 3216, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 3The University of Western Australia, Perth, WA 6009, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Corresponding author: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='ghafarian@deakin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='au F Lexure joints have dominant superiority over traditional mechanical joints in precision engineering including mi- cromanipulation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Flexure-based parallel micro- manipulators benefit from the advantages of both flexure joints and parallel-kinematics configurations, and addition- ally utilize the important characteristics of micromanipulators such as frictionless motion, absence of mechanical play and backlash, and no need for lubrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' These features are important and effective for various micro/nano positioning and nano-alignment applications, and it is not surprising that flexure-based parallel micro/nano manipulation systems stand out among others as the key element in the ultra-precision technologies [1]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Wei and Xu [11] proposed a force- sensing cell microinjector based on a single-axis compliant small-stiffness mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Yong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [12] presented single- and dual-stage vertical positioners for high-speed piezoelectric nanopositioning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [13] presented a decoupled piezo-driven XY micro/nano positioning system with a travel range of ±27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='7 × ±26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6µm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [14] presented a custom made atomic force microscopy (AFM) which was integrated with a 3-DOF XYZ parallel-kinematics piezo-driven micromanipulator for high-speed imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [15] presented the design and analysis of two par- allel compliant piezo-driven 3-DOF micro/nano positioning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Meanwhile, the inverse Bouc-Wen (BW) model was applied as a feedforward hysteresis compensator in the feed- forward/feedback hybrid control method to compensate for the hysteresis of piezoelectric actuators (PEAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The proposed micromanipulator exhibited small translational and rotational workspaces of ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1×±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2×±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µm3 and ±112×±52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5× ±48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µrad3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [16] proposed the design of a 3-PRR (prismatic-revolute-revolute) XYθ micromanipula- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The Scott-Russell (SR) mechanism was utilized in the pro- posed design to magnify the displacement of the PEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [17] presented a 6-DOF piezo-driven micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' As a result of having no amplification mechanism, the workspace of the micromanipulator was very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [18] demonstrated the design, modeling, and experimental analysis of a piezo-driven XY micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The experimental results illustrated that the XY micromanipulator had a working range of ±75 × ±73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µm2 with the resolution of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='128µm and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='143µm in the X- and Y-directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Among the flexure-based parallel micromanipulators, the com- pliant Zθxθy type micro/nano positioning systems have be- come the research focus due to their important advantages for in out-of-plane positioning tasks [19]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Qu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [23] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='05358v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='RO] 13 Jan 2023 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 1: Monolithic Zθxθy parallel micromanipulator presented the design, modeling and test of a piezo-driven θxθy flexure-based micro/nano positioning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The exper- imental results indicated that the developed micromanipulator could achieve a workspace of ±515µrad × ±460µrad about its two working axes with a resolution of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [24] introduced the mechanical design, modeling and experimental tests of a large-angle Zθxθy macromanipulator driven by four small air-gap voice coil actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The proposed system could achieve rotational-motion ranges of ±41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='59mrad and ±41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='13mrad in the working axes, for which the Z- mode frequency was 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6Hz and the rotational ones were 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='45Hz and 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='09Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The motion resolution of the macromanipulator was ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='67µrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [25] presented an active vibration control system which was con- structed based on a non-monolithic 3-DOF Zθxθy micro/nano manipulation system with an in-plane dimension of 160mm (diameter) and an out-of-plane height of 60mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Cao and Chen [26] demonstrated the development of a system identifica- tion model for a commercially-available 3-DOF piezo-driven Zθxθy micromanipulator (P-558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='TCD, Physik Instrumente).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The system was driven by four PEAs and had a motion range of ±25µm×±250µrad×±250µrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The quasi-static analysis of a non-monolithic compliant tripod system for micro/nano positioning applications was presented by Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The proposed micromanipulator had an overall positioning range of ±41µm × ±330µrad × ±385µrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Considering the above- mentioned studies, a compliant monolithic 3-DOF piezo- driven micromanipulator was introduced by the authors with a larger workspace and fine resolution capable of executing three out-of-plane motions, one translation and two rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The structure of the proposed monolithic micromanipulator was optimized completely to have a maximum working range and bandwidth frequency higher than 100Hz [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Regardless of the types of micromanipulators used to perform micro/nano manipulation tasks, an effective motion tracking control strategy can improve the tracking performances of the system significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In addition, disturbances such as cross- coupling, parametric uncertainties, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' can practically affect and degrade the performance of a precision positioning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' TABLE I: Mechanical and physical properties of ABS Symbol Quantity Value ν Poisson’s ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='35 ρ Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='9087(g/cm3) E Young’s modulus 2200(MPa) σyield Tensile yield strength 31(MPa) σultimate Tensile ultimate strength 55(MPa) Therefore, designing and utilizing a disturbance observer- based control methodology to be able to estimate and com- pensate the effect of disturbances for achieving high precision applications is very beneficial [29]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [32] introduced a nonlinear disturbance observer (NDO) for robotic manipulators for various purposes such as friction compensa- tion, independent joint control, sensorless torque control, and fault diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Furthermore, the global exponential stability of the proposed NDO was guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Lau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [33] pre- sented an enhanced adaptive robust disturbance observer-based motion tracking control methodology for tracking a desired motion trajectory in the presence of unknown or uncertain system’s parameters, nonlinearities including hysteresis, and disturbances in the motion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The proposed control methodology was applied in a semi-automated hand-held ear surgical device for the treatment of Otitis Media with Effusion (OME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Motivated by the previous work [28], an experimental study of a large range piezo-driven spatial compliant monolithic parallel Zθxθy micro/nano manipulation system with a fine res- olution is presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Monolithically manufactured designs are very important for micro/nano applications and they are preferable in comparison with the assembled manip- ulation structures because of elimination of the unwanted fea- tures that affect a smooth and accurate nano-resolution manip- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Other advantages of the proposed micromanipulator are low manufacturing and material cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Nano-meter/radian resolution, large amplification ratio, repeatability, and stability are ensured due to the characteristics of the proposed mono- lithic micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' To investigate the motion range and decouple the micromanipulator’s motions, the inverse kinemat- ics is obtained using FEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The performances of the developed micromanipulator are investigated in the real-time experiments via three feedback control methodologies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Proportional- Integral-Derivative control (PID), sliding mode control (SMC), and nonlinear disturbance observer-based sliding mode control (SMC-NDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The role of the developed NDO is to compensate for the uncertain disturbances in the real-time experiments to achieve high precision manipulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In the end, the ex- perimental results, including frequency, resolution, and several complex trajectory motion tracking analyses are presented, and precise manipulations can be guaranteed by the developed monolithic micromanipulator and control methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' MECHANICAL DESIGN As presented in Figure 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' the structure of the micromanip- ulator consists of a fixed base platform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' six leaf-flexure-based parallelogram mechanisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' three flexure-based Scott-Russell amplification mechanisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' three flexure-based spherical joint Input Force 3-DOF Monolithic Micromanipulator MovingPlatform Input Force Input Force Flexure-Based Spherical Joint FixedBasePlatform Flexure-Based Scott-Russell AmplificationMechanism Leaf-Flexure-Based Parallelogram Mechanism3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 2: Reachable workspace of the developed micromanipulator modules, and a moving platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Two sets of leaf parallel- ograms are incorporated into the input and output points of the Scott-Russell amplification mechanism as prismatic joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' This linearizes the motion of the Scott-Russell mechanism and increases the micromanipulator’s stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Because PEAs produce a very small displacement as a proportion of their length, mechanical displacement amplification is inevitably required for large displacement applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The Scott-Russell amplification mechanism is a well established mechanical amplifier [16], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Additionally, it has the advantage of transforming into a horizontal input to a vertical output which is ideal for the compactness of the designed Zθxθy micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The platform needs to generate rotational motions along two different axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' A spherical joint, which is capable of rotating in three different axes is adopted for the connection between the platform and the manipulation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In comparison with previous designs, here the PEAs are placed outside of the micromanipulator not inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Thus, different sizes of PEAs can be used for the proposed monolithic design without affecting the geometry and overall dimensions of the structural design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The overall dimensions of the proposed de- sign are 201mm, 180mm, 75mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Developments in 3D printing enable complex geometries to be easily and economically manufactured in multiple materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Therefore, the proposed design is fabricated using high density and high accuracy 3D-printing from Acrylonitrile Butadiene Styrene (ABS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The mechanical and physical properties of ABS are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' WORKSPACE ANALYSIS The usable workspace is limited by material stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In order to maintain the stability, repeatability, and capabilities of the micromanipulator in precise manipulation, it is very important that the applied stress on the micromanipulator due to the load remains less than its tensile yield strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Using the stress and safety factor analyses in FEA software (ANSYS), the maximum Von-Mises stress and the minimum safety factor occur when the input displacement of 90µm is applied to the micromanipulator as the input of the three PEAs simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The values of maximum Von-Mises stress and minimum safety factor corresponding to the 90µm inputs were 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='585MPa and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='71, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' It is worth noting that the obtained minimum safety factor was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='71, as the safety factor must always be greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Therefore, more input force/displacement could have been applied to obtain a larger workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' However, the value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='71 is considered as a lower limit for the safety factor of the micromanipulator to avoid some deficiencies including creep, fatigue, and mechanical failure of the micromanipulator in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Three PEAs (Physik Instrumente P-843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='60) with a maxi- mum displacement of 90µm were assumed to be used for analysing the workspace based on the FEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The inverse kine- matics of the micromanipulator was computed by evaluating the motion of the system across 80 inputs, spanning the full input range, and performing a regression on the resulting outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The best fit inverse kinematics is computed as: J−1 = � � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0110 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1877 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0095 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0053 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0093 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0056 � � (1) The resultant workspace of computational analysis is pre- sented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The values of motions along Z, θx, and θy axes are ±238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µm, ±4830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µrad, and ±5486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2µrad, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Since the determinant of the Jacobian matrix is non-zero, therefore there is no singularity in the workspace of the micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' CONTROLLER DESIGN The accuracy of a manipulation task is very dependent on eliminating undesired disturbances affecting the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Therefore, in this section, establishment of control method- ologies will be presented for the utilization in the experi- mental study to overcome this challenging issue for precise micro/nano manipulation applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Sliding Mode Control (SMC) In general, flexure-based micromanipulators are structures comprising solid links and flexure hinges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The monolithic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 3: Block diagram of the proposed SMC-NDO control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='methodology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Rx (μrad) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Ry (μrad) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Ry (μrad) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='3000 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='z(μm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='z (μm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Rx (μrad)[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Plant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Trajectory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Inverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='[9a] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Sliding Mode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Piezo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Capacitive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Generator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Kinematics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Amplifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Sensors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='+q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Nonlinear Disturbance Observer4 construction reduces assembly errors and guarantees high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' A lumped parameter dynamic model for a flexure- based micromanipulator can be given by: mq ¨q + cq ˙q + kqq = dq + uq (2) where q represents the three output motions of the microma- nipulator (Z, θx, θy), and mq, cq, and kq are the equivalent mass, damping and stiffness in the corresponding axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' dq and uq are the lumped disturbance and control action in a given axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' A state-space representation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (2) can be obtained by considering x1 = q and x2 = ˙q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The tracking error between the actual and desired motions can be introduced as follows: e = x1 − x1des (3) There are two steps in the design of an SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The first step is designing a sliding surface so that the plant restricted to the sliding surface has a desired system response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' This means Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 4: Schematic diagram of sensing, control, and experimen- tal research facility a b c Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 5: Modal analysis of the micromanipulator: (a) Z-resonant frequency (b) θx-resonant frequency (c) θy-resonant frequency the state variables of the plant dynamics are constrained to satisfy another set of equations which define the so-called switching surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The second step is constructing a switched feedback gains necessary to drive the plant’s state trajectory to the sliding surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' These strategies are built on the generalized Lyapunov stability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In this study, a PID sliding surface was selected to reduce the steady-state error and improve the transient response speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The selected sliding surface is given by: s = λpe + λi � t 0 edt + λd ˙e (4) where λp, λi, and λd are positive proportional, integral, and derivative parameters to be selected, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The SMC Physik Instrumente AmplifierE-509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='C3A h52 255 255 3-AxisPiezo Controller Physik Instrumente E-500 500 500 540 Elliot ScientificMDE263 3-axis micropositioners (PhysikInstrumente) Capacitive Sensor D-050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='00 (Physik Instrumente) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='8 MicromanipulatorElliotScientificMIDE263 axismicropositioners Micromanipulator P-843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='60PEAs (Physik Instrumente) Capacitive Sensor D-050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='00 (Physik Instrumente) Base Platform mounted on a vibration-isolated optical table20 10 Magnitude (dB) 10 Experimental Estimated 20 101 102 Frequency (Hz)20 10 Magnitude (dB) 10 Experimental Estimated 20 101 102 Frequency (Hz)30 Experimental Estimated 25 Magnitude (dB) 20 15 10 5 0 10 1 102 Frequency (Hz)5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 6: System response to the applied staircase input law is intended to force the tracking error e to approach the sliding surface and then proceed to the origin along the sliding surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Hence, the sliding surface needs to be stable, and therefore yields to the fact that ˙s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Additionally, the control action of the SMC approach consists of two parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' equivalent control and switching control as follows: u = ueq + usw (5) Thus, it can be realized that to be able to force the actual motions to converge to the desired ones, the equivalent control and switching control need to be as follows: ueq = cx2 + kx1 + m¨x1des − m λi λd e − m λp λd ˙e − ˆd usw = −a1s − a2sgn(s) (6) where a1 and a2 are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The term ˆd is the estimation of the disturbance which will be derived later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The deviation between the actual and estimated disturbance is defined by, ˜d, and can be given as follows: ˜d = d − ˆd (7) Lyapunov stability function is used to verify the design performance of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In order to have a stable controller, it is required that V must be bounded, which means that it Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 7: Star trajectory tracking results is important to prove ˙V ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Therefore, the derivative of Lyapunov function is derived as follows: ˙V = s ˙s = s{λd m usw + λd m d − λd m ˆd} = s{−k1s −k2sgn(s) + λd m ˜d} = −k1s2 − k2|s| +λd m ˜ds ≤ −k1s2 − |s|(k2 − λd m ˜d) (8) where k1 and k2 are equal to λda1 m and λda2 m , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The stability of the designed controller is guaranteed, if the inequality k2 ≥ λd m ˜d is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Furthermore, the discontinuity of the signum function in the SMC law can cause chattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Therefore, in order to avoid this problem, the discontinuous signum function is replaced by the contin- uous hyperbolic tangent function (tanh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In other words, the function is used as an approximator of the signum function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Nonlinear Disturbance Observer (NDO) An NDO is designed to estimate the undesired disturbance dq, introduced in the dynamic model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' dq represents all disturbances that may be caused by unmodeled dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The objective is to design an observer such that the estima- tion ˆd yielded by the observer exponentially approaches the disturbance d under any q and ˙q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The dynamic model of the micromanipulator, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (2), can be rewritten in the general description of a form as follows: ˙x = f(x) + g1(x)u + g2(x)d (9) where x is a representation of the state variables x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Besides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' functions f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' and g2 can also be found to be as follows: g1(x) = g2(x) = 1 mq f(x) = − cq mq x2 − kq mq x1 (10) The NDO is designed according to the following formula- tions [34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [35]: ˆd = z + p(x) ˙z = −l(x)g2(x)z − l(x){g2(x)p(x) + f(x) + g1(x)u} (11) where ˆd and z are the estimate of the undesired disturbance and the internal state of the nonlinear observer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 5 PID SMC 4 SMC-NDO Desired 3 (μrad) 2 1 JLI 0 0 2 4 6 8 10 12 14 16 18 20 22 t (s)6 PID SMC 5 SMC-NDO Desired 4 (μrad) 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 t (s)600 PID SMC 400 SMC-NDO Desired 200 (μrad) 0 200 400 600 20 15 10 5 0 5 10 15 20 Z (μm)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='12 PID SMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1 SMC-NDO Desired 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='08 uwh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='06 (ur) N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='04 15141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='02 0 2 4 6 8 10 12 14 16 18 20 22 t (s)6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 8: Experimental control tracking errors in the star trajectory and p(x) is a nonlinear function to be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The nonlinear observer gain l(x) is defined as follows: l(x) = ∂p(x) ∂x (12) It can be shown that ˆd approaches d exponentially if p(x) is chosen such that: ˙˜d + ∂p(x) ∂x g2(x) ˜d = 0 (13) Therefore, if p(x) is chosen as lx2, it can be realized from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (13) that the designed NDO yields to following form in the time domain: ˜d(t) = ˜d(0)exp(− l mt) (14) According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (14), if l is chosen to be a positive constant, when time goes to infinity the disturbance estimation error, ˜d, converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Therefore, the designed NDO is globally exponentially stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Robustness Analysis of SMC-NDO The robustness of the proposed SMC-NDO is proven in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The feedback control of the system described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 9: RMRL logo trajectory tracking results (2) is robust, if control law (5) is applied and it satisfies the conditions of: lim s→0+ ˙s < 0 , lim s→0− ˙s > 0 (15) The above-mentioned conditions illustrate that if the phase trajectory around s = 0 points to the s hyper-plane, then it will enter the sliding mode when the hyper-plane s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' By substituting control law (5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (2), the equation of motion of the closed-loop system is obtained as: ¨q = 1 mq � ˜dq − a1s − a2tanh(s) � + � ¨qdes − λi λd e − λp λd ˙e � (16) Moreover, by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (16) into the differentiation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (4), the dynamic sliding mode equation is obtained as: ˙s = λp ˙e + λie + λd¨e = λp ˙e + λie + λd(¨q − ¨qdes) ≈ ˜dq − a1s − a2tanh(s) (17) Utilizing the two conditions (15), and the stability condition that was obtained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='A, (a2 ≥ ˜d), it is trivial that the robustness conditions are satifisfied as following: lim s→0+ ˙s = ˜dq − a2tanh(s) < 0 lim s→0− ˙s = ˜dq − a2tanh(s) > 0 (18) Thus, the phase trajectory around s = 0 enters the sliding mode in the hyper-plane and s = 0, so the PID sliding surface (4) is: s = (λp + λis−1 + λds)e = 0 (19) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (19) and having the knowledge of ˙e = ˙x2 − ˙x2des, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (9) is rewritten as following: x1 = x1des + e x2 = x2des − λp λd e − λi λd s−1e (20) Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (20) becomes the state equation when the system enters the sliding mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' We know that the second-order system Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (2) can be expressed by the first-order state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (20) and that the dynamic characteristics of the system are independent of d(t), therefore, SMC-NDO is highly robust when the system enters the sliding mode, thereby ensuring that the system exhibits a good dynamic response and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='25 PID SMC SMC-NDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2 (wn) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='05 0 0 2 4 6 8 10 12 14 16 18 20 22 t (s)9 PID 8 SMC SMC-NDO 7 6 5 Error e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 t (s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4 PID SMC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2 SMC-NDO ,-axis (μrad) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2 0 0 2 4 6 8 10 12 14 16 18 20 22 t (s)600 PID SMC SMC-NDO 400 Desired 200 (μrad) 0 200 400 600 009- 400 200 0 200 400 600 a, (μrad)7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 10: Experimental control tracking errors in the RMRL logo trajectory D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Time Convergence Stability Analysis of SMC-NDO The time convergence of the sliding mode equation is investigated in this section when the disturbance observer and chattering reduction methods are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' [36] Assume that a continuous definite function V (t) ≥ 0 satisfies the subsequent differential inequality ˙V (t)+ β1V α(t) + β2V (t) ≤ 0, where β1, β2, and α are the positive constants, and the range of α is between 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Based on which the function V (t) converges to zero in finite time given as: ts = 1 β2(1 − α) ln β2V (1−α)(0) + β1 β1 (21) Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Consider system described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' If the surface manifold and control law are chosen according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (4) and (6), then sliding mode enforcement along the sliding surface, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (4), can be validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' As a result, the system states will converge to the desired reference trajectories and tracking errors will converge to zero in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Proof of Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Consider the Lyapunov candidate func- tions of the form as: V = 1 2s2 (22) Taking the time derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (22) and utilizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (17) yield to: ˙V = s{ ˜d − a1s − a2tanh(s)} ≤ −a1s2 − a2|s| (23) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 11: 3D-Archimedean spiral trajectory tracking results Let β1 = √ 2a2, β2 = 2a1, and α = 1 2, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (23) can be rewritten as: ˙V (t) + β1V α(t) + β2V (t) ≤ 0 (24) Then, according to Lemma, the differential inequality (24) yields the finite-settling time as represented by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (21), and that completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Finally, Figure 3 provides the information on how the SMC- NDO was implemented in the real-time experiment to achieve remarkable performances from the micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' EXPERIMENTS AND RESULTS Experimental tests were carried out to investigate the me- chanical and dynamic performances of the developed flexure- based micromanipulator, as well as the efficiency of the proposed control methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The schematic diagram of the experimental set-up is presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The ba- sic operation of the flexure-based micromanipulator was to change the drive voltage from the voltage control unit (am- plifier module from Physik Instrumente) to the three PEAs from Physik Instrumente (model P-843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='60) to further drive the micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' During installation of the PEAs, pre- compression forces were applied to keep the actuators’ tip and the micromanipulator in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' These PEAs are multilayer PZT stacked ceramic translators capable of 90µm displace- ment corresponding to a range of operating voltage from 0 to 100V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The position of the moving platform was measured by three capacitive sensors (Physik Instrumente D-050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The TABLE II: Control parameters Description Parameter Value PID PID constants kp 50 ki 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5e6 kd 0 SMC PID sliding surface constants λp 50 λi 13e6 λd 100 Switching control constants a1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5 a2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6 SMC-NDO Nonlinear observer gain l 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='09 PID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='08 SMC SMC-NDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='07 (wr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='06 z-axis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='04 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='01 0 0 2 6 8 10 12 14 16 18 20 22 t(s)8 PID SMC SMC-NDO 6 axis (μrad) 5 3 0 2 4 6 8 10 12 14 16 18 20 22 t (s)25 PID SMC SMC-NDO 20 axis (μrad) 15 Error 10 5 0 0 2 4 6 8 10 12 14 16 18 20 22 t (s)450~ 400~ 350~ 300~ 250~ (μrad) 200~ 150~ PID 100 (pejr) 168 SMC 166 SMC-NDO 164 50 =Desired 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='72 105 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='74, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='76, 104 50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='78 Z (μm) ox (μrad) 400 200 15 10 0 0 200 5 10 , (μrad) 400 15 Z (μm)8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 12: Experimental control tracking errors in the 3D-Archimedean spiral trajectory capacitive sensors had a circular active area with a 4 mm radius surrounded by an annular guard ring, with a specified working range of 50µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The electronic interface (Physik Instrumente E-509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='C3A) to the capacitive sensors provided a distance measurement as an analog voltage, which was recorded at the control computer with a 16-bit analog-to-digital converter (ADC) and at the rate of 10kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In order to reduce the external vibration disturbances, all experimental tests were performed on a vibration-isolated optical table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' System Identification System identification of the developed micromanipulator was conducted to investigate the dynamic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Hence, a sinusoidal sweep signal with the frequency linearly varying from 1Hz to 250Hz was applied to the PEAs input, one at a time, in order to excite the modes of vibration of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The frequency responses to these inputs are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Resonance was observed in the response along the Z, θx, and θy axes occured at 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4Hz, 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1Hz, and 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Finally, three continuous-time transfer functions were identified from the frequency response along each motion axis and formulated as follows: TFz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='774e05 s2 + 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='44s + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='08e05 (25) TFθx = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='792e05 s2 + 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='16s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='186e06 (26) TFθy = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='454e06 s2 + 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='13s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='346e06 (27) Considering the transfer functions (15)-(17), mq, cq, and kq can easily be calculated for implementing into to the SMC and SMC-NDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In the next stage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' closed-loop experiment, the control parameters were found by empirical rules based on observation of the response to reference and disturbance changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' These selected control parameters for the PID, SMC, and SMC-NDO control schemes are presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Resolution characterization of the system To test the closed-loop system positioning resolution, the staircase response of the positioner was measured along each working axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The magnitude of each step was chosen to be 20nm for the translational axis and 900nrad for the rotational axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The obtained results from the three control techniques are presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The best resolution result was captured utilizing the SMC-NDO control technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The minimum resolution was found to be 8nm, 500nrad, and 460nrad along the Z, θx, and θy axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The positioning resolution is limited by the measurement noise in motion axes, quantization in the measurement of rotation, the resolution of capacitive sensors, and the output resolution of the data acquisition card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Motion tracking characterization of the system To further illustrate the motion abilities of the developed micromanipulator alongside with the designed control ap- proaches, three complex curves tracking experiments were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' In which, a star, Robotics and Mechatronics Re- search Laboratory (RMRL) logo, and 3D-Archimedean spiral were selected and designed as the target tracking trajectories due to their high level of complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The tracking results are presented in Figures 7 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The results of the trajectory tracking and tracking errors of the star trajectory implementing the three proposed control schemes are presented in Figures 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The best tracking error was obtained by implementing SMC-NDO and the maximum values of error were 14nm for the Z-axis, 700nrad and 494nrad for the θx and θy axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Figure 7 illustrates the projection of the tracked trajectory on the ZY- plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The RMRL logo was designed as a symbol of the most complex trajectory to verify the performance of the microma- nipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The results of the trajectory tracking and tracking errors are presented in Figures 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The projection of the designed trajectory on the XY-plane is presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' According to the results, the micromanipulator was capable of tracking the designed logo with high precision and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' It must be noted that by using sensors with a higher resolution, more accurate motion tracking can be achieved even with the same control methodologies proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The tracking results of the 3D-Archimedean spiral are pre- sented in Figures 11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' These two figures provide the difference between the desired and actual trajectories while implementing different control methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Same as the previous trajectories tracking, the SMC-NDO control scheme was observed to outperform the SMC and PID controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2 PID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='18 SMC SMC-NDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='14 (wr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='12 Z-axis ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1 rror 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='02 0 0 2 4 6 8 10 12 14 16 18 20 22 t(s)PID SMC 6 SMC-NDO 5 axis (μrad) 4 3 Error 2 0 0 2 4 6 8 10 12 14 16 18 20 22 t (s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='8 PID 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6 SMC SMC-NDO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4 v-axis (μrad) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='8 rror 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2 0 0 2 4 6 8 10 12 14 16 18 20 22 t (s)9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 13: Hysteresis cancellation of different control methodologies TABLE III: Summary of the numerical results of trajectory tracking and hysteresis compensation Controller Z-axis θx-axis θy-axis Trajectory tracking RMSE (µm) %Improvement (SMC-NDO/PID) RMSE (µrad) %Improvement (SMC-NDO/PID) RMSE (µrad) %Improvement (SMC-NDO/PID) Star PID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1094 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='9298 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='3903 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='92 SMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1012 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='6336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='3596 SMC-NDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='3615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1291 RMRL logo PID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0239 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='9472 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='61 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='8958 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='07 SMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0184 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2886 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='9058 SMC-NDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='3652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='7942 3D-Archimedean spiral PID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0819 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='7101 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5296 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='25 SMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0660 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4345 SMC-NDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='0026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='1258 Hysteresis compensation %Hysteresis max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' %Improvement (SMC-NDO/PID & SMC) %Hysteresis max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' %Improvement (SMC-NDO/PID & SMC) %Hysteresis max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' %Improvement (SMC-NDO/PID & SMC) PID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='49 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='44 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='54 100 SMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='44 SMC-NDO 0 0 0 Also, the SMC provided a better tracking performance than the PID control scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Considering SMC-NDO, the maximum tracking error along the Z, θx, and θy axes were observed to be 11nm, 644nrad, and 650nrad, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' On the other hand, with respect to the output displacement and rotations, there were within error of %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='045 along the Z-axis, %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='088 along the θx-axis and %0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='148 along the θy-axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Hysteresis Compensation Hysteresis is an undesired effect for precise manipulation using a piezo-actuated micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Hence, a series of sinusoidal inputs were generated and applied to the developed micromanipulator to investigate the capability of the proposed control methodologies for hysteresis compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The results are presented in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' It can be observed that the hystere- sis effect was compensated/eliminated utilizing the SMC-NDO control methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Finally, the performances of micromanipulator and control methodologies regarding very complex motion tracking and hysteresis compensation are summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Accord- ing to this table, the maximum and minimum improvements of %100 and %67 were achieved, respectively, by implementing the SMC-NDO control methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The minimum bound of the improvement will increase if a smooth time-transitioned trajectory is considered to track, which is the case in many practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' CONCLUSION A monolithic parallel Zθxθy micromanipulator with large range, high resolution, and high bandwidth frequency was presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' This positioning system utilized flexure-based com- ponents to produce desired motions in three out-of-plane directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The linearized displacement relation between the actuation space and the output Cartesian space of the mi- cromanipulator was established utilizing FEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Hence, the micromanipulator’s workspace was obtained implementing the inverse kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Additionally, stress level in the flexure hinges was found to be below the yield point of the ma- terial (ABS) under the maximum input of the PEAs, thus verifying the repeatability and stability of the developed mi- cromanipulator for the precise manipulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' An SMC control methodology with nonlinear disturbance observer was introduced and utilized for the experimental evaluation of the micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' This control methodology was established to track very complex desired motion trajectories with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' It was also designed to accommodate system para- metric uncertainties, cross-axis coupling, and nonlinearities including unknown disturbances and the hysteresis effect in the Zθxθy micromanipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' The stability and robustness of the proposed closed-loop control technique (SMC-NDO) were proved, and the convergence of the position tracking errors to zero was guaranteed by the established system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Furthermore, an experimental facility was established to investigate the effectiveness of control methodologies and micromanipulator’s high precision positioning capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Based on the recorded experimental results, the positioning system showed very fine position and orientation resolutions of ±4nm, ±250nrad, and ±230nrad throughout the micromanipulator’s motion range of ±238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µm × ±4830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='5µrad × ±5486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2µrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Finally, based on the root mean square error (RMSE) analysis of trajectory 20 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='3 5 (un) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2 desired 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4 XLabel N 5 10 PID 15 SMC SMC-NDO 20 20 15 10 5 0 5 10 15 20 Z (μm) actual600 400 200 (μrad) 2 cesired 0 2 4 6 8 + 200 400 PID SMC SMC-NDO 600 600 400 200 0 200 400 600 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' (μrad) 4 actual600 400 N 0 200 (μrad) 2 cesired 4 2 0 2 200 400 PID SMC SMC-NDO 600 600 400 200 0 200 400 600 (μrad) y10 motion tracking and maximum hysteresis investigation, high resolution, high accuracy, hysteresis elimination, and conse- quently the effectiveness of the SMC-NDO control methodol- ogy were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was funded by the Australian Research Council (ARC) Discovery Project (DP) grant, and the Australian Research Council (ARC) Linkage Infrastructure, Equipment and Facilities (LIEF) grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' CONFLICT OF INTEREST The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Ghafarian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content='4960080 [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Huo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Liang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' Tian, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE4T4oBgHgl3EQf9w5t/content/2301.05358v1.pdf'} +page_content=' 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file mode 100644 index 0000000000000000000000000000000000000000..188977b2dface4b2705b28af9739944fa29dd31a --- /dev/null +++ b/OtFIT4oBgHgl3EQfeCvq/content/tmp_files/2301.11273v1.pdf.txt @@ -0,0 +1,2004 @@ +AlignGraph: A Group of +Generative Models for Graphs +Kimia Shayestehfard +Dana Brooks +Stratis Ioannidis∗ +Abstract +It is challenging for generative models to learn a dis- +tribution over graphs because of the lack of permuta- +tion invariance: nodes may be ordered arbitrarily across +graphs, and standard graph alignment is combinatorial +and notoriously expensive. We propose AlignGraph, a +group of generative models that combine fast and effi- +cient graph alignment methods with a family of deep +generative models that are invariant to node permuta- +tions. +Our experiments demonstrate that our frame- +work successfully learns graph distributions, outper- +forming competitors by 25% − 560% in relevant per- +formance scores. +1 +Introduction +Graph generative models have applications across do- +mains like chemistry, neuroscience and engineering. +Generative models learn a distribution over graphs, and +are used to subsequently sample from this distribution. +They can, e.g., be used to predict interfaces between +proteins during drug design and discovery [1, 2], or to +perform hypothesis testing and simulation for social net- +works, when collecting real graphs is difficult [3, 4]. +Traditional generative models for graphs such as +the Barabási-Albert [5], Erdös-Rényi [6], and stochastic +block models [7] generate graphs with provable formal +properties but which often lack realism. For example, +the Erdös-Rényi model produces graphs with a light- +tailed degree distribution [5, 8], while the Barabási- +Albert model fails to generate graphs with a high clus- +tering coefficient [9]. Deep generative models such as +variational autoencoders [10] and graph recurrent neu- +ral networks [11, 12] have shown great potential in learn- +ing distributions from graph datasets, at greater fidelity +than traditional models. However, learning a distribu- +tion of graphs over a dataset poses a significant chal- +lenge because of the lack of permutation invariance, +since graph nodes may be subject to arbitrary permuta- +tions across graphs: the correspondence between nodes +in different graph samples may be a priori unknown. +∗{kshayestehfard, brooks, ioannidis}@ece.neu.edu, Electrical and +Computer Engineering Department, Northeastern University, Boston, +MA, USA. +This is a problem because state-of-the-art genera- +tive models, like the ones listed above, rely on latent +node embeddings. +Such embeddings vary drastically +even under nearly isomorphic graphs [13]. In turn, this +can hamper the fidelity of the graph generation process +significantly. Note that this is a much harder setting +than, e.g., images or text, where inputs have a canonical +orientation. Finding the correspondence between graph +nodes is a notoriously hard problem [14, 15, 11, 16], and +it is exacerbated when the number of sampled graphs is +large. To that end, we propose AlignGraph, a group of +permutation invariant graph alignment methods com- +bined with their application to a group of base genera- +tive models. Our main contributions are as follows: +1. AlignGraph +incorporates +convex +graph +multi- +distance methods in training to achieve permuta- +tion invariance. We use these tools both as means +to construct alignment in a tractable fashion as well +as to create soft penalties in training. +2. AlignGraph is a general flexible framework. It can +be applied to a broad class of base generative mod- +els, enhancing their permutation invariance. +We +demonstrate this here by applying it to graph recur- +rent neural networks [11], gated recurrent attention +networks [12] and variational autoencoders [10] as +our base generative models. +3. We propose three methods that can be parallelized +to accelerate graph multi-distances. +Leveraging +parallelism, our methods speed up computation +by a 40× factor, while maintaining the alignment +accuracy and, in some cases, improving it. +4. We conduct experiments on both synthetic and real +data, showing that AlignGraph outperforms both +our base and other competitor models. We define +two performance scores to measure accuracy. We +then show that our model achieves 25% − 250% +improvement in those scores over base models and +62.5% − 4000% improvement over other competi- +tors. +2 +Related Work +Graph Embeddings. Graph embeddings map nodes +into a lower-dimensional space and have been ap- +arXiv:2301.11273v1 [cs.SI] 26 Jan 2023 + +plied to link prediction [10, 17, 16], node classification +[18, 19, 17, 20], and clustering [20]. Many of the state-of- +the-art graph embedding algorithms capture the relative +position of nodes on the embedding space [13]. Because +of the non-convexity of training objectives and the exis- +tence of multiple local minima, even isomorphic graphs +can map to completely different embeddings using the +same embedding algorithm [13]. This is further exac- +erbated when graphs are near-isomorphic (i.e., differ in +a few edges) as well as when the embeddings are ran- +domized [13]. Embeddings play a central role in graph +generative models (see Sec. 3.3), but lack of permuta- +tion invariance can introduce significant distortions. +Deep Generative Models. +Deep generative mod- +els can be categorized into three groups: +generative +adversarial networks (GANs), variational autoencoders +(VAEs), and auto-regressive models. +NetGAN [16] +learns the distribution of biased random walks over a +single graph. GraphVAE [14] and NEVAE [15] use VAEs +linking node embeddings to edges. GraphRNN [11] is +an auto-regressive model that constructs a graph se- +quentially over nodes and edges. +GRAN [12] is an- +other auto-regressive model that uses graph neural net- +works (GNNs) with an attention mechanism to generate +a block of nodes and edges sequentially. +Several of these methods contain techniques to par- +tially deal with permutation invariance. For example, +GraphVAE [14] uses an approximate graph matching to +penalize misalignment between each input graph and its +corresponding reconstructed graph. +NEVAE [15] and +GraphRNN [11] use a breadth-first-search node order- +ing scheme and GRAN [12] marginalizes over a family of +canonical node orderings to handle permutation invari- +ance. However, none of these methods address permu- +tation invariance by finding a consistent node ordering +across sampled graphs. In comparison, the graph align- +ment approach we introduce here does exactly this; in +addition, it is generic and can be applied to the broad +group of base generative models listed above to enhance +their permutation invariance (see also Sec. 6). +Graph distances. +Classic methods to compute the +distance between graphs include the edit distance [21, +22] and the maximum common subgraph distance [23, +24]. Although they are metrics, they are hard to com- +pute. Bento & Ioannidis [25] recently introduced a fam- +ily of metrics for graph distances that is computationally +tractable but limited to computing the distance between +two graphs. To compute distances among a larger group +of graphs, it is important that the distance function sat- +isfies alignment consistency [26]. There are works on +multi-distances that enforce this constraint [27, 28, 29]; +however, none of these methods satisfy generalizations +of the metric properties. Gromov-Wasserstein Learning +(GWL), proposed by Xu et al. [30] satisfies both of these +properties. However, GWL has cubic complexity and is +not applicable to recurrent neural networks. Recently, +two approaches were proposed by Kiss et al. [31] and +Safavi & Bento [32] to measure the distance among a +group of graphs; we describe both in detail in Sec 3.2. +Both satisfy alignment consistency and a generalization +of metric properties [31]. However, both are also slow +when applied to a large number of graphs. +We propose a framework to accelerate these two +graph multi-distance algorithms, by leveraging paral- +lelization and graph coarsening [33]. Graph coarsening +has been used in community detection [34, 35], graph +embeddings [36] and alignment between two graphs [37]. +We coarsen graphs using K-means clustering, and in- +corporate this accelerated graph alignment method into +our framework to address permutation invariance. To +the best of our knowledge, we are the first to accelerate +graph multi-distances by graph coarsening. +3 +Background +3.1 +Minimum Distance between two Graphs. +Let G = (V, E) be an undirected graph with node set +V = [m] ≡ {1, 2, . . . , m} and edge set E ⊆ [m] × [m], +represented by adjacency matrix A ∈ {0, 1}m×m. The +entries of this adjacency matrix are indexed by the +nodes in V . We denote the set that contains all such +matrices by Ω ⊆ Rm×m. Consider two graphs GA = +(V, EA), GB = (V, EB) with adjacency matrices A, B ∈ +Ω. +One way to measure the distance between these +two graphs is to find an alignment between nodes and +compute an edge discrepancy (i.e., edit distance [38, 21]) +between them. An alignment can be represented by a +permutation matrix P ∈ Pm, where: +Pm ≜ {P ∈ {0, 1}m×m; P1 = 1, P T 1 = 1}. +(3.1) +However, finding such an alignment is generally compu- +tationally intractable [25, 39]. Bento & Ioannidis [25] +introduce a distance function dS : Ω2 �−→ R, defined as: +dS(A, B) = min +P ∈Wm ∥AP − PB∥ + βtr(P T DA,B), +(3.2) +where β > 0 is a positive regularization parameter, +∥ · ∥ is a matrix norm, tr is the trace operator, matrix +DA,B ∈ Rm×m represents the dissimilarity between +nodes across the two graphs, and matrix P is a doubly +stochastic alignment matrix, that is, P ∈ Wm, where +Wm ≜ {P ∈ [0, 1]m×m; P1 = 1, P T 1 = 1}. +(3.3) +Matrix DA,B is generally a distance matrix, where each +element represents the pairwise distances between the +embeddings or features of nodes across two graphs. + +For example, for two matrices of graph embeddings +ZA ∈ Rm×d and ZB ∈ Rm×d that map nodes of a graph +into a lower-dimensional space, i.e. d < m, DA,B is: +DA,B = [Da,b]a∈V,b∈V ∈ Rm×m, and +(3.4a) +Da,b = ||zA +a − zB +b ||2, +∀ a ∈ V, b ∈ V, +(3.4b) +where zA +a +indicates the a-th row of matrix ZA, zB +b +indicates the b-th row of ZB. +Intuitively, the first +term in Eq. (3.2) is a probabilistic mapping between +nodes of two graphs and the second term penalizes the +dissimilarity between the embeddings of the nodes that +are mapped to each other. Eq. (3.2) is a pseudometric +and a convex optimization problem [25], and thus can +be computed efficiently via standard techniques. +3.2 +Minimum Distance among n Graphs. Con- +sider a distance function d(Gi, Gj) like Eq. (3.2) that +induces a (possibly stochastic) alignment matrix Pij be- +tween two pairs of graphs. To compute the minimum +distance between a group of n > 2 graphs, one could +simply generalize the distance function d(Gi, Gj) to mul- +tiple graphs, via d(G1, G2, . . . , Gn) = � +i,j∈[n] d(Gi, Gj). +However, such a generalization does not guarantee the +joint alignment between multiple graphs: that is, if Pij +aligns Gi with Gj, and Pjl aligns Gj with Gl, the align- +ment matrix Pil should keep the consistency of align- +ments under transitivity, i.e., Pil = PijPjl [32]. This +property is known as alignment consistency. +We de- +scribe next two distance functions that induce align- +ments that satisfy this property. +3.2.1 +Fermat Distance. Let d(A, B) be a metric +for two graphs such that d : Ω2 �−→ R. +Then the +Fermat distance function [31] associated with d is the +map of dF : Ωn �−→ R defined by: dF (A1, A2, . . . , An) = +min +A0∈Ω +�n +i=1 d(Ai, A0), capturing the distance among a +set of graphs. If d is a metric then the Fermat distance +function induced by d is a so-called n-metric [32]. The +Fermat distance function induced by Eq. (3.2) is: +dF (A1, . . . , An) = +min +A0∈Ω +Pi∈Wm,∀i∈[n] +�n +i=1 G(Pi, A0; Ai), +(3.5) +where D = 0 and Wm represents the set of doubly +stochastic matrices and G : Wm × Ω × Ω �−→ R is: +G(Pi, A0; Ai) = ∥AiPi − PiA0∥. +(3.6) +Graph G0, corresponding to A0, represents the center +of set G. +The Fermat distance function in Eq. (3.5) +is a pseudo n-metric [32]. This optimization problem is +non-convex; nonetheless, it can be solved approximately +via alternating minimization (AM): Eq. (3.5) then re- +duces to solving two alternating convex optimization +problems. The first one has nm2 parameters and nm2 +constraints. The second problem reduces to n optimiza- +tion problems with m2 parameters and m2 constraints. +More details regarding AM iterations can be found in +App. A. +3.2.2 +G-align distance. Safavi & Bento [32] intro- +duce the G-align distance function, a convex function +that satisfies metric properties and alignment consis- +tency. Consider the map dG : Ωn �−→ R defined by: +dG(A1, . . . , An) = min +Pij∈S +1 +2 +� +i,j∈[n] G(Pij; Ai, Aj), +(3.7) +where G(Pij; Ai, Aj) is given by Eq. (3.6) (with D = 0), +S = {{Pij}i,j∈[n] : Pij ∈ Pm, ∀i, j ∈ [n], +PilPlj = Pij, ∀i, j, l ∈ [n], Pii = I, ∀i ∈ [n]}, +(3.8) +where Pm is the set of permutation matrices and +PilPlj = Pij captures alignment consistency. +Let P ∈ Rnm×nm be a matrix with n2 blocks such +that the (i, j)-th block is Pij, i.e.: +P = +� +� +I +P12 P13 ... P1n +P21 +I +P23 ... P2n +... +... +... +... +... +Pn1 Pn2 Pn3 ... +I +� +� . +(3.9) +Safavi & Bento [32] prove that the alignment consis- +tency is equivalent to P ⪰ 0 (see Lemma 4 in Safavi & +Bento [32]). By relaxing the permutation matrices con- +straint in (3.7) to a doubly stochastic constraint, the +G-align distance function is as follows: +dG(A1, . . . , An) = +min +Pij∈Wm, +Pii=I,P ⪰0 +1 +2 +� +i,j∈[n] +G(Pij; Ai, Aj). +(3.10) +This is a pseudo n-metric (see Theorem 5 and Remark 4 +in Safavi and Bento [32]) and a convex optimization +problem with O(n2m2) variables and n constraints. In +practice, this problem can be solved via optimization +toolboxes such as CVXPY [40] as well as the Frank- +Wolfe algorithm (FW) [41]; the latter is outlined in +App. B. +Despite convexity, the quadratic nature of P (in +terms of n and m) in G-align distance and Pi ∈ Wm, +i ∈ [n] (in terms of m) in Fermat distance makes these +computations expensive. We address this in Section 5. +3.3 +Graph Generative Models. Given a set of +undirected graphs G = {G1, G2, . . . , Gn} sampled from +p(G), for each graph Gi(Vi, Ei), ∀i ∈ [n], we denote the +adjacency matrices by Ai ∈ Rm×m and feature matrices +by Xi ∈ Rm×f. Features could be either one-hot node +indicator vectors or consist of graph characteristics, + +such as, e.g., node degrees. The nodes of each graph are +mapped to a latent embedding space via a deep neural +network, parameterized by φ [20]: +Zi = fφ(Ai, Xi), +(3.11) +where Zi = {zi1, zi2, · · · , zim} denotes the hidden node +representations and φ represents parameters of the deep +neural network encoder. The decoder parameterized by +θ takes the hidden representations and reconstructs the +adjacency matrix, i.e.: +ˆAi = gθ(Zi), +(3.12) +where ˆAi is the estimated adjacency matrix. Both the +encoder and decoder can be randomized, and induce a +distribution pφ,θ over graphs. A loss often used to train +parameters over graphs is the negative log likelihood: +L(φ, θ; A, X) = − �n +i=1 log pφ,θ(Ai), +(3.13) +where A = {A1, A2, . . . , An} denotes the set of adja- +cency matrices and X = {X1, X2, . . . , Xn} represents +the set of feature matrices. +Several existing generative models can be described +using the general framework described by Eq. (3.11)- +(3.13). GraphRNN [11] is an auto-regressive model that +generates node embeddings sequentially: fφ (3.11) is +an RNN that encodes the states of graph generated so +far, and gθ (3.12) is a Gated Recurrent Unit (GRU) +model that outputs the distribution of the next node’s +adjacency vector. GRAN [12] is also an auto-regressive +model that generates the graph in a block by block ba- +sis. +In this model, fφ (3.11) is a GRU that uses an +attention-weighted sum over the neighborhood of each +node to produce the corresponding node embedding and +gθ (3.12) models the probability of generating edges +in a block comprising multiple rows of a graph adja- +cency matrix via a mixture of Bernoulli distributions. +In VAE [10], fφ (3.11) is a probabilistic encoding de- +noted by qφ(Zi|Ai, Xi). There is a prior over the latent +variables pz(Zi) ∼ N(0, I) and gθ (3.12) is defined as +the inner product between latent variables. +The loss +function (3.13) is further approximated by a variational +lower bound of the log-likelihood [42]. All these models +can be trained by minimizing the loss (3.13) via stan- +dard gradient methods. +4 +AlignGraph +We present AlignGraph, our framework for enhancing +the permutation invariance of base generative models. +AlignGraph can be applied to any base generative model +of the form given by Eq. (3.11)- (3.13): we indeed apply +it to GraphRNN [11], GRAN [12] and VAE [10] in Sec. 6. +We consider three AlignGraph variants, described next. +4.1 +G-align-Single. We begin by aligning sampled +graphs. +To do this, we first compute P by solving +the problem in Eq. (3.10) [32]. We take the first block +column of P , i.e., {Pi1}n +i=1, and project each Pi1 ∈ Wm +onto the set of permutation matrices, i.e.: +˜ +Pi1 = ΠPm(Pi1), +(4.14) +where ΠPm is the orthogonal projection to Pm. This +can be done in polynomial time with the Hungarian +algorithm [43]. Given these permutation matrices, we +align all graphs and features with the first graph, via: +˜Ai = ˜ +Pi1 +T Ai ˜ +Pi1, +˜ +Xi = ˜ +Pi1 +T Xi +∀i ∈ [n]. +(4.15) +Note that, by alignment consistency +(3.8), this could +be done on any block column; our selection of {Pi1}n +i=1 +is arbitrary. Given the alignment matrices { ˜ +Pi1}i∈[n], +the adjacency matrices A and feature matrices X, we +aim to solve the following optimization problem: +min +φ,θ +1 +n +�n +i=1 L(φ, θ; ˜Ai, ˜ +Xi), +(4.16) +where φ and θ are the DNN parameters and L(·) is +the loss defined in Eq. (3.11)-(3.13). Eq. (4.16) can be +solved via stochastic gradient descent (SGD). +4.2 +G-align-Double. In our second approach, we +(a) compute a central graph across the graph set and (b) +enforce that this graph and aligned graphs are jointly +embeddable in the same space. +To that end, we use +two base generative models combined with the G-align +distance function. We again compute P from the G- +align distance [32] by solving Eq. (3.10) and project +each Pi1 ∈ Wm onto the set of permutation matrices +{ ˜ +Pi1}i∈n via (4.14). +We then align the adjacency +matrices and feature matrices as in Eq. (4.15). We then +estimate the center graph G0 of set G, i.e. the graph +which has the minimum distance from all the graphs in +the graph set via: +min +ˆ +A0j,k,∈[0,1],∀j,k∈[m] +�n +i=1 ∥ ˜Ai − ˆ +A0∥. +(4.17) +Prob. (4.17) is convex and can be solved via standard +methods: the objective has n terms, and the problem +has m2 parameters and O(m2) constraints. +Since +ˆA0j,k ∈ [0, 1], ∀j, k ∈ [m], we binarize the elements +of the adjacency matrix for G0 by using a threshold. +Once we estimate G0, given the permutation matrices +{ ˜ +Pi1}i∈n and the graph set G with one-hot encoding +feature matrices X, we train two generative models of +the chosen type. +We train the first with G0 and the +second with all aligned {Gi}n +i=1 . We train the generative + +models jointly, by penalizing the distance between the +embeddings of Gi and G0, i.e.: +min +Φ,Θ +1 +n +�n +i=1[L(φ, θ; ˜Ai, ˜ +Xi) + β tr(D( ˜Zi, Z0))] ++ L(φ0, θ0, A0, X0), +(4.18) +where β > 0 is a positive regularization parameter, L(·) +is a loss function of a base generative model defined +in Eq. (3.11)-(3.13), Φ = {φ0, φ} and Θ = {θ0, θ} are +the generative models parameters, Z0, {Zi}i∈[n] ∈ Rm×d +are the hidden representation of nodes and D is given +by Eq. (3.4b). Note that the trace enforces the joint +embeddability of all graphs with the central graph. +The objective in Eq. (4.18) again can be minimized via +SGD. After training we take only the generative model +parameterized by φ, θ to generate new graphs. +4.3 +Fermat-Double. In this model, we combine the +Fermat distance function with two similar-structure +generative models. +We first use the Fermat distance +function defined in Eq. (3.5) to estimate graph align- +ment matrices {Pi}i∈[n] and G0 via alternating mini- +mization. Then, we project each Pi ∈ Wm onto the set +of permutation matrices { ˜Pi}i∈n via Eq. (4.14). Given +the graph set G, the center graph G0 and alignment +matrices { ˜Pi}i∈[n], we train the two generative mod- +els jointly. We train the first with G0 and the second +with the aligned {Gi}i∈[n]. We minimize the distance +between the embeddings of these two generative models +by solving the following optimization problem: +min +Φ,Θ +1 +n +�n +i=1[L(φ, θ; ˜Ai, ˜ +Xi) + β tr(D( ˜Zi, Z0))] ++ L(φ0, θ0, A0, X0), +(4.19) +where β +> +0 is a positive regularization parame- +ter, L(·) is again a loss function of a base genera- +tive model defined in Eq. (3.11)-(3.13), Φ = {φ0, φ} +and Θ = {θ0, θ} are the generative models parameters, +Z0, {Zi}i∈[n] ∈ Rm×d are graph embeddings and D is +given in Eq. (3.4b). We again solve Eq. (4.19) w.r.t. Φ +and Θ via SGD. After training, we again use only the +generative model parameterized by φ and θ to generate +graphs. +4.4 +Extensions. Our +proposed +graph +alignment +methods are not limited to graphs with equal numbers +of nodes; they can be readily extended to collections +of graphs with a variable number of nodes by employ- +ing one of several ways to add “dummy” nodes such +that all graphs have equal number of nodes [25]. +A +simple solution is to first find the maximum number +of nodes mmax in the graph set and then expand all +graphs with |Vi| < mmax, i ∈ [n] by adding “dummy” +nodes such that all graphs have mmax nodes. In the ex- +panded graphs “dummy” nodes are connected to each +other as well the actual nodes by edges with a small +weight (e.g., 0.01) to differentiate these edges from the +edges connecting the actual nodes. +5 +Accelerated Multi-Distances. +In both Fermat distance and G-align distance, as the +number n of graphs grows, alignment becomes more +computationally expensive. We propose three methods +to accelerate multi-distance algorithms. +All methods +produce a final center graph, G0out; once this is com- +puted, all the graphs in G can be aligned with G0out (and +each other) via Eq. (3.2). We describe these methods +assuming alignment happens via the G-align distance, +but the methods extend, mutatis mutandis, to Fermat +distance as well, by replacing Eq. 3.10 with Eq. 3.5. We +provide pseudocode for all three methods in App. E. +G-Parallel: Grouping and Parallelizing Graphs. +This method has a recursive structure, comprising +O(logKn) stages, where K ∈ N. In each stage, we apply +the same three-step procedure on a smaller set of graphs, +starting from the full set of graphs in the training set. In +the first step, we divide the set of graphs into a collection +of smaller groupings of size K ≪ n. In the second step, +we compute the alignment via Eq. (3.10) within each +group. +In the third step, we output a center graph, +computed via Eq. (4.17), for each group. Note that the +operations in the second and third steps can happen +in parallel. The procedure then executes recursively on +the (smaller) set of center graphs. The output of the +final stage is a single center graph, G0out. We note that, +for Eq. (3.5), Eq. (3.10), and Eq. (4.17), computing +alignments over K ≪ n rather than n graphs yields +significant performance dividends even serially, because +the execution cost is super-quadratic in the number of +graphs. The total number of such K-graph problems we +compute is O( n +K ). +C-Serial: Coarsening Graphs. In this method, we +create coarsened graphs [33] by partitioning each graph +into c ∈ N clusters via clustering algorithm such as +K-means. +In short, the nodes in a coarsened graph +are super-nodes representing all nodes in the original +graphs’ clusters. The weighted edges are the unions of +edges connecting two clusters in the original graph. We +next compute the graph alignment across the coarsened +graphs, via Eq. (3.10). Having mapped clusters to each +other across graphs, we refine alignments: +we align +the nodes within the clusters via Eq. (3.10) on a per- +cluster basis. +This yields a global alignment; finally, +we construct a center graph by computing the center +for the clusters and the edges connecting the clusters +via Eq. (4.17). +In this method, we need to compute + +|V |ave +|E|ave +n +Alignment alg. +Community (small) +45 +98 +100 +G-Parallel +Community (large) +150 +2727 +100 +CG-Parallel +Grid +36 +265 +100 +G-Parallel +Ego-Citeseer +35 +65 +100 +G-Parallel +Ego-B-A (small) +118 +298 +100 +CG-Parallel +Ego-B-A (large) +1028 +1471 +68 +CG-Parallel +Protein +117 +280 +100 +CG-Parallel +Table 1: Dataset summary including average number of nodes +and edges and number of graphs in the graph set, along with the +algorithm used to compute graph alignment. For smaller graphs +(with |V |ave < 50 ) we use the G-Parallel method. +For larger +graphs, to further accelerate computing the graph alignment, we +use CG-Parallel. For all parallel alignment algorithms we use a +single machine with 40 CPUs. +distances over O(n) graphs again but of size O(c), with +the refinement involving nc pairwise alignments of size, +approximately, m/c, assuming clusters of equal size. +CG-Parallel: Coarsening, Grouping and Paral- +lelizing. Similar to G-Parallel, this method is recur- +sive and in each stage we apply the same procedure on +a smaller set of graph. We just change what happens +in each stage compared to G-Parallel. Again, similar +to G-Parallel, in each stage we first divide graphs into +smaller groupings. In each of these smaller groups, we +compute the center graphs exactly the same way we did +in C-serial, i.e., by coarsening graphs, computing the +alignments via Eq. (3.10), computing the center graph +by computing the center of clusters and edges connect- +ing clusters via Eq. (4.17). The procedure then executes +recursively on the (smaller) set of center graphs. The +output of the final stage is a center graph, G0out, for the +whole set. The total number of stages in this method is +O(logKn). The total number of such K-graph problems +we compute is O( n +K ) with the refinement involving Kc +pairwise alignments of size, approximately, m/c, assum- +ing clusters of equal size. +6 +Experimental Setup +6.1 +Datasets. We perform experiments on both syn- +thetic and real datasets with varying numbers of nodes +and edges, using the code in [11]. +Community. +We generate two community graphs, +with +three-communities +from +the +stochastic +block +model [11]. The first graph has |V | = 45 total nodes +and [5, 15, 17] nodes in the communities. The second +has |V | = 150 total nodes and [40, 50, 60] nodes in the +communities. In both graphs, each community is gener- +ated by the Erdős-Rényi model (E-R) [6]. The probabil- +ity for edge creation in each community is p = 0.7. For +the smaller graph 0.05|V | inter-community edges were +added and for the large community graph 0.005|V | inter- +community edges were added u.a.r. In order to build the +graph set, we generate 100 random graphs by randomly +permuting the graph and then add noise by randomly +removing and re-adding 10% of edges, selected u.a.r. +Grid. We construct a 2-D grid graph with |V | = 36 +nodes. As above, we generate 100 graphs by randomly +permuting the graph and again add noise by randomly +removing and re-adding 10% of edges, u.a.r. +Ego-B-A (small). We generate 100 graphs with |V | = +950 nodes using the Barabási-Albert model. +During +the generation of each graph, each node in a graph is +connected to 5 existing nodes. We then construct 1−hop +ego graphs with |V | ∈ [100 − 130] nodes. +Ego-B-A (large). We generate 68 graphs using the +Barabási-Albert model. Each graph has |V | = 75500 +nodes such that each node is connected to 5 existing +nodes during generation. In the next step, we construct +1−hop ego graphs with |V | ∈ [1000 − 1050] nodes. +Ego-Citeseer. Similar to +[11, 44], we construct 100 +3-hop ego graphs from the Citeseer network [45], with +|V | ∈ [30 − 40] nodes. +Protein. +Similar to +[11, 12], we select 100 protein +graphs from a protein dataset [46] with |V | ∈ [100, 130] +nodes. The nodes in these graphs represent amino acids +and the edges are placed between all pairs of nodes that +are less than 6 Angstroms apart. +Table 1 summarizes each dataset as well as graph set +size and the methods used to compute graph alignments. +In all datasets, we use CVXPY [40] as our solver; addi- +tional implementation details can be found in App. E. +6.2 +Algorithms. We compare our methods against +three +base +generative +models, +GraphRNN +[11], +GRAN [12], and VAE [10] and two competitors, Graph- +VAE [14] and DeepGMG [47]. +Additional details on +baseline algorithms are in App. F. We compare these +baselines to all three versions of AlignGraph described +in Sec. 4, where for each of our algorithms we test +with three base generative models (GraphRNN [11], +GRAN [12], VAE [10]). Our code is publicly available.1 +6.3 +Performance Metrics . +In all experiments we +take 80% of the full set of graphs for training and use the +rest for testing. We train our generative models on the +training set, and use them to generate a set of synthetic +graphs, whose properties we then compare to graphs in +the test set to evaluate whether the generated graphs +are likely to have come from the same distribution as +the test set. We use two performance metrics to assess +the quality of the generated graphs. In both metrics, +we first calculate a set of summary statistics from each +individual graph (e.g., degree distribution, clustering co- +1https://github.com/neu-spiral/AlignGraph + +efficient, etc.); we summarise these statistics in App. C. +Then we compare the distributions of these statistics be- +tween the generated and test graphs w.r.t. two metrics. +The first is the smmd score: this score, proposed by You +et al. [11], measures the maximum mean discrepency +(MMD) between two distributions of graph statistics. +The smmd takes values in [0, 1] (the smaller the better). +We calculate an average MMD across all the statistics; +a formal definition can be found in App. D. +The second performance score is the smvr score: +this measures the squared difference between the mean +values of the two distributions, rescaled by the variance +of the value over the ground truth graphs. This score +takes values in [0, ∞] (the smaller the better). Again, +we average this across all statistics (see also App. D). +We also report the time it took to compute graph +alignments, ta, and the total training time of generative +models, ttr. We compute graph alignment only once and +pre-align graphs before training our AlignGraph models. +We measure ta and ttr to have a fair comparison between +the improvement we might get in smmd and smvr and +the cost of this improvement in terms of the total time +consumed by each model. +To evaluate the performance of our accelerated +multi-distances, we measure the accuracy of alignments. +For this purpose, we first compute graph alignment and +the center graph via Eq. (3.5) for Fermat distance and +Eq. (3.10) and Eq. (4.17) for G-align distance. +We +then align the graphs in the graph set w.r.t G0 and +evaluate the distance of G0 from the graph set via +d0 = 1 +n +�n +i=1 +∥P T +i AiPi−A0∥ +∥A1∥ +(smaller is better). +6.4 +Results +Accelerated Multi-distances Speed and Accu- +racy. We investigate the impact of our methods on run- +ning time and on the accuracy of graph alignment com- +putation on a graph set with 12 3−community graphs +of |V | = 45 nodes. In Fig. 1a and Fig. 1c we report the +total time to compute the alignment using G-align dis- +tance and Fermat distance, respectively. These figures +demonstrate that our proposed methods reduces the +computation time by 40 times. In Fig. 1b and Fig. 1d +we compute d0 via Eq. 6.3. Our results illustrate that +our acceleration methods improve the accuracy of esti- +mated center graphs. Since G-Parallel and CG-Parallel +have the best trade offs for the running time and ac- +curacy, we use these two methods to compute graph +alignment in our next experiments. +Evaluating the Generated Graphs. Table 2 sum- +marizes the performance scores smmd and smvr on all +7 datasets. +Our experiments show that our model +achieves 25% − 250% accuracy improvement over base +(a) Running time (in seconds) us- +ing G-align distance and applying +our proposed methods to G-align +distance. +(b) Accuracy of G-align distance +and applying our proposed accel- +erated multi-distances to Fermat +distance. +(c) Same as part (a) but for Fer- +mat distance. +(d) Same as part (b) but for Fer- +mat distance. +Figure 1: Computation time and accuracy of computing the graph +alignment in community graphs given the baselines and our three +accelerated multi-distances, G-Parallel (40 CPUs), CG-Parallel +(40 CPUs), and C-Serial. G-align distance has better performance +compared to Fermat distance. +Moreover, due to the clustered +structure of community graphs clustering and grouping graphs in +CG-Parallel also improves the accuracy. +models and 62.5%−4000% improvement over other com- +petitors. In some datasets, such as Community graphs +and Protein graphs, G-align-Double and Fermat-Double +that jointly train two similar structure generative mod- +els produce the best performance scores. +In the ma- +jority of the experiments, applying our frameworks to +either base GraphRNN or base GRAN leads to the best +performance scores. +However, there is no clear win- +ner between these two base generative models. Our re- +sults in Table 2 illustrate that our accelerated multi- +distances methods scale well to larger graphs and are +compatible with large datasets with |V | > 1000. More- +over, comparing the ttr of G-align-Single (GraphRNN) +and G-align-Single (GRAN) models with their baselines +demonstrate that our models are 4.21% − 44% faster. +This happens due to the pre-alignment of graphs in our +models. On the other hand, the ta/trm ratio for G-align- +Single (GraphRNN) and G-align-Single (GRAN) mod- +els ranges from 0.89% to 150%, where 150% belongs to +the alignment of our largest dataset, Ego-B-A (large). +While this pre-alignment took 70 minutes, it led to at +least 83% improvement in the performance scores. +Impact of Graph Perturbation. We investigate the +impact of graph perturbation on the performance of +our models by perturbing edges in the 3-community +graphs dataset with |V | = 45. +The perturbation + +800 +running time(s) +600 +400 +200 +0 +G-align +C-Serial +G-Parallel +CG-Parallel1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +G-align +C-Serial +G-Parallel +CG-Parallel2500 +2000 +time(s) +1500 +running +1000 +500 +0 +Fermat +C-Serial +G-Parallel +CG-Parallel1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Fermat +C-Serial +G-Parallel +CG-ParallelCommunity Graphs +Grid Graphs +Ego − Citeseer Graphs +Ego − B − A Graphs +Protein Graphs +(|V |ave, |E|ave) +(|V |ave, |E|ave) +(|V |ave, |E|ave) +(|V |ave, |E|ave) +(|V |ave, |E|ave) +(|V |ave, |E|ave) +(|V |ave, |E|ave) +(45, 98) +(150, 2727) +(36, 265) +(35, 65) +(118, 298) +(1028, 1471) +(117, 280) +smmd +smvr +ttr(min) ta(min) +smmd +smvr +ttr(min) ta(min) +smmd smvr ttr(min) ta(min) +smmd +smvr ttr(min) ta(min) +smmd +smvr +ttr(min) ta(min) +smmd +smvr +ttr(min) ta(min) smmd +smvr +ttr(min) ta(min) +GraphVAE +0.20 +612.17 5229.60 +0 +− +− +− +− +0.13 13.39 3827.79 +0 +0.04 +0.66 4084.49 +0 +− +− +− +− +− +− +− +− +− +− +− +− +DeepGMG +0.15 +1180.66 2771.26 +0 +− +− +− +− +0.18 +7.16 2771.41 +0 +0.01 +0.92 2771.47 +0 +− +− +− +− +− +− +− +− +− +− +− +− +VAE +0.18 +895.42 +0.13 +0 +0.22 +6475.40 +0.86 +0 +0.24 66.09 +0.06 +0 +0.06 +24.22 +0.07 +0 +0.16 +375.55 +0.18 +0 +0.25 +22110.65 +26.64 +0 +0.12 +8.05 +19.40 +0 +GraphRNN +0.08 +50.11 +45.30 +0 +0.14 +1270.27 +213.16 +0 +0.10 11.21 +45.73 +0 +0.005 +0.19 +52.45 +0 +0.018 +2.78 +75.86 +0 +− +− +− +− +0.06 +1.28 +75.19 +0 +GRAN +0.017 +26.98 +77.33 +0 +0.16 4611223.41 525.76 +0 +0.12 27.64 +76.69 +0 +0.009 +0.70 +76.17 +0 +0.011 +0.60 +19.54 +0 +0.011 +31.26 +50.44 +0 +0.07 +1.61 +22.03 +0 +G-align-Single (VAE) +0.16 +357.20 +9.72 +25.5 +0.19 +4983.63 +72.70 +50.41 +0.20 12.52 +5.04 +5.08 +0.02 +5.37 +8.32 +8.86 +0.10 +63.56 +16.76 +2.99 +0.24 +22131.28 +70.45 +70.36 +0.15 +4.47 +22.31 +4.7 +G-align-Double (VAE) +0.09 +8.76 +9.12 +25.5 +0.16 +488.93 +126.35 +50.41 +0.17 39.72 +6.97 +5.08 +0.04 +2.81 +8.41 +8.86 +0.03 +38.08 +108.29 +2.99 +− +− +− +− +0.03 +0.90 +137.26 +4.7 +Fermat-Double (VAE) +0.09 +1543.08 +8.84 +6.33 +0.18 +138.39 +77.90 +77.90 +0.18 45.19 +5.70 +3.18 +0.02 +13.37 +7.21 +5.69 +0.07 +380.99 +95.44 +125.4 +− +− +− +− +0.04 +0.93 +85.30 +121.81 +G-align-Single (GraphRNN) +0.04 +92.49 +41.42 +25.5 +0.12 +1268.49 +211.58 +50.41 +0.09 6.72 +46.86 +5.08 +0.002 0.12 +41.10 +8.86 +0.007 +1.05 +76.50 +2.99 +− +− +− +− +0.07 +1.74 +67.10 +4.7 +G-align-Double (GraphRNN) +0.06 +116.28 +190.87 +25.5 +0.13 +796.20 +1919.36 +50.41 +0.12 +8.04 +140.63 +5.08 +0.002 0.08 +204.85 +8.86 +0.009 +0.97 +1266.28 +2.99 +− +− +− +− +0.05 +1.26 +1251.21 +4.7 +Fermat-Double (GraphRNN) +0.05 +40.41 +152.45 +6.33 +0.13 +764.67 +1816.18 +77.90 +0.19 12.55 105.59 +3.18 +0.004 0.05 +133.41 +5.69 +0.007 +0.85 +1250.64 +125.4 +− +− +− +− +0.04 +1.55 +1277.53 121.81 +G-align-Single (GRAN) +0.012 +24.75 +73.57 +25.5 +0.12 +27974.71 +473.26 +50.41 +0.08 12.81 +53.0 +5.08 +0.005 +0.20 +47.74 +8.86 +0.04 +6.60 +18.75 +2.99 +0.006 +6.86 +45.56 +70.36 +0.13 +3.68 +19.92 +4.7 +G-align-Double (GRAN) +0.14 +1171.44 175.35 +25.5 +0.19 +15481.15 +962.48 +50.41 +0.13 45.77 191.80 +5.08 +0.008 +6.90 +223.56 +8.86 +0.008 +0.56 +846.33 +2.99 +− +− +− +− +0.07 +2.88 +843.62 +4.7 +Fermat-Double (GRAN) +0.13 +928.87 +166.14 +6.33 +0.04 +40.6 +1039.26 +77.90 +0.20 46.03 198.68 +3.18 +0.03 +24.55 206.43 +5.69 +0.11 +9.32 +785.09 +125.4 +− +− +− +− +0.13 178.12 835.23 +121.81 +Table 2: Comparison of two performance scores for synthetic and real graphs graphs. |V |ave is the average number nodes and |E|ave +is the average number of edges in the graph set. ttr indicates the total time to train generative models and ta is the total time to +compute graph alignment using either G-align distance or Fermat distance. (−) indicates an out of memory failure. Overall, applying +our frameworks to base RNN or base GRAN leads to better performance scores compared to baselines. +(a) +(b) . +Figure 2: Sensitivity of 3 base generative models combined with +G-align-Single to noise for community graphs with |V | = 45. x- +axis: the percentage of edges perturbed, y-axis : smmd (left), +smvr (right). +G-align-Single (GraphRNN) shows better overall +performance, however, in all models there is a direct relation +between the perturbation factor and the performance drop. +factor ρ is defined as the percentage of edges that we +randomly remove and re-add u.a.r. +The ρ values are +set to [10, 20, 50, 100] in our experiments. We note that +with ρ < 20%, graphs still have community structures. +In the extreme however, with a ρ = 100%, graphs +are effectively Erdös-Rényi and, thus, their statistics +differ significantly from those of the test set. +We +compute smmd and smvr for graphs generated with these +perturbation factors. Fig. 2 illustrates the performance +of the G-align-Single model using GraphRNN, GRAN +and VAE as base generative models. +G-align-Single +(GraphRNN) and G-align-Single (GRAN) models have +relatively good smmd compared to G-align-Single (VAE) +when ρ < 50%. At ρ = 10%, G-align-Single (GRAN) +has the best performance which is exactly inline with +the results we have in Table 2. As the noise increases, +G-align-Single (GraphRNN) shows more robustness to +noise compared to the other two models. As expected, +all models are adversely affected when ρ > 50%. +7 +Conclusion +We present a group of models that learn distributions +of graphs. 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Diamond and S. Boyd, +“CVXPY: A Python- +embedded modeling language for convex optimization,” +JMLR, 2016. +[41] M. Frank, P. Wolfe et al., “An algorithm for quadratic +programming,” Naval Research Logistics Quarterly, +1956. +[42] D. P. Kingma and M. Welling, “Auto-Encoding Varia- +tional Bayes,” in ICLR, 2014. +[43] H. W. Kuhn, “The hungarian method for the assign- +ment problem,” Naval Research Logistics Quarterly, +1955. +[44] C. Tran, W.-Y. Shin, A. Spitz, and M. Gertz, “Deepnc: +Deep generative network completion,” IEEE TPAMI / +PAMI, 2020. +[45] P. Sen, G. Namata, M. Bilgic, L. Getoor, B. Galligher, +and T. Eliassi-Rad, “Collective classification in net- +work data,” AI magazine, 2008. +[46] P. D. Dobson and A. J. Doig, “Distinguishing enzyme +structures from non-enzymes without alignments,” +Journal of molecular biology, 2003. +[47] Y. +Li, +O. +Vinyals, +C. +Dyer, +R. +Pascanu, +and +P. W. Battaglia, “Learning deep generative models of +graphs,” CoRR, vol. abs/1803.03324, 2018. +[48] S. Boyd, S. P. Boyd, and L. Vandenberghe, Convex +optimization. +Cambridge University Press, 2004. +[49] F. Pedregosa et al., “Scikit-learn: Machine learning in +Python,” JMLR, 2011. + +A +Alternating Minimization. +At each iteration t ∈ N, we update A0 and {Pi}i∈[n] as +follows: +A.1 +Updating A0. Given that {Pi}i∈[n] is fixed and +D = 0, minimizing Eq. (3.5) w.r.t A0 leads to the +following problem: +min +A0∈Rm×m +n +� +i=1 +∥AiP (t−1) +i +− P (t−1) +i +A(t) +0 ∥ +(A.1) +This problem is convex and at step t ∈ N can be +solved via convex optimization. +Once we solve this +optimization problem, we set a threshold to binarize +the elements of A0. +A.2 +Updating {Pi}i∈[n]. Given that A0 is fixed and +D = 0, let LP ({Pi}(t) +i∈[n]) be the loss function at step +t ∈ N. +(A.2) +LP ({Pi}(t) +i∈[n])) = �n +i=1 ∥AiP (t) +i +− P (t) +i +A(t) +0 ∥ +Minimizing Eq. (3.5) w.r.t {Pi}i∈[n] leads to the follow- +ing problem. +min +Pi∈Wm LP ({Pi}(t) +i∈[n]) +(A.3) +This step is convex. It can be solved via optimization +toolboxes such as CVXPY [40] or efficient algorithms +such as Frank-Wolfe algorithm [41]. Frank-Wolfe algo- +rithm is explained in details in the Section B. +B +Frank Wolfe. +The objective function in Eq. (A.2) can be solved via +Frank-Wolfe algorithm. +Frank-Wolfe is an iterative +algorithm that solves the problem through a sequence +of linear programs (LPs). This algorithm starts from a +feasible P 0 ∈ Wm, e.g. , the identity matrix I and in +each iteration t ∈ N proceeds as follows: +S(t) = +arg min +Sij∈Wm,Sii=I,S⪰0 +tr(ST , ∇P LP (P (t))) +(B.4a) +P (t+1) = (1 − γt)P (t) + γtS(t), +(B.4b) +where γt is the step size and can be set to e.g. +2 +t+2 or +determined by line search [48] as follows: +(B.5) +γt = arg +min +γt∈[0,1]LP (((1 − γt)P (t) + γtS(t)) +C +Table of metrics. +In the Table 3 we provide the lists of metrics we +measured in the experiments and their description. +Notation +Description +D.D +Graphs degree distribution +C.C +distribution of clustering coefficient of nodes +for each graph in the graph set +ASRT: +assortativity, Pearson correlation coefficient of +degree between pairs of linked nodes +TRI: +number of triangles for each graph in the graph set +WG.C: +wedge count, number of wedges for each graph +in the graph set +CL.C: +claw count, number of claws for each graph +in the graph set +Table 3: Summary of metrics. +D +Performance scores. +In order to calculate MMD2 , let a function f belong to a +unit ball in a reproducing kernel Hilbert space (RKHS) +H, f ∈ H, and k be the kernel. The MMD2 between +two sets of samples {xi}N +i=1∼iidp and {yi}N +i=1∼iidq from +distributions p and q is computed as follows: +(D.6) +MMD2 = +1 +N(N − 1) +�N +i=1 +�N +j̸=i(k(xi, xj) + k(yi, yj)) +− 1 +N 2 +�N +i=1 +�N +j=1(k(xi, yj) + k(xj, yi)) +The performance of MMD2 depends on choice of +the kernel. +Here we use Gaussian-Wasserstein RBF +kernel k(x, y) = e− W (p,q)2 +2σ2 +, where W(p, q) is the first +Wasserstein distance. The k(x, y) function is bounded, +k(x, y) ∈ [0, 1] and therefore MMD2 ∈ [0, 2]. +smmd score. +Combining MMD2 of all metrics we +measured, we present smmd score to assess the overall +quality of generated graphs. +(D.7) +smmd = +1 +12(MMD2(D.D) + MMD2(C.C) + MMD2(ASRT) ++ MMD2(TRI) + MMD2(WG.C) + MMD2(CL.C)) +Note that smmd ∈ [0, 1]. The smaller this score, the +smaller the distance between the generated graphs and +test set. +smvr score. +Our second performance score, smvr is + +Algorithm 1: G-Parallel: Grouping and Paralleliz- +ing Graphs +Input: G = {G1, G2, ..., Gn}, K : number of graphs in +each group. +Output: G0out : the center graph. +for k = {0, 1, 2, · · · , [ n +K ]} do +˜ +Gk = {G1+k×K, G2+k×K, · · · , GK+k×K} +˜ +Gk = align( ˜ +Gk) +˜G0 +k = center( ˜ +Gk) +end +˜G = { ˜G0 +k, for k = {0, 1, 2, · · · , [ N +K ]}} +if [ n +K ] > K then +while [ n +K ] > K do +n = [ n +K ] +G = ˜G +for k = {0, 1, 2, · · · , [ n +K ]} do +˜ +Gk = {G1+k×K, G2+k×K, · · · , GK+k×K} +˜ +Gk = align( ˜ +Gk) +˜G0 +k = center( ˜ +Gk) +end +˜G = { ˜G0 +kfor k = {0, 1, 2, · · · , [ n +K ]}} +end +end +Gout = ˜G +G0out = center(Gout) +formulated as follows: +(D.8) +smvr = 1 +6( +(µr(D.D) − µg(D.D))2 +σ2r (D.D) ++ (µr(ASRT) − µg(ASRT))2 +σ2r (ASRT) ++ (µr(C.C) − µg(C.C))2 +σ2r (C.C) ++ (µr(TRI) − µg(TRI))2 +σ2r (TRI) ++ (µr(WG.C) − µg(WG.C))2 +σ2 +t (WG.C) ++ +(µr(CL.C) − µg(CL.C))2 +σ2r (CL.C) +), +where µr, µg and σ2 +t represent mean value for the +reference set, mean value for the generated set and +variance of the reference set, respectively. +E +Accelerating Multi-distances +In Alg. 1 we explain how to compute the final center +graph by G-Parallel. We describe C-Serial algorithm is +in details in Alg. 2 and the detail of CG-Parallel are in +Alg. 3. Table 4 shows summary of datasets, acceleration +methods and solvers used to compute graph alignment. +Algorithm 2: C-Serial: Coarsening Graphs +Input: G += {G1, G2, ..., Gn}, c : number of clusters in +graphs. +Output: G0out : the center graph. +for l = {0, 1, 2, · · · , n} do +ˆGl = coarsen(Gl) +end +ˆG = align({ ˆG1, ˆG2, · · · , ˆGn}) +for l = {0, 1, 2, · · · , n} do +align clusters given their alignment in the coarsened +graphs. +end +compute center of clusters. +compute center of edges connecting clusters. +˜G0 +k: build given the center of clusters and center of edges +connecting clusters. +Algorithm 3: CG-Parallel: Coarsening, Grouping +and Parallelizing Graphs +Input: G = {G1, G2, ..., Gn}, K : number of graphs in each +group, c : number of clusters in graphs. +Output: G0out : the center graph. +if [ n +K ] > K then +while [ n +K ] > K do +for k = {0, 1, 2, · · · , [ n +K ]} do +˜ +Gk = {G1+k×K, G2+k×K, · · · , GK+k×K} +for l = {0, 1, 2, · · · , K} do +ˆGl+k×K = coarsen(Gl+k×K) +end +ˆ +Gk = align({ ˆG1+k×K, ˆG2+k×K, · · · , ˆGK+k×K}) +for l = {0, 1, 2, · · · , K} do +align clusters given their alignment in the +coarsened graphs. +end +compute center of clusters. +compute center of edges connecting clusters. +˜G0 +k: build given the center of clusters and center +of edges connecting clusters. +end +˜G = { ˜G0 +k, for k = {0, 1, 2, · · · , [ N +K ]}} +end +end +Gout = ˜G +G0out = center(Gout) +|V |ave +|E|ave +n +Alignment alg. +Solver (Fermat) +Solver (G-align) +Community (small) +45 +98 +100 +G-Parallel +CVXPY + AM +CVXPY +Community (large) +150 +2727 +100 +CG-Parallel +CVXPY + AM +CVXPY +Grid +36 +265 +100 +G-Parallel +CVXPY + AM +CVXPY +Ego-Citeseer +35 +65 +100 +G-Parallel +CVXPY + AM +CVXPY +Ego-B-A (small) +118 +298 +100 +CG-Parallel +CVXPY + AM +CVXPY +Ego-B-A (large) +1028 +1471 +68 +CG-Parallel +CVXPY + AM +CVXPY +Table 4: Dataset summary including average number of nodes +and edges and number of graphs in the graph set, along with the +algorithm and solvers used to compute graph alignment in Fermat +and G-align distance. +For smaller graphs (with |V |ave < 50 +) we use the G-Parallel method. +For larger graphs, to further +accelerate computing the graph alignment we use CG-Parallel. +By using these acceleration techniques, all alignment problems +can be solved by CVXPY. +F +Implementation Details +We compared the performance our models against five +different deep baseline described below. +GraphRNN. You et al. [11] proposes a framework +based on graph neural networks. +This model uses +a graph-level RNN to add a new node to a node + +sequence each time step and an edge-level RNN to +model the generation process of nodes and edges. The +reference code for this model is provided by the authors +and we followed their recommendation for setting the +hyperparameters. +GRAN. Liao et al. [12] proposes a graph recurrent at- +tention framework. This model uses an attention-based +GNN and generates a block of graphs that consists of +multiple rows of graph adjacency matrices conditioned +on the previously generated blocks of the graph and uses +a group canonical node ordering, e.g., DFS and BFS to +address node ordering problem. +VAE. Kipf & Welling [10] propose a variational autoen- +coder that is characterized by a probabilistic inference +model that maps observed data to a latent representa- +tion, a prior distribution over the latent variables and +a probabilistic generative model. We randomly pick a +graph with a random node ordering from graph set G +and train a VAE to generate graphs. +GraphVAE. Simonovsky & Komodakis [14] propose +a variational autoencoder that outputs a probabilistic +fully-connected graph and uses a graph matching algo- +rithm to align graph to the ground truth. GraphVAE +outputs a graph with adjacency matrix, node attributes +and edge attributes. +We adapt it to our problem by +using one-hot representations of the features. The en- +coder is a graph convolutional network and the decoder +is a multi-layer perception. We used code for this model +from [11] and set the hyperparameters based on recom- +mendations made in [14]. +DeepGMG. Li et al. [47] introduce a generative model +for graphs that generates graphs in a sequential manner. +It generates one node at a time and connects each +node to the partial graph already generated by creating +edges one by one. +We used the implementation in +[11] and the hyperparameters were set based on the +recommendations made in [47]. +We take 80% of graphs for training and the rest for +the test sets. +During testing, GraphRNN model and +GRAN model generate graphs directly. However, the +output of the VAE decoder is an adjacency matrix with +elements in the range of [0, 1]. We binarize the adja- +cency matrix by applying a threshold, τ. +We find τ +by comparing two sets of graphs, the ones generated +from the VAE and 20% of the graphs in the training +set, and computing two scores, which we denote smmd +and smvr (see Sec. 6.3) to measure the distance between +these two sets for a range of values of τ. +We chose +the value of τ that returns the smallest smmd as our +optimal threshold in testing. +In all our AlignGraph +models, we pre-compute the graph alignment for all +datasets and use the aligned graphs for training the gen- +erative models. We use GraphRNN [11], GRAN [12] and +VAE [10] as our base generative models. We followed +the instructions given in [11] and +[12] to set the hy- +perparameters in GraphRNN [11] and GRAN [12] and +for the VAE [10] we used the hyperparameters given +in [10] and set the learning rate to 0.001. +(We note +that VAE[18] here refers to the model proposed by Kipf +& Welling and is different from GraphVAE[14] by Si- +monovsky & Komodakis. In all models, the hidden di- +mensions {Zi}i∈[n] of small graphs are set to 16. For +medium graphs (|V | ∈ [100 − 500] ) the hidden dimen- +sions are 64 and the hidden dimensions of the large +graphs (|V | > 500) are set to 256. In all experiments, +The node features are one-hot indicator vectors. The +AlignGraph architectures are implemented in Python3 +using Tensorflow and Pytorch. +We implemented the +solution of the constrained optimization problems in +Section 4 via CVXPY. We implemented all solvers in +Python3. For clustering graphs we use Scikit-learn [49]. +All experiments are carried out on a Tesla V100 GPU +with 32 GB memory and 5120 cores. +G-Parallel and +CG-Parallel methods parallelizes the computations us- +ing python multiprocessing package. For both of these +parallel graph alignment algorithms we use a single ma- +chine with 40 CPUs. + diff --git a/OtFIT4oBgHgl3EQfeCvq/content/tmp_files/load_file.txt b/OtFIT4oBgHgl3EQfeCvq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aad982ba2c7fd93d78dc9954f1a066c321ce1d32 --- /dev/null +++ b/OtFIT4oBgHgl3EQfeCvq/content/tmp_files/load_file.txt @@ -0,0 +1,1364 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf,len=1363 +page_content='AlignGraph: A Group of Generative Models for Graphs Kimia Shayestehfard Dana Brooks Stratis Ioannidis∗ Abstract It is challenging for generative models to learn a dis- tribution over graphs because of the lack of permuta- tion invariance: nodes may be ordered arbitrarily across graphs, and standard graph alignment is combinatorial and notoriously expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We propose AlignGraph, a group of generative models that combine fast and effi- cient graph alignment methods with a family of deep generative models that are invariant to node permuta- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our experiments demonstrate that our frame- work successfully learns graph distributions, outper- forming competitors by 25% − 560% in relevant per- formance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1 Introduction Graph generative models have applications across do- mains like chemistry, neuroscience and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Generative models learn a distribution over graphs, and are used to subsequently sample from this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' They can, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', be used to predict interfaces between proteins during drug design and discovery [1, 2], or to perform hypothesis testing and simulation for social net- works, when collecting real graphs is difficult [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Traditional generative models for graphs such as the Barabási-Albert [5], Erdös-Rényi [6], and stochastic block models [7] generate graphs with provable formal properties but which often lack realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For example, the Erdös-Rényi model produces graphs with a light- tailed degree distribution [5, 8], while the Barabási- Albert model fails to generate graphs with a high clus- tering coefficient [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Deep generative models such as variational autoencoders [10] and graph recurrent neu- ral networks [11, 12] have shown great potential in learn- ing distributions from graph datasets, at greater fidelity than traditional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' However, learning a distribu- tion of graphs over a dataset poses a significant chal- lenge because of the lack of permutation invariance, since graph nodes may be subject to arbitrary permuta- tions across graphs: the correspondence between nodes in different graph samples may be a priori unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' ∗{kshayestehfard, brooks, ioannidis}@ece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='edu, Electrical and Computer Engineering Department, Northeastern University, Boston, MA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This is a problem because state-of-the-art genera- tive models, like the ones listed above, rely on latent node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Such embeddings vary drastically even under nearly isomorphic graphs [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In turn, this can hamper the fidelity of the graph generation process significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Note that this is a much harder setting than, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', images or text, where inputs have a canonical orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Finding the correspondence between graph nodes is a notoriously hard problem [14, 15, 11, 16], and it is exacerbated when the number of sampled graphs is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' To that end, we propose AlignGraph, a group of permutation invariant graph alignment methods com- bined with their application to a group of base genera- tive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our main contributions are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' AlignGraph incorporates convex graph multi- distance methods in training to achieve permuta- tion invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We use these tools both as means to construct alignment in a tractable fashion as well as to create soft penalties in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' AlignGraph is a general flexible framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' It can be applied to a broad class of base generative mod- els, enhancing their permutation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We demonstrate this here by applying it to graph recur- rent neural networks [11], gated recurrent attention networks [12] and variational autoencoders [10] as our base generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We propose three methods that can be parallelized to accelerate graph multi-distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Leveraging parallelism, our methods speed up computation by a 40× factor, while maintaining the alignment accuracy and, in some cases, improving it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We conduct experiments on both synthetic and real data, showing that AlignGraph outperforms both our base and other competitor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We define two performance scores to measure accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We then show that our model achieves 25% − 250% improvement in those scores over base models and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5% − 4000% improvement over other competi- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2 Related Work Graph Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Graph embeddings map nodes into a lower-dimensional space and have been ap- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11273v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='SI] 26 Jan 2023 plied to link prediction [10, 17, 16], node classification [18, 19, 17, 20], and clustering [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Many of the state-of- the-art graph embedding algorithms capture the relative position of nodes on the embedding space [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Because of the non-convexity of training objectives and the exis- tence of multiple local minima, even isomorphic graphs can map to completely different embeddings using the same embedding algorithm [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This is further exac- erbated when graphs are near-isomorphic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', differ in a few edges) as well as when the embeddings are ran- domized [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Embeddings play a central role in graph generative models (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='3), but lack of permuta- tion invariance can introduce significant distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Deep Generative Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Deep generative mod- els can be categorized into three groups: generative adversarial networks (GANs), variational autoencoders (VAEs), and auto-regressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' NetGAN [16] learns the distribution of biased random walks over a single graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GraphVAE [14] and NEVAE [15] use VAEs linking node embeddings to edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GraphRNN [11] is an auto-regressive model that constructs a graph se- quentially over nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GRAN [12] is an- other auto-regressive model that uses graph neural net- works (GNNs) with an attention mechanism to generate a block of nodes and edges sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Several of these methods contain techniques to par- tially deal with permutation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For example, GraphVAE [14] uses an approximate graph matching to penalize misalignment between each input graph and its corresponding reconstructed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' NEVAE [15] and GraphRNN [11] use a breadth-first-search node order- ing scheme and GRAN [12] marginalizes over a family of canonical node orderings to handle permutation invari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' However, none of these methods address permu- tation invariance by finding a consistent node ordering across sampled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In comparison, the graph align- ment approach we introduce here does exactly this;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' in addition, it is generic and can be applied to the broad group of base generative models listed above to enhance their permutation invariance (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Graph distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Classic methods to compute the distance between graphs include the edit distance [21, 22] and the maximum common subgraph distance [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Although they are metrics, they are hard to com- pute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Bento & Ioannidis [25] recently introduced a fam- ily of metrics for graph distances that is computationally tractable but limited to computing the distance between two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' To compute distances among a larger group of graphs, it is important that the distance function sat- isfies alignment consistency [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' There are works on multi-distances that enforce this constraint [27, 28, 29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' however, none of these methods satisfy generalizations of the metric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Gromov-Wasserstein Learning (GWL), proposed by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [30] satisfies both of these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' However, GWL has cubic complexity and is not applicable to recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Recently, two approaches were proposed by Kiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [31] and Safavi & Bento [32] to measure the distance among a group of graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' we describe both in detail in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Both satisfy alignment consistency and a generalization of metric properties [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' However, both are also slow when applied to a large number of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We propose a framework to accelerate these two graph multi-distance algorithms, by leveraging paral- lelization and graph coarsening [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Graph coarsening has been used in community detection [34, 35], graph embeddings [36] and alignment between two graphs [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We coarsen graphs using K-means clustering, and in- corporate this accelerated graph alignment method into our framework to address permutation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' To the best of our knowledge, we are the first to accelerate graph multi-distances by graph coarsening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3 Background 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1 Minimum Distance between two Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Let G = (V, E) be an undirected graph with node set V = [m] ≡ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , m} and edge set E ⊆ [m] × [m], represented by adjacency matrix A ∈ {0, 1}m×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The entries of this adjacency matrix are indexed by the nodes in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We denote the set that contains all such matrices by Ω ⊆ Rm×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Consider two graphs GA = (V, EA), GB = (V, EB) with adjacency matrices A, B ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' One way to measure the distance between these two graphs is to find an alignment between nodes and compute an edge discrepancy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', edit distance [38, 21]) between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' An alignment can be represented by a permutation matrix P ∈ Pm, where: Pm ≜ {P ∈ {0, 1}m×m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' P1 = 1, P T 1 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1) However, finding such an alignment is generally compu- tationally intractable [25, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Bento & Ioannidis [25] introduce a distance function dS : Ω2 �−→ R, defined as: dS(A, B) = min P ∈Wm ∥AP − PB∥ + βtr(P T DA,B), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2) where β > 0 is a positive regularization parameter, ∥ · ∥ is a matrix norm, tr is the trace operator, matrix DA,B ∈ Rm×m represents the dissimilarity between nodes across the two graphs, and matrix P is a doubly stochastic alignment matrix, that is, P ∈ Wm, where Wm ≜ {P ∈ [0, 1]m×m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' P1 = 1, P T 1 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='3) Matrix DA,B is generally a distance matrix, where each element represents the pairwise distances between the embeddings or features of nodes across two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For example, for two matrices of graph embeddings ZA ∈ Rm×d and ZB ∈ Rm×d that map nodes of a graph into a lower-dimensional space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' d < m, DA,B is: DA,B = [Da,b]a∈V,b∈V ∈ Rm×m, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4a) Da,b = ||zA a − zB b ||2, ∀ a ∈ V, b ∈ V, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4b) where zA a indicates the a-th row of matrix ZA, zB b indicates the b-th row of ZB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Intuitively, the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2) is a probabilistic mapping between nodes of two graphs and the second term penalizes the dissimilarity between the embeddings of the nodes that are mapped to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2) is a pseudometric and a convex optimization problem [25], and thus can be computed efficiently via standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2 Minimum Distance among n Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Con- sider a distance function d(Gi, Gj) like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2) that induces a (possibly stochastic) alignment matrix Pij be- tween two pairs of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' To compute the minimum distance between a group of n > 2 graphs, one could simply generalize the distance function d(Gi, Gj) to mul- tiple graphs, via d(G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , Gn) = � i,j∈[n] d(Gi, Gj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' However, such a generalization does not guarantee the joint alignment between multiple graphs: that is, if Pij aligns Gi with Gj, and Pjl aligns Gj with Gl, the align- ment matrix Pil should keep the consistency of align- ments under transitivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', Pil = PijPjl [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This property is known as alignment consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We de- scribe next two distance functions that induce align- ments that satisfy this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1 Fermat Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Let d(A, B) be a metric for two graphs such that d : Ω2 �−→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Then the Fermat distance function [31] associated with d is the map of dF : Ωn �−→ R defined by: dF (A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , An) = min A0∈Ω �n i=1 d(Ai, A0), capturing the distance among a set of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' If d is a metric then the Fermat distance function induced by d is a so-called n-metric [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The Fermat distance function induced by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2) is: dF (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , An) = min A0∈Ω Pi∈Wm,∀i∈[n] �n i=1 G(Pi, A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Ai), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5) where D = 0 and Wm represents the set of doubly stochastic matrices and G : Wm × Ω × Ω �−→ R is: G(Pi, A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Ai) = ∥AiPi − PiA0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='6) Graph G0, corresponding to A0, represents the center of set G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The Fermat distance function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5) is a pseudo n-metric [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This optimization problem is non-convex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' nonetheless, it can be solved approximately via alternating minimization (AM): Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5) then re- duces to solving two alternating convex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The first one has nm2 parameters and nm2 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The second problem reduces to n optimiza- tion problems with m2 parameters and m2 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' More details regarding AM iterations can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2 G-align distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Safavi & Bento [32] intro- duce the G-align distance function, a convex function that satisfies metric properties and alignment consis- tency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Consider the map dG : Ωn �−→ R defined by: dG(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , An) = min Pij∈S 1 2 � i,j∈[n] G(Pij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Ai, Aj), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='7) where G(Pij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Ai, Aj) is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='6) (with D = 0), S = {{Pij}i,j∈[n] : Pij ∈ Pm, ∀i, j ∈ [n], PilPlj = Pij, ∀i, j, l ∈ [n], Pii = I, ∀i ∈ [n]}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='8) where Pm is the set of permutation matrices and PilPlj = Pij captures alignment consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Let P ∈ Rnm×nm be a matrix with n2 blocks such that the (i, j)-th block is Pij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' : P = � � I P12 P13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' P1n P21 I P23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' P2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Pn1 Pn2 Pn3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' I � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='9) Safavi & Bento [32] prove that the alignment consis- tency is equivalent to P ⪰ 0 (see Lemma 4 in Safavi & Bento [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' By relaxing the permutation matrices con- straint in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='7) to a doubly stochastic constraint, the G-align distance function is as follows: dG(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , An) = min Pij∈Wm, Pii=I,P ⪰0 1 2 � i,j∈[n] G(Pij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Ai, Aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10) This is a pseudo n-metric (see Theorem 5 and Remark 4 in Safavi and Bento [32]) and a convex optimization problem with O(n2m2) variables and n constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In practice, this problem can be solved via optimization toolboxes such as CVXPY [40] as well as the Frank- Wolfe algorithm (FW) [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' the latter is outlined in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Despite convexity, the quadratic nature of P (in terms of n and m) in G-align distance and Pi ∈ Wm, i ∈ [n] (in terms of m) in Fermat distance makes these computations expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We address this in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='3 Graph Generative Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Given a set of undirected graphs G = {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , Gn} sampled from p(G), for each graph Gi(Vi, Ei), ∀i ∈ [n], we denote the adjacency matrices by Ai ∈ Rm×m and feature matrices by Xi ∈ Rm×f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Features could be either one-hot node indicator vectors or consist of graph characteristics, such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', node degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The nodes of each graph are mapped to a latent embedding space via a deep neural network, parameterized by φ [20]: Zi = fφ(Ai, Xi), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11) where Zi = {zi1, zi2, · · · , zim} denotes the hidden node representations and φ represents parameters of the deep neural network encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The decoder parameterized by θ takes the hidden representations and reconstructs the adjacency matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' : ˆAi = gθ(Zi), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='12) where ˆAi is the estimated adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Both the encoder and decoder can be randomized, and induce a distribution pφ,θ over graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' A loss often used to train parameters over graphs is the negative log likelihood: L(φ, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' A, X) = − �n i=1 log pφ,θ(Ai), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='13) where A = {A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , An} denotes the set of adja- cency matrices and X = {X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , Xn} represents the set of feature matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Several existing generative models can be described using the general framework described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11)- (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GraphRNN [11] is an auto-regressive model that generates node embeddings sequentially: fφ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11) is an RNN that encodes the states of graph generated so far, and gθ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='12) is a Gated Recurrent Unit (GRU) model that outputs the distribution of the next node’s adjacency vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GRAN [12] is also an auto-regressive model that generates the graph in a block by block ba- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In this model, fφ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11) is a GRU that uses an attention-weighted sum over the neighborhood of each node to produce the corresponding node embedding and gθ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='12) models the probability of generating edges in a block comprising multiple rows of a graph adja- cency matrix via a mixture of Bernoulli distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In VAE [10], fφ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11) is a probabilistic encoding de- noted by qφ(Zi|Ai, Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' There is a prior over the latent variables pz(Zi) ∼ N(0, I) and gθ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='12) is defined as the inner product between latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The loss function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='13) is further approximated by a variational lower bound of the log-likelihood [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' All these models can be trained by minimizing the loss (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='13) via stan- dard gradient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 4 AlignGraph We present AlignGraph, our framework for enhancing the permutation invariance of base generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' AlignGraph can be applied to any base generative model of the form given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11)- (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='13): we indeed apply it to GraphRNN [11], GRAN [12] and VAE [10] in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We consider three AlignGraph variants, described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1 G-align-Single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We begin by aligning sampled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' To do this, we first compute P by solving the problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We take the first block column of P , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', {Pi1}n i=1, and project each Pi1 ∈ Wm onto the set of permutation matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' : ˜ Pi1 = ΠPm(Pi1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='14) where ΠPm is the orthogonal projection to Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This can be done in polynomial time with the Hungarian algorithm [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Given these permutation matrices, we align all graphs and features with the first graph, via: ˜Ai = ˜ Pi1 T Ai ˜ Pi1, ˜ Xi = ˜ Pi1 T Xi ∀i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='15) Note that, by alignment consistency (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='8), this could be done on any block column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' our selection of {Pi1}n i=1 is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Given the alignment matrices { ˜ Pi1}i∈[n], the adjacency matrices A and feature matrices X, we aim to solve the following optimization problem: min φ,θ 1 n �n i=1 L(φ, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' ˜Ai, ˜ Xi), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='16) where φ and θ are the DNN parameters and L(·) is the loss defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='16) can be solved via stochastic gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2 G-align-Double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In our second approach, we (a) compute a central graph across the graph set and (b) enforce that this graph and aligned graphs are jointly embeddable in the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' To that end, we use two base generative models combined with the G-align distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We again compute P from the G- align distance [32] by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10) and project each Pi1 ∈ Wm onto the set of permutation matrices { ˜ Pi1}i∈n via (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We then align the adjacency matrices and feature matrices as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We then estimate the center graph G0 of set G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' the graph which has the minimum distance from all the graphs in the graph set via: min ˆ A0j,k,∈[0,1],∀j,k∈[m] �n i=1 ∥ ˜Ai − ˆ A0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='17) Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='17) is convex and can be solved via standard methods: the objective has n terms, and the problem has m2 parameters and O(m2) constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Since ˆA0j,k ∈ [0, 1], ∀j, k ∈ [m], we binarize the elements of the adjacency matrix for G0 by using a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Once we estimate G0, given the permutation matrices { ˜ Pi1}i∈n and the graph set G with one-hot encoding feature matrices X, we train two generative models of the chosen type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We train the first with G0 and the second with all aligned {Gi}n i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We train the generative models jointly, by penalizing the distance between the embeddings of Gi and G0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' : min Φ,Θ 1 n �n i=1[L(φ, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' ˜Ai, ˜ Xi) + β tr(D( ˜Zi, Z0))] + L(φ0, θ0, A0, X0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='18) where β > 0 is a positive regularization parameter, L(·) is a loss function of a base generative model defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='13), Φ = {φ0, φ} and Θ = {θ0, θ} are the generative models parameters, Z0, {Zi}i∈[n] ∈ Rm×d are the hidden representation of nodes and D is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Note that the trace enforces the joint embeddability of all graphs with the central graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The objective in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='18) again can be minimized via SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' After training we take only the generative model parameterized by φ, θ to generate new graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='3 Fermat-Double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In this model, we combine the Fermat distance function with two similar-structure generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We first use the Fermat distance function defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5) to estimate graph align- ment matrices {Pi}i∈[n] and G0 via alternating mini- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Then, we project each Pi ∈ Wm onto the set of permutation matrices { ˜Pi}i∈n via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Given the graph set G, the center graph G0 and alignment matrices { ˜Pi}i∈[n], we train the two generative mod- els jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We train the first with G0 and the second with the aligned {Gi}i∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We minimize the distance between the embeddings of these two generative models by solving the following optimization problem: min Φ,Θ 1 n �n i=1[L(φ, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' ˜Ai, ˜ Xi) + β tr(D( ˜Zi, Z0))] + L(φ0, θ0, A0, X0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='19) where β > 0 is a positive regularization parame- ter, L(·) is again a loss function of a base genera- tive model defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='11)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='13), Φ = {φ0, φ} and Θ = {θ0, θ} are the generative models parameters, Z0, {Zi}i∈[n] ∈ Rm×d are graph embeddings and D is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We again solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='19) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Φ and Θ via SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' After training, we again use only the generative model parameterized by φ and θ to generate graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4 Extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our proposed graph alignment methods are not limited to graphs with equal numbers of nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' they can be readily extended to collections of graphs with a variable number of nodes by employ- ing one of several ways to add “dummy” nodes such that all graphs have equal number of nodes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' A simple solution is to first find the maximum number of nodes mmax in the graph set and then expand all graphs with |Vi| < mmax, i ∈ [n] by adding “dummy” nodes such that all graphs have mmax nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In the ex- panded graphs “dummy” nodes are connected to each other as well the actual nodes by edges with a small weight (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='01) to differentiate these edges from the edges connecting the actual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 5 Accelerated Multi-Distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In both Fermat distance and G-align distance, as the number n of graphs grows, alignment becomes more computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We propose three methods to accelerate multi-distance algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' All methods produce a final center graph, G0out;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' once this is com- puted, all the graphs in G can be aligned with G0out (and each other) via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We describe these methods assuming alignment happens via the G-align distance, but the methods extend, mutatis mutandis, to Fermat distance as well, by replacing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We provide pseudocode for all three methods in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' G-Parallel: Grouping and Parallelizing Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This method has a recursive structure, comprising O(logKn) stages, where K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In each stage, we apply the same three-step procedure on a smaller set of graphs, starting from the full set of graphs in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In the first step, we divide the set of graphs into a collection of smaller groupings of size K ≪ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In the second step, we compute the alignment via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10) within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In the third step, we output a center graph, computed via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='17), for each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Note that the operations in the second and third steps can happen in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The procedure then executes recursively on the (smaller) set of center graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The output of the final stage is a single center graph, G0out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We note that, for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='17), computing alignments over K ≪ n rather than n graphs yields significant performance dividends even serially, because the execution cost is super-quadratic in the number of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The total number of such K-graph problems we compute is O( n K ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' C-Serial: Coarsening Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In this method, we create coarsened graphs [33] by partitioning each graph into c ∈ N clusters via clustering algorithm such as K-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In short, the nodes in a coarsened graph are super-nodes representing all nodes in the original graphs’ clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The weighted edges are the unions of edges connecting two clusters in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We next compute the graph alignment across the coarsened graphs, via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Having mapped clusters to each other across graphs, we refine alignments: we align the nodes within the clusters via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10) on a per- cluster basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This yields a global alignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' finally, we construct a center graph by computing the center for the clusters and the edges connecting the clusters via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In this method, we need to compute |V |ave |E|ave n Alignment alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Community (small) 45 98 100 G-Parallel Community (large) 150 2727 100 CG-Parallel Grid 36 265 100 G-Parallel Ego-Citeseer 35 65 100 G-Parallel Ego-B-A (small) 118 298 100 CG-Parallel Ego-B-A (large) 1028 1471 68 CG-Parallel Protein 117 280 100 CG-Parallel Table 1: Dataset summary including average number of nodes and edges and number of graphs in the graph set, along with the algorithm used to compute graph alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For smaller graphs (with |V |ave < 50 ) we use the G-Parallel method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For larger graphs, to further accelerate computing the graph alignment, we use CG-Parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For all parallel alignment algorithms we use a single machine with 40 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' distances over O(n) graphs again but of size O(c), with the refinement involving nc pairwise alignments of size, approximately, m/c, assuming clusters of equal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' CG-Parallel: Coarsening, Grouping and Paral- lelizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Similar to G-Parallel, this method is recur- sive and in each stage we apply the same procedure on a smaller set of graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We just change what happens in each stage compared to G-Parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Again, similar to G-Parallel, in each stage we first divide graphs into smaller groupings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In each of these smaller groups, we compute the center graphs exactly the same way we did in C-serial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', by coarsening graphs, computing the alignments via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10), computing the center graph by computing the center of clusters and edges connect- ing clusters via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The procedure then executes recursively on the (smaller) set of center graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The output of the final stage is a center graph, G0out, for the whole set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The total number of stages in this method is O(logKn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The total number of such K-graph problems we compute is O( n K ) with the refinement involving Kc pairwise alignments of size, approximately, m/c, assum- ing clusters of equal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 6 Experimental Setup 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1 Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We perform experiments on both syn- thetic and real datasets with varying numbers of nodes and edges, using the code in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We generate two community graphs, with three-communities from the stochastic block model [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The first graph has |V | = 45 total nodes and [5, 15, 17] nodes in the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The second has |V | = 150 total nodes and [40, 50, 60] nodes in the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In both graphs, each community is gener- ated by the Erdős-Rényi model (E-R) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The probabil- ity for edge creation in each community is p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For the smaller graph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='05|V | inter-community edges were added and for the large community graph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='005|V | inter- community edges were added u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In order to build the graph set, we generate 100 random graphs by randomly permuting the graph and then add noise by randomly removing and re-adding 10% of edges, selected u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We construct a 2-D grid graph with |V | = 36 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' As above, we generate 100 graphs by randomly permuting the graph and again add noise by randomly removing and re-adding 10% of edges, u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Ego-B-A (small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We generate 100 graphs with |V | = 950 nodes using the Barabási-Albert model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' During the generation of each graph, each node in a graph is connected to 5 existing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We then construct 1−hop ego graphs with |V | ∈ [100 − 130] nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Ego-B-A (large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We generate 68 graphs using the Barabási-Albert model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Each graph has |V | = 75500 nodes such that each node is connected to 5 existing nodes during generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In the next step, we construct 1−hop ego graphs with |V | ∈ [1000 − 1050] nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Ego-Citeseer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Similar to [11, 44], we construct 100 3-hop ego graphs from the Citeseer network [45], with |V | ∈ [30 − 40] nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Similar to [11, 12], we select 100 protein graphs from a protein dataset [46] with |V | ∈ [100, 130] nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The nodes in these graphs represent amino acids and the edges are placed between all pairs of nodes that are less than 6 Angstroms apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Table 1 summarizes each dataset as well as graph set size and the methods used to compute graph alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In all datasets, we use CVXPY [40] as our solver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' addi- tional implementation details can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2 Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We compare our methods against three base generative models, GraphRNN [11], GRAN [12], and VAE [10] and two competitors, Graph- VAE [14] and DeepGMG [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Additional details on baseline algorithms are in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We compare these baselines to all three versions of AlignGraph described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 4, where for each of our algorithms we test with three base generative models (GraphRNN [11], GRAN [12], VAE [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our code is publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='3 Performance Metrics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In all experiments we take 80% of the full set of graphs for training and use the rest for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We train our generative models on the training set, and use them to generate a set of synthetic graphs, whose properties we then compare to graphs in the test set to evaluate whether the generated graphs are likely to have come from the same distribution as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We use two performance metrics to assess the quality of the generated graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In both metrics, we first calculate a set of summary statistics from each individual graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', degree distribution, clustering co- 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='com/neu-spiral/AlignGraph efficient, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' we summarise these statistics in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Then we compare the distributions of these statistics be- tween the generated and test graphs w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' two metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The first is the smmd score: this score, proposed by You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [11], measures the maximum mean discrepency (MMD) between two distributions of graph statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The smmd takes values in [0, 1] (the smaller the better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We calculate an average MMD across all the statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' a formal definition can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The second performance score is the smvr score: this measures the squared difference between the mean values of the two distributions, rescaled by the variance of the value over the ground truth graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This score takes values in [0, ∞] (the smaller the better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Again, we average this across all statistics (see also App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We also report the time it took to compute graph alignments, ta, and the total training time of generative models, ttr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We compute graph alignment only once and pre-align graphs before training our AlignGraph models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We measure ta and ttr to have a fair comparison between the improvement we might get in smmd and smvr and the cost of this improvement in terms of the total time consumed by each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' To evaluate the performance of our accelerated multi-distances, we measure the accuracy of alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For this purpose, we first compute graph alignment and the center graph via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5) for Fermat distance and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='17) for G-align distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We then align the graphs in the graph set w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='t G0 and evaluate the distance of G0 from the graph set via d0 = 1 n �n i=1 ∥P T i AiPi−A0∥ ∥A1∥ (smaller is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4 Results Accelerated Multi-distances Speed and Accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We investigate the impact of our methods on run- ning time and on the accuracy of graph alignment com- putation on a graph set with 12 3−community graphs of |V | = 45 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1c we report the total time to compute the alignment using G-align dis- tance and Fermat distance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' These figures demonstrate that our proposed methods reduces the computation time by 40 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1d we compute d0 via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our results illustrate that our acceleration methods improve the accuracy of esti- mated center graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Since G-Parallel and CG-Parallel have the best trade offs for the running time and ac- curacy, we use these two methods to compute graph alignment in our next experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Evaluating the Generated Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Table 2 sum- marizes the performance scores smmd and smvr on all 7 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our experiments show that our model achieves 25% − 250% accuracy improvement over base (a) Running time (in seconds) us- ing G-align distance and applying our proposed methods to G-align distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (b) Accuracy of G-align distance and applying our proposed accel- erated multi-distances to Fermat distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (c) Same as part (a) but for Fer- mat distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (d) Same as part (b) but for Fer- mat distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Figure 1: Computation time and accuracy of computing the graph alignment in community graphs given the baselines and our three accelerated multi-distances, G-Parallel (40 CPUs), CG-Parallel (40 CPUs), and C-Serial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' G-align distance has better performance compared to Fermat distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Moreover, due to the clustered structure of community graphs clustering and grouping graphs in CG-Parallel also improves the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' models and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5%−4000% improvement over other com- petitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In some datasets, such as Community graphs and Protein graphs, G-align-Double and Fermat-Double that jointly train two similar structure generative mod- els produce the best performance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In the ma- jority of the experiments, applying our frameworks to either base GraphRNN or base GRAN leads to the best performance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' However, there is no clear win- ner between these two base generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our re- sults in Table 2 illustrate that our accelerated multi- distances methods scale well to larger graphs and are compatible with large datasets with |V | > 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' More- over, comparing the ttr of G-align-Single (GraphRNN) and G-align-Single (GRAN) models with their baselines demonstrate that our models are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='21% − 44% faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This happens due to the pre-alignment of graphs in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' On the other hand, the ta/trm ratio for G-align- Single (GraphRNN) and G-align-Single (GRAN) mod- els ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='89% to 150%, where 150% belongs to the alignment of our largest dataset, Ego-B-A (large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' While this pre-alignment took 70 minutes, it led to at least 83% improvement in the performance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Impact of Graph Perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We investigate the impact of graph perturbation on the performance of our models by perturbing edges in the 3-community graphs dataset with |V | = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The perturbation 800 running time(s) 600 400 200 0 G-align C-Serial G-Parallel CG-Parallel1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='0 G-align C-Serial G-Parallel CG-Parallel2500 2000 time(s) 1500 running 1000 500 0 Fermat C-Serial G-Parallel CG-Parallel1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='0 Fermat C-Serial G-Parallel CG-ParallelCommunity Graphs Grid Graphs Ego − Citeseer Graphs Ego − B − A Graphs Protein Graphs (|V |ave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' |E|ave) (|V |ave,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='12 835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='23 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='81 Table 2: Comparison of two performance scores for synthetic and real graphs graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' |V |ave is the average number nodes and |E|ave is the average number of edges in the graph set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' ttr indicates the total time to train generative models and ta is the total time to compute graph alignment using either G-align distance or Fermat distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (−) indicates an out of memory failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Overall, applying our frameworks to base RNN or base GRAN leads to better performance scores compared to baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (a) (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Figure 2: Sensitivity of 3 base generative models combined with G-align-Single to noise for community graphs with |V | = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' x- axis: the percentage of edges perturbed, y-axis : smmd (left), smvr (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' G-align-Single (GraphRNN) shows better overall performance, however, in all models there is a direct relation between the perturbation factor and the performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' factor ρ is defined as the percentage of edges that we randomly remove and re-add u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The ρ values are set to [10, 20, 50, 100] in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We note that with ρ < 20%, graphs still have community structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In the extreme however, with a ρ = 100%, graphs are effectively Erdös-Rényi and, thus, their statistics differ significantly from those of the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We compute smmd and smvr for graphs generated with these perturbation factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2 illustrates the performance of the G-align-Single model using GraphRNN, GRAN and VAE as base generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' G-align-Single (GraphRNN) and G-align-Single (GRAN) models have relatively good smmd compared to G-align-Single (VAE) when ρ < 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' At ρ = 10%, G-align-Single (GRAN) has the best performance which is exactly inline with the results we have in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' As the noise increases, G-align-Single (GraphRNN) shows more robustness to noise compared to the other two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' As expected, all models are adversely affected when ρ > 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 7 Conclusion We present a group of models that learn distributions of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our method is generic with respect to the generative model employed, performs better than the competitors, and enhances permutation invariant and robustness to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Acknowledgments The authors gratefully acknowledge support by the National Science Foundation (grants IIS-1741197, CCF- 1750539) and Google via GCP credit support.' 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optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Cambridge University Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [49] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', “Scikit-learn: Machine learning in Python,” JMLR, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' A Alternating Minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' At each iteration t ∈ N, we update A0 and {Pi}i∈[n] as follows: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1 Updating A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Given that {Pi}i∈[n] is fixed and D = 0, minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='t A0 leads to the following problem: min A0∈Rm×m n � i=1 ∥AiP (t−1) i − P (t−1) i A(t) 0 ∥ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1) This problem is convex and at step t ∈ N can be solved via convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Once we solve this optimization problem, we set a threshold to binarize the elements of A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2 Updating {Pi}i∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Given that A0 is fixed and D = 0, let LP ({Pi}(t) i∈[n]) be the loss function at step t ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2) LP ({Pi}(t) i∈[n])) = �n i=1 ∥AiP (t) i − P (t) i A(t) 0 ∥ Minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='t {Pi}i∈[n] leads to the follow- ing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' min Pi∈Wm LP ({Pi}(t) i∈[n]) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='3) This step is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' It can be solved via optimization toolboxes such as CVXPY [40] or efficient algorithms such as Frank-Wolfe algorithm [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Frank-Wolfe algo- rithm is explained in details in the Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' B Frank Wolfe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The objective function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2) can be solved via Frank-Wolfe algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Frank-Wolfe is an iterative algorithm that solves the problem through a sequence of linear programs (LPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This algorithm starts from a feasible P 0 ∈ Wm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' , the identity matrix I and in each iteration t ∈ N proceeds as follows: S(t) = arg min Sij∈Wm,Sii=I,S⪰0 tr(ST , ∇P LP (P (t))) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4a) P (t+1) = (1 − γt)P (t) + γtS(t), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='4b) where γt is the step size and can be set to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2 t+2 or determined by line search [48] as follows: (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='5) γt = arg min γt∈[0,1]LP (((1 − γt)P (t) + γtS(t)) C Table of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In the Table 3 we provide the lists of metrics we measured in the experiments and their description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Notation Description D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='D Graphs degree distribution C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C distribution of clustering coefficient of nodes for each graph in the graph set ASRT: assortativity, Pearson correlation coefficient of degree between pairs of linked nodes TRI: number of triangles for each graph in the graph set WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C: wedge count, number of wedges for each graph in the graph set CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C: claw count, number of claws for each graph in the graph set Table 3: Summary of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' D Performance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In order to calculate MMD2 , let a function f belong to a unit ball in a reproducing kernel Hilbert space (RKHS) H, f ∈ H, and k be the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The MMD2 between two sets of samples {xi}N i=1∼iidp and {yi}N i=1∼iidq from distributions p and q is computed as follows: (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='6) MMD2 = 1 N(N − 1) �N i=1 �N j̸=i(k(xi, xj) + k(yi, yj)) − 1 N 2 �N i=1 �N j=1(k(xi, yj) + k(xj, yi)) The performance of MMD2 depends on choice of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Here we use Gaussian-Wasserstein RBF kernel k(x, y) = e− W (p,q)2 2σ2 , where W(p, q) is the first Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The k(x, y) function is bounded, k(x, y) ∈ [0, 1] and therefore MMD2 ∈ [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' smmd score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Combining MMD2 of all metrics we measured, we present smmd score to assess the overall quality of generated graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='7) smmd = 1 12(MMD2(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='D) + MMD2(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C) + MMD2(ASRT) + MMD2(TRI) + MMD2(WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C) + MMD2(CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C)) Note that smmd ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The smaller this score, the smaller the distance between the generated graphs and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' smvr score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Our second performance score, smvr is Algorithm 1: G-Parallel: Grouping and Paralleliz- ing Graphs Input: G = {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', Gn}, K : number of graphs in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Output: G0out : the center graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' for k = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [ n K ]} do ˜ Gk = {G1+k×K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' G2+k×K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GK+k×K} ˜ Gk = align( ˜ Gk) ˜G0 k = center( ˜ Gk) end ˜G = { ˜G0 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' for k = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [ N K ]}} if [ n K ] > K then while [ n K ] > K do n = [ n K ] G = ˜G for k = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [ n K ]} do ˜ Gk = {G1+k×K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' G2+k×K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GK+k×K} ˜ Gk = align( ˜ Gk) ˜G0 k = center( ˜ Gk) end ˜G = { ˜G0 kfor k = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [ n K ]}} end end Gout = ˜G G0out = center(Gout) formulated as follows: (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='8) smvr = 1 6( (µr(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='D) − µg(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='D))2 σ2r (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='D) + (µr(ASRT) − µg(ASRT))2 σ2r (ASRT) + (µr(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C) − µg(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C))2 σ2r (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C) + (µr(TRI) − µg(TRI))2 σ2r (TRI) + (µr(WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C) − µg(WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C))2 σ2 t (WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C) + (µr(CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C) − µg(CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C))2 σ2r (CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='C) ), where µr, µg and σ2 t represent mean value for the reference set, mean value for the generated set and variance of the reference set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' E Accelerating Multi-distances In Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 1 we explain how to compute the final center graph by G-Parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We describe C-Serial algorithm is in details in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 2 and the detail of CG-Parallel are in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Table 4 shows summary of datasets, acceleration methods and solvers used to compute graph alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Algorithm 2: C-Serial: Coarsening Graphs Input: G = {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', Gn}, c : number of clusters in graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Output: G0out : the center graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' for l = {0, 1, 2, · · · , n} do ˆGl = coarsen(Gl) end ˆG = align({ ˆG1, ˆG2, · · · , ˆGn}) for l = {0, 1, 2, · · · , n} do align clusters given their alignment in the coarsened graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' end compute center of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' compute center of edges connecting clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' ˜G0 k: build given the center of clusters and center of edges connecting clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Algorithm 3: CG-Parallel: Coarsening, Grouping and Parallelizing Graphs Input: G = {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', Gn}, K : number of graphs in each group, c : number of clusters in graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Output: G0out : the center graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' if [ n K ] > K then while [ n K ] > K do for k = {0, 1, 2, · · · , [ n K ]} do ˜ Gk = {G1+k×K, G2+k×K, · · · , GK+k×K} for l = {0, 1, 2, · · · , K} do ˆGl+k×K = coarsen(Gl+k×K) end ˆ Gk = align({ ˆG1+k×K, ˆG2+k×K, · · · , ˆGK+k×K}) for l = {0, 1, 2, · · · , K} do align clusters given their alignment in the coarsened graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' end compute center of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' compute center of edges connecting clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' ˜G0 k: build given the center of clusters and center of edges connecting clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' end ˜G = { ˜G0 k, for k = {0, 1, 2, · · · , [ N K ]}} end end Gout = ˜G G0out = center(Gout) |V |ave |E|ave n Alignment alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Solver (Fermat) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Solver (G-align) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Community (small) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='98 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='G-Parallel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY + AM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Community (large) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='2727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CG-Parallel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY + AM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Grid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='265 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='G-Parallel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY + AM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Ego-Citeseer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='G-Parallel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY + AM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Ego-B-A (small) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='118 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='298 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CG-Parallel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY + AM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Ego-B-A (large) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1028 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='1471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CG-Parallel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY + AM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='CVXPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='Table 4: Dataset summary including average number of nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='and edges and number of graphs in the graph set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' along with the algorithm and solvers used to compute graph alignment in Fermat and G-align distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For smaller graphs (with |V |ave < 50 ) we use the G-Parallel method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For larger graphs, to further accelerate computing the graph alignment we use CG-Parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' By using these acceleration techniques, all alignment problems can be solved by CVXPY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' F Implementation Details We compared the performance our models against five different deep baseline described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GraphRNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [11] proposes a framework based on graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This model uses a graph-level RNN to add a new node to a node sequence each time step and an edge-level RNN to model the generation process of nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The reference code for this model is provided by the authors and we followed their recommendation for setting the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [12] proposes a graph recurrent at- tention framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' This model uses an attention-based GNN and generates a block of graphs that consists of multiple rows of graph adjacency matrices conditioned on the previously generated blocks of the graph and uses a group canonical node ordering, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=', DFS and BFS to address node ordering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Kipf & Welling [10] propose a variational autoen- coder that is characterized by a probabilistic inference model that maps observed data to a latent representa- tion, a prior distribution over the latent variables and a probabilistic generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We randomly pick a graph with a random node ordering from graph set G and train a VAE to generate graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GraphVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Simonovsky & Komodakis [14] propose a variational autoencoder that outputs a probabilistic fully-connected graph and uses a graph matching algo- rithm to align graph to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' GraphVAE outputs a graph with adjacency matrix, node attributes and edge attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We adapt it to our problem by using one-hot representations of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The en- coder is a graph convolutional network and the decoder is a multi-layer perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We used code for this model from [11] and set the hyperparameters based on recom- mendations made in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' DeepGMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' [47] introduce a generative model for graphs that generates graphs in a sequential manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' It generates one node at a time and connects each node to the partial graph already generated by creating edges one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We used the implementation in [11] and the hyperparameters were set based on the recommendations made in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We take 80% of graphs for training and the rest for the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' During testing, GraphRNN model and GRAN model generate graphs directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' However, the output of the VAE decoder is an adjacency matrix with elements in the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We binarize the adja- cency matrix by applying a threshold, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We find τ by comparing two sets of graphs, the ones generated from the VAE and 20% of the graphs in the training set, and computing two scores, which we denote smmd and smvr (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='3) to measure the distance between these two sets for a range of values of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We chose the value of τ that returns the smallest smmd as our optimal threshold in testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In all our AlignGraph models, we pre-compute the graph alignment for all datasets and use the aligned graphs for training the gen- erative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We use GraphRNN [11], GRAN [12] and VAE [10] as our base generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We followed the instructions given in [11] and [12] to set the hy- perparameters in GraphRNN [11] and GRAN [12] and for the VAE [10] we used the hyperparameters given in [10] and set the learning rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' (We note that VAE[18] here refers to the model proposed by Kipf & Welling and is different from GraphVAE[14] by Si- monovsky & Komodakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In all models, the hidden di- mensions {Zi}i∈[n] of small graphs are set to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For medium graphs (|V | ∈ [100 − 500] ) the hidden dimen- sions are 64 and the hidden dimensions of the large graphs (|V | > 500) are set to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' In all experiments, The node features are one-hot indicator vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' The AlignGraph architectures are implemented in Python3 using Tensorflow and Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We implemented the solution of the constrained optimization problems in Section 4 via CVXPY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' We implemented all solvers in Python3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For clustering graphs we use Scikit-learn [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' All experiments are carried out on a Tesla V100 GPU with 32 GB memory and 5120 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' G-Parallel and CG-Parallel methods parallelizes the computations us- ing python multiprocessing package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} +page_content=' For both of these parallel graph alignment algorithms we use a single ma- chine with 40 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFIT4oBgHgl3EQfeCvq/content/2301.11273v1.pdf'} diff --git a/P9FJT4oBgHgl3EQfJCz4/vector_store/index.faiss b/P9FJT4oBgHgl3EQfJCz4/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b4c93907b26b4185da86ed35953fd714d998e2da --- /dev/null +++ b/P9FJT4oBgHgl3EQfJCz4/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:66a259e5a0a5e5dd1af772650d56c2daef1031896309a296f50ef58cbccbabec +size 5832749 diff --git a/PdAzT4oBgHgl3EQfWvxw/content/2301.01306v1.pdf b/PdAzT4oBgHgl3EQfWvxw/content/2301.01306v1.pdf new file mode 100644 index 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b/QdFKT4oBgHgl3EQfhi5s/content/tmp_files/2301.11838v1.pdf.txt @@ -0,0 +1,1950 @@ +Quantum-enhanced quantum Monte Carlo: an industrial view +Maximilian Amsler1, Peter Deglmann2,3, Matthias Degroote4, Michael P. Kaicher3, Matthew +Kiser5,6, Michael K¨uhn3, Chandan Kumar7, Andreas Maier8, Georgy Samsonidze9, +Anna Schroeder10,11, Michael Streif4,∗, Davide Vodola2, and Christopher Wever1 +QUTAC Material Science Working Group +(Dated: January 30, 2023) +In this work, we test a recently developed method to enhance classical auxiliary-field quantum +Monte Carlo (AFQMC) calculations with quantum computers against examples from chemistry and +material science, representatives of classes of industry-relevant systems. As molecular test cases, we +calculate the energy curve of H4 and relative energies of ozone and singlet molecular oxygen with +respect to triplet molecular oxygen, which are industrially relevant in organic oxidation reactions. +We find that trial wave functions beyond single Slater determinants improve the performance of +AFQMC and allow to generate energies close to chemical accuracy compared to full configuration +interaction (FCI) or experimental results. +As a representative for material science we study a +quasi-1D Fermi-Hubbard model derived from CuBr2, a compound displaying electronic structure +properties analogous to cuprates. We find that trial wave functions with both, significantly larger +fidelities and lower energies over a Hartree-Fock solution, do not necessarily lead to better AFQMC +results. +I. +INTRODUCTION +Recent years have shown significant advancements +in the field of quantum computing, both in build- +ing more powerful quantum hardware with an ever- +increasing number of qubits and lower error rates, as well +as in developing quantum algorithms to solve problems +in optimization [1], machine learning [2, 3], and in cryp- +tography [4]. Finding solutions to classically intractable +problems in quantum chemistry and material science is +often touted as the first application of future quantum +computers in industry [5, 6]. +With improvements in quantum hardware and quan- +tum algorithms, the search for areas in industry where +quantum computing could provide an economic or tech- +∗ +Corresponding +author: +michael.streif@boehringer- +ingelheim.com +1 Corporate Sector Research and Advance Engineering, Robert +Bosch +GmbH, +Robert-Bosch-Campus +1, +71272 +Renningen, +Germany +2 BASF SE, Quantum Chemistry, Carl-Bosch-Str. +38, 67063 +Ludwigshafen, Germany +3 BASF Digital Solutions GmbH, Next Generation Computing, +Pfalzgrafenstr. 1, 67056, Ludwigshafen, Germany +4 Quantum Lab, Boehringer Ingelheim, Ingelheim am Rhein, +Germany +5 Volkswagen AG, Ungererstr. 69, 80805 Munich, Germany +6 TUM School of Natural Sciences, Technical University of +Munich, Boltzmannstr. 10, 85748 Garching, Germany +7 BMW Group, New Technology and Innovation, Parkring +19-23, 85748, Garching, Munich, Germany +8 Munich Re AG, Munich, Germany +9 Robert Bosch LLC, Research and Technology Center, Sunny- +vale, CA 94085, USA +10 Merck KGaA, Frankfurter Straße 250, 64293 Darmstadt, +Germany +11 +Quantum Computing Group, +Department of Computer +Science, Technical University of Darmstadt, Mornewegstraße +30, 64293 Darmstadt, Germany +nological advantage over current approaches has gath- +ered much attention in the past few years. +While in +academic studies, much weight is given to demonstrat- +ing the superior scaling of a quantum algorithm over a +respective classical counterpart in terms of gate complex- +ity in the large-problem-size-limit [7, 8], in an industrial +setting, a quantum advantage is reached when the use of +a quantum device allows to improve processes, cut down +costs, or design new products. Since fully error-corrected +quantum computers which can execute quantum algo- +rithms with provable speedups over classical algorithms +are years away, it is intriguing to address the question +of whether a quantum advantage in Noisy-Intermediate +Scale Quantum (NISQ) devices can be found for indus- +trial purposes. +Even though NISQ devices are limited to short quan- +tum circuits, they might provide some advantage over +classical algorithms, as computations in classically in- +tractable regions of the Hilbert space are possible even +with modest quantum resources [3, 9]. A promising class +of NISQ algorithms are variational quantum algorithms, +such as the variational quantum eigensolver (VQE) for +chemistry problems [10, 11]. The VQE is a hybrid al- +gorithm, meaning that the computational task is split +between a classical and a quantum processor, while the +quantum processor is used only to estimate the energy of +a given quantum state manipulated by a set of variational +parameters, updated by the classical computer. Due to +the accumulation of errors with increasing problem size +and run time, the largest demonstration of such algo- +rithms has been limited to a few tens of qubits [12, 13], +even though quantum computers with hundreds of qubits +exist [14]. +Various classical post-processing steps have +been introduced to mitigate the effects of noise [15]. +However, the required classical computational overhead +resulting from such techniques often nullifies any pos- +sible advantage [16]. Moreover, the classical optimiza- +tion of the variational parameters can suffer from vanish- +arXiv:2301.11838v1 [quant-ph] 27 Jan 2023 + +2 +ing gradients, known as the barren plateau phenomenon +[17], which can prevent the classical optimization rou- +tine from finding the global optimum. In addition, there +is some numerical evidence that suggests that the gate- +error probabilities needed to generate variational states +that describe the ground state of certain molecules within +chemical accuracy can lie below the gate-error probabili- +ties required by most quantum error-correction protocols +[18]. +The observation of those practical challenges in many +numerical simulations and experiments indicates that +VQE or related variational algorithms alone are unlikely +to generate a quantum advantage in the future. +It is +therefore important to explore if other algorithms can ex- +ploit the computational power present in NISQ devices +[19–21], and to benchmark the readiness of such new al- +gorithms for industry applications. +A +promising +avenue +is +given +by +classical +post- +processing techniques, such as using the output to cal- +culate interaction energies [22, 23], improving the energy +estimates using neural networks [24], or in quantum sub- +space expansions [25–27]. In Ref. [13], results on H4 and +a small periodic model for the carbon allotrope diamond +obtained with a NISQ device suggest that such output +can also be used as a trial wave function to guide classi- +cal quantum Monte Carlo (QMC) calculations, more pre- +cisely auxiliary-field quantum Monte Carlo (AFQMC). +In this work, we apply AFQMC to a selection of +industry-relevant problems. As a molecular test case, we +calculate relative energies of ozone and singlet molecular +oxygen with respect to triplet molecular oxygen using ex- +perimental geometries from Refs. [28, 29] and compare to +experimental results [29, 30]. To connect to typical in- +dustrial workflows, where geometries are optimized with +DFT, we compare the AFQMC energies obtained from +experimental geometries and from DFT-optimized ge- +ometries. As an example from material science, we calcu- +late the ground state energy of a one-dimensional CuBr2 +chain, mapped to a low-dimensional Hubbard model. We +use trial wave functions obtained from classical meth- +ods and VQE circuits. We compare the performance of +AFQMC guided by those trial wave functions to mean- +field- and other standard quantum chemistry methods. +From an industrial perspective, it is important to iden- +tify the model errors and to determine necessary improve- +ments to a given method. +About QUTAC: +To investigate the potential impact +of quantum computing for industrial applications, thir- +teen leading German companies are cooperating inside +the Quantum Technology & Applications Consortium +(QUTAC). The goal of this collective effort is to evaluate +the latest quantum algorithms against industry-relevant +applications and provide guidance on needed quantum +developments toward industrial applicability. +II. +AUXILIARY FIELD QUANTUM MONTE +CARLO (AFQMC) +The AFQMC algorithm is an ab-initio method that +allows the use of any one-particle basis of size M to +project out the ground state of a strongly interact- +ing fermionic system by performing a random walk in +the space of fermionic Gaussian states [31]. Fermionic +Gaussian states are exponentials of Hermitian quadratic +fermionic operators [32] and build the basis of AFQMC in +both the space of Slater determinants and Hartree-Fock +Bogoliubov states [31, 33–37]. +Projector methods use the property that the solution of +the imaginary time Schr¨odinger equation of the Hamilto- +nian H asymptotically approaches the ground state |Ψ0⟩, +|Ψ0⟩ = lim +τ→∞ |Ψ(τ)⟩ = lim +τ→∞ +e−(H−E0)τ |ΨI⟩ +� +⟨ΨI|e−2(H−E0)τ|ΨI⟩ +, (1) +where E0 is the unknown ground state energy, |ΨI⟩ is +the initial state, and we assume ⟨ΨI|Ψ0⟩ ̸= 0. Since E0 +is in principle unknown, it is replaced by various adap- +tive estimators in the AFQMC algorithm [33]. Classical +methods have so far not been able to solve Eq. (1) ef- +ficiently for strongly interacting systems, and therefore +one has to resort to approximate methods. The core idea +of AFQMC is to transform the imaginary time propaga- +tor of a quartic operator into an integral over a quadratic +operator, whose action on a fermionic Gaussian state can +be computed efficiently [37–39]. This transformation is +realized through a Hubbard-Stratonovich transformation +[40]. The resulting integral over matrix exponentials of +quadratic operators is then solved in a Monte Carlo fash- +ion [31]. +One hallmark problem of fermionic systems which ap- +pears in QMC approaches is the so-called sign or phase +problem, which is caused by the anticommutation rela- +tions of fermions and phase accumulated in imaginary +propagation, respectively [31, 41]. This results in an ex- +ponential divergence of the variance of the estimator of +the energy in the k-th step of a Monte Carlo simulation +of NW walkers, where the energy estimator is defined as +E(k) ≃ +�NW +w=1 Wk,weiθk,wEloc(Ψk,w) +�NW +w=1 Wk,weiθk,w +, +(2) +where Wk,w and θk,w denote the amplitude and phase of +the w-th walker Ψk,w at the k-th iteration, +Eloc(Ψk,w) = ⟨ΨT |H|Ψk,w⟩ +⟨ΨT |Ψk,w⟩ +(3) +is the local energy, and |ΨT ⟩ is the trial wave func- +tion. The latter controls the evolution of the simulation +and combined with a phaseless approximation tames the +phase problem at the expense of an introduced bias in +the energy estimator [31]. To be efficiently computable +on classical computers, the class of wave functions that + +3 +can be used as trial states (or walkers) has been limited +to linear combinations of (non-orthogonal) Slater deter- +minants or Hartree-Fock Bogoliubov states. +However, +quantum computers allow us to probe trial wave func- +tions outside this class and thus possibly improve the +performance of classical AFQMC calculations, as first re- +alized by Ref. [13]. In the next section, we describe the +classical and quantum trial wave functions used in this +work. +III. +AFQMC TRIAL WAVE FUNCTIONS +A. +Variational Quantum Eigensolver (VQE) +The VQE [10] is a variational quantum algorithm tai- +lored to find the ground states of a Hamiltonian. The +VQE uses a parameterized quantum circuit or ansatz, +and classically optimizes the parameters θ of the circuit +with the goal to minimize a cost function, more specifi- +cally, the expectation value of the Hamiltonian ⟨Ψ|H|Ψ⟩. +It fulfills the variational principle +E = ⟨Ψ|H|Ψ⟩ +⟨Ψ|Ψ⟩ +≥ E0, +(4) +where E0 is the exact ground state energy. Equality holds +when |Ψ⟩ = |Ψ0⟩. To keep the number of parameters θ +and the depth of the circuit as small as possible, the +ansatz is typically tailored specifically to the problem +[42, 43]. +A popular ansatz for chemistry problems is the uni- +tary coupled-cluster ansatz [44], which we introduce in +Appendix A. Experimental retrieval of the expectation +value from quantum hardware can be implemented effi- +ciently by employing schemes such as shadow tomogra- +phy [45], basis rotation groupings [46], or by optimizing +the number of commuting terms [47], which reduces the +number of required measurements. +The efficient optimization of the parameters θ is +an open field of research. +While the well-established +gradient-free optimizer COBYLA [48] is a popular choice +during the prototyping phase, a quantum-aware opti- +mizer is inevitable to avoid issues associated with noisy +quantum hardware. For example, the quantum natural +gradient descent [49] achieves faster convergence by con- +sidering the geometric information of quantum states. +B. +Matrix product states +Matrix product states (MPS) provide an efficient +parametrization of one-dimensional quantum states in +terms of matrices with dimensions bounded by the non- +negative integer bond dimension χ [50]. The bond di- +mension can be viewed as a parameter that controls the +amount of entanglement and thus the degree of expres- +sivity of an MPS. MPS are variational states that can +approximate ground states of a many-body Hamiltonian +when used within the density matrix renormalization +group (DMRG) method that can capture the entangle- +ment structure of strongly correlated wave functions [51]. +In this work, we use an MPS representation of the +ground state of the Fermi-Hubbard model as a trial wave +function |ΨT ⟩ in AFQMC. We generate MPS with dif- +ferent bond dimensions χ by employing the DMRG al- +gorithm implemented in the library ITensor [52]. Choos- +ing different bond dimensions for the MPS provides trial +states that approximate the true ground state |Ψ0⟩ in a +controlled manner. This allows us to understand the ef- +fect of the trial states on the resulting estimated energy +provided by AFQMC. +IV. +RESULTS +A. +Molecular systems +In the following, we use AFQMC with classical and +quantum trial wave functions to calculate (i) the ground +state energies of H4 in a rectangular shape for varying +side lengths and (ii) the relative energies of ozone and +singlet molecular oxygen with respect to triplet molec- +ular oxygen. +For all AFQMC calculations we use ipie +[53] with computational details summarized in the Ap- +pendix in Table V. To generate the Hamiltonian inte- +grals and classical trial wave functions we used PySCF +[54]. To generate the VQE trial wave function, cirq [55] +was used. As a quantum trial wave function we use a +UCCSD-VQE ansatz applied to a single Slater determi- +nant, see Appendix A for additional information. +To +allow for a practical setup, we initialise the VQE param- +eters with classical CCSD amplitudes [44] and treat the +optimisation of the VQE as a black box using scipy’s [56] +implementation of the COBYLA optimiser. +1. +H4 Square +As a first benchmark, we study H4 in a minimal basis +STO-3G in a rectangular shape with the geometry given +by +H4 :(H, (0, 0, 0)), (H, (0, 0, a)), +(H, (a, 0, 0)), (H, (a, 0, a)) +where we vary the side length a from 0.85˚A to 2.5˚A and +aim to find the ground state energy of the singlet state. +The H4 molecular system is a commonly used bench- +mark for classical and quantum electronic structure al- +gorithms due to its relatively small size but still present +correlation effects [13, 36, 57–59]. Due to near degen- +eracy of the ground state, both the static and dynamic +correlations are relevant for an accurate ground state en- +ergy calculation [58, 60]. Moreover, this system was used +as the first benchmark in Google’s AFQMC landmark + +4 +1.9 +1.8 +1.7 +1.6 +1.5 +1.4 +Energy [Hartree] +RHF +FCI +VQE +AFQMC(RHF) +AFQMC(VQE) +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 +Side length [Å] +0 +1 +2 +Error [kcal/mol] +FIG. 1. (top) Energy surface of the H4 square with varying +side length of the square. +For this calculation, a minimal +basis STO-3G was used. +It should be noted that the FCI +and VQE curve are hard to distinguish because of the scale of +the y-axis and their similarity to the AFQMC(VQE) results. +(bottom) The error of the ground state estimate of plain VQE +(purple) and AFQMC(VQE) (green) w.r.t. the true ground +state energy. +The error bars stem from the statistical MC +errors. +paper [13]. Here, we extend this benchmark by calculat- +ing the potential energy curve and using a different VQE +ansatz. +We start by running a VQE circuit for each side length +using eight qubits. We subsequently use the optimized +VQE output as a trial wave function in an AFQMC cal- +culation. +In Fig. 1(a) we show AFQMC energies together with +restricted Hartree-Fock (RHF) and VQE energies and +display the exact ground state energy as a reference. A +single-determinant RHF trial wave function is insufficient +to guide the AFQMC to the ground state energy of the +system [36]. However, the VQE and its AFQMC(VQE) +post-processed result can barely be discerned from the +exact FCI result in this figure. As shown in Fig. 1(b), +VQE alone does not generate results within chemical ac- +curacy for most side lengths. +Applying AFQMC as a +post-processing procedure to the VQE results improves +the energy estimates. However, even for this small exam- +ple using a noiseless simulation with access to the per- +fect state vector, AFQMC cannot generate results within +chemical accuracy for all side lengths. +2. +Ozone and molecular oxygen +As a second benchmark, we investigate the perfor- +mance of AFQMC for calculating energy differences be- +tween species (relative energies). We calculate the rela- +tive energies ∆E(x) of ozone and singlet molecular oxy- +gen with respect to the triplet ground state of molecular +oxygen. While of significant industrial relevance, many +standard quantum chemistry methods fail to estimate +relative energies and energetic differences between spin +states within chemical accuracy. The calculation of the +latter with AFQMC was studied in [61, 62]. +In the following, we compare AFQMC results to results +obtained from standard quantum chemistry methods and +experimental values from [29, 30]. We denote the error +with respect to the experiment by ∆∆E(x) = ∆E(x) − +exp. value(x). In the case of molecular oxygen respective +experimental values are directly available from the liter- +ature [29] whereas for ozone the available experimental +values refer to enthalpies of formation at absolute zero +temperature, ∆Hf +0K(O3), [30] that inherently include a +zero-point vibrational energy (ZPVE). To compare our +calculated results to those experimental values, we sub- +tract the ZPVE from the enthalpy of formation according +to: exp. value(O3) = ∆Hf +0K(O3) − ∆ZPVE(O3) wherein +the ZPVEs are calculated at the density functional +theory (DFT) level (B3LYP/def2-QZVPP) resulting in +ZPVE(3O2) = 0.0037 Ha and ZPVE(O3) = 0.0064 Ha +yielding a correction of ∆ZPVE(O3) = 2.2 kJ/mol. +Molecular dioxygen O2, in particular singlet oxygen +1O2, which is formed by electronic excitation of the +air constituent triplet oxygen 3O2, is a highly reactive +molecular species [63, 64] involved in various chemistries +such as desired as well as undesired photochemical reac- +tions [64–66], ene reactions [67], and organic oxidation +reactions in general. +Singlet molecular oxygen is also +responsible for damage in biological materials [68, 69]. +Ozone (O3) is another gaseous, highly reactive form of +oxygen that is typically formed in the atmosphere, e.g., +by a photochemical reaction catalyzed by NOx. Ozone +undergoes a fast chemical reaction with C-C double bond +containing materials, such as rubbers, finally cleaving the +C-C double bond and is thus responsible for significant +material damage worldwide [70]. Due to these reasons - +their reactivity and abundance - the modifications of oxy- +gen make for a compelling use-case application of high- +level accurate quantum chemical methods. +Relative energies of singlet molecular oxygen and ozone +with respect to triplet molecular oxygen are of particu- +lar interest to understand the stabilities of the respective +competing species. More generally, an accurate predic- +tion of energy differences, such as relative stabilities, re- +action energies, and activation energies, is important in +an industrial context as they can be related to exper- +imentally observable thermodynamic and kinetic prop- +erties of molecules and chemical reactions. Previously, +there have been theoretical and computational studies +on the aforementioned three molecular forms of oxygen +[71–77] revealing that singlet molecular oxygen and to +an even larger extent ozone are challenging systems for +single reference electronic structure methods. +We base our calculations on experimental geometries +from Refs. [28, 29]. +As a comparison, we use geome- +tries optimized with DFT at the B3LYP/def2-QZVPP +level [78, 79] resulting in the geometries given in (B2). + +5 +HF [Ha] +VQE [Ha] +CAS [Ha] +AFQMC energy -225.2901(12) -225.2940(5) -225.2966(4) +TABLE I. AFQMC results on the ground state energy of +ozone using a HF, VQE and CAS trial wave function. For +all calculations cc-pVQZ was used as basis. The VQE and +CAS trial wave function were obtained using an (12e, 9o) ac- +tive space built from CASSCF MOs. +We find that our DFT geometries overestimate the bond +length of singlet and triplet molecular oxygen by roughly +0.01˚A and 0.003˚A respectively, with respect to the ex- +periment. For ozone, we find an absolute difference of +0.002˚A in the bond length rO−O and a difference of 0◦09′ +in the bond angle θO−O−O. It was furthermore studied +in Table IV how large the scatter of structural parame- +ters and ZPVEs is when applying different levels of the- +ory at which the computation of energy gradients is well +established; here, it turned out, that the differences are +small between different classes of density functionals (e.g. +ZPVEs vary between 1 and 2 kJ/mol), whereas HF and +MP2 predictions are partially far away from DFT and +the experimental values. +To have consistent active spaces across all molecules, +the molecular orbitals with predominantly atomic 2p +character are selected, resulting in (8e, 6o) active spaces +for the singlet- and triplet molecular oxygen systems and +a (12e, 9o) active space for ozone. To further improve +the choice of active space orbitals for AFQMC, we run +additional CASSCF calculations using the PySCF pro- +gram package [54] and base our trial wave function gen- +eration on the resulting CASSCF orbitals. We note that +such active spaces should be able to capture a large frac- +tion of the correlations present in the system, while still +would allow for experimental implementations on cur- +rently available quantum hardware. +To benchmark the performance of AFQMC using a +quantum trial wave function, we focus on the ozone +system. We first run a UCCSD-VQE (detailed in Ap- +pendix A) inside the (12e, 9o) active space. The result- +ing wave function together with the exact solution inside +this active space (CAS) and the Hartree-Fock solution (in +canonical MOs) is input as trial wave function to bench- +mark the performance of AFQMC. In Table I, we report +the AFQMC results on the experimental geometries of +ozone. We find that the energy of AFQMC with a VQE +trial wavefunction lies between AFQMC with a HF and +a CAS trial wavefunction. +In the following, we assume that future first-generation +quantum computers can generate the exact ground state +wave function in a small active space. We use the CAS +trial wave functions to benchmark the performance of +AFQMC when calculating relative energies. We follow +Refs. [61, 80] and calculate total energies using the cc- +pVTZ and cc-pVQZ basis [81]. We extrapolate the cor- +relation energies to find the total energies in the complete +basis set (CBS) limit, see Appendix C for more details, +and use the CBS total energies to calculate the relative +UHF +B3LYP +CCSD(T) +CASSCF +NEVPT2 +AFQMC +−75 +−50 +−25 +0 +25 +50 +75 +100 +Error w.r.t experiment [kJ/mol] +-23.0 +-52.0 +28.7 +-8.5 +-3.4 +5.7 +81.0 +40.0 +8.1 +9.6 +-10.3 +1.2 +Error in relative energies w.r.t. 3O2 (experimental geometries) +∆∆E(1O2) +∆∆E(O3) +FIG. 2. Comparison of the error in relative energies of singlet +molecular oxygen and ozone with respect to triplet molecular +oxygen using experimental geometries given in Eq. B1 with +respect to experimental data. +We report the results from +different computational methods obtained using the cc-pVQZ +(for UHF) and def2-QZVPP (for B3LYP) basis set and results +extrapolated to the CBS limit (otherwise). +The CASSCF +and NEVPT2 calculations were carried out in a (12e, 9o) +active space for ozone and (8e, 6o) active spaces for singlet +and triplet molecular oxygen. The CASSCF wave function +was used as a trial wave function in the AFQMC calculation. +The error in AFQMC stems from the statistical MC error. +energies. In Fig. 2, we show the results in comparison to +the experimental values. We find that while commonly +used methods such as DFT (B3LYP), CCSD(T), and +CASSCF alone are not able to reach chemical accuracy +(4 kJ/mol), the AFQMC results, ∆E(1O2) = 100.4 ± 2.4 +kJ/mol and ∆E(O3) = 144.4 ± 2.7 kJ/mol, are within +or close to chemical accuracy with respect to the experi- +mental results. +However, this comparison has to be taken with a grain +of salt. +First, for all calculations, the statistical error +of AFQMC itself is of the order of a few kJ/mol, mak- +ing quantitative comparisons between two calculations, +which are needed to calculate relative energies, difficult. +For calculations of chemical reactions with more than one +educt and product, the propagation of errors would get +more severe, for example, in redox reactions, SN2 reac- +tions and many more. Second, we only use two points +(cc-pVTZ and cc-pVQZ energies) to extrapolate to the +CBS limit. +Future calculations using larger basis sets +(cc-pVXZ with X ≥ 5) could be used to benchmark +the CBS result and to improve it further. +Also, the +scheme employed here to extrapolate to the CBS limit +is one example of many possible choices [80, 82], result- +ing in a choice-supportive bias. +Using the same level +of theory for calculating the ZPVEs as for calculating +the total energies might improve the results in the case +of ozone. However, it would not be expected that this +would strongly alter the observed trends. +This would +come at an increased algorithmic cost, as it would re- +quire accurate estimating forces and ZPVEs in AFQMC. +Lastly, the experimental values themselves are associated +with errors. We note that, from an industrial perspec- +tive, CBS extrapolations are rarely performed due to + +6 +time constraints. When comparing the plain cc-pVQZ +results to the experimental value, we find that the error +of AFQMC increases by 4 kJ/mol, see Table VI. When +comparing to cc-pVQZ results using DFT-optimized ge- +ometries as commonly done in the industry, we find rel- +ative energies of ∆E(1O2) = 104.9 ± 1.2 kJ/mol and +∆E(O3) = 147.1±1.4 kJ/mol, comparable to the results +obtained with experimental geometries. +B. +Extended systems +Material systems exhibiting effects of strong corre- +lation ranging from metal-insulator transitions to half- +metallicity and spin-charge separation are of interest to +various technological applications. A particularly inter- +esting class of materials are cuprate high-temperature su- +perconductors, the physics of which are believed to stem +from a single correlated d band in the low-energy spec- +trum [83, 84]. A minimal model to describe such a system +is the one-band Hubbard model. Still, extensions of it by +generalized Hubbard-like Hamiltonian are more refined +and, e.g., take explicitly into account the effect of the +oxygen p-orbitals [85]. The general atomic structure of +cuprates consists of one or several planes of CuO2 in a +square lattice stacked along the z-direction, interspersed +with layers of guest atoms which act as carrier donors. It +is commonly believed that the superconductivity is con- +fined to the 2D-CuO2 planes and that all relevant spin +and charge carriers reside in those substructures [86]. +Analogous to cuprates, copper-bromides consist of +similar structural building blocks where Cu2+ are sur- +rounded by four Br– (instead of O2– in cuprates) in a +square planar arrangement, which in the latter however +form chains instead of 2D sheets. Due to this structural +and chemical similarity, both materials systems share +related electronic properties [87], with a single dx2−y2 +band crossing the Fermi level. Emerging antiferromag- +netic [88, 89] and multiferroic properties [90–92] have +sparked recent interest in these materials classes, with +potential applications in a wide range of magnetoelectric +devices. +Native CuBr2 crystallizes in a polymeric structure with +C2/m-symmetry, where square planar units of CuBr4 +(one Cu surrounded by four Br atoms) are linked together +to form chains, packed parallel along one direction [93]. +These chains interact weakly with each other, with the +main electronic properties governed by the intra-chain +interaction. +Here, we map CuBr2 to an effective, but +fictitious 1D model system and compute its properties +based on a one-band Hubbard model. We construct the +quasi-1D structural model of CuBr2 by isolating a single +chain and placing it in a cell where periodic images of the +chains are separated by vacuum of 10 and 5 ˚A in the lat- +eral and vertical direction, respectively (see Fig. 3a). The +band structure and density of states (DOS) are given in +Fig. 3b. The Cu–Br distance and Cu–Br–Cu angle within +the chain are retained at experimental values of 2.398˚A +and 92.35◦, respectively. +To build the effective model, we use DFT as imple- +mented in the Quantum ESPRESSO package [94] us- +ing the Perdew-Burke-Ernzerhof approximation to the +exchange-correlation functional [95] and norm-conserving +pseudopotentials [96]. +We employ a plane-wave cutoff +energy of 100 Ry in conjunction with a k-points mesh +at a density of 0.2/˚A to obtain converged results. +A +single-orbital tight-binding model is constructed by Wan- +nierizing the dx2−y2 band crossing the Fermi level us- +ing the Wannier90 package [97]. The resulting Wannier +orbital is shown in Fig. 6a, which was converted with +wan2respack [98] to serve as input for RESPACK [99] +to obtain the screened Coulomb interaction parameters +based on the constrained random phase approximation +(cRPA). We include 83 virtual orbitals in addition to +the 17 occupied states, resulting in sufficiently converged +screened interaction parameters. +After keeping only the transfer integrals up to the +3rd nearest neighbors (see Appendix D), the resulting +parametrized Hubbard-like Hamiltonian is +HH = − tx +� +iσ +a† +i,σai+1,σ + H.c. +− txx +� +i,σ +a† +i,σai+2,σ + H.c. +− txxx +� +i,σ +a† +i,σai+3,σ + H.c. +− U +2 +� +iσ +niσ + U +� +i +ni,↑ni,↓ +(5) +where a† +i,σ (ai,σ) creates (annihilates) an electron with +spin σ on the lattice site labelled by i and niσ = a† +i,σai,σ. +The explicit values of the parameters tx, txx, txxx and U +(in eV) are reported in Table II. Note that the spu- +rious hopping terms txxxx, ty, tz, and txy are small +and justify their neglect in our model. +Since the ra- +tio U/ max (t) ≈ 28 ≫ 1, the Hamiltonian (5) ap- +proaches the regime of a spin-1/2 Heisenberg antiferro- +magnet [100, 101]. +The number of lattice sites in the +periodic lattice is denoted as L, and we use open bound- +ary conditions in the simulations. The Hamiltonian HH +conserves the number of electrons and the total spin, and +all simulations are performed at half-filling with balanced +spin, N↑ = N↓ = L/2. In the following, we study how +the fidelity and energy of a trial wave function impact the +energy obtained from an AFQMC simulation of Eq. (5). +We compute the ground state energy of HH via +AFQMC using ipie [53] with computational details sum- +marized in Table V. We consider two lattice lengths, +L = 6 and L = 10, and use different wave functions +as trial states |ΨT ⟩: (i) a Hartree-Fock (HF) mean-field +state obtained from an imaginary time evolution of a +Slater determinant following the work of [39]; (ii) a MPS +wave function with bond dimension χ = 4, 8, 16, 32 ob- +tained via DMRG. For L = 6, the MPS at half filling in +the zero spin sector generates 400 Slater determinants, + +7 +(a) +x +y +z +(b) +FIG. 3. (a) Structural model of the quasi-1D chain of CuBr2. The blue and brown spheres denote the Cu and Br atoms, +respectively, while the periodic images are separated by 10 ˚A and 5 ˚A in the lateral (green arrow) and vertical (blue arrow) +directions, respectively. The arrows indicate the hopping distance along the chain, ranging from the nearest (red arrow) to the +4th nearest-neighbor (yellow arrow). (b) The band structure of the CuBr2 chain on the left, together with its density of states +(DOS) on the right. Note, that the energy is shifted such that the Fermi level is at value zero. The single wannierized band is +shown in blue. The inset shows the Brillouin zone, with the irreducible portion outlined by the blue lines and the labels of the +special k-points. +µ = t(000) tx = t(100) txx = t(200) txxx = t(300) txxxx = t(400) ty = t(010) tz = t(001) txy = t(110) U(000, 0) J(000, 0) +−1.0987 +0.0478 +0.1570 +0.0339 +0.0059 +0.0019 +−0.0046 +−0.0005 +4.15 +4.15 +TABLE II. Interaction parameters for the Cu:dx2−y2 orbital of CuBr2 with respect to lattice vectors t(xyz) in Miller index +notation, in eV. +all of which are included as a trial wave function. For +L = 10, since including all generated Slater determinants +of the MPS state would be too demanding. Instead, we +sample Slater determinants with a weight > 5 × 10−3, +resulting in a trial wave function with up to ∼ 300 Slater +determinants. The initial walkers for the AFQMC imag- +inary time propagation are chosen as the HF state of (i) +for both trial states. +In Fig. 4 we plot the energy defined in Eq. (2) for (a)- +(b) a HF trial wave function and (c)-(d) four different +MPS states with bond dimensions χ = 4, 8, 16, 32 as a +function of the block number for L = 6 (left panels) and +L = 10 (right panels). For the mean-field HF and the +χ = 4, 8 MPS states, after an initial equilibration phase, +the energies reach the exact value. For the higher fidelity +MPS with χ = 16, 32, we do not observe any initial equi- +libration phase and the local energy oscillates from the +beginning around the exact value. This observation can +signify that the MPS are already very close to the exact +ground state of HH. +To establish a link between the quality of the trial wave +function and its effect on the performance of AFQMC, +we show the respective fidelities of the trial states and +the resulting AFQMC energy estimates in Table III and +Fig. 5. In Fig. 5(a) we show the fidelities of the differ- +ent trial states with respect to the exact ground state +of HH for the two lattice lengths L = 6 and L = 10. +We observe that even though MPS are generally more +expressive than a single Slater determinant, one requires +a certain amount of entanglement, i.e. sufficiently large +bond dimension χ, to improve the fidelity over the HF +mean-field solution for the Hamiltonian (5). In Fig. 5(b)- +(c), we plot the difference of the energy estimator defined +in Eq. (2) with respect to the true ground state energy +E0 for system sizes L = 6 and L = 10 and different trial +wave functions. +From Table III we find that even though the HF state +possesses a lower fidelity than most MPS trial states, +its performance in AFQMC is comparable to the perfor- +mance with a MPS trial state of the largest employed +bond dimension. Also, we find that starting with a trial +state with a better energy does not guarantee an im- +proved AFQMC result over a trial state with inferior +energy. +This is an indicator that both the fidelity (or +related measures) and energy may not be the only quan- +tities that characterizes the “goodness” of a trial wave +function in AFQMC, but that other properties such as +symmetry properties of a trial wave function are impor- +tant, as found in Ref. [102, 103]. Such symmetries are +present in the HF state, but, generally, not in the state +sampled from an MPS output. + +C +XX +XXX +XXXX8 +12.52 +12.51 +HF - Energy [eV] +(a) +L = 6 +HF +20.90 +20.88 +20.86 +(b) +L = 10 +HF +100 +101 +102 +103 +104 +Block +20.00 +17.50 +15.00 +12.50 +DMRG - Energy [eV] +(c) +103 +104 +Block +12.55 +12.50 += 4 += 8 += 16 += 32 +100 +101 +102 +103 +104 +Block +40.00 +30.00 +20.00 +(d) +103 +104 +Block +21.2 +21.0 += 4 += 8 += 16 += 32 +FIG. 4. Energy estimator of the Fermi-Hubbard model as a function of the AFQMC projection steps for different trial wave +functions. Panels (a) and (b) show the energy estimator obtained from AFQMC using a mean-field HF state as a trial wave +function. +Panels (c) and (d) show the results for the energy estimator obtained from AFQMC when an MPS with bond +dimension χ is used as a trial wave function. +L +Trial +| ⟨ΨT |Ψ0⟩ |2 +E0 − ETrial +E0 − EAFQMC +6 +HF +0.225 +−2.40 × 10−2 +−3.53 × 10−4 +χ = 4 +0.108 +−3.70 × 10−2 +−8.84 × 10−4 +χ = 8 +0.359 +−9.85 × 10−3 +−8.02 × 10−4 +χ = 16 +0.753 +−9.24 × 10−4 +−2.02 × 10−4 +χ = 32 +1.000 +−6.89 × 10−7 +−6.88 × 10−6 +10 +HF +0.101 +−4.37 × 10−2 +5.10 × 10−4 +χ = 4 +0.001 +−8.05 × 10−2 +−1.45 × 10−2 +χ = 8 +0.006 +−5.18 × 10−2 +−1.28 × 10−2 +χ = 16 +0.340 +−8.89 × 10−3 +3.65 × 10−4 +χ = 32 +0.765 +−8.77 × 10−4 +5.97 × 10−4 +TABLE III. Summary of the results for the AFQMC simu- +lations of the Hubbard Hamiltonian HH for systems of size +L = 6 and 10. Here, ETrial = ⟨ΨT |HH|ΨT ⟩ is the expecta- +tion value of the energy of the trial wave function, E0 is the +exact ground state energy of HH, EAFQMC is the estimate +of the energy from the AFQMC. All energies are given in eV. +The fidelities (second column) and the energy differences (last +column) are also shown in Fig. 5. +V. +CONCLUSION +In this work, we applied AFQMC with classical and +quantum trial wave functions to calculate i) the poten- +tial energy curve of H4, ii) relative energies of ozone and +singlet molecular oxygen with respect to triplet molec- +ular oxygen, and iii) total energies of a CuBr2 system +mapped to a low-dimensional Fermi-Hubbard model. +For H4, we found that using a VQE trial wave func- +tion (with an energy within a few kcal/mol to the exact +ground state energy) does not allow AFQMC to reach +chemical accuracy for all side lengths. +For calculating +the total energy of ozone, we considered three different +trial wave functions. +We found that the AFQMC en- +ergies utilizing a VQE trial wave function lie between +AFQMC energies using a HF or a CAS trial wave func- +tion. For calculating the relative energies of ozone and +singlet molecular oxygen with respect to triplet molecular +oxygen, we used AFQMC with CAS trial wave functions, +yielding relative energies within or close to chemical accu- +racy. When using DFT-optimized geometries, commonly +used in industry, we found comparable results. +One source of error is the calculation of the ZPVE, +required making the connection with the experimental +values. Calculating the zero-point vibrational energy in +AFQMC could potentially improve the results. However, +this would come at an additional algorithmic cost and +make the method less applicable in today’s industrial +workflows. +As an example from material science, we provided +a +non-trivial +quasi-1D +Fermi-Hubbard +Hamiltonian, +Eq. (5), describing a single chain of CuBr2, which ex- +hibits similar electronic structure features as cuprates. +We found that na¨ıve trial wave functions obtained from +sampling from the output of a DMRG calculation that +possesses both a better energy and a better fidelity over +a simple HF mean-field state do not necessarily lead to +better AFQMC results. This suggests that quantum trial +wave functions should be physically motivated and re- +spect the expected symmetries of the Hamiltonian. In +addition, we find that a simple HF wave function ob- +tained from the method of Ref. [39] already provides a +trial wave function for which AFQMC can give an energy +estimate with an absolute difference of ∼ 10−4 eV to the +FCI results for this system. Future studies of CuBr2 (and +related systems) should follow the strategies of [37, 102– +104] and investigate the effect of using different Hubbard- +Stratonovich transformations, or walkers based on gen- +eralized Hartree Fock or Hartree-Fock Bogoliubov states, + +9 +HF +0.0 +0.5 +1.0 +Fidelities += 4 += 8 += 16 += 32 +(a) +L = 6 +L = 10 +HF +15 +10 +5 +0 +×10 +4 += 4 += 8 += 16 += 32 +(b) +exact +energy estimate +HF +15 +10 +5 +0 +×10 +3 += 4 += 8 += 16 += 32 +(c) +Comparison to exact diagonalization [eV] +L = 10 +L = 6 +exact +energy estimate +FIG. 5. +(a) Fidelities of the different trial wave functions +with respect to the exact ground state of HH for two lattice +lengths L = 6 and L = 10. HF denotes a mean-field state +obtained according to [39]. χ = 4, 8, 16, 32 denote MPS op- +timized via DMRG with bond dimension χ. (b)-(c) Energy +estimates from AFQMC for the Hubbard model using differ- +ent trial wave functions for L = 6 and L = 10. The plots +show the difference between the converged AFQMC energy +averaged over the last 2000 blocks and the energy computed +via exact diagonalization. The dashed horizontal line denotes +the numerically exact ground state value. See Table III for +numerical values. +and the effect of different bases on the AFQMC calcula- +tions. Regarding the demands on a quantum algorithm, +it will be crucial that a quantum computer can provide a +trial wave function that respects the symmetry of the +ground state of the problem Hamiltonian. +Strategies +for designing a quantum state on a quantum computer +could follow adiabatic state preparation-inspired varia- +tional Ans¨atze of [105], or center around providing trial +states inspired to solve the Heisenberg Hamiltonian in +the large−U limit [100]. +In general, a further field of investigation is to un- +derstand better the role of the trial wave function with +respect to the performance of (classical) AFQMC cal- +culations. +Specifically, it remains an open question +what properties a trial wave function should possess in +AFQMC to generate results within chemical accuracy. +For example, whether a high fidelity with respect to +the ground state is more important than a low energy. +This insight would help to develop specific quantum algo- +rithms tailored to generate good trial wave functions for +AFQMC and to understand for which systems a quantum +trial wave function would yield an advantage over classi- +cally accessible trial wave functions. Regarding the state +preparation of the trial wave function on NISQ hardware, +the effect of noise on AFQMC results remains a topic for +further investigation. +While AFQMC has seen significant improvements over +the last two decades [36], to become a useful tool in the +industry, its classical or future quantum-enhanced imple- +mentation has to be incorporated into current industrial +workflows, made easier to use and show consistent im- +provement over currently used quantum chemistry meth- +ods. The work presented by Ref. [13, 106, 107] and here +are the first steps in this direction. +ACKNOWLEDGMENTS +We thank Fionn Malone and Joonho Lee for insightful +discussions and early access to the AFQMC python pack- +age ipie [53]. MA, CW, and GS thank Takashi Koretsune +and Kazuma Nakamura for fruitful discussions concern- +ing cRPA calculations. We thank Joonho Lee, Nikolaj +Moll and Clemens Utschig-Utschig for their feedback on +the manuscript. +[1] E. Farhi, J. Goldstone, and S. 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Phys. +70, 1039 (1998). +Appendix A: VQE UCCSD ansatz +The UCC-VQE ansatz [44] originated from the Coupled Cluster theory [108], where the wave function is represented +as +|Ψ⟩ = eT |HF⟩ , +(A1) +with T being the excitation operator given by +T = +η +� +i=1 +Ti, +(A2) +T1 = +� +i∈occ +a∈virt +ti +aa† +aai, +(A3) +T2 = +� +i>j∈occ +a>b∈virt +tij +aba† +aa† +baiaj , +(A4) +. . . +(A5) +Depending on the truncation order of the series, the CC method is called CCSD (single and double excitation) or +CCSDT (single, double, and triple), and so forth. A unitary version of Eq. (A1) [109] +|Ψ⟩ = eT −T † |HF⟩ , +(A6) +allows serving as a VQE ansatz on a quantum computer, where the amplitudes t are variational parameters. To +implement the unitary on a quantum computer, first-order Trotterization formulas [110, 111] can be used, resulting +in +U(t) ≈ UTrott(t) = +� +i +eti(τi−τ † +i ) . +(A7) +Bravyi-Kitaev or Jordan Wigner transformations [112] convert the operators τi into products of Pauli gates native to +quantum hardware. One aspect that should be mentioned regarding the practical application of the UCCSD ansatz is +the parameter initialization. While the VQE optimization usually converges even with the Hartree-Fock state as the +starting point (i.e., initializing all parameters with zero), one can significantly reduce the number of training iterations +by taking results from perturbation methods, e.g., MP2 [113], or CCSD calculation as the initial parameter guess. +Recent findings suggest that initialization of the parameters through CCSD leads to significantly better results than +initializing through MP2 [114]. +Appendix B: Molecular structures +For the calculations on singlet and triplet molecular oxygen, and ozone presented in Sec. IV A 2, Table VI and +Table VII, we use experimental geometries from [28, 29]: +3O2 : +x/˚A +y/˚A +z/˚A +O +0.0 +0.0 +0.603 760 +O +0.0 +0.0 +−0.603 760 +1O2 : O +0.0 +0.0 +0.607 800 +O +0.0 +0.0 +−0.607 800 +(B1) +O3 : +O +0.0 +0.0 +0.0 +O +0.0 +0.0 +1.271 700 0 +O +1.138 385 0 +0.0 +1.838 534 0 + +13 +HF +MP2 +BP86 +B3LYP M06-2X +3O2 +reference +UHF +UHF +UKS +UKS +UKS +rO−O [pm] +115.79 121.90 +122.03 +120.43 +118.79 +ZPVE [kJ/mol] 11.78 +8.84 +9.24 +9.78 +10.55 +1O2 +reference +UHF +RHF +UKS +UKS +UKS +rO−O [pm] +115.61 124.34 +122.12 +120.46 +118.73 +ZPVE [kJ/mol] 11.80 +7.72 +9.20 +9.75 +10.54 +O3 +reference +UHF +RHF +RKS (=UKS) +UKS +UKS +rO−O [pm] +127.54 127.81 +127.55 +127.41 +125.69 +θO−O−O [◦] +118.14 116.76 +118.14 +116.55 +116.05 +ZPVE [kJ/mol] 17.66 +24.69 +17.63 +16.87 +18.57 +TABLE IV. A benchmark on the method dependence of structural parameters as well as zero-point vibrational energies (ZPVE). +All calculations are performed using the cc-pVQZ basis. +as well as geometries optimized at the density functional theory (DFT) level (UB3LYP/def2-QZVPP), resulting in +the xyz-structures: +3O2 : +x/˚A +y/˚A +z/˚A +O +0.0 +0.0 +0.602 012 0 +O +0.0 +0.0 +−0.602 012 0 +1O2 : O +0.0 +0.0 +0.602 100 8 +O +0.0 +0.0 +−0.602 100 8 +(B2) +O3 : +O +3.194 622 9 +3.905 696 2 +0.0 +O −2.104 635 3 +3.246 733 9 +0.0 +O −1.027 881 8 +3.927 080 0 +0.0 +Appendix C: Complete Basis Set (CBS) Extrapolation +To extrapolate the total energies of the singlet and triplet molecular oxygen, and ozone, we follow [80, 115]. We first +subtract the RHF/R(O)HF energies from the results of the correlated methods energies (AFQMC(CAS), CASSCF, +CCSD(T) and NEVPT2) to extrapolate the correlation energy using the function [80] +EX +corr = E∞ +corr + aX−3 , +(C1) +with a being a free parameter and X denoting the cardinal number (in cc-pVXZ). As in Ref. [61], we only use the +total energies from X = {T, Q} together with the cc-pV5Z RHF/R(O)HF energy as CBS Hartree Fock energy E∞ +HF . +Appendix D: Hubbard model Hamiltonian +The Hamiltonian of the generalized (multi-orbital) Fermi-Hubbard model in the second-quantization form is given +by [99, 116–121]: +H = +� +ijmm′σ +tmm′ (Rj − Ri) a† +imσajm′σ + +� +imm′σσ′ +� +Umm′ (0, 0) − Jmm′ (0, 0) δσσ′� +nimσnim′σ′, +(D1) +where i and j are lattice indices, Ri and Rj are lattice vectors, m and m′ are orbital indices, σ and σ′ are spin indices, +a† +imσ and aimσ are creation and annihilation operators, and nimσ = a† +imσaimσ is the particle number operator. The +transfer integral and direct and exchange Coulomb integrals are given by: +tmm′ (Rj − Ri) = +� +φ∗ +im (r) HMF (r) φjm′ (r) dr, +(D2) +Umm′ (Rj − Ri, ω) = +�� +φ∗ +im (r) φim (r) W (r, r′, ω) φ∗ +jm′ (r′) φjm′ (r′) dr dr′, +(D3) +Jmm′ (Rj − Ri, ω) = +�� +φ∗ +im (r) φjm′ (r) W (r, r′, ω) φ∗ +jm′ (r′) φim (r′) dr dr′, +(D4) + +14 +H4 +O2 & O3 Hubbard system (CuBr2) +Number of walkers NW +103 +103 +103 +Number of blocks NB +103 +104 +104 +Time steps per block +10 +10 +10 +Equilibration time (in blocks) 100 +1000 +3000 +Time step ∆τ (a.u.) +0.005 0.005 +0.005 +TABLE V. Parameters used in the AFQMC calculations. Parameters not reported here are set to the default setting of ipie +[53]. +(a) +(b) +FIG. 6. (a) Isosurface plot of the Wannier orbital of CuBr2 with dx2−y2 character, centered on a Cu atom (b) Frequency- +dependent direct Coulomb integral for the Cu:dx2−y2. The value of UH for our Hubbard model was taken as the static limit of +the real part of U(ω), i.e., UH = lim +ω→0 ℜ [U(ω)]. +where r and r′ are spatial coordinates, ω is frequency, φim are Wannier orbitals, HMF is the mean-field Hamiltonian, +and W is the screened Coulomb interaction potential calculated within the constrained random phase approximation +(cRPA) [99]. Note that tmm (0) are the orbital energies, which can be omitted in the case of degenerate orbitals. Also +note that Umm (0, ω) = Jmm (0, ω), which ensures the Pauli exclusion principle in Eq. (D1). +Based on the Hamiltonian of the generalized (multi-orbital) Fermi-Hubbard model, the Wannier orbital and the +frequency-dependent direct Coulomb integral plots of the Cu:dx2−y2 orbital of CuBr2 are shown in Fig. 6a and 6b, +respectively, and the interaction parameters are given in Table II. A comparison of these values suggests that transfer +integrals up to the 3rd nearest neighbor (i.e., tx, txx, and txxx) must be included in the model. The terms txxxx, ty, +tz, and txy are small and are thus neglected. + +15 +Basis Method +E(3O2) [Ha] +E(1O2) [Ha] +E(O3) [Ha] +∆E(1O2) +[kJ/mol] +∆E(O3) [kJ/mol] +Experiment [29, 30] +- +- +- +94.7 +143.2 +cc-pVDZ +R(O)HF +-149.6083 +-149.5414 +-224.2657 +175.7 +385.3 +CCSD(T) +-149.9892 +-149.9405 +-224.9149 +127.9 +180.9 +CASSCF +-149.7087 +-149.6757 +-224.4976 +86.7 +171.9 +NEVPT2 +-149.9579 +-149.9209 +-224.8759 +96.9 +159.8 +AFQMC(HF) +-149.9827(7) +-149.9403(6) +-224.9116(14) +111.2(25) +164.0(41) +AFQMC(CAS) +-149.9912(3) +-149.9500(2) +-224.9188(4) +108.0(10) +178.3(13) +cc-pVTZ +R(O)HF +-149.6528 +-149.5877 +-224.3405 +171.1 +364.4 +CCSD(T) +-150.1536 +-150.1058 +-225.1700 +125.4 +158.5 +CASSCF +-149.7519 +-149.7192 +-224.5688 +86.1 +155.2 +NEVPT2 +-150.1138 +-150.0782 +-225.1179 +93.5 +138.8 +AFQMC(HF) +-150.1494(7) +-150.1055(8) +-225.1651(10) +115.3(28) +155.1(31) +AFQMC(CAS) +-150.1554(4) +-150.1146(3) +-225.1740(4) +106.9(13) +155.1(15) +cc-pVQZ/def2-QZVPP +R(O)HF +-149.6643 +-149.5995 +-224.3585 +170.0 +362.1 +UPBE +-150.2569 +-150.2431 +-225.3366 +36.3 +128.0 +UB3LYP +-150.3376 +-150.3214 +-225.4366 +42.7 +183.2 +CCSD(T) +-150.2343 +-150.1871 +-225.2929 +124.0 +153.8 +CASSCF +-149.7633 +-149.7306 +-224.5866 +85.9 +153.2 +NEVPT2 +-150.1928 +-150.1577 +-224.2378 +92.0 +134.8 +AFQMC(HF) +-150.2279(7) +-150.1880(7) +-225.2901(12) +104.7(27) +135.8(36) +AFQMC(CAS) +-150.2354(4) +-150.1962(3) +-225.2966(4) +102.9(13) +148.4(14) +CBS +CCSD(T) +-150.2879 +-150.2409 +-225.3742 +123.4 +151.3 +CASSCF +-149.7662 +-149.7334 +-224.5911 +86.3 +152.8 +NEVPT2 +-150.2450 +-150.2102 +-225.3169 +91.4 +132.9 +AFQMC(CAS) +-152.2884(7) +-150.2502(6) +-225.3776(8) +100.4(24) +144.4(27) +TABLE VI. Results using the experimental geometries given in (B1) for various methods and basis sets together with the +extrapolated results to the complete basis set (CBS) limit. For all AFQMC(CAS), NEVPT2, and CASSCF calculations, (8e, +6o) and (12e, 9o) active spaces were used for singlet and triplet molecular oxygen, and ozone, respectively. All other calculations +were done in the canonical MO basis (R(O)HF). For ozone the experimental value was corrected by the ZPVEs as described +in the main text. + +16 +Method +E(3O2) [Ha] +E(1O2) [Ha] +E(O3) [Ha] +∆E(1O2) +[kJ/mol] +∆E(O3) [kJ/mol] +Experiment [29, 30] +- +- +- +94.7 +143.2 +R(O)HF +-149.6649 +-149.6018 +-224.3578 +165.6 +366.4 +UPBE +-150.2568 +-150.2428 +-225.3366 +36.9 +127.6 +UB3LYP +-150.3377 +-150.3215 +-225.4366 +42.4 +183.5 +CCSD(T) +-150.2344 +-150.1869 +-225.2929 +124.7 +154.0 +CASSCF +-149.7632 +-149.7301 +-224.5867 +87.0 +152.8 +NEVPT2 +-150.1928 +-150.1576 +-225.2378 +92.4 +134.7 +AFQMC(CAS) +-150.2356(4) +-150.1957(3) +-225.2974(4) +104.9(12) +147.1(14) +TABLE VII. Results using the DFT-optimized geometries given in (B2) for various methods using the def2-QZVPP basis set +for DFT (B3LYP and PBE) and cc-pVQZ otherwise. For all AFQMC(CAS), NEVPT2, and CASSCF calculations, (8e, 6o) +and (12e, 9o) active spaces were used for singlet and triplet molecular oxygen, and ozone, respectively. All other calculations +were done in the canonical MO basis (R(O)HF). For ozone, the experimental value was corrected by the ZPVEs as described +in the main text. + diff --git a/QdFKT4oBgHgl3EQfhi5s/content/tmp_files/load_file.txt b/QdFKT4oBgHgl3EQfhi5s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..afa961fbc036bd141a39f60afb46c565c322e263 --- /dev/null +++ b/QdFKT4oBgHgl3EQfhi5s/content/tmp_files/load_file.txt @@ -0,0 +1,1705 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf,len=1704 +page_content='Quantum-enhanced quantum Monte Carlo: an industrial view Maximilian Amsler1, Peter Deglmann2,3, Matthias Degroote4, Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Kaicher3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Matthew Kiser5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Michael K¨uhn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Chandan Kumar7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Andreas Maier8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Georgy Samsonidze9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Anna Schroeder10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Michael Streif4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Davide Vodola2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' and Christopher Wever1 QUTAC Material Science Working Group (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 2023) In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' we test a recently developed method to enhance classical auxiliary-field quantum Monte Carlo (AFQMC) calculations with quantum computers against examples from chemistry and material science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' representatives of classes of industry-relevant systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As molecular test cases, we calculate the energy curve of H4 and relative energies of ozone and singlet molecular oxygen with respect to triplet molecular oxygen, which are industrially relevant in organic oxidation reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We find that trial wave functions beyond single Slater determinants improve the performance of AFQMC and allow to generate energies close to chemical accuracy compared to full configuration interaction (FCI) or experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As a representative for material science we study a quasi-1D Fermi-Hubbard model derived from CuBr2, a compound displaying electronic structure properties analogous to cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We find that trial wave functions with both, significantly larger fidelities and lower energies over a Hartree-Fock solution, do not necessarily lead to better AFQMC results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' INTRODUCTION Recent years have shown significant advancements in the field of quantum computing, both in build- ing more powerful quantum hardware with an ever- increasing number of qubits and lower error rates, as well as in developing quantum algorithms to solve problems in optimization [1], machine learning [2, 3], and in cryp- tography [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Finding solutions to classically intractable problems in quantum chemistry and material science is often touted as the first application of future quantum computers in industry [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' With improvements in quantum hardware and quan- tum algorithms, the search for areas in industry where quantum computing could provide an economic or tech- ∗ Corresponding author: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='streif@boehringer- ingelheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='com 1 Corporate Sector Research and Advance Engineering, Robert Bosch GmbH, Robert-Bosch-Campus 1, 71272 Renningen, Germany 2 BASF SE, Quantum Chemistry, Carl-Bosch-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 38, 67063 Ludwigshafen, Germany 3 BASF Digital Solutions GmbH, Next Generation Computing, Pfalzgrafenstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 1, 67056, Ludwigshafen, Germany 4 Quantum Lab, Boehringer Ingelheim, Ingelheim am Rhein, Germany 5 Volkswagen AG, Ungererstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 69, 80805 Munich, Germany 6 TUM School of Natural Sciences, Technical University of Munich, Boltzmannstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Germany 7 BMW Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' New Technology and Innovation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Parkring 19-23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 85748,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Germany 8 Munich Re AG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Germany 9 Robert Bosch LLC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Research and Technology Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Sunny- vale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' CA 94085,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' USA 10 Merck KGaA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Frankfurter Straße 250,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 64293 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Germany 11 Quantum Computing Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Department of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Technical University of Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Mornewegstraße 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 64293 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Germany nological advantage over current approaches has gath- ered much attention in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' While in academic studies, much weight is given to demonstrat- ing the superior scaling of a quantum algorithm over a respective classical counterpart in terms of gate complex- ity in the large-problem-size-limit [7, 8], in an industrial setting, a quantum advantage is reached when the use of a quantum device allows to improve processes, cut down costs, or design new products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Since fully error-corrected quantum computers which can execute quantum algo- rithms with provable speedups over classical algorithms are years away, it is intriguing to address the question of whether a quantum advantage in Noisy-Intermediate Scale Quantum (NISQ) devices can be found for indus- trial purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Even though NISQ devices are limited to short quan- tum circuits, they might provide some advantage over classical algorithms, as computations in classically in- tractable regions of the Hilbert space are possible even with modest quantum resources [3, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A promising class of NISQ algorithms are variational quantum algorithms, such as the variational quantum eigensolver (VQE) for chemistry problems [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The VQE is a hybrid al- gorithm, meaning that the computational task is split between a classical and a quantum processor, while the quantum processor is used only to estimate the energy of a given quantum state manipulated by a set of variational parameters, updated by the classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Due to the accumulation of errors with increasing problem size and run time, the largest demonstration of such algo- rithms has been limited to a few tens of qubits [12, 13], even though quantum computers with hundreds of qubits exist [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Various classical post-processing steps have been introduced to mitigate the effects of noise [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' However, the required classical computational overhead resulting from such techniques often nullifies any pos- sible advantage [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Moreover, the classical optimiza- tion of the variational parameters can suffer from vanish- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='11838v1 [quant-ph] 27 Jan 2023 2 ing gradients, known as the barren plateau phenomenon [17], which can prevent the classical optimization rou- tine from finding the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In addition, there is some numerical evidence that suggests that the gate- error probabilities needed to generate variational states that describe the ground state of certain molecules within chemical accuracy can lie below the gate-error probabili- ties required by most quantum error-correction protocols [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The observation of those practical challenges in many numerical simulations and experiments indicates that VQE or related variational algorithms alone are unlikely to generate a quantum advantage in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' It is therefore important to explore if other algorithms can ex- ploit the computational power present in NISQ devices [19–21], and to benchmark the readiness of such new al- gorithms for industry applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A promising avenue is given by classical post- processing techniques, such as using the output to cal- culate interaction energies [22, 23], improving the energy estimates using neural networks [24], or in quantum sub- space expansions [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [13], results on H4 and a small periodic model for the carbon allotrope diamond obtained with a NISQ device suggest that such output can also be used as a trial wave function to guide classi- cal quantum Monte Carlo (QMC) calculations, more pre- cisely auxiliary-field quantum Monte Carlo (AFQMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In this work, we apply AFQMC to a selection of industry-relevant problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As a molecular test case, we calculate relative energies of ozone and singlet molecular oxygen with respect to triplet molecular oxygen using ex- perimental geometries from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [28, 29] and compare to experimental results [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To connect to typical in- dustrial workflows, where geometries are optimized with DFT, we compare the AFQMC energies obtained from experimental geometries and from DFT-optimized ge- ometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As an example from material science, we calcu- late the ground state energy of a one-dimensional CuBr2 chain, mapped to a low-dimensional Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We use trial wave functions obtained from classical meth- ods and VQE circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We compare the performance of AFQMC guided by those trial wave functions to mean- field- and other standard quantum chemistry methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' From an industrial perspective, it is important to iden- tify the model errors and to determine necessary improve- ments to a given method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' About QUTAC: To investigate the potential impact of quantum computing for industrial applications, thir- teen leading German companies are cooperating inside the Quantum Technology & Applications Consortium (QUTAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The goal of this collective effort is to evaluate the latest quantum algorithms against industry-relevant applications and provide guidance on needed quantum developments toward industrial applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' AUXILIARY FIELD QUANTUM MONTE CARLO (AFQMC) The AFQMC algorithm is an ab-initio method that allows the use of any one-particle basis of size M to project out the ground state of a strongly interact- ing fermionic system by performing a random walk in the space of fermionic Gaussian states [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Fermionic Gaussian states are exponentials of Hermitian quadratic fermionic operators [32] and build the basis of AFQMC in both the space of Slater determinants and Hartree-Fock Bogoliubov states [31, 33–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Projector methods use the property that the solution of the imaginary time Schr¨odinger equation of the Hamilto- nian H asymptotically approaches the ground state |Ψ0⟩, |Ψ0⟩ = lim τ→∞ |Ψ(τ)⟩ = lim τ→∞ e−(H−E0)τ |ΨI⟩ � ⟨ΨI|e−2(H−E0)τ|ΨI⟩ , (1) where E0 is the unknown ground state energy, |ΨI⟩ is the initial state, and we assume ⟨ΨI|Ψ0⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Since E0 is in principle unknown, it is replaced by various adap- tive estimators in the AFQMC algorithm [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Classical methods have so far not been able to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (1) ef- ficiently for strongly interacting systems, and therefore one has to resort to approximate methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The core idea of AFQMC is to transform the imaginary time propaga- tor of a quartic operator into an integral over a quadratic operator, whose action on a fermionic Gaussian state can be computed efficiently [37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' This transformation is realized through a Hubbard-Stratonovich transformation [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The resulting integral over matrix exponentials of quadratic operators is then solved in a Monte Carlo fash- ion [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' One hallmark problem of fermionic systems which ap- pears in QMC approaches is the so-called sign or phase problem, which is caused by the anticommutation rela- tions of fermions and phase accumulated in imaginary propagation, respectively [31, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' This results in an ex- ponential divergence of the variance of the estimator of the energy in the k-th step of a Monte Carlo simulation of NW walkers, where the energy estimator is defined as E(k) ≃ �NW w=1 Wk,weiθk,wEloc(Ψk,w) �NW w=1 Wk,weiθk,w , (2) where Wk,w and θk,w denote the amplitude and phase of the w-th walker Ψk,w at the k-th iteration, Eloc(Ψk,w) = ⟨ΨT |H|Ψk,w⟩ ⟨ΨT |Ψk,w⟩ (3) is the local energy, and |ΨT ⟩ is the trial wave func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The latter controls the evolution of the simulation and combined with a phaseless approximation tames the phase problem at the expense of an introduced bias in the energy estimator [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To be efficiently computable on classical computers, the class of wave functions that 3 can be used as trial states (or walkers) has been limited to linear combinations of (non-orthogonal) Slater deter- minants or Hartree-Fock Bogoliubov states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' However, quantum computers allow us to probe trial wave func- tions outside this class and thus possibly improve the performance of classical AFQMC calculations, as first re- alized by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In the next section, we describe the classical and quantum trial wave functions used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' AFQMC TRIAL WAVE FUNCTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Variational Quantum Eigensolver (VQE) The VQE [10] is a variational quantum algorithm tai- lored to find the ground states of a Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The VQE uses a parameterized quantum circuit or ansatz, and classically optimizes the parameters θ of the circuit with the goal to minimize a cost function, more specifi- cally, the expectation value of the Hamiltonian ⟨Ψ|H|Ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' It fulfills the variational principle E = ⟨Ψ|H|Ψ⟩ ⟨Ψ|Ψ⟩ ≥ E0, (4) where E0 is the exact ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Equality holds when |Ψ⟩ = |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To keep the number of parameters θ and the depth of the circuit as small as possible, the ansatz is typically tailored specifically to the problem [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A popular ansatz for chemistry problems is the uni- tary coupled-cluster ansatz [44], which we introduce in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Experimental retrieval of the expectation value from quantum hardware can be implemented effi- ciently by employing schemes such as shadow tomogra- phy [45], basis rotation groupings [46], or by optimizing the number of commuting terms [47], which reduces the number of required measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The efficient optimization of the parameters θ is an open field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' While the well-established gradient-free optimizer COBYLA [48] is a popular choice during the prototyping phase, a quantum-aware opti- mizer is inevitable to avoid issues associated with noisy quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For example, the quantum natural gradient descent [49] achieves faster convergence by con- sidering the geometric information of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Matrix product states Matrix product states (MPS) provide an efficient parametrization of one-dimensional quantum states in terms of matrices with dimensions bounded by the non- negative integer bond dimension χ [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The bond di- mension can be viewed as a parameter that controls the amount of entanglement and thus the degree of expres- sivity of an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' MPS are variational states that can approximate ground states of a many-body Hamiltonian when used within the density matrix renormalization group (DMRG) method that can capture the entangle- ment structure of strongly correlated wave functions [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In this work, we use an MPS representation of the ground state of the Fermi-Hubbard model as a trial wave function |ΨT ⟩ in AFQMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We generate MPS with dif- ferent bond dimensions χ by employing the DMRG al- gorithm implemented in the library ITensor [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Choos- ing different bond dimensions for the MPS provides trial states that approximate the true ground state |Ψ0⟩ in a controlled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' This allows us to understand the ef- fect of the trial states on the resulting estimated energy provided by AFQMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Molecular systems In the following, we use AFQMC with classical and quantum trial wave functions to calculate (i) the ground state energies of H4 in a rectangular shape for varying side lengths and (ii) the relative energies of ozone and singlet molecular oxygen with respect to triplet molec- ular oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For all AFQMC calculations we use ipie [53] with computational details summarized in the Ap- pendix in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To generate the Hamiltonian inte- grals and classical trial wave functions we used PySCF [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To generate the VQE trial wave function, cirq [55] was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As a quantum trial wave function we use a UCCSD-VQE ansatz applied to a single Slater determi- nant, see Appendix A for additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To allow for a practical setup, we initialise the VQE param- eters with classical CCSD amplitudes [44] and treat the optimisation of the VQE as a black box using scipy’s [56] implementation of the COBYLA optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' H4 Square As a first benchmark, we study H4 in a minimal basis STO-3G in a rectangular shape with the geometry given by H4 :(H, (0, 0, 0)), (H, (0, 0, a)), (H, (a, 0, 0)), (H, (a, 0, a)) where we vary the side length a from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='85˚A to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5˚A and aim to find the ground state energy of the singlet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The H4 molecular system is a commonly used bench- mark for classical and quantum electronic structure al- gorithms due to its relatively small size but still present correlation effects [13, 36, 57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Due to near degen- eracy of the ground state, both the static and dynamic correlations are relevant for an accurate ground state en- ergy calculation [58, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Moreover, this system was used as the first benchmark in Google’s AFQMC landmark 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 Energy [Hartree] RHF FCI VQE AFQMC(RHF) AFQMC(VQE) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 Side length [Å] 0 1 2 Error [kcal/mol] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (top) Energy surface of the H4 square with varying side length of the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For this calculation, a minimal basis STO-3G was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' It should be noted that the FCI and VQE curve are hard to distinguish because of the scale of the y-axis and their similarity to the AFQMC(VQE) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (bottom) The error of the ground state estimate of plain VQE (purple) and AFQMC(VQE) (green) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' the true ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The error bars stem from the statistical MC errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' paper [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Here, we extend this benchmark by calculat- ing the potential energy curve and using a different VQE ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We start by running a VQE circuit for each side length using eight qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We subsequently use the optimized VQE output as a trial wave function in an AFQMC cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 1(a) we show AFQMC energies together with restricted Hartree-Fock (RHF) and VQE energies and display the exact ground state energy as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A single-determinant RHF trial wave function is insufficient to guide the AFQMC to the ground state energy of the system [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' However, the VQE and its AFQMC(VQE) post-processed result can barely be discerned from the exact FCI result in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 1(b), VQE alone does not generate results within chemical ac- curacy for most side lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Applying AFQMC as a post-processing procedure to the VQE results improves the energy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' However, even for this small exam- ple using a noiseless simulation with access to the per- fect state vector, AFQMC cannot generate results within chemical accuracy for all side lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Ozone and molecular oxygen As a second benchmark, we investigate the perfor- mance of AFQMC for calculating energy differences be- tween species (relative energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We calculate the rela- tive energies ∆E(x) of ozone and singlet molecular oxy- gen with respect to the triplet ground state of molecular oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' While of significant industrial relevance, many standard quantum chemistry methods fail to estimate relative energies and energetic differences between spin states within chemical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The calculation of the latter with AFQMC was studied in [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In the following, we compare AFQMC results to results obtained from standard quantum chemistry methods and experimental values from [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We denote the error with respect to the experiment by ∆∆E(x) = ∆E(x) − exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' value(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In the case of molecular oxygen respective experimental values are directly available from the liter- ature [29] whereas for ozone the available experimental values refer to enthalpies of formation at absolute zero temperature, ∆Hf 0K(O3), [30] that inherently include a zero-point vibrational energy (ZPVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To compare our calculated results to those experimental values, we sub- tract the ZPVE from the enthalpy of formation according to: exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' value(O3) = ∆Hf 0K(O3) − ∆ZPVE(O3) wherein the ZPVEs are calculated at the density functional theory (DFT) level (B3LYP/def2-QZVPP) resulting in ZPVE(3O2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0037 Ha and ZPVE(O3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0064 Ha yielding a correction of ∆ZPVE(O3) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Molecular dioxygen O2, in particular singlet oxygen 1O2, which is formed by electronic excitation of the air constituent triplet oxygen 3O2, is a highly reactive molecular species [63, 64] involved in various chemistries such as desired as well as undesired photochemical reac- tions [64–66], ene reactions [67], and organic oxidation reactions in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Singlet molecular oxygen is also responsible for damage in biological materials [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Ozone (O3) is another gaseous, highly reactive form of oxygen that is typically formed in the atmosphere, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=', by a photochemical reaction catalyzed by NOx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Ozone undergoes a fast chemical reaction with C-C double bond containing materials, such as rubbers, finally cleaving the C-C double bond and is thus responsible for significant material damage worldwide [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Due to these reasons - their reactivity and abundance - the modifications of oxy- gen make for a compelling use-case application of high- level accurate quantum chemical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Relative energies of singlet molecular oxygen and ozone with respect to triplet molecular oxygen are of particu- lar interest to understand the stabilities of the respective competing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' More generally, an accurate predic- tion of energy differences, such as relative stabilities, re- action energies, and activation energies, is important in an industrial context as they can be related to exper- imentally observable thermodynamic and kinetic prop- erties of molecules and chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Previously, there have been theoretical and computational studies on the aforementioned three molecular forms of oxygen [71–77] revealing that singlet molecular oxygen and to an even larger extent ozone are challenging systems for single reference electronic structure methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We base our calculations on experimental geometries from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As a comparison, we use geome- tries optimized with DFT at the B3LYP/def2-QZVPP level [78, 79] resulting in the geometries given in (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 5 HF [Ha] VQE [Ha] CAS [Ha] AFQMC energy -225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2901(12) -225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2940(5) -225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2966(4) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' AFQMC results on the ground state energy of ozone using a HF, VQE and CAS trial wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For all calculations cc-pVQZ was used as basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The VQE and CAS trial wave function were obtained using an (12e, 9o) ac- tive space built from CASSCF MOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We find that our DFT geometries overestimate the bond length of singlet and triplet molecular oxygen by roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='01˚A and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='003˚A respectively, with respect to the ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For ozone, we find an absolute difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='002˚A in the bond length rO−O and a difference of 0◦09′ in the bond angle θO−O−O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' It was furthermore studied in Table IV how large the scatter of structural parame- ters and ZPVEs is when applying different levels of the- ory at which the computation of energy gradients is well established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' here, it turned out, that the differences are small between different classes of density functionals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' ZPVEs vary between 1 and 2 kJ/mol), whereas HF and MP2 predictions are partially far away from DFT and the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To have consistent active spaces across all molecules, the molecular orbitals with predominantly atomic 2p character are selected, resulting in (8e, 6o) active spaces for the singlet- and triplet molecular oxygen systems and a (12e, 9o) active space for ozone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To further improve the choice of active space orbitals for AFQMC, we run additional CASSCF calculations using the PySCF pro- gram package [54] and base our trial wave function gen- eration on the resulting CASSCF orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We note that such active spaces should be able to capture a large frac- tion of the correlations present in the system, while still would allow for experimental implementations on cur- rently available quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To benchmark the performance of AFQMC using a quantum trial wave function, we focus on the ozone system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We first run a UCCSD-VQE (detailed in Ap- pendix A) inside the (12e, 9o) active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The result- ing wave function together with the exact solution inside this active space (CAS) and the Hartree-Fock solution (in canonical MOs) is input as trial wave function to bench- mark the performance of AFQMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In Table I, we report the AFQMC results on the experimental geometries of ozone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We find that the energy of AFQMC with a VQE trial wavefunction lies between AFQMC with a HF and a CAS trial wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In the following, we assume that future first-generation quantum computers can generate the exact ground state wave function in a small active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We use the CAS trial wave functions to benchmark the performance of AFQMC when calculating relative energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We follow Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [61, 80] and calculate total energies using the cc- pVTZ and cc-pVQZ basis [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We extrapolate the cor- relation energies to find the total energies in the complete basis set (CBS) limit, see Appendix C for more details, and use the CBS total energies to calculate the relative UHF B3LYP CCSD(T) CASSCF NEVPT2 AFQMC −75 −50 −25 0 25 50 75 100 Error w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='t experiment [kJ/mol] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 Error in relative energies w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 3O2 (experimental geometries) ∆∆E(1O2) ∆∆E(O3) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Comparison of the error in relative energies of singlet molecular oxygen and ozone with respect to triplet molecular oxygen using experimental geometries given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' B1 with respect to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We report the results from different computational methods obtained using the cc-pVQZ (for UHF) and def2-QZVPP (for B3LYP) basis set and results extrapolated to the CBS limit (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The CASSCF and NEVPT2 calculations were carried out in a (12e, 9o) active space for ozone and (8e, 6o) active spaces for singlet and triplet molecular oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The CASSCF wave function was used as a trial wave function in the AFQMC calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The error in AFQMC stems from the statistical MC error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 2, we show the results in comparison to the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We find that while commonly used methods such as DFT (B3LYP), CCSD(T), and CASSCF alone are not able to reach chemical accuracy (4 kJ/mol), the AFQMC results, ∆E(1O2) = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 kJ/mol and ∆E(O3) = 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 kJ/mol, are within or close to chemical accuracy with respect to the experi- mental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' However, this comparison has to be taken with a grain of salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' First, for all calculations, the statistical error of AFQMC itself is of the order of a few kJ/mol, mak- ing quantitative comparisons between two calculations, which are needed to calculate relative energies, difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For calculations of chemical reactions with more than one educt and product, the propagation of errors would get more severe, for example, in redox reactions, SN2 reac- tions and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Second, we only use two points (cc-pVTZ and cc-pVQZ energies) to extrapolate to the CBS limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Future calculations using larger basis sets (cc-pVXZ with X ≥ 5) could be used to benchmark the CBS result and to improve it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Also, the scheme employed here to extrapolate to the CBS limit is one example of many possible choices [80, 82], result- ing in a choice-supportive bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Using the same level of theory for calculating the ZPVEs as for calculating the total energies might improve the results in the case of ozone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' However, it would not be expected that this would strongly alter the observed trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' This would come at an increased algorithmic cost, as it would re- quire accurate estimating forces and ZPVEs in AFQMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Lastly, the experimental values themselves are associated with errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We note that, from an industrial perspec- tive, CBS extrapolations are rarely performed due to 6 time constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' When comparing the plain cc-pVQZ results to the experimental value, we find that the error of AFQMC increases by 4 kJ/mol, see Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' When comparing to cc-pVQZ results using DFT-optimized ge- ometries as commonly done in the industry, we find rel- ative energies of ∆E(1O2) = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 kJ/mol and ∆E(O3) = 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 kJ/mol, comparable to the results obtained with experimental geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Extended systems Material systems exhibiting effects of strong corre- lation ranging from metal-insulator transitions to half- metallicity and spin-charge separation are of interest to various technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A particularly inter- esting class of materials are cuprate high-temperature su- perconductors, the physics of which are believed to stem from a single correlated d band in the low-energy spec- trum [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A minimal model to describe such a system is the one-band Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Still, extensions of it by generalized Hubbard-like Hamiltonian are more refined and, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=', take explicitly into account the effect of the oxygen p-orbitals [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The general atomic structure of cuprates consists of one or several planes of CuO2 in a square lattice stacked along the z-direction, interspersed with layers of guest atoms which act as carrier donors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' It is commonly believed that the superconductivity is con- fined to the 2D-CuO2 planes and that all relevant spin and charge carriers reside in those substructures [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Analogous to cuprates, copper-bromides consist of similar structural building blocks where Cu2+ are sur- rounded by four Br– (instead of O2– in cuprates) in a square planar arrangement, which in the latter however form chains instead of 2D sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Due to this structural and chemical similarity, both materials systems share related electronic properties [87], with a single dx2−y2 band crossing the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Emerging antiferromag- netic [88, 89] and multiferroic properties [90–92] have sparked recent interest in these materials classes, with potential applications in a wide range of magnetoelectric devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Native CuBr2 crystallizes in a polymeric structure with C2/m-symmetry, where square planar units of CuBr4 (one Cu surrounded by four Br atoms) are linked together to form chains, packed parallel along one direction [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' These chains interact weakly with each other, with the main electronic properties governed by the intra-chain interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Here, we map CuBr2 to an effective, but fictitious 1D model system and compute its properties based on a one-band Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We construct the quasi-1D structural model of CuBr2 by isolating a single chain and placing it in a cell where periodic images of the chains are separated by vacuum of 10 and 5 ˚A in the lat- eral and vertical direction, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The band structure and density of states (DOS) are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The Cu–Br distance and Cu–Br–Cu angle within the chain are retained at experimental values of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='398˚A and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='35◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To build the effective model, we use DFT as imple- mented in the Quantum ESPRESSO package [94] us- ing the Perdew-Burke-Ernzerhof approximation to the exchange-correlation functional [95] and norm-conserving pseudopotentials [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We employ a plane-wave cutoff energy of 100 Ry in conjunction with a k-points mesh at a density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2/˚A to obtain converged results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A single-orbital tight-binding model is constructed by Wan- nierizing the dx2−y2 band crossing the Fermi level us- ing the Wannier90 package [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The resulting Wannier orbital is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 6a, which was converted with wan2respack [98] to serve as input for RESPACK [99] to obtain the screened Coulomb interaction parameters based on the constrained random phase approximation (cRPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We include 83 virtual orbitals in addition to the 17 occupied states, resulting in sufficiently converged screened interaction parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' After keeping only the transfer integrals up to the 3rd nearest neighbors (see Appendix D), the resulting parametrized Hubbard-like Hamiltonian is HH = − tx � iσ a† i,σai+1,σ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' − txx � i,σ a† i,σai+2,σ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' − txxx � i,σ a† i,σai+3,σ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' − U 2 � iσ niσ + U � i ni,↑ni,↓ (5) where a† i,σ (ai,σ) creates (annihilates) an electron with spin σ on the lattice site labelled by i and niσ = a† i,σai,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The explicit values of the parameters tx, txx, txxx and U (in eV) are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Note that the spu- rious hopping terms txxxx, ty, tz, and txy are small and justify their neglect in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Since the ra- tio U/ max (t) ≈ 28 ≫ 1, the Hamiltonian (5) ap- proaches the regime of a spin-1/2 Heisenberg antiferro- magnet [100, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The number of lattice sites in the periodic lattice is denoted as L, and we use open bound- ary conditions in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The Hamiltonian HH conserves the number of electrons and the total spin, and all simulations are performed at half-filling with balanced spin, N↑ = N↓ = L/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In the following, we study how the fidelity and energy of a trial wave function impact the energy obtained from an AFQMC simulation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We compute the ground state energy of HH via AFQMC using ipie [53] with computational details sum- marized in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We consider two lattice lengths, L = 6 and L = 10, and use different wave functions as trial states |ΨT ⟩: (i) a Hartree-Fock (HF) mean-field state obtained from an imaginary time evolution of a Slater determinant following the work of [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (ii) a MPS wave function with bond dimension χ = 4, 8, 16, 32 ob- tained via DMRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For L = 6, the MPS at half filling in the zero spin sector generates 400 Slater determinants, 7 (a) x y z (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (a) Structural model of the quasi-1D chain of CuBr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The blue and brown spheres denote the Cu and Br atoms, respectively, while the periodic images are separated by 10 ˚A and 5 ˚A in the lateral (green arrow) and vertical (blue arrow) directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The arrows indicate the hopping distance along the chain, ranging from the nearest (red arrow) to the 4th nearest-neighbor (yellow arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (b) The band structure of the CuBr2 chain on the left, together with its density of states (DOS) on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Note, that the energy is shifted such that the Fermi level is at value zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The single wannierized band is shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The inset shows the Brillouin zone, with the irreducible portion outlined by the blue lines and the labels of the special k-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' µ = t(000) tx = t(100) txx = t(200) txxx = t(300) txxxx = t(400) ty = t(010) tz = t(001) txy = t(110) U(000, 0) J(000, 0) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0046 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0005 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='15 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Interaction parameters for the Cu:dx2−y2 orbital of CuBr2 with respect to lattice vectors t(xyz) in Miller index notation, in eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' all of which are included as a trial wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For L = 10, since including all generated Slater determinants of the MPS state would be too demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Instead, we sample Slater determinants with a weight > 5 × 10−3, resulting in a trial wave function with up to ∼ 300 Slater determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The initial walkers for the AFQMC imag- inary time propagation are chosen as the HF state of (i) for both trial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 4 we plot the energy defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (2) for (a)- (b) a HF trial wave function and (c)-(d) four different MPS states with bond dimensions χ = 4, 8, 16, 32 as a function of the block number for L = 6 (left panels) and L = 10 (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For the mean-field HF and the χ = 4, 8 MPS states, after an initial equilibration phase, the energies reach the exact value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For the higher fidelity MPS with χ = 16, 32, we do not observe any initial equi- libration phase and the local energy oscillates from the beginning around the exact value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' This observation can signify that the MPS are already very close to the exact ground state of HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To establish a link between the quality of the trial wave function and its effect on the performance of AFQMC, we show the respective fidelities of the trial states and the resulting AFQMC energy estimates in Table III and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 5(a) we show the fidelities of the differ- ent trial states with respect to the exact ground state of HH for the two lattice lengths L = 6 and L = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We observe that even though MPS are generally more expressive than a single Slater determinant, one requires a certain amount of entanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' sufficiently large bond dimension χ, to improve the fidelity over the HF mean-field solution for the Hamiltonian (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 5(b)- (c), we plot the difference of the energy estimator defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (2) with respect to the true ground state energy E0 for system sizes L = 6 and L = 10 and different trial wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' From Table III we find that even though the HF state possesses a lower fidelity than most MPS trial states, its performance in AFQMC is comparable to the perfor- mance with a MPS trial state of the largest employed bond dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Also, we find that starting with a trial state with a better energy does not guarantee an im- proved AFQMC result over a trial state with inferior energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' This is an indicator that both the fidelity (or related measures) and energy may not be the only quan- tities that characterizes the “goodness” of a trial wave function in AFQMC, but that other properties such as symmetry properties of a trial wave function are impor- tant, as found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [102, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Such symmetries are present in the HF state, but, generally, not in the state sampled from an MPS output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' C XX XXX XXXX8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='52 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='51 HF - Energy [eV] (a) L = 6 HF 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='90 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='88 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='86 (b) L = 10 HF 100 101 102 103 104 Block 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='00 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='50 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='00 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='50 DMRG - Energy [eV] (c) 103 104 Block 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='55 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='50 = 4 = 8 = 16 = 32 100 101 102 103 104 Block 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='00 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='00 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='00 (d) 103 104 Block 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 = 4 = 8 = 16 = 32 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Energy estimator of the Fermi-Hubbard model as a function of the AFQMC projection steps for different trial wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Panels (a) and (b) show the energy estimator obtained from AFQMC using a mean-field HF state as a trial wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Panels (c) and (d) show the results for the energy estimator obtained from AFQMC when an MPS with bond dimension χ is used as a trial wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' L Trial | ⟨ΨT |Ψ0⟩ |2 E0 − ETrial E0 − EAFQMC 6 HF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='225 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='40 × 10−2 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='53 × 10−4 χ = 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='108 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='70 × 10−2 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='84 × 10−4 χ = 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='359 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='85 × 10−3 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='02 × 10−4 χ = 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='753 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='24 × 10−4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='02 × 10−4 χ = 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='000 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='89 × 10−7 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='88 × 10−6 10 HF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='101 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='37 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='10 × 10−4 χ = 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='001 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='05 × 10−2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='45 × 10−2 χ = 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='006 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='18 × 10−2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='28 × 10−2 χ = 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='340 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='89 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='65 × 10−4 χ = 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='765 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='77 × 10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='97 × 10−4 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Summary of the results for the AFQMC simu- lations of the Hubbard Hamiltonian HH for systems of size L = 6 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Here, ETrial = ⟨ΨT |HH|ΨT ⟩ is the expecta- tion value of the energy of the trial wave function, E0 is the exact ground state energy of HH, EAFQMC is the estimate of the energy from the AFQMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' All energies are given in eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The fidelities (second column) and the energy differences (last column) are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' CONCLUSION In this work, we applied AFQMC with classical and quantum trial wave functions to calculate i) the poten- tial energy curve of H4, ii) relative energies of ozone and singlet molecular oxygen with respect to triplet molec- ular oxygen, and iii) total energies of a CuBr2 system mapped to a low-dimensional Fermi-Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For H4, we found that using a VQE trial wave func- tion (with an energy within a few kcal/mol to the exact ground state energy) does not allow AFQMC to reach chemical accuracy for all side lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For calculating the total energy of ozone, we considered three different trial wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We found that the AFQMC en- ergies utilizing a VQE trial wave function lie between AFQMC energies using a HF or a CAS trial wave func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For calculating the relative energies of ozone and singlet molecular oxygen with respect to triplet molecular oxygen, we used AFQMC with CAS trial wave functions, yielding relative energies within or close to chemical accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' When using DFT-optimized geometries, commonly used in industry, we found comparable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' One source of error is the calculation of the ZPVE, required making the connection with the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Calculating the zero-point vibrational energy in AFQMC could potentially improve the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' However, this would come at an additional algorithmic cost and make the method less applicable in today’s industrial workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As an example from material science, we provided a non-trivial quasi-1D Fermi-Hubbard Hamiltonian, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (5), describing a single chain of CuBr2, which ex- hibits similar electronic structure features as cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We found that na¨ıve trial wave functions obtained from sampling from the output of a DMRG calculation that possesses both a better energy and a better fidelity over a simple HF mean-field state do not necessarily lead to better AFQMC results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' This suggests that quantum trial wave functions should be physically motivated and re- spect the expected symmetries of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In addition, we find that a simple HF wave function ob- tained from the method of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [39] already provides a trial wave function for which AFQMC can give an energy estimate with an absolute difference of ∼ 10−4 eV to the FCI results for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Future studies of CuBr2 (and related systems) should follow the strategies of [37, 102– 104] and investigate the effect of using different Hubbard- Stratonovich transformations, or walkers based on gen- eralized Hartree Fock or Hartree-Fock Bogoliubov states, 9 HF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 Fidelities = 4 = 8 = 16 = 32 (a) L = 6 L = 10 HF 15 10 5 0 ×10 4 = 4 = 8 = 16 = 32 (b) exact energy estimate HF 15 10 5 0 ×10 3 = 4 = 8 = 16 = 32 (c) Comparison to exact diagonalization [eV] L = 10 L = 6 exact energy estimate FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (a) Fidelities of the different trial wave functions with respect to the exact ground state of HH for two lattice lengths L = 6 and L = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' HF denotes a mean-field state obtained according to [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' χ = 4, 8, 16, 32 denote MPS op- timized via DMRG with bond dimension χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (b)-(c) Energy estimates from AFQMC for the Hubbard model using differ- ent trial wave functions for L = 6 and L = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The plots show the difference between the converged AFQMC energy averaged over the last 2000 blocks and the energy computed via exact diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The dashed horizontal line denotes the numerically exact ground state value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' See Table III for numerical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' and the effect of different bases on the AFQMC calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Regarding the demands on a quantum algorithm, it will be crucial that a quantum computer can provide a trial wave function that respects the symmetry of the ground state of the problem Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Strategies for designing a quantum state on a quantum computer could follow adiabatic state preparation-inspired varia- tional Ans¨atze of [105], or center around providing trial states inspired to solve the Heisenberg Hamiltonian in the large−U limit [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' In general, a further field of investigation is to un- derstand better the role of the trial wave function with respect to the performance of (classical) AFQMC cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Specifically, it remains an open question what properties a trial wave function should possess in AFQMC to generate results within chemical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For example, whether a high fidelity with respect to the ground state is more important than a low energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' This insight would help to develop specific quantum algo- rithms tailored to generate good trial wave functions for AFQMC and to understand for which systems a quantum trial wave function would yield an advantage over classi- cally accessible trial wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Regarding the state preparation of the trial wave function on NISQ hardware, the effect of noise on AFQMC results remains a topic for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' While AFQMC has seen significant improvements over the last two decades [36], to become a useful tool in the industry, its classical or future quantum-enhanced imple- mentation has to be incorporated into current industrial workflows, made easier to use and show consistent im- provement over currently used quantum chemistry meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The work presented by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [13, 106, 107] and here are the first steps in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Fionn Malone and Joonho Lee for insightful discussions and early access to the AFQMC python pack- age ipie [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' MA, CW, and GS thank Takashi Koretsune and Kazuma Nakamura for fruitful discussions concern- ing cRPA calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We thank Joonho Lee, Nikolaj Moll and Clemens Utschig-Utschig for their feedback on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Farhi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Goldstone, and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Posternak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Yamasaki, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Andersen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' B 74, 125120 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [119] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Castellani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Natoli, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Ranninger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 12 Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' B 18, 4945 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [120] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Fr´esard and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Kotliar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' B 56, 12909 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [121] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Imada, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Fujimori, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Tokura, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 70, 1039 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Appendix A: VQE UCCSD ansatz The UCC-VQE ansatz [44] originated from the Coupled Cluster theory [108], where the wave function is represented as |Ψ⟩ = eT |HF⟩ , (A1) with T being the excitation operator given by T = η � i=1 Ti, (A2) T1 = � i∈occ a∈virt ti aa† aai, (A3) T2 = � i>j∈occ a>b∈virt tij aba† aa† baiaj , (A4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (A5) Depending on the truncation order of the series, the CC method is called CCSD (single and double excitation) or CCSDT (single, double, and triple), and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A unitary version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (A1) [109] |Ψ⟩ = eT −T † |HF⟩ , (A6) allows serving as a VQE ansatz on a quantum computer, where the amplitudes t are variational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' To implement the unitary on a quantum computer, first-order Trotterization formulas [110, 111] can be used, resulting in U(t) ≈ UTrott(t) = � i eti(τi−τ † i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (A7) Bravyi-Kitaev or Jordan Wigner transformations [112] convert the operators τi into products of Pauli gates native to quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' One aspect that should be mentioned regarding the practical application of the UCCSD ansatz is the parameter initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' While the VQE optimization usually converges even with the Hartree-Fock state as the starting point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=', initializing all parameters with zero), one can significantly reduce the number of training iterations by taking results from perturbation methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=', MP2 [113], or CCSD calculation as the initial parameter guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Recent findings suggest that initialization of the parameters through CCSD leads to significantly better results than initializing through MP2 [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Appendix B: Molecular structures For the calculations on singlet and triplet molecular oxygen, and ozone presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' IV A 2, Table VI and Table VII, we use experimental geometries from [28, 29]: 3O2 : x/˚A y/˚A z/˚A O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='603 760 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='603 760 1O2 : O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='607 800 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='607 800 (B1) O3 : O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='271 700 0 O 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='138 385 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='838 534 0 13 HF MP2 BP86 B3LYP M06-2X 3O2 reference UHF UHF UKS UKS UKS rO−O [pm] 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='79 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='90 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='03 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='43 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='79 ZPVE [kJ/mol] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='78 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='84 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='24 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='78 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='55 1O2 reference UHF RHF UKS UKS UKS rO−O [pm] 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='61 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='34 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='12 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='46 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='73 ZPVE [kJ/mol] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='72 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='54 O3 reference UHF RHF RKS (=UKS) UKS UKS rO−O [pm] 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='54 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='81 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='55 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='41 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='69 θO−O−O [◦] 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='14 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='76 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='14 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='55 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='05 ZPVE [kJ/mol] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='66 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='69 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='63 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='87 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='57 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A benchmark on the method dependence of structural parameters as well as zero-point vibrational energies (ZPVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' All calculations are performed using the cc-pVQZ basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' as well as geometries optimized at the density functional theory (DFT) level (UB3LYP/def2-QZVPP), resulting in the xyz-structures: 3O2 : x/˚A y/˚A z/˚A O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='602 012 0 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='602 012 0 1O2 : O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='602 100 8 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='602 100 8 (B2) O3 : O 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='194 622 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='905 696 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 O −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='104 635 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='246 733 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 O −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='027 881 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='927 080 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 Appendix C: Complete Basis Set (CBS) Extrapolation To extrapolate the total energies of the singlet and triplet molecular oxygen, and ozone, we follow [80, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' We first subtract the RHF/R(O)HF energies from the results of the correlated methods energies (AFQMC(CAS), CASSCF, CCSD(T) and NEVPT2) to extrapolate the correlation energy using the function [80] EX corr = E∞ corr + aX−3 , (C1) with a being a free parameter and X denoting the cardinal number (in cc-pVXZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' As in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' [61], we only use the total energies from X = {T, Q} together with the cc-pV5Z RHF/R(O)HF energy as CBS Hartree Fock energy E∞ HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Appendix D: Hubbard model Hamiltonian The Hamiltonian of the generalized (multi-orbital) Fermi-Hubbard model in the second-quantization form is given by [99,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 116–121]: H = � ijmm′σ tmm′ (Rj − Ri) a† imσajm′σ + � imm′σσ′ � Umm′ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 0) − Jmm′ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 0) δσσ′� nimσnim′σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (D1) where i and j are lattice indices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Ri and Rj are lattice vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' m and m′ are orbital indices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' σ and σ′ are spin indices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' a† imσ and aimσ are creation and annihilation operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' and nimσ = a† imσaimσ is the particle number operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The transfer integral and direct and exchange Coulomb integrals are given by: tmm′ (Rj − Ri) = � φ∗ im (r) HMF (r) φjm′ (r) dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (D2) Umm′ (Rj − Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' ω) = �� φ∗ im (r) φim (r) W (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' ω) φ∗ jm′ (r′) φjm′ (r′) dr dr′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (D3) Jmm′ (Rj − Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' ω) = �� φ∗ im (r) φjm′ (r) W (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' ω) φ∗ jm′ (r′) φim (r′) dr dr′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (D4) 14 H4 O2 & O3 Hubbard system (CuBr2) Number of walkers NW 103 103 103 Number of blocks NB 103 104 104 Time steps per block 10 10 10 Equilibration time (in blocks) 100 1000 3000 Time step ∆τ (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='005 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Parameters used in the AFQMC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Parameters not reported here are set to the default setting of ipie [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (a) Isosurface plot of the Wannier orbital of CuBr2 with dx2−y2 character, centered on a Cu atom (b) Frequency- dependent direct Coulomb integral for the Cu:dx2−y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The value of UH for our Hubbard model was taken as the static limit of the real part of U(ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=', UH = lim ω→0 ℜ [U(ω)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' where r and r′ are spatial coordinates, ω is frequency, φim are Wannier orbitals, HMF is the mean-field Hamiltonian, and W is the screened Coulomb interaction potential calculated within the constrained random phase approximation (cRPA) [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Note that tmm (0) are the orbital energies, which can be omitted in the case of degenerate orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Also note that Umm (0, ω) = Jmm (0, ω), which ensures the Pauli exclusion principle in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' (D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Based on the Hamiltonian of the generalized (multi-orbital) Fermi-Hubbard model, the Wannier orbital and the frequency-dependent direct Coulomb integral plots of the Cu:dx2−y2 orbital of CuBr2 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 6a and 6b, respectively, and the interaction parameters are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' A comparison of these values suggests that transfer integrals up to the 3rd nearest neighbor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=', tx, txx, and txxx) must be included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' The terms txxxx, ty, tz, and txy are small and are thus neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 15 Basis Method E(3O2) [Ha] E(1O2) [Ha] E(O3) [Ha] ∆E(1O2) [kJ/mol] ∆E(O3) [kJ/mol] Experiment [29, 30] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 cc-pVDZ R(O)HF 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6083 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5414 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2657 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3 CCSD(T) 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9892 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9405 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9149 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 CASSCF 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7087 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6757 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4976 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 NEVPT2 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9579 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9209 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8759 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8 AFQMC(HF) 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9827(7) 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9403(6) 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9116(14) 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2(25) 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0(41) AFQMC(CAS) 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9912(3) 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1 364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 CCSD(T) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1536 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1058 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1700 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5 CASSCF 149.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 CCSD(T) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2343 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1871 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2929 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8 CASSCF 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7633 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7306 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5866 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 NEVPT2 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1928 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1577 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2378 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8 AFQMC(HF) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2279(7) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1880(7) 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2901(12) 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7(27) 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2409 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3742 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3 CASSCF 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7662 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7334 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5911 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8 NEVPT2 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2450 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2102 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3169 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 AFQMC(CAS) 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2884(7) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2502(6) 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3776(8) 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4(24) 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4(27) TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Results using the experimental geometries given in (B1) for various methods and basis sets together with the extrapolated results to the complete basis set (CBS) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For all AFQMC(CAS), NEVPT2, and CASSCF calculations, (8e, 6o) and (12e, 9o) active spaces were used for singlet and triplet molecular oxygen, and ozone, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' All other calculations were done in the canonical MO basis (R(O)HF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For ozone the experimental value was corrected by the ZPVEs as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' 16 Method E(3O2) [Ha] E(1O2) [Ha] E(O3) [Ha] ∆E(1O2) [kJ/mol] ∆E(O3) [kJ/mol] Experiment [29, 30] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2 R(O)HF 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6649 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6018 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3578 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6 366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 UPBE 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2568 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2428 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3366 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='6 UB3LYP 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3377 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='3215 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4366 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5 CCSD(T) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2344 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1869 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2929 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 CASSCF 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7632 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7301 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='5867 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='0 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='8 NEVPT2 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1928 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1576 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2378 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='4 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='7 AFQMC(CAS) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2356(4) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1957(3) 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='2974(4) 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='9(12) 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content='1(14) TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' Results using the DFT-optimized geometries given in (B2) for various methods using the def2-QZVPP basis set for DFT (B3LYP and PBE) and cc-pVQZ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For all AFQMC(CAS), NEVPT2, and CASSCF calculations, (8e, 6o) and (12e, 9o) active spaces were used for singlet and triplet molecular oxygen, and ozone, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' All other calculations were done in the canonical MO basis (R(O)HF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} +page_content=' For ozone, the experimental value was corrected by the ZPVEs as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFKT4oBgHgl3EQfhi5s/content/2301.11838v1.pdf'} diff --git a/RNE0T4oBgHgl3EQfUQBU/content/tmp_files/2301.02247v1.pdf.txt b/RNE0T4oBgHgl3EQfUQBU/content/tmp_files/2301.02247v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..908389c351d030aad794fdf54fbba4fff902f03d --- /dev/null +++ b/RNE0T4oBgHgl3EQfUQBU/content/tmp_files/2301.02247v1.pdf.txt @@ -0,0 +1,1097 @@ +Quantum metric unveils defect freezing in non-Hermitian systems +Karin Sim,1 Nicol`o Defenu,1 Paolo Molignini,2, 3 and R. Chitra1 +1Institute for Theoretical Physics, ETH Z¨urich, 8093 Zurich, Switzerland +2Cavendish Laboratory, University of Cambridge, +19 J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom +3Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden +(Dated: January 9, 2023) +Nonhermiticity in quantum Hamiltonians leads to non-unitary time evolution and possibly com- +plex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. +In this work, we study the dynamics of an exactly solvable non-Hermitian system, hosting both +PT -symmetric and PT -broken modes subject to a linear quench. +Employing a fully consistent +framework, in which the Hilbert space is endowed with a nontrivial dynamical metric, we analyze +the dynamics of the generated defects. In contrast to Hermitian systems, our study reveals that +PT -broken time evolution leads to defect freezing and hence the violation of quantum adiabaticity. +Additionally, no Kibble-Zurek scaling regime in the quasi-adiabatic limit exists in our model. This +physics necessitates the quantum metric framework, as it is missed by the oft used approach of +normalizing quantities by the time-dependent norm of the state. Our results are relevant for a wide +class of experimental systems. +Introduction. Non-Hermitian Hamiltonians [1–3] pro- +vide a framework to explore a complex array of out-of- +equilibrium phenomena. Far from being a purely math- +ematical pursuit, non-Hermitian descriptions have been +employed widely in both classical and quantum systems. +Arguably, the most well known examples include the +study of non-Hermitian spin chains in the context of +the Kardar-Parisi-Zhang equation [4] and the localiza- +tion of particles in an imaginary vector potential, used +to explain the depinning of vortex lines in a supercon- +ductor [5]. More recently, the field has seen a dramatic +revival courtesy of effective descriptions of Lindbladian +dynamics in dissipative systems [6], continuously moni- +tored systems [7, 8], amplification in optomechanical sys- +tems [9], quantum sensors [10], and more. Nonhermiticity +has unveiled a plethora of interesting phenomena, such as +quantum phase transitions without gap closure [11, 12], +anomalous behaviors of quantum emitters [13], tachyonic +physics [14, 15] and unconventional topology [16–18] to +name a few. Interest in non-Hermitian systems is further +enchanced by the concomitant experimental realizations +in diverse platforms: optical systems [16, 19], semicon- +ductor microcavities [20] and acoustic systems [21] in the +presence of drive and dissipation. +Non-Hermitian +Hamiltonians +which +preserve +PT - +symmetry (i.e., the combined operation of parity and +time reversal) [22–24] constitute a special class of systems +possessing a real spectrum, prompting their interpreta- +tion as a natural extension to conventional quantum me- +chanics [25]. When PT -symmetry is spontaneously bro- +ken, exceptional points (EPs) arise where the eigenvalues +become complex-valued, a topic of much theoretical [26– +28] and experimental [29] interest. +Conventionally, the +Hamiltonian is given by a Hermitian operator which plays +the dual role of both the energy operator and the gen- +erator of time translations [30]. However, nonhermiticity +leads to non-unitary time evolution and possibly com- +plex energy eigenvalues, both of which imply that the +Hamiltonian loses this dual role [25, 31]. Consequently, +nonhermiticity ushers in new challenges to fundamental +concepts in conventional quantum mechanics, necessitat- +ing a more general framework. +Multiple approaches are used to tackle the aforemen- +tioned issues and compute observables. +Foremost is +biorthogonal quantum mechanics [32–34], which has been +widely studied in the context of PT -symmetric Hamilto- +nians, though its application is limited to a subset of non- +Hermitian Hamiltonians. +More often, time-dependent +probabilities [35] and observables [19, 36, 37] are explic- +itly normalized by the non-conserved norm of the states +in an ad hoc manner. As we shall see in this work, this +method can fail to capture salient aspects of the physics. +A more robust method to study non-Hermitian systems is +based on considering the Hilbert space as non-stationary +and endowed with a non-trivial time-dependent met- +ric [31, 38–40]. +It can be regarded as a generalization +of biorthogonal quantum mechanics [32] encompassing +spontaneous PT -broken scenarios as well. +This met- +ric framework presents a consistent formulation of non- +Hermitian quantum mechanics. +It has been adopted +to recover fundamental theorems of quantum informa- +tion [41], as well as being especially relevant for the evo- +lution of entanglement [42, 43]. +Quantum quenches and driving have emerged as tools +of choice to explore the non-trivial dynamics of quan- +tum systems [44–47]. +The richness of the emergent +phenomenology in Hermitian systems naturally behoves +the study of quantum quenches in non-Hermitian sys- +tems [34, 48–52]. +A famous example of non-trivial dy- +namics concerns topological defects generated when a +coupling is quenched across a quantum critical point [53]. +The Kibble-Zurek scaling predicts that the defect den- +arXiv:2301.02247v1 [quant-ph] 5 Jan 2023 + +2 +sity scales as a power law with quench time, where the +exponents are determined by the static critical expo- +nents [54, 55]. Using the wavefunction normalization ap- +proach, recent work predicted a modified Kibble-Zurek +scaling when a system is quenched across EPs [19, 37], +thereby recovering adiabaticity. +On the other hand, +breakdown of adiabaticity was seen experimentally in +dissipative superconducting qubits governed by effective +non-Hermitian Hamiltonians [56]. In this Letter, using an +exactly solvable non-Hermitian model, we rigorously in- +vestigate the fundamental question of whether quantum +adiabaticity survives. We show that the metric plays a +crucial role in the violation of quantum adiabaticity when +EPs are traversed adiabatically. A mere normalization of +physical quantities by the norm of the time-evolved state +completely fails to capture this fundamental aspect. +Metric framework. We begin by introducing the met- +ric framework. The inner product in a Hilbert space is +defined via its metric ρ(t) as ⟨·, ·⟩ρ(t). For a system de- +scribed by a Hermitian Hamiltonian, the metric is static +and is the identity operator. +In the case of a time- +dependent non-Hermitian Hamiltonian H(t), the metric +of the Hilbert space develops a non-trivial time evolu- +tion, even in the PT -symmetric regimes [31, 57]. +The +dynamics of the Hilbert space Hρ(t) is encoded in the +time evolution of the metric ρ(t), given by [31, 39] +i ˙ρ(t) = H†(t)ρ(t) − ρ(t)H(t), +(1) +where the overdot denotes time derivative. +Provided +that a solution to Eq. +(1) can be found [31], we can +map the system to a Hermitian Hamiltonian h(t) = +η(t)H(t)η−1(t) + i ˙η(t)η−1(t), where we have introduced +the square-root decomposition of the positive-definite +metric, ρ(t) = η†(t)η(t). The Hamiltonian h(t) acts in +a different Hilbert space H [31, 57], where the nonher- +miticity is encoded in the dynamics of η(t). +The time evolution of the states |ψ(t)⟩ in Hρ(t) +and |Ψ(t)⟩ in H +is governed by the time-dependent +Schr¨odinger equation (TDSE) +i d +dt|ψ(t)⟩ = H(t)|ψ(t)⟩ +i d +dt|Ψ(t)⟩ = h(t)|Ψ(t)⟩ +(2) +where the unitarity of the evolution is conserved in both +representations, since ⟨ψ(t)|ρ(t)|ψ(t)⟩ = ⟨Ψ(t)|Ψ(t)⟩ = 1 +at all times t [31]. +The states are related by |Ψ(t)⟩ = +η(t)|ψ(t)⟩. Under this formalism, the expectation value +of an operator ˆo : H → H is given by +⟨O(t)⟩metric = ⟨Ψ(t)|ˆo|Ψ(t)⟩ = ⟨ψ(t)|ρ(t) ˆO(t)|ψ(t)⟩ (3) +where +ˆO(t) : Hρ(t) → Hρ(t) is defined as +ˆO(t) = +η−1(t)ˆoη(t). In contrast, the expectation of ˆo calculated +from a simple normalization by the time-dependent norm +is given by +⟨O(t)⟩norm = ⟨ψ(t)|ˆo|ψ(t)⟩ +⟨ψ(t)|ψ(t)⟩ . +(4) +as was done, for example, in Ref. [37]. +Exactly solvable model. +To highlight the nontrivial +role played by the metric, we consider an exactly solv- +able model of effective two level systems parameterised +by momentum k. This is given by the Hamiltonian [34] +Hk(t) = kσx + iγσy + Ftσz +(5) +where σi denotes the Pauli matrices and F, k, γ ∈ R. +Eq. (5) is a generalization of the Hamiltonian presented +in Ref. [58] and realized experimentally in Ref. [59], by +adding a real drive term Ft and applying a basis rota- +tion. In our case, the non-Hermitian term γ corresponds +to the imaginary tachyon mass [58] and the parameter +k is the momentum. +The dimensionless term +γ2 +F +sets +the scale for the extent of nonhermiticity in our model. +Note that we recover a purely Hermitian Hamiltonian by +setting γ = 0. PT -symmetry is realised in our model +by the operators P = σy and T = −iσyK where K is +complex conjugation, such that [Hk, PT ] = 0. At the +EP, spontaneous breaking of this symmetry occurs and +the states are no longer eigenstates of the PT +opera- +tor. The instantaneous eigenvalues of Eq. (5) are given +by E±,k(t) = ± +� +F 2t2 + k2 − γ2, as shown in Fig. 1. +By tuning the momentum k and the imaginary mass γ, +our Hamiltonian permits us to study the evolution of +two different types of modes: those that undergo fully +PT -symmetric evolution, |k| ≥ |γ| and those that pass +through EPs during their evolution, |k| < |γ|. +The dynamics of our model is exactly solvable through +Eqs. (1) and (2), making our model ideal for illustrat- +ing an accurate description of non-Hermitian physics. In +analogy to the Hermitian Landau-Zener problem [60], we +time-evolve the system between Hermitian initial and end +points, which are given by the asymptotic limits t → ±∞. +The uniqueness of the metric ρk(t) is ensured by the Her- +mitian initial condition, ρk(t → −∞) = 1 valid for all k. +Using the exact solution for ρk(t), we can map our prob- +lem to a Hermitian Hamiltonian hk(t), where the dynam- +ical richness of ρk(t) is directly encoded in the dynamics +of hk(t) [61]. +In contrast to the original Hamiltonian Hk(t), we find +that hk(t) does not describe a linear quench, where the +extent of its departure from a linear quench regime is +dictated by the parameters +γ2 +F +and δ = +k2−γ2 +2F +. +This +modified dynamics due to the metric directly influences +the evolution of the state |Ψ(t)⟩k, defined in Eq. (2), for a +certain parameter regime. For k ≫ γ, i.e. very weak non- +hermiticity, the departure from a linear quench is rather +insignificant and |Ψ(t)⟩k and |ψ(t)⟩k,norm ≡ +|ψ(t)⟩k +∥|ψ(t)⟩k∥ are +in good agreement with each other, as shown in Fig. 2(a). + +3 +FIG. 1. The instantaneous spectrum of the non-Hermitian +Hamiltonian given by Eq (5) as a function of time, where +γ = 1 and γ2 +F = 2.5. The static system has exceptional points +at k = ±γ. For k = 0.2γ, the solid and dashed lines indicate +the real and imaginary parts, respectively. Our model allows +us to track both PT -broken and PT -symmetric evolution. +However, this equivalence breaks down when k ∼ γ (even +when PT -symmetry is not broken) and in the PT -broken +regime |k| < γ, as shown in Figs. 2 (b)-(d). Curiously, +for the critical value k = γ, the evolution of the state +|Ψ(t)⟩k is entirely due to the metric. Consequently, the +state |ψ(t)⟩k,norm stays at the north pole of the Bloch +sphere and does not evolve, as shown in Fig. 2 (c). An- +other striking difference concerns the k ↔ −k symmetry: +|Ψ(t)⟩k=|Ψ(t)⟩−k ∀k but this symmetry is in general not +respected by |ψ(t)⟩k,norm. This asymmetry in the norm +method, which stems from the fact that H†(t) ̸= H(t), +is clearly seen for k = ±γ. For k = γ, the time evolution +of |ψ(t)⟩k,norm only involves the upper level such that +|ψ(t)⟩k=γ ∝ (1, 0)T . This does not hold for |ψ(t)⟩k=−γ, +which involves a transition between the levels. +This +asymmetry is not present in |Ψ(t)⟩k=±γ as the metric +dynamics restores the correct symmetry by taking into +account the states evolved using both H(t) and H†(t) in +the construction of the metric [61]. To summarize, Fig. 2 +shows that the metric substantially impacts the time evo- +lution, even for PT -symmetric evolution close to the EP. +Spin expectation. The very different state trajectories +predicted by the two methods lead to different spin ex- +pectation values ⟨σz(t)⟩k,metric and ⟨σz(t)⟩k,norm, calcu- +lated from Eqs. (3) and (4) by setting ˆo = σz [61]. We find +that ⟨σz(t)⟩k,norm is not symmetric under the individual +replacement of k → −k or γ → −γ, but is only invariant +under the combined replacement of these two variables. +On the other hand, ⟨σz(t)⟩k,metric is invariant under ei- +ther of these replacements, reflecting the symmetry of +the instantaneous spectrum E±,k(t). +The exact results for the spin expectation values in +FIG. 2. +The evolution of the normalized states |Ψ(t)⟩k = +ηk(t)|ψ(t)⟩k (blue) and |ψ(t)⟩k,norm ≡ +|ψ(t)⟩k +∥|ψ(t)⟩k∥ (red) on the +Bloch sphere for (a) k = 2γ, (b) k = 1.1γ, (c) k = γ and +(d) k = 0.2γ, c.f. +Fig. 1. +Here γ = 1, +γ2 +F += 2.5 which is +far from the adiabatic limit, and the evolution is between the +asymptotic initial state at the north pole (black dot) and a +distant end point at t = +80 +√ +F . +For k ≫ γ, the dynamics +of |Ψ(t)⟩k and |ψ(t)⟩k,norm are in good agreement with each +other, see (a). However, this is not true for k ≈ γ even for PT - +symmetric evolution, see (b). For k = γ, the dynamics of the +system is completely due to the metric, as |ψ(t)⟩k,norm stays +at the north pole and does not evolve in time, see (c). The +discrepancy between |Ψ(t)⟩k and |ψ(t)⟩k,norm is significant for +PT -broken evolution too, see (d). +the asymptotic limit, ⟨σz(∞)⟩ ≡ ⟨σz(t → ∞)⟩ obtained +using both formalisms, are given by [61] +⟨σz(∞)⟩k,metric = (2k2 − γ2)e−2πδ − k2 +k2 − γ2e−2πδ +⟨σz(∞)⟩k,norm = 2ke−2πδ − k + γ +2γe−2πδ + k − γ +(6) +where the different regimes of nonhermiticity are dictated +by the magnitude of γ2 +F . In the limit γ → 0, we recover +the time evolution under a Hermitian Hamiltonian. In +this case, both ⟨σz(∞)⟩k,metric and ⟨σz(∞)⟩k,norm con- +verge to the standard Landau-Zener result 2e−2πδ0 − 1 +where δ0 = k2 +2F [60]. +Thus, our study reveals the necessity to explicitly con- +sider the non-trivial dynamics induced by the metric in +order to obtain a correct description of non-Hermitian +Hamiltonian dynamics in all parameter regimes. +The +metric is essential in ensuring that the spin expectation + +() +-3 +0 +3 +VFt +k = 2 +k= +k = 1.1 +/- +k = 0.2a +(b) +(c) +(d) +—[亚(t)>k +—[b(t)k,norm4 +FIG. 3. The asymptotic value of the spin expectation values, +given by Eq. (6), in the adiabatic limit F → 0 (here γ2 +F = 400 +and γ = 1). The shaded areas show the defect contribution +from the PT -broken modes. Although the behavior of the +PT -symmetric modes is accurately captured by both meth- +ods, we see that the effect of defect freezing is only captured +when the metric is taken into account. This is a direct con- +sequence of the odd parity of ⟨σz(∞)⟩k,norm with respect to +k. +fulfills certain symmetry requirements arising from the +instantaneous spectrum. It is worth noting that, for a +limited subset of initial conditions and Hamiltonian pa- +rameters, the two approaches may still produce similar +results, see Fig. 2 (a). +Adiabatic limit. We now turn to the adiabatic limit +F → 0. +For γ = 0, the adiabatic limit is the regime +where we recover universal dynamics and Kibble-Zurek +scaling. +This scaling is verifiable in experiments, and, +due to universality, is unaffected by any modification of +the dynamical protocol nor of the microscopic details of +the model. For the non-Hermitian case where γ ̸= 0, we +first remark that the adiabatic limit corresponds to the +regime of strong nonhermiticity γ2 +F → ∞ in our model. +The presence or absence of the aforementioned correct +symmetry in physical observables, as obtained from the +metric vs. the normalization methods, leads to a direct +physical consequence in this limit. +In analogy to the Hermitian Landau-Zener and Kibble- +Zurek problem, the defects are defined as the excitations +which move away from the south pole of the Bloch sphere. +Note that the south pole of the Bloch sphere corresponds +to the ground state of the Hermitian end point. +The +density of defects is then given by [37] +Σz = ΣPT s +z ++ ΣPT b +z +ΣPT s/b +z += +� +k∈PT s/b +dk +2π lim +F →0⟨σz(∞)⟩k +(7) +where PT s and PT b indicate the contributions from the +modes undergoing PT -symmetric and PT -broken evolu- +tion, |k| ≥ γ and |k| < γ, respectively. The asymptotic +expression ⟨σz(∞)⟩k is given by Eq. (6). +For the PT - +broken modes, the metric and the norm methods predict +starkly different asymptotic behaviors in the adiabatic +limit. We obtain ⟨σz(∞)⟩k,metric → 1 − 2k2 +γ2 , consistent +with the k ↔ −k symmetry. On the other hand, using +the norm method, we obtain ⟨σz(∞)⟩k,norm → k +γ , which +is anti-symmetric with respect to k. +This is shown in +Fig. 3. The contribution of the PT -broken modes to the +defect density is thus +� +ΣPT b +z +� +metric = γ +3π +� +ΣPT b +z +� +norm = 0. +(8) +The non-zero defect contribution from the PT -broken +modes shows that defects are generated when a system is +driven across an exceptional point, no matter how slow +the drive is, thus violating quantum adiabaticity. This is +in stark contrast to the Hermitian case where the defect +density tends to zero as F → 0 [60], and is consistent +with the findings of recent experimental work [56]. This +is because non-Hermitian systems are inherently out of +equilibrium. However, this defect freezing effect is not +captured if we do not take the dynamics of the metric +into account. +This is a direct consequence of the odd +parity of ⟨σz(∞)⟩k,norm with respect to k. +We saw in Fig. +2(b) that, away from the adiabatic +limit, the time-evolved state |Ψ(t)⟩k shows non-trivial +behavior even for PT -symmetric modes. +However, a +clear distinction in the behaviors between PT -symmetric +and PT -broken modes is recovered in the adiabatic +limit. This is shown in Fig. 3. For the PT -symmetric +modes, the metric and the norm methods predict the +same asymptotic behaviors: ⟨σz(∞)⟩k → −1 and thus +ΣPT s +z += γ +π − 1. In this limit, the PT -symmetric modes +are pinned to the south pole of the Bloch sphere, where +the term γ +π in ΣPT s +z +shows a reduction in the fraction of +spins pointing to the south pole compared to the Hermi- +tian case. We emphasize that these are not the defects. +In addition to the violation of quantum adiabaticity, +there is no Kibble-Zurek scaling regime in this system, +in contrast to the prediction in Ref. [37]. Indeed, con- +ventional many-body systems are expected to display a +power-law scaling of the defects generated after a slow +ramp across a critical point. For a generic spin system, +this would mean σz = −1 + O(F θ) leading to a defect +density ∼ F θ, where θ depends on the critical exponents +at equilibrium. +For an infinite ensemble of Hermitian +two-level systems, one has θ = 1 +2 [53, 62, 63]. The case of +a non-Hermitian drive has been studied in Ref. [37] using +the normalization approach, yielding a modified Kibble- +Zurek scaling with θ = +2 +3. +In contrast, for the static +non-Hermitian term under study here, the Kibble-Zurek +scaling is wiped out and the density of defects freezes +to a rate-independent value ∼ γ, which survives even in +the adiabatic limit F → 0. In fact, the asymptotic limit +given by Eq. (8) is valid for F ≪ 1, such that there is no +F-dependence in the defect density for several orders of +magnitudes of small F. It is worth noting that, while the +rate-independent result in Eq. (8) is rather remarkable + +a +(0z(0))k,metric +(b) +(αz(00))>k,norm +1 +0 +0 +0 +VF +V +k +VF +VF +/F5 +for an ensemble of two-level systems, a similar violation +of Kibble-Zurek scaling has already been observed when +crossing infinitely degenerate critical points [64–66]. +Conclusion. Our work shows that quantum adiabatic- +ity is violated and Kibble-Zurek scaling is lost in the +presence of nonhermiticity. +Defects are created purely +by the PT -broken modes, which survive even in the adi- +abatic quench limit. +This is consistent with the spec- +tral coalescence at the EPs leading to ambiguity across a +quench. The normalization approach completely misses +this fundamental feature, as it fails to reflect the correct +symmetry of the observables. Our results can be exper- +imentally verified in a variety of photonic and phononic +platforms where non-Hermitian drives can be directly im- +plemented. For example, the evolution of the metric can +be directly engineered using single-photon interferome- +try [19] and parametric amplification [67]. +Many open +questions regarding the dynamics of non-Hermitian sys- +tems remain, in particular, the post-quench spread of +correlation and the putative violation of Lieb-Robinson +bounds [11, 36, 49]. +Acknowledgments. +This work is supported by the +Deutsche Forschungsgemeinschaft (DFG, German Re- +search Foundation) under Germany’s Excellence Strategy +EXC2181/1-390900948 (the Heidelberg STRUCTURES +Excellence Cluster) and a Simons Investigator Award. +The authors would like to thank G. M. Graf for numer- +ous fruitful discussions and E. Bergholtz for comments +on our manuscript. +[1] Y. Ashida, +Z. Gong, and M. Ueda, Non-hermitian +physics, Advances in Physics 69, 249 (2020). +[2] E. J. 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A +99, 063834 (2019). + +Supplemental Material: Quantum metric unveils defect freezing in non-Hermitian +systems +Karin Sim,1 Nicol`o Defenu,1 Paolo Molignini,2, 3 and R. Chitra1 +1Institute for Theoretical Physics, ETH Z¨urich, 8093 Zurich, Switzerland +2Cavendish Laboratory, University of Cambridge, +19 J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom +3Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden +(Dated: January 9, 2023) +SOLUTION TO THE TIME-DEPENDENT SCHR¨ODINGER EQUATION +The time evolution of each k-mode |ψ(t)⟩k in the Hilbert space Hρ(t) is governed by the time-dependent Schr¨odinger +equation (TDSE) +i d +dt|ψ(t)⟩k = Hk(t)|ψ(t)⟩k, +(S.1) +where the Hamiltonian Hk(t) = kσx + iγσy + Ftσz [1] is as given in Eqn. (5) of the main text. +We take the initial state to be the ground state of the initial Hamiltonian, |ψ(t → −∞)⟩k = (eiϕk, 0)T , where ϕk is +an irrelevant global phase. Defining +fk(t) = D−iδ +� +−e +iπ +4 √ +2Ft +� +gk(t) = D−iδ−1 +� +−e +iπ +4 √ +2Ft +� +(S.2) +where Dν(z) is the parabolic cylinder function [2] and δ = k2−γ2 +2F +is dimensionless, we find the time-evolved state to +be +|ψ(t)⟩k = e− πδ +4 +� +e− iπ +4 fk(t) +− (k−γ) +√ +2F gk(t) +� +. +(S.3) +In particular, we note that the state and its bare norm |ψ(t)⟩k ̸= |ψ(t)⟩−k and ⟨ψ(t)|ψ(t)⟩k ̸= ⟨ψ(t)|ψ(t)⟩−k do not +reflect the k ↔ −k symmetry. +TIME EVOLUTION OF THE METRIC ρ(t) +The dynamics of the Hilbert space Hρ(t) is encoded in the time evolution of the metric ρ(t), given by [3–7] +i ˙ρ(t) = H†(t)ρ(t) − ρ(t)H(t), +(S.4) +where the overdot denotes time derivative. +To solve Eqn. (S.4) for a general non-Hermitian Hamiltonian H(t) of a two-level system, we find two linearly +independent solutions to the TDSE +i d +dt|φi(t)⟩ = H†(t)|φi(t)⟩, +i = 1, 2 +(S.5) +which describes the dynamics under the Hermitian conjugate, H†(t). +The metric ρ(t) is then given by +ρ(t) = +2 +� +i=1 +|φi(t)⟩⟨φi(t)| +(S.6) +which satisfies Eqn. (S.4) by construction. +arXiv:2301.02247v1 [quant-ph] 5 Jan 2023 + +2 +For our model, the initial value of the metric is given by ρk(t → −∞) = 1 for all k since we have a Hermitian starting +point. We thus solve Eqn. (S.5) with the initial conditions |φ1(t → −∞)⟩k = (1, 0)T and |φ2(t → −∞)⟩k = (0, 1)T +up to irrelevant global phases. This gives +|φ1(t)⟩k = e− πδ +4 +� +e− iπ +4 fk(t) +− (k+γ) +√ +2F gk(t) +� +, +|φ2(t)⟩k = e− πδ +4 +� k−γ +√ +2F g∗ +k(t) +e +iπ +4 f ∗ +k(t) +� +. +(S.7) +Since ρk(t) is Hermitian by construction, we can express it in terms of the Pauli matrices +ρk(t) = ρ0,k(t)1 + +� +j=x,y,z +ρj,k(t)σj +(S.8) +where its components are given by +ρ0,k(t) = e− πδ +2 +� +|fk(t)|2 + +�k2 + γ2 +2F +� +|gk(t)|2 +� +ρx,k(t) = − 2γ +√ +2F +e− πδ +2 Re +� +e +iπ +4 f ∗ +k(t)gk(t) +� +ρy,k(t) = − 2γ +√ +2F +e− πδ +2 Im +� +e +iπ +4 f ∗ +k(t)gk(t) +� +ρz,k(t) = −kγ +F e− πδ +2 |gk(t)|2 +(S.9) +where Re, Im denote the real and imaginary parts of the functions. +Using the identity +e− πδ +2 � +|fk(t)|2 + δ|gk(t)|2� += 1, +(S.10) +we see that unitary evolution is recovered in the Hilbert space Hρ(t) , since ⟨ψ(t)|ρk(t)|ψ(t)⟩k = 1 at all times. We +also recover ρk(t) = 1 in the Hermitian case γ = 0. +MAPPING TO HERMITIAN h(t) +We can also map the system to a stationary Hilbert space H +described by the Hermitian Hamiltonian [7] +hk(t) = ηk(t)Hk(t)η−1 +k (t) + i ˙ηk(t)η−1 +k (t), +(S.11) +where we have introduced the square-root decomposition of the metric, ρk(t) = η† +k(t)ηk(t). +The time-evolved state in H +is given by +i d +dt|Ψ(t)⟩k = hk(t)|Ψ(t)⟩k +(S.12) +which is related to |ψ(t)⟩k by |Ψ(t)⟩k = ηk(t)|ψ(t)⟩k. +In the Hermitian case γ = 0, the time-evolved states satisfy |Ψ(t)⟩k = |ψ(t)⟩k up to a global phase. Although ηk(t) +needs not be unique, this imposes some constraints on its choice. +In our model, this is satisfied if we choose a Hermitian ηk(t) = η† +k(t), such that [8] +ηk(t) = θk(t) +2 +1 + +� +j=x,y,z +ρj,k(t) +θk(t) σj +(S.13) +where +θk(t) = +� +ρ0,k(t) + +� +ρ2 +0,k(t) − 1 + +� +ρ0,k(t) − +� +ρ2 +0,k(t) − 1 +(S.14) + +3 +and ρj,k(t), j = 0, x, y, z are given in Eqn. (S.9). With this choice of ηk(t), we recover ηk(t) = 1 for all k in the +Hermitian case γ = 0. +Using Eqns. (S.11) and (S.13), we obtain +hk(t) = k +� +1 + γ2 +F ∆hx(t) +� +σx + +√ +F +�√ +Ft + γ2 +F ∆hz(t) +� +σz +(S.15) +where we recover hk(t)|γ=0 = Hk(t)|γ=0 = kσx + Ftσz in the Hermitian case γ = 0. +The non-Hermitian contributions to hk(t) are proportional to the dimensionless parameter γ2 +F which is a measure +of the extent of non-Hermiticity. The dimensionless non-Hermitian correction terms are given by +∆hx(t) = −1 +2 +�|fk(t)|2 +|gk(t)|2 + k2 +2F +�−1 +∆hz(t) = +1 +√ +2 +� +Re(e +iπ +4 f ∗ +k(t)gk(t)) +|fk(t)|2 + k2 +2F |gk(t) +2 +| +� +(S.16) +which can be completely parameterized by δ and γ2 +F by writing k2 +2F = δ + γ2 +2F . +From Eqns. +(S.15) and (S.16), we see that hk(t) picks up a complicated time dependence in the presence of +non-Hermiticity. The extent of departure from the original linear quench is controlled by the parameters δ and γ2 +F . +SPIN EXPECTATION +Setting ˆo = σz and ˆO(t) = η−1 +k (t)σzηk(t) ≡ ˜σz(t) in Eq. (3) of the main text, the spin expectation value under the +metric formalism is given by +⟨σz(t)⟩k,metric = ⟨Ψ(t)|σz|Ψ(t)⟩k = ⟨ψ(t)|ρ(t)˜σz(t)|ψ(t)⟩k += ⟨ψ(t)|η† +k(t)σzηk(t)|ψ(t)⟩k. +(S.17) +Substituting Eqns. (S.3) and (S.13) into Eqn. (S.17), we obtain +⟨σz(t)⟩k,metric = +2 + +� +2k2−γ2 +kγ +� +ρz,k(t) +1 + ρ0,k(t) +. +(S.18) +Using the asymptotic expressions +lim +t→∞|fk(t)|2 = e− 3πδ +2 +lim +t→∞|gk(t)|2 = e +πδ +2 +δ (1 − e−2πδ) +(S.19) +and Eqn. (S.9), we obtain Eqn. (6) in the main text. +The same producedure can be done for ⟨σz(t)⟩k,norm using Eq. (4) of the main text and Eq. (S.3). The asymptotic +expression, Eq. (6) in the main text, is then obtained by using Eq. (S.19). +In particular, in the adiabatic limit F → 0 with a finite γ, the parameter δ → ±∞ with the sign depending on the +sign of k2 −γ2. This restores the clear distinction in the behaviors between the PT -broken and PT -symmetric modes +in the adiabatic limit. +[1] X. Shen, F. Wang, Z. Li, and Z. Wu, Phys. Rev. A 100, 062514 (2019), URL https://link.aps.org/doi/10.1103/ +PhysRevA.100.062514. +[2] M. Abramowitz and I. A. E. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical +Tables, vol. 9 (Dover, New York, 1972), ISBN 0486612724, ch. 19. +[3] F. Scholtz, H. Geyer, and F. Hahne, Annals of Physics 213, 74 (1992), ISSN 0003-4916, URL https://www.sciencedirect. +com/science/article/pii/000349169290284S. + +4 +[4] H. B. Geyer, W. D. Heiss, and F. G. Scholtz, Canadian Journal of Physics 86, 1195 (2008), URL https://doi.org/10. +1139/p08-060. +[5] A. Fring and T. 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Frith, Time-dependence in non-hermitian quantum systems (2020), URL https://arxiv.org/abs/2002.01977. + diff --git a/RNE0T4oBgHgl3EQfUQBU/content/tmp_files/load_file.txt b/RNE0T4oBgHgl3EQfUQBU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d67dd60a857df3e3ce64ea2ff88b09a36bf4963 --- /dev/null +++ b/RNE0T4oBgHgl3EQfUQBU/content/tmp_files/load_file.txt @@ -0,0 +1,795 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf,len=794 +page_content='Quantum metric unveils defect freezing in non-Hermitian systems Karin Sim,1 Nicol`o Defenu,1 Paolo Molignini,2, 3 and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Chitra1 1Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' ETH Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 8093 Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Switzerland 2Cavendish Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 19 J J Thomson Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Cambridge CB3 0HE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' United Kingdom 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Stockholm University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' AlbaNova University Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 106 91 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Sweden (Dated: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2023) Nonhermiticity in quantum Hamiltonians leads to non-unitary time evolution and possibly com- plex energy eigenvalues,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' which can lead to a rich phenomenology with no Hermitian counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In this work, we study the dynamics of an exactly solvable non-Hermitian system, hosting both PT -symmetric and PT -broken modes subject to a linear quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Employing a fully consistent framework, in which the Hilbert space is endowed with a nontrivial dynamical metric, we analyze the dynamics of the generated defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In contrast to Hermitian systems, our study reveals that PT -broken time evolution leads to defect freezing and hence the violation of quantum adiabaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Additionally, no Kibble-Zurek scaling regime in the quasi-adiabatic limit exists in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This physics necessitates the quantum metric framework, as it is missed by the oft used approach of normalizing quantities by the time-dependent norm of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Our results are relevant for a wide class of experimental systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Non-Hermitian Hamiltonians [1–3] pro- vide a framework to explore a complex array of out-of- equilibrium phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Far from being a purely math- ematical pursuit, non-Hermitian descriptions have been employed widely in both classical and quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Arguably, the most well known examples include the study of non-Hermitian spin chains in the context of the Kardar-Parisi-Zhang equation [4] and the localiza- tion of particles in an imaginary vector potential, used to explain the depinning of vortex lines in a supercon- ductor [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' More recently, the field has seen a dramatic revival courtesy of effective descriptions of Lindbladian dynamics in dissipative systems [6], continuously moni- tored systems [7, 8], amplification in optomechanical sys- tems [9], quantum sensors [10], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Nonhermiticity has unveiled a plethora of interesting phenomena, such as quantum phase transitions without gap closure [11, 12], anomalous behaviors of quantum emitters [13], tachyonic physics [14, 15] and unconventional topology [16–18] to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Interest in non-Hermitian systems is further enchanced by the concomitant experimental realizations in diverse platforms: optical systems [16, 19], semicon- ductor microcavities [20] and acoustic systems [21] in the presence of drive and dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Non-Hermitian Hamiltonians which preserve PT - symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=', the combined operation of parity and time reversal) [22–24] constitute a special class of systems possessing a real spectrum, prompting their interpreta- tion as a natural extension to conventional quantum me- chanics [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' When PT -symmetry is spontaneously bro- ken, exceptional points (EPs) arise where the eigenvalues become complex-valued, a topic of much theoretical [26– 28] and experimental [29] interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Conventionally, the Hamiltonian is given by a Hermitian operator which plays the dual role of both the energy operator and the gen- erator of time translations [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' However, nonhermiticity leads to non-unitary time evolution and possibly com- plex energy eigenvalues, both of which imply that the Hamiltonian loses this dual role [25, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Consequently, nonhermiticity ushers in new challenges to fundamental concepts in conventional quantum mechanics, necessitat- ing a more general framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Multiple approaches are used to tackle the aforemen- tioned issues and compute observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Foremost is biorthogonal quantum mechanics [32–34], which has been widely studied in the context of PT -symmetric Hamilto- nians, though its application is limited to a subset of non- Hermitian Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' More often, time-dependent probabilities [35] and observables [19, 36, 37] are explic- itly normalized by the non-conserved norm of the states in an ad hoc manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' As we shall see in this work, this method can fail to capture salient aspects of the physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' A more robust method to study non-Hermitian systems is based on considering the Hilbert space as non-stationary and endowed with a non-trivial time-dependent met- ric [31, 38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' It can be regarded as a generalization of biorthogonal quantum mechanics [32] encompassing spontaneous PT -broken scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This met- ric framework presents a consistent formulation of non- Hermitian quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' It has been adopted to recover fundamental theorems of quantum informa- tion [41], as well as being especially relevant for the evo- lution of entanglement [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Quantum quenches and driving have emerged as tools of choice to explore the non-trivial dynamics of quan- tum systems [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The richness of the emergent phenomenology in Hermitian systems naturally behoves the study of quantum quenches in non-Hermitian sys- tems [34, 48–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' A famous example of non-trivial dy- namics concerns topological defects generated when a coupling is quenched across a quantum critical point [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The Kibble-Zurek scaling predicts that the defect den- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='02247v1 [quant-ph] 5 Jan 2023 2 sity scales as a power law with quench time, where the exponents are determined by the static critical expo- nents [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Using the wavefunction normalization ap- proach, recent work predicted a modified Kibble-Zurek scaling when a system is quenched across EPs [19, 37], thereby recovering adiabaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' On the other hand, breakdown of adiabaticity was seen experimentally in dissipative superconducting qubits governed by effective non-Hermitian Hamiltonians [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In this Letter, using an exactly solvable non-Hermitian model, we rigorously in- vestigate the fundamental question of whether quantum adiabaticity survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We show that the metric plays a crucial role in the violation of quantum adiabaticity when EPs are traversed adiabatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' A mere normalization of physical quantities by the norm of the time-evolved state completely fails to capture this fundamental aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Metric framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We begin by introducing the met- ric framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The inner product in a Hilbert space is defined via its metric ρ(t) as ⟨·, ·⟩ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For a system de- scribed by a Hermitian Hamiltonian, the metric is static and is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In the case of a time- dependent non-Hermitian Hamiltonian H(t), the metric of the Hilbert space develops a non-trivial time evolu- tion, even in the PT -symmetric regimes [31, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The dynamics of the Hilbert space Hρ(t) is encoded in the time evolution of the metric ρ(t), given by [31, 39] i ˙ρ(t) = H†(t)ρ(t) − ρ(t)H(t), (1) where the overdot denotes time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Provided that a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (1) can be found [31], we can map the system to a Hermitian Hamiltonian h(t) = η(t)H(t)η−1(t) + i ˙η(t)η−1(t), where we have introduced the square-root decomposition of the positive-definite metric, ρ(t) = η†(t)η(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The Hamiltonian h(t) acts in a different Hilbert space H [31, 57], where the nonher- miticity is encoded in the dynamics of η(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The time evolution of the states |ψ(t)⟩ in Hρ(t) and |Ψ(t)⟩ in H is governed by the time-dependent Schr¨odinger equation (TDSE) i d dt|ψ(t)⟩ = H(t)|ψ(t)⟩ i d dt|Ψ(t)⟩ = h(t)|Ψ(t)⟩ (2) where the unitarity of the evolution is conserved in both representations, since ⟨ψ(t)|ρ(t)|ψ(t)⟩ = ⟨Ψ(t)|Ψ(t)⟩ = 1 at all times t [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The states are related by |Ψ(t)⟩ = η(t)|ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Under this formalism, the expectation value of an operator ˆo : H → H is given by ⟨O(t)⟩metric = ⟨Ψ(t)|ˆo|Ψ(t)⟩ = ⟨ψ(t)|ρ(t) ˆO(t)|ψ(t)⟩ (3) where ˆO(t) : Hρ(t) → Hρ(t) is defined as ˆO(t) = η−1(t)ˆoη(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In contrast, the expectation of ˆo calculated from a simple normalization by the time-dependent norm is given by ⟨O(t)⟩norm = ⟨ψ(t)|ˆo|ψ(t)⟩ ⟨ψ(t)|ψ(t)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (4) as was done, for example, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Exactly solvable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' To highlight the nontrivial role played by the metric, we consider an exactly solv- able model of effective two level systems parameterised by momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This is given by the Hamiltonian [34] Hk(t) = kσx + iγσy + Ftσz (5) where σi denotes the Pauli matrices and F, k, γ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (5) is a generalization of the Hamiltonian presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' [58] and realized experimentally in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' [59], by adding a real drive term Ft and applying a basis rota- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In our case, the non-Hermitian term γ corresponds to the imaginary tachyon mass [58] and the parameter k is the momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The dimensionless term γ2 F sets the scale for the extent of nonhermiticity in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Note that we recover a purely Hermitian Hamiltonian by setting γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' PT -symmetry is realised in our model by the operators P = σy and T = −iσyK where K is complex conjugation, such that [Hk, PT ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' At the EP, spontaneous breaking of this symmetry occurs and the states are no longer eigenstates of the PT opera- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The instantaneous eigenvalues of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (5) are given by E±,k(t) = ± � F 2t2 + k2 − γ2, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' By tuning the momentum k and the imaginary mass γ, our Hamiltonian permits us to study the evolution of two different types of modes: those that undergo fully PT -symmetric evolution, |k| ≥ |γ| and those that pass through EPs during their evolution, |k| < |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The dynamics of our model is exactly solvable through Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (1) and (2), making our model ideal for illustrat- ing an accurate description of non-Hermitian physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In analogy to the Hermitian Landau-Zener problem [60], we time-evolve the system between Hermitian initial and end points, which are given by the asymptotic limits t → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The uniqueness of the metric ρk(t) is ensured by the Her- mitian initial condition, ρk(t → −∞) = 1 valid for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Using the exact solution for ρk(t), we can map our prob- lem to a Hermitian Hamiltonian hk(t), where the dynam- ical richness of ρk(t) is directly encoded in the dynamics of hk(t) [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In contrast to the original Hamiltonian Hk(t), we find that hk(t) does not describe a linear quench, where the extent of its departure from a linear quench regime is dictated by the parameters γ2 F and δ = k2−γ2 2F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This modified dynamics due to the metric directly influences the evolution of the state |Ψ(t)⟩k, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (2), for a certain parameter regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For k ≫ γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' very weak non- hermiticity, the departure from a linear quench is rather insignificant and |Ψ(t)⟩k and |ψ(t)⟩k,norm ≡ |ψ(t)⟩k ∥|ψ(t)⟩k∥ are in good agreement with each other, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The instantaneous spectrum of the non-Hermitian Hamiltonian given by Eq (5) as a function of time, where γ = 1 and γ2 F = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The static system has exceptional points at k = ±γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='2γ, the solid and dashed lines indicate the real and imaginary parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Our model allows us to track both PT -broken and PT -symmetric evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' However, this equivalence breaks down when k ∼ γ (even when PT -symmetry is not broken) and in the PT -broken regime |k| < γ, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2 (b)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Curiously, for the critical value k = γ, the evolution of the state |Ψ(t)⟩k is entirely due to the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Consequently, the state |ψ(t)⟩k,norm stays at the north pole of the Bloch sphere and does not evolve, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' An- other striking difference concerns the k ↔ −k symmetry: |Ψ(t)⟩k=|Ψ(t)⟩−k ∀k but this symmetry is in general not respected by |ψ(t)⟩k,norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This asymmetry in the norm method, which stems from the fact that H†(t) ̸= H(t), is clearly seen for k = ±γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For k = γ, the time evolution of |ψ(t)⟩k,norm only involves the upper level such that |ψ(t)⟩k=γ ∝ (1, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This does not hold for |ψ(t)⟩k=−γ, which involves a transition between the levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This asymmetry is not present in |Ψ(t)⟩k=±γ as the metric dynamics restores the correct symmetry by taking into account the states evolved using both H(t) and H†(t) in the construction of the metric [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' To summarize, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2 shows that the metric substantially impacts the time evo- lution, even for PT -symmetric evolution close to the EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Spin expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The very different state trajectories predicted by the two methods lead to different spin ex- pectation values ⟨σz(t)⟩k,metric and ⟨σz(t)⟩k,norm, calcu- lated from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (3) and (4) by setting ˆo = σz [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We find that ⟨σz(t)⟩k,norm is not symmetric under the individual replacement of k → −k or γ → −γ, but is only invariant under the combined replacement of these two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' On the other hand, ⟨σz(t)⟩k,metric is invariant under ei- ther of these replacements, reflecting the symmetry of the instantaneous spectrum E±,k(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The exact results for the spin expectation values in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The evolution of the normalized states |Ψ(t)⟩k = ηk(t)|ψ(t)⟩k (blue) and |ψ(t)⟩k,norm ≡ |ψ(t)⟩k ∥|ψ(t)⟩k∥ (red) on the Bloch sphere for (a) k = 2γ, (b) k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='1γ, (c) k = γ and (d) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='2γ, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Here γ = 1, γ2 F = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='5 which is far from the adiabatic limit, and the evolution is between the asymptotic initial state at the north pole (black dot) and a distant end point at t = 80 √ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For k ≫ γ, the dynamics of |Ψ(t)⟩k and |ψ(t)⟩k,norm are in good agreement with each other, see (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' However, this is not true for k ≈ γ even for PT - symmetric evolution, see (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For k = γ, the dynamics of the system is completely due to the metric, as |ψ(t)⟩k,norm stays at the north pole and does not evolve in time, see (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The discrepancy between |Ψ(t)⟩k and |ψ(t)⟩k,norm is significant for PT -broken evolution too, see (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' the asymptotic limit, ⟨σz(∞)⟩ ≡ ⟨σz(t → ∞)⟩ obtained using both formalisms, are given by [61] ⟨σz(∞)⟩k,metric = (2k2 − γ2)e−2πδ − k2 k2 − γ2e−2πδ ⟨σz(∞)⟩k,norm = 2ke−2πδ − k + γ 2γe−2πδ + k − γ (6) where the different regimes of nonhermiticity are dictated by the magnitude of γ2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In the limit γ → 0, we recover the time evolution under a Hermitian Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In this case, both ⟨σz(∞)⟩k,metric and ⟨σz(∞)⟩k,norm con- verge to the standard Landau-Zener result 2e−2πδ0 − 1 where δ0 = k2 2F [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Thus, our study reveals the necessity to explicitly con- sider the non-trivial dynamics induced by the metric in order to obtain a correct description of non-Hermitian Hamiltonian dynamics in all parameter regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The metric is essential in ensuring that the spin expectation () 3 0 3 VFt k = 2 k= k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='1 /- k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='2a (b) (c) (d) —[亚(t)>k —[b(t)k,norm4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The asymptotic value of the spin expectation values, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (6), in the adiabatic limit F → 0 (here γ2 F = 400 and γ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The shaded areas show the defect contribution from the PT -broken modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Although the behavior of the PT -symmetric modes is accurately captured by both meth- ods, we see that the effect of defect freezing is only captured when the metric is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This is a direct con- sequence of the odd parity of ⟨σz(∞)⟩k,norm with respect to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' fulfills certain symmetry requirements arising from the instantaneous spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' It is worth noting that, for a limited subset of initial conditions and Hamiltonian pa- rameters, the two approaches may still produce similar results, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Adiabatic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We now turn to the adiabatic limit F → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For γ = 0, the adiabatic limit is the regime where we recover universal dynamics and Kibble-Zurek scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This scaling is verifiable in experiments, and, due to universality, is unaffected by any modification of the dynamical protocol nor of the microscopic details of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For the non-Hermitian case where γ ̸= 0, we first remark that the adiabatic limit corresponds to the regime of strong nonhermiticity γ2 F → ∞ in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The presence or absence of the aforementioned correct symmetry in physical observables, as obtained from the metric vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' the normalization methods, leads to a direct physical consequence in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In analogy to the Hermitian Landau-Zener and Kibble- Zurek problem, the defects are defined as the excitations which move away from the south pole of the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Note that the south pole of the Bloch sphere corresponds to the ground state of the Hermitian end point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The density of defects is then given by [37] Σz = ΣPT s z + ΣPT b z ΣPT s/b z = � k∈PT s/b dk 2π lim F →0⟨σz(∞)⟩k (7) where PT s and PT b indicate the contributions from the modes undergoing PT -symmetric and PT -broken evolu- tion, |k| ≥ γ and |k| < γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The asymptotic expression ⟨σz(∞)⟩k is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For the PT - broken modes, the metric and the norm methods predict starkly different asymptotic behaviors in the adiabatic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We obtain ⟨σz(∞)⟩k,metric → 1 − 2k2 γ2 , consistent with the k ↔ −k symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' On the other hand, using the norm method, we obtain ⟨σz(∞)⟩k,norm → k γ , which is anti-symmetric with respect to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The contribution of the PT -broken modes to the defect density is thus � ΣPT b z � metric = γ 3π � ΣPT b z � norm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (8) The non-zero defect contribution from the PT -broken modes shows that defects are generated when a system is driven across an exceptional point, no matter how slow the drive is, thus violating quantum adiabaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This is in stark contrast to the Hermitian case where the defect density tends to zero as F → 0 [60], and is consistent with the findings of recent experimental work [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This is because non-Hermitian systems are inherently out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' However, this defect freezing effect is not captured if we do not take the dynamics of the metric into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This is a direct consequence of the odd parity of ⟨σz(∞)⟩k,norm with respect to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We saw in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2(b) that, away from the adiabatic limit, the time-evolved state |Ψ(t)⟩k shows non-trivial behavior even for PT -symmetric modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' However, a clear distinction in the behaviors between PT -symmetric and PT -broken modes is recovered in the adiabatic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For the PT -symmetric modes, the metric and the norm methods predict the same asymptotic behaviors: ⟨σz(∞)⟩k → −1 and thus ΣPT s z = γ π − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In this limit, the PT -symmetric modes are pinned to the south pole of the Bloch sphere, where the term γ π in ΣPT s z shows a reduction in the fraction of spins pointing to the south pole compared to the Hermi- tian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We emphasize that these are not the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In addition to the violation of quantum adiabaticity, there is no Kibble-Zurek scaling regime in this system, in contrast to the prediction in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Indeed, con- ventional many-body systems are expected to display a power-law scaling of the defects generated after a slow ramp across a critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For a generic spin system, this would mean σz = −1 + O(F θ) leading to a defect density ∼ F θ, where θ depends on the critical exponents at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For an infinite ensemble of Hermitian two-level systems, one has θ = 1 2 [53, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The case of a non-Hermitian drive has been studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' [37] using the normalization approach, yielding a modified Kibble- Zurek scaling with θ = 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In contrast, for the static non-Hermitian term under study here, the Kibble-Zurek scaling is wiped out and the density of defects freezes to a rate-independent value ∼ γ, which survives even in the adiabatic limit F → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In fact, the asymptotic limit given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (8) is valid for F ≪ 1, such that there is no F-dependence in the defect density for several orders of magnitudes of small F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' It is worth noting that, while the rate-independent result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (8) is rather remarkable a (0z(0))k,metric (b) (αz(00))>k,norm 1 0 0 0 VF V k VF VF /F5 for an ensemble of two-level systems, a similar violation of Kibble-Zurek scaling has already been observed when crossing infinitely degenerate critical points [64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Our work shows that quantum adiabatic- ity is violated and Kibble-Zurek scaling is lost in the presence of nonhermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Defects are created purely by the PT -broken modes, which survive even in the adi- abatic quench limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This is consistent with the spec- tral coalescence at the EPs leading to ambiguity across a quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The normalization approach completely misses this fundamental feature, as it fails to reflect the correct symmetry of the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Our results can be exper- imentally verified in a variety of photonic and phononic platforms where non-Hermitian drives can be directly im- plemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' For example, the evolution of the metric can be directly engineered using single-photon interferome- try [19] and parametric amplification [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Many open questions regarding the dynamics of non-Hermitian sys- tems remain, in particular, the post-quench spread of correlation and the putative violation of Lieb-Robinson bounds [11, 36, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Re- search Foundation) under Germany’s Excellence Strategy EXC2181/1-390900948 (the Heidelberg STRUCTURES Excellence Cluster) and a Simons Investigator Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The authors would like to thank G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' M.' metadata={'source': 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crossing of an infinitely degenerate critical point, Annales Henri Poincar´e 18, 1755 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' [66] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Defenu, Quantum adiabatic cycles and their break- down, Communications Physics 4, 150 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' [67] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Wang and A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Chitra1 1Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' ETH Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 8093 Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Switzerland 2Cavendish Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 19 J J Thomson Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Cambridge CB3 0HE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' United Kingdom 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Stockholm University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' AlbaNova University Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 106 91 Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Sweden (Dated: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' 2023) SOLUTION TO THE TIME-DEPENDENT SCHR¨ODINGER EQUATION The time evolution of each k-mode |ψ(t)⟩k in the Hilbert space Hρ(t) is governed by the time-dependent Schr¨odinger equation (TDSE) i d dt|ψ(t)⟩k = Hk(t)|ψ(t)⟩k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='1) where the Hamiltonian Hk(t) = kσx + iγσy + Ftσz [1] is as given in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (5) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We take the initial state to be the ground state of the initial Hamiltonian, |ψ(t → −∞)⟩k = (eiϕk, 0)T , where ϕk is an irrelevant global phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Defining fk(t) = D−iδ � −e iπ 4 √ 2Ft � gk(t) = D−iδ−1 � −e iπ 4 √ 2Ft � (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='2) where Dν(z) is the parabolic cylinder function [2] and δ = k2−γ2 2F is dimensionless, we find the time-evolved state to be |ψ(t)⟩k = e− πδ 4 � e− iπ 4 fk(t) − (k−γ) √ 2F gk(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='3) In particular, we note that the state and its bare norm |ψ(t)⟩k ̸= |ψ(t)⟩−k and ⟨ψ(t)|ψ(t)⟩k ̸= ⟨ψ(t)|ψ(t)⟩−k do not reflect the k ↔ −k symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' TIME EVOLUTION OF THE METRIC ρ(t) The dynamics of the Hilbert space Hρ(t) is encoded in the time evolution of the metric ρ(t), given by [3–7] i ˙ρ(t) = H†(t)ρ(t) − ρ(t)H(t), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='4) where the overdot denotes time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' To solve Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='4) for a general non-Hermitian Hamiltonian H(t) of a two-level system, we find two linearly independent solutions to the TDSE i d dt|φi(t)⟩ = H†(t)|φi(t)⟩, i = 1, 2 (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='5) which describes the dynamics under the Hermitian conjugate, H†(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The metric ρ(t) is then given by ρ(t) = 2 � i=1 |φi(t)⟩⟨φi(t)| (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='6) which satisfies Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='4) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='02247v1 [quant-ph] 5 Jan 2023 2 For our model, the initial value of the metric is given by ρk(t → −∞) = 1 for all k since we have a Hermitian starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We thus solve Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='5) with the initial conditions |φ1(t → −∞)⟩k = (1, 0)T and |φ2(t → −∞)⟩k = (0, 1)T up to irrelevant global phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This gives |φ1(t)⟩k = e− πδ 4 � e− iπ 4 fk(t) − (k+γ) √ 2F gk(t) � , |φ2(t)⟩k = e− πδ 4 � k−γ √ 2F g∗ k(t) e iπ 4 f ∗ k(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='7) Since ρk(t) is Hermitian by construction, we can express it in terms of the Pauli matrices ρk(t) = ρ0,k(t)1 + � j=x,y,z ρj,k(t)σj (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='8) where its components are given by ρ0,k(t) = e− πδ 2 � |fk(t)|2 + �k2 + γ2 2F � |gk(t)|2 � ρx,k(t) = − 2γ √ 2F e− πδ 2 Re � e iπ 4 f ∗ k(t)gk(t) � ρy,k(t) = − 2γ √ 2F e− πδ 2 Im � e iπ 4 f ∗ k(t)gk(t) � ρz,k(t) = −kγ F e− πδ 2 |gk(t)|2 (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='9) where Re, Im denote the real and imaginary parts of the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Using the identity e− πδ 2 � |fk(t)|2 + δ|gk(t)|2� = 1, (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='10) we see that unitary evolution is recovered in the Hilbert space Hρ(t) , since ⟨ψ(t)|ρk(t)|ψ(t)⟩k = 1 at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' We also recover ρk(t) = 1 in the Hermitian case γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' MAPPING TO HERMITIAN h(t) We can also map the system to a stationary Hilbert space H described by the Hermitian Hamiltonian [7] hk(t) = ηk(t)Hk(t)η−1 k (t) + i ˙ηk(t)η−1 k (t), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='11) where we have introduced the square-root decomposition of the metric, ρk(t) = η† k(t)ηk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The time-evolved state in H is given by i d dt|Ψ(t)⟩k = hk(t)|Ψ(t)⟩k (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='12) which is related to |ψ(t)⟩k by |Ψ(t)⟩k = ηk(t)|ψ(t)⟩k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In the Hermitian case γ = 0, the time-evolved states satisfy |Ψ(t)⟩k = |ψ(t)⟩k up to a global phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Although ηk(t) needs not be unique, this imposes some constraints on its choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In our model, this is satisfied if we choose a Hermitian ηk(t) = η† k(t), such that [8] ηk(t) = θk(t) 2 1 + � j=x,y,z ρj,k(t) θk(t) σj (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='13) where θk(t) = � ρ0,k(t) + � ρ2 0,k(t) − 1 + � ρ0,k(t) − � ρ2 0,k(t) − 1 (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='14) 3 and ρj,k(t), j = 0, x, y, z are given in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' With this choice of ηk(t), we recover ηk(t) = 1 for all k in the Hermitian case γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' Using Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='11) and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='13), we obtain hk(t) = k � 1 + γ2 F ∆hx(t) � σx + √ F �√ Ft + γ2 F ∆hz(t) � σz (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='15) where we recover hk(t)|γ=0 = Hk(t)|γ=0 = kσx + Ftσz in the Hermitian case γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The non-Hermitian contributions to hk(t) are proportional to the dimensionless parameter γ2 F which is a measure of the extent of non-Hermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The dimensionless non-Hermitian correction terms are given by ∆hx(t) = −1 2 �|fk(t)|2 |gk(t)|2 + k2 2F �−1 ∆hz(t) = 1 √ 2 � Re(e iπ 4 f ∗ k(t)gk(t)) |fk(t)|2 + k2 2F |gk(t) 2 | � (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='16) which can be completely parameterized by δ and γ2 F by writing k2 2F = δ + γ2 2F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' From Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='15) and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='16), we see that hk(t) picks up a complicated time dependence in the presence of non-Hermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The extent of departure from the original linear quench is controlled by the parameters δ and γ2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' SPIN EXPECTATION Setting ˆo = σz and ˆO(t) = η−1 k (t)σzηk(t) ≡ ˜σz(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (3) of the main text, the spin expectation value under the metric formalism is given by ⟨σz(t)⟩k,metric = ⟨Ψ(t)|σz|Ψ(t)⟩k = ⟨ψ(t)|ρ(t)˜σz(t)|ψ(t)⟩k = ⟨ψ(t)|η† k(t)σzηk(t)|ψ(t)⟩k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='17) Substituting Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='3) and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='13) into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='17), we obtain ⟨σz(t)⟩k,metric = 2 + � 2k2−γ2 kγ � ρz,k(t) 1 + ρ0,k(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='18) Using the asymptotic expressions lim t→∞|fk(t)|2 = e− 3πδ 2 lim t→∞|gk(t)|2 = e πδ 2 δ (1 − e−2πδ) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='19) and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='9), we obtain Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (6) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The same producedure can be done for ⟨σz(t)⟩k,norm using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (4) of the main text and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' The asymptotic expression, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (6) in the main text, is then obtained by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' In particular, in the adiabatic limit F → 0 with a finite γ, the parameter δ → ±∞ with the sign depending on the sign of k2 −γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' This restores the clear distinction in the behaviors between the PT -broken and PT -symmetric modes in the adiabatic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE0T4oBgHgl3EQfUQBU/content/2301.02247v1.pdf'} +page_content=' [1] X.' metadata={'source': 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Taranets1, Hangjie Ji2, +and Marina Chugunova3 +1 Institute of Applied Mathematics and Mechanics of the NASU, +G.Batyuka Str. 19, 84100, Sloviansk, Ukraine +2 Department of Mathematics, North Carolina State University, +Raleigh, NC 27695, USA +3 Claremont Graduate University, 150 E. 10th Str., Claremont, +CA 91711, USA +E-mail: taranets_r@yahoo.com, hangjie_ji@ncsu.edu, and +marina.chugunova@cgu.edu +10 January 2023 +Abstract. +This paper analytically investigates the solutions to +a control-volume model for liquid films flowing down a vertical +fibre. The dynamic evolution of the free surface is governed by a +coupled degenerate nonlinear PDE system for the fluid film radius +and the axial velocity. We prove the existence of weak solutions to +the coupled system based on the application of a priori estimates +derived for energy-entropy functionals. +Existence of travelling +wave solutions to the system is also established. Numerical studies +are presented to illustrate the derived analytical results for both +the dynamic PDE solutions and the travelling wave structures. +Keywords: Thin films; travelling waves; fourth-order parabolic partial +differential equations +arXiv:2301.02720v1 [math.AP] 6 Jan 2023 + +Travelling waves of a control-volume fibre coating model +2 +1. Introduction +Thin liquid films flowing down a vertical fibre have attracted many +interests in the past decades due to their importance in a variety of +engineering applications, including heat and mass exchangers, thermal +desalination, and vapor and CO2 capturing [27, 28, 35–37]. +These +liquid films are fundamentally driven by Rayleigh-Plateau instability +and gravity modulation, spontaneously exhibiting complex interfacial +instability and pattern formation [20, 21, 24, 29]. +Previous experimental works have found that the downstream flow +dynamics and pattern formation highly depend on the flow rate, fibre +radius, liquid properties, and inlet geometries. +Specifically, three +typical flow regimes have been extensively studied [7, 10, 18, 25, 34]. +At high flow rates, the convective instability dominates the system and +irregular droplet coalescence occur frequently. For lower flow rates, +the Rayleigh-Plateau regime emerges where stable travelling droplets +move at a constant speed. +If flow rates are further reduced, the +isolated droplet regime occurs where widely-spaced droplets coexist +with small amplitude wave patterns. Similar regime transitions can +also be triggered by varying the nozzle diameters or imposing a gradient +to the liquid property along the fibre [8, 12, 13]. A good understanding +of these dynamics is critical to the design and control of engineering +systems that involve a stable train of travelling droplets. +In the low Reynolds number limit, classical lubrication models +have been developed for the dynamics of falling viscous liquid +films along an axisymmetric cylindrical fibre. +Under the long-wave +approximation, the resultant evolution equations are a family of +fourth-order degenerate parabolic PDEs for the fluid film thickness +[5, 10, 11, 15, 19]. +These models incorporate gravity and both +stabilizing and destabilizing roles of surface tension by characterizing +the axial and azimuthal curvature of the free surface. +Numerical +and analytical investigations for this type of models also revealed + +Travelling waves of a control-volume fibre coating model +3 +the dependence of the droplet dynamics on the substrate effects and +external physical fields [12, 14, 22]. +For higher flow rates and for fluid films near the nozzle where +inertial effects are significant, systems of coupled equations for both +the film thickness and the local flow rate have also been investigated +[13, 25, 26, 32]. +These models include inertia effects based on a +weighted-residual integeral boundary layer approach by assuming a +local velocity profile. More recently, Ruan et al. [23] proposed a new +framework for liquid films flowing down a fibre using a control-volume +approach. Their model expresses the conservation of mass and axial +momentum via the coupled dimensionless equation for the fluid film +radius h(x, t) and the mean axial velocity u(x, t), where the momentum +equation is +ut + a +�u2 +2 +� +x ++ b κx = c [(h2 − 1)ux]x +h2 − 1 ++ 1 − +u +g(h) +(1.1) +and the mass conservation equation is +2hht + a[u(h2 − 1)]x = 0, +(1.2) +where the dimensionless parameter a represents the square of the +Froude number, b is the reciprocal of the Bond number, c represents +the ratio of axial viscous to gravitational forces, and g(h) represents +the axial velocity profile. The film thickness is given by h(x, t)−1, and +κ represents the combined azimuthal and streamwise curvatures of the +free surface, +κ = +1 +h(1 + h2 +x)1/2 − +� +hx +(1 + h2 +x)1/2 +� +x +. +(1.3) +Furthermore, by taking different forms of g(h), the model (1.1)–(1.2) +corresponds to different flow regimes. For the high Reynolds number +regime, we have the plug flow model with +g(h) = h2 − 1 +(1.4) + +Travelling waves of a control-volume fibre coating model +4 +that assumes a uniform velocity in the cross section with a viscous +drag force on the fluid. For the low Reynolds number case, the fully- +developed laminar velocity profile is assumed, with +g(h) = I(h) +h2 − 1, +(1.5) +where I(h) = +1 +16[4h4 ln(h) + (h2 − 1)(1 − 3h2)]. +While extensive modelling works have been carried out for falling +liquid films, relatively less research [14] have focused on establishing +analytical understanding of the developed models. In this work, we will +analytically investigate the coupled equations (1.1)–(1.2), with a focus +on the travelling wave solutions. Energy and entropy estimates will be +constructed to establish the existence of weak solutions to the problem. +Similar analytical techniques were also applied to other models (for +example, see [2–4, 14]). +The rest of the paper is structured as follows. +In section 2, we +formulate the problem statement. In section 3, we show the existence +of weak solutions to the problem via energy and entropy estimates. +Section 4 presents a detailed discussion on travelling wave solutions. +Numerical studies are presented in section 5 for both the plug flow and +the laminar flow cases, followed by concluding remarks in section 6. +2. Problem statement +We study the following initial boundary value problem: +ut + a +�u2 +2 +� +x ++ b κx = c [(h2 − 1)ux]x +h2 − 1 ++ 1 − +u +g(h) in QT, +(2.1) +2hht + a[u(h2 − 1)]x = 0 in QT, +(2.2) +u and h are |Ω| − periodic, +(2.3) +u(x, 0) = u0(x), h(x, 0) = h0(x), +(2.4) + +Travelling waves of a control-volume fibre coating model +5 +where Ω ⊂ R1 is an open interval, QT := Ω × (0, T), a, b, c are non- +negative constants, and +κ = f(hx)h−1 − f 3(hx)hxx, +f(z) = (1 + z2)− 1 +2, +Φ(z) = +1 +f(z), Φ′(z) = zf(z), Φ′′(z) = f 3(z), +g(h) = h2 − 1 or g(h) = I(h) +h2 − 1, +where +I(h) := +1 +16[4h4 ln(h) + (h2 − 1)(1 − 3h2)]. +Let +v = h2 − 1. +Then we can rewrite (2.1) and (2.2) in the following form: +ut + a +�u2 +2 +� +x ++ b κx = c (v ux)x +v ++ 1 − +u +g(h) +in QT, +(2.5) +vt + a(u v)x = 0 +in QT. +(2.6) +Integrating (2.6) in Qt, we find that v(x, t) satisfies the conservation +of mass, +� +Ω +v(x, t) dx = +� +Ω +v0(x) dx := M > 0 ∀ t ⩾ 0. +(2.7) +Furthermore, we assume that the initial data (v0, u0) satisfy +h0 ⩾ 1, i. e. v0 := h2 +0 − 1 ⩾ 0, for all x ∈ ¯Ω; √v0 ∈ H1(Ω); +h0Φ(h0,x), − log(v0), v0u2 +0 ∈ L1(Ω). +(2.8) +Definition 2.1. A pair (v, u) is a weak solution to (2.5)–(2.6) with +periodic boundary conditions and initial conditions (v0, u0) if 0 ⩽ v ∈ +C( ¯QT) and u satisfy the regularity properties +√v ∈ L∞(0, T; H1(Ω)); − log(v), vu2 ∈ L∞(0, T; L1(Ω)); +(2.9) + +Travelling waves of a control-volume fibre coating model +6 +hΦ(hx) ∈ L∞(0, T; L1(Ω)); χ{|hx|<∞}h−1f(hx) ∈ L1(QT); +(2.10) +χ{|hx|<∞} +� +hf 3(hx)hxx; χ{v>0} +√vux, +� +v +g(h)u ∈ L2(QT), +(2.11) +and the following holds +�� +QT +vφt dxdt + +� +Ω +v0φ(x, 0) dx + +�� +QT +uvφx dxdt = 0, +�� +QT +uvψt dxdt + +� +Ω +u0v0ψ(x, 0) dx + a +�� +QT +χ{v>0}vu2ψx dxdt+ +b +�� +QT +χ{|hx|<∞}κvxψ dxdt + b +�� +QT +χ{|hx|<∞}κvψx dxdt− +c +�� +QT +χ{v>0}vuxψx dxdt + +�� +QT +� +v − +uv +g(h) +� +ψ dxdt = 0 +for all φ ∈ C∞ +c ( ¯QT) and ψ ∈ C∞ +c ( ¯QT) such that φ(x, T) = ψ(x, T) = 0. +Based on the definition 2.1, we will establish the existence of weak +solutions to the problem and prove the following theorem: +Theorem 1. Let the initial data (v0, u0) satisfy (2.7)–(2.8), and T > 0. +Then there exists a weak solution (v, u) in the sense of Definition 2.1. +Moreover, the sets {v(., t) = 0} and {|hx(., t)| = ∞} have measure zero +for any t ∈ [0, T]. +3. Existence of weak solutions +In this section, we will introduce the energy and entropy functionals +for the problem and show their estimates in subsections 3.1 and 3.2. +The proof of key results in Lemma 3.1 and Lemma 3.2 follows the work +of Kitavtsev et al. [17]. + +Travelling waves of a control-volume fibre coating model +7 +3.1. Energy estimate +Let us denote the energy functional by +E(t) := 1 +2 +� +Ω +(vu2 + 4b +a hΦ(hx)) dx. +Lemma 3.1 (Energy inequality). Let (v, u) be sufficiently smooth +solution to the system (2.5)–(2.6) with periodic boundary conditions, +then (v, u) satisfy the following inequality +E(T) + c +�� +QT +vu2 +x dxdt + +�� +QT +u2v +g(h) dxdt ⩽ C0(T), +(3.1) +where C0(T) = (E +1 +2(0) + +√ +2M +2 T)2. +Proof of Lemma 3.1. Multiplying (2.5) by uv and integrating over Ω, +we have +� +Ω +uvut dx + a +� +Ω +uv( u2 +2 )x dx + b +� +Ω +uvκx dx = +c +� +Ω +u(vux)x dx + +� +Ω +uv(1 − +u +g(h)) dx. +(3.2) +Since the first two integrals on the left-hand-side of (3.2) satisfy +� +Ω +uvut dx + a +� +Ω +uv( u2 +2 )x dx = +� +Ω +v( u2 +2 )t dx − a +� +Ω +(uv)x +u2 +2 dx = +� +Ω +v( u2 +2 )t dx + +� +Ω +vt +u2 +2 dx = 1 +2 +d +dt +� +Ω +vu2 dx, + +Travelling waves of a control-volume fibre coating model +8 +and the third integral on the left-hand-side of (3.2) satisfies +b +� +Ω +uvκx dx = −b +� +Ω +(uv)xκ dx = b +a +� +Ω +vtκ dx = +2b +a +� +Ω +hht(f(hx)h−1 − f 3(hx)hxx) dx = 2b +a +� +Ω +(htf(hx) − hhtΦ′′(hx)hxx) dx = +2b +a +� +Ω +(htf(hx) − hht(Φ′(hx))x) dx = 2b +a +� +Ω +(htf(hx) + (hht)xΦ′(hx)) dx = +2b +a +� +Ω +(htf(hx) + hthxΦ′(hx) + hhxtΦ′(hx)) dx = +2b +a +� +Ω +(htf(hx) + hth2 +xf(hx) + h(Φ(hx))t) dx = +2b +a +� +Ω +(htΦ(hx) + h(Φ(hx))t) dx = 2b +a +d +dt +� +Ω +hΦ(hx) dx, +then from (3.2) it follows that +d +dtE(t) + c +� +Ω +vu2 +x dx + +� +Ω +u2v +g(h) dx = +� +Ω +uv dx. +(3.3) +Taking into account (2.7) and +� +Ω +uv dx ⩽ M +1 +2 +�� +Ω +vu2 dx +� 1 +2, +(3.4) +and after integrating (3.3) in time, we obtain the energy estimate +(3.1). +3.2. Entropy estimate +We define two entropy-like functionals for the plug flow and laminar +flow models separately. For the plug flow model with g(h) = v, we + +Travelling waves of a control-volume fibre coating model +9 +denote the entropy functional by +S1(u, h) := 1 +2 +� +Ω +� +v(u + c +a +vx +v )2 + 4b +a hΦ(hx) + 2c +a2(v − log(v)) +� +dx +For the laminar flow model with g(h) = I(h) +v , we define the entropy +functional as +S2(u, h) := 1 +2 +� +Ω +� +v(u + +c +a+ϵ0 +vx +v )2 + +c2ϵ0 +a(a+ϵ0)2 +v2 +x +v + 4b +a hΦ(hx)+ +4c +a(a+ϵ0)(8v − G(h)) +� +dx +where G′(h) = +h +g(h) and ϵ0 > 0. We will prove the following entropy +estimates for S1(u, h) and S2(u, h). +Lemma 3.2 (Entropy inequality). Let (v, u) be sufficiently smooth +solution to the system (2.5)–(2.6) with periodic boundary conditions, +then (v, u) satisfy the following inequality +S1(u, h) + 2bc +a +�� +QT +hf 3(hx)h2 +xx dxdt + +�� +QT +u2 dxdt+ +2bc +a +�� +QT +h−1f(hx) dxdt ⩽ C1(T) if g(h) = v, +(3.5) +S2(u, h) + +2bc +a+ϵ0 +�� +QT +hf 3(hx)h2 +xx dxdt + +�� +QT +u2v2 +I(h) dxdt+ +2bc +a+ϵ0 +�� +QT +h−1f(hx) dxdt ⩽ C2(T) if g(h) = I(h) +v , +(3.6) +where +C1(T) := S1(u0, h0) + +T +� +0 +(c C0(t) + (2M) +1 +2C +1 +2 +0 (t)) dt, + +Travelling waves of a control-volume fibre coating model +10 +C2(T) := S2(u0, h0) + +T +� +0 +( a c +a+ϵ0C0(t) + (2M) +1 +2C +1 +2 +0 (t) + +16c +ϵ0(a+ϵ0)M) dt. +Remark 3.1. Using the estimate (a + b)2 ⩾ ϵa2 − +ϵ +1−ϵb2 for any +ϵ ∈ [0, 1), from (3.5) and (3.1), we deduce that for the plug flow model, +� +Ω +v2 +x +v dx ⩽ ( a +c)2� 1 +1−ϵ +� +Ω +v u2 dx + 2 +ϵC1(T) +� +⩽ C3(T), +(3.7) +where +C3(T) := ( a +c)2� 2 +1−ϵC0(T) + 2 +ϵC1(T) +� +. +(3.8) +From (3.7) it follows that (v +1 +2)x ∈ L∞(0, T; L2(Ω)). The same can be +shown in the case of laminar flows with g(h) = I(h) +v +as well. +Proof of Lemma 3.2. Multiplying (2.5) by vx and integrating over Ω, +we have +� +Ω +(ut + auux)vx dx + b +� +Ω +vxκx dx = +− c +� +Ω +vux( vx +v )x dx − +� +Ω +u vx +g(h) dx. +(3.9) +For the first integral on the left-hand-side of (3.9), we have +� +Ω +(ut + auux)vx dx = +� +Ω +(utvx + a(uv)xux − a vu2 +x) dx = +� +Ω +(utvx − vtux − a vu2 +x) dx = +� +Ω +(utvx + uvxt − a vu2 +x) dx = +d +dt +� +Ω +uvx dx − a +� +Ω +vu2 +x dx. + +Travelling waves of a control-volume fibre coating model +11 +To handle the first integral on the right-hand-side of (3.9), we deduce +d +dt +� +Ω +v2 +x +v dx = 2 +� +Ω +vxvxt +v +dx − +� +Ω +v2 +xvt +v2 dx = −2 +� +Ω +( vx +v )xvt dx− +� +Ω +v2 +x +v2 vt dx = 2a +� +Ω +( vx +v )x(uv)xdx + a +� +Ω +v2 +x +v2 (uv)x dx = +2a +� +Ω +( vx +v )x(uv)xdx − 2a +� +Ω +( vx +v )xuvxdx = 2a +� +Ω +vux( vx +v )xdx. +For the second integral on the left-hand-side of (3.9), we have +b +� +Ω +vxκx dx = −b +� +Ω +vxx(f(hx)h−1 − f 3(hx)hxx) dx = +− 2b +� +Ω +(h2 +x + hhxx)(f(hx)h−1 − f 3(hx)hxx) dx = +− 2b +� +Ω +h−1h2 +xf(hx) dx − 2b +� +Ω +f(hx)(1 − f 2(hx)h2 +x)hxx dx+ ++ 2b +� +Ω +hf 3(hx)h2 +xx dx = −2b +� +Ω +h−1h2 +xf(hx) dx − 2b +� +Ω +f 3(hx)hxx dx+ +2b +� +Ω +hf 3(hx)h2 +xx dx = −2b +� +Ω +h−1h2 +xf(hx) dx + 2b +� +Ω +hf 3(hx)h2 +xx dx. +Then from (3.9), it follows that +d +dt +� +Ω +(uvx + +c +2a +v2 +x +v ) dx + 2b +� +Ω +hf 3(hx)h2 +xx dx = +a +� +Ω +vu2 +x dx + 2b +� +Ω +h−1h2 +xf(hx) dx − +� +Ω +u vx +g(h) dx. +(3.10) + +Travelling waves of a control-volume fibre coating model +12 +Multiplying (3.10) by c +a and using (3.3), we arrive at +d +dt +� +Ω +( c +auvx + +c2 +2a2 +v2 +x +v ) dx + d +dtE(t) + 2bc +a +� +Ω +hf 3(hx)h2 +xx dx+ +� +Ω +u2v +g(h) dx = 2bc +a +� +Ω +h−1h2 +xf(hx) dx − c +a +� +Ω +u vx +g(h) dx + +� +Ω +uv dx. +(3.11) +Note that +� +Ω +( c +auvx + +c2 +2a2 +v2 +x +v ) dx + 1 +2 +� +Ω +vu2 dx = 1 +2 +� +Ω +v(u + c +a +vx +v )2 dx, +c +a +� +Ω +u vx +g(h) dx = c +a +� +Ω +(uv)x +g(h) dx − c +a +� +Ω +v +g(h)ux dx = +− c +a2 +� +Ω +vt +g(h) dx − c +a +� +Ω +v +g(h)ux dx = − c +a2 +d +dt +� +Ω +log(v) dx if g(h) = v. +Therefore, for the plug flow model with g(h) = v, (3.11) has the form +1 +2 +d +dt +� +Ω +� +v(u + c +a +vx +v )2 + 4b +a hΦ(hx) − 2c +a2 log(v) +� +dx+ +2bc +a +� +Ω +hf 3(hx)h2 +xx dx + +� +Ω +u2 dx + 2bc +a +� +Ω +h−1f(hx) dx = +2bc +a +� +Ω +h−1Φ(hx) dx + +� +Ω +uv dx +(3.12) +If we set g(h) = I(h) +v +for the laminar flow model, then by using +� +Ω +u vx +g(h) dx = − 2 +a +� +Ω +hht +g(h) dx − +� +Ω +v2 +I(h)ux dx = +− 2 +a +d +dt +� +Ω +G(h) dx − +� +Ω +v2 +I(h)ux dx, where G′(h) = +h +g(h), + +Travelling waves of a control-volume fibre coating model +13 +from (3.10) we get +d +dt +� +Ω +(uvx + +c +2a +v2 +x +v ) dx + 2b +� +Ω +hf 3(hx)h2 +xx dx ⩽ (a + ϵ0) +� +Ω +vu2 +x dx+ +2b +� +Ω +h−1h2 +xf(hx) dx + 2 +a +d +dt +� +Ω +G(h) dx + +1 +4ϵ0 +� +Ω +v3 +I2(h) dx, +(3.13) +where ϵ0 > 0. Multiplying (3.13) by +c +a+ϵ0 and using (3.3), we arrive at +1 +2 +d +dt +� +Ω +� +v(u + +c +a+ϵ0 +vx +v )2 + +c2ϵ0 +a(a+ϵ0)2 +v2 +x +v + 4b +a hΦ(hx) − +4c +a(a+ϵ0)G(h) +� +dx+ +2bc +a+ϵ0 +� +Ω +hf 3(hx)h2 +xx dx + +� +Ω +u2v2 +I(h) dx + +2bc +a+ϵ0 +� +Ω +h−1f(hx) dx ⩽ +2bc +a+ϵ0 +� +Ω +h−1Φ(hx) dx + +� +Ω +uv dx + +c +4ϵ0(a+ϵ0) +� +Ω +v3 +I2(h) dx. +(3.14) +Note that v − log(v) ⩾ 1 and 8v − G(h) ⩾ 16 for all v ⩾ 0 (as +v +I(h) → 8 +when h → 1 and +v +I(h) → 0 when h → +∞ ). Then, due to (2.7), (3.12) +and (3.14) can be rewritten in the form +d +dtS1(u, h) + 2bc +a +� +Ω +hf 3(hx)h2 +xx dx + +� +Ω +u2 dx + 2bc +a +� +Ω +h−1f(hx) dx = +2bc +a +� +Ω +h−1Φ(hx) dx + +� +Ω +uv dx if g(h) = v, +(3.15) +d +dtS2(u, h) + +2bc +a+ϵ0 +� +Ω +hf 3(hx)h2 +xx dx + +� +Ω +u2v2 +I(h) dx+ +2bc +a+ϵ0 +� +Ω +h−1f(hx) dx ⩽ +2bc +a+ϵ0 +� +Ω +h−1Φ(hx) dx + +� +Ω +uv dx+ +c +4ϵ0(a+ϵ0) +� +Ω +v3 +I2(h) dx if g(h) = I(h) +v . +(3.16) + +Travelling waves of a control-volume fibre coating model +14 +Taking into account (3.4), +� +Ω +h−1Φ(hx) dx ⩽ +� +Ω +hΦ(hx) dx for v ⩾ 0, +� +Ω +v3 +I2(h) dx ⩽ 64M for v ⩾ 0 +and (3.3), from (3.15) and (3.16) we obtain (3.5) and (3.6). +3.3. Approximate problem +For given δ > 0, η > 0 and ε > 0, we consider the following +approximate system +(v u)t + a [(uv + ηv4vxxx)u]x + b v κx = c (v ux)x + v +� +1 − +u +g(h) +� +− δuxxxx + ε a[p(v)]x − ε a vvxxxxx, +(3.17) +vt + a(u v)x = −a η +� +v4vxxx +� +x , +(3.18) +u and v are |Ω| − periodic, +(3.19) +u(x, 0) = uεη,0(x), v(x, 0) = vεη,0(x), +(3.20) +where uεη,0 ∈ H1(Ω) and 0 < vεη,0 ∈ H2(Ω) such that +uε,0(x) → u0(x) strongly in L2(Ω), vε,0(x) ⩾ v0(x) + εθ, θ ∈ (0, 1 +2), +vε,0(x) → v0(x) strongly in W 1 +1 (Ω) ∩ C(¯Ω), +ε +1 +2vε,0xx(x) → 0 strongly in L2(Ω), +and +p(z) = 1 +2z−2, g(h) = v or g(h) = |I(h)| +v . + +Travelling waves of a control-volume fibre coating model +15 +Lemma 3.3. Let u ∈ L2(0, T; H2(Ω)) be a periodic function. For any +0 < vηε,0(x) ∈ H2(Ω), the problem (3.18)—(3.19) has a unique weak +positive solution v ∈ C +3 +2 , 3 +8( ¯QT) such that +vt ∈ L2(QT), v ∈ L∞(0, T; H2(Ω)), v ∈ L2(0, T; H4(Ω)), +(3.21) +� +Ω +v dx = +� +Ω +vηε,0 dx =: Mηε > 0, and v satisfies (3.18) a. e. in QT. +Moreover, there exists a constant C > 1 depending on η such that +1 +C ⩽ v ⩽ C. +Proof of Lemma 3.3. The main line of proof follows the approach in +[1]. We omit details and restrict the discussion only to the key elements +of the proof. +First of all, we approximate (3.18) by +vt + a([u]αv)x = −a η((v4 + β)vxxx)x, +(3.22) +where β > 0 and [u]α denotes a smooth approximation of u such that +[u]α → u strongly in L2(0, T; H2(Ω)) as α → 0. +We also approximate vηε,0 in the H2-norm by C4+γ functions vβηε,0, +satisfying (3.18), and replace (3.20) by +v(x, 0) = vβεη,0(x). +(3.23) +Using the parabolic Schauder estimates from [31], one can generalise +[9, Theorem 6.3, p. 302] and show that the problem (3.22)–(3.23) has a +unique classical solution vβα ∈ C +4+γ,1+ γ +4 +x,t +(Ω×[0, τβα]) for some τβα > 0. +Next, for simplicity, we will write v instead vβα. Multiplying (3.22) + +Travelling waves of a control-volume fibre coating model +16 +by −vxx and integrating by parts, we have +1 +2 +d +dt +� +Ω +v2 +x dx + ηa +� +Ω +(v4 + β)v2 +xxx dx = +a +� +Ω +([u]αv)xvxx dx = −a +� +Ω +([u]α)xxvvx dx − 3a +2 +� +Ω +([u]α)xv2 +x dx ⩽ +a sup +Ω +|v| +�� +Ω +([u]α)2 +xx dx +� 1 +2�� +Ω +v2 +x dx +� 1 +2 + 3a +2 sup +Ω +|([u]α)x| +� +Ω +v2 +x dx ⩽ +�� +Ω +([u]α)2 +xx dx +� 1 +2� +5a|Ω| +1 +2 +2 +� +Ω +v2 +x dx + Mβηε +|Ω| +�� +Ω +v2 +x dx +� 1 +2� +, +whence +� +Ω +v2 +x dx + ηa +�� +QT +(v4 + β)v2 +xxx dxdt ⩽ +� +∥vβηε,0x∥2+ Mβηε +|Ω| +T +� +0 +∥([u]α)xx∥2e +− 5a|Ω| +1 +2 +2 +t� +0 +∥([u]α)xx∥2 ds +dt +�2 +e +5a|Ω| +1 +2 +T� +0 +∥([u]α)xx∥2 dt +. +(3.24) +Due to (3.24), we deduce that ∥vβ∥C +1 +2 , 1 +8 ( ¯QT ) is uniformly bounded with +respect to α, β and τβα. For any fixed values of β and α, by [9, Theorem +9.3, p. 316], we can to extend the solution vβ step-by-step to all of QT +for any T > 0. +Let us denote by Gβ(z) ⩾ 0 such that +Gβ(z) = +z +� +1 +y +� +1 +dsdy +s4+β, +G′′ +β(z) = +1 +z4+β. + +Travelling waves of a control-volume fibre coating model +17 +Multiplying (3.18) by G′ +β(v) and integrating by parts, we arrive at +d +dt +� +Ω +Gβ(v) dx + ηa +� +Ω +v2 +xx dx = −a +� +Ω +([u]α)x(vG′ +β(v) − Gβ(v)) dx ⩽ +aC sup +Ω +|([u]α)x| +� +Ω +Gβ(v) dx, +whence +� +Ω +Gβ(v) dx + ηa +�� +QT +v2 +xx dxdt ⩽ e +aC +T� +0 +sup +Ω +|([u]α)x| dt � +Ω +Gβ(vβηε,0) dx. +(3.25) +Due to (3.24) and (3.25), similar to the proof of [1, Theorem 4.1, p.190], +after taking β → 0, we obtain the global existence of a unique positivity +classical solution v0α for any α > 0. Moreover, +1 +C ⩽ v0α ⩽ C < ∞, +where C > 1 is independent of α. +For the limit process α → 0, we need the following a priori estimate. +Multiplying (3.22) with β = 0 by vxxxx and integrating by parts, we + +Travelling waves of a control-volume fibre coating model +18 +have +1 +2 +d +dt +� +Ω +v2 +xx dx + ηa +� +Ω +v4v2 +xxxx dx = −a +� +Ω +([u]αv)xvxxxx dx− +4ηa +� +Ω +v3vxvxxxvxxxx dx ⩽ a +�� +Ω +v4v2 +xxxx dx +� 1 +2�� +Ω +([u]αv)2 +x +v4 +dx +� 1 +2+ +4ηa +�� +Ω +v4v2 +xxxx dx +� 1 +2�� +Ω +v2v2 +xv2 +xxx dx +� 1 +2 ⩽ +ηa +2 +� +Ω +v4v2 +xxxx dx + a +η +� +Ω +([u]αv)2 +x +v4 +dx + 16aη +� +Ω +v2v2 +xv2 +xxx dx ⩽ +ηa +2 +� +Ω +v4v2 +xxxx dx + 2a +η +� +∥v−1∥2 +∞∥([u]α)x∥2 +2 + ∥v−1∥4 +∞∥vx∥2 +∞∥[u]α∥2 +2 +� ++ +16aη∥v−1∥2 +∞∥vx∥2 +∞ +� +Ω +v4v2 +xxx dx, +whence +d +dt∥vxx∥2 +2 + ηa +� +Ω +v4v2 +xxxx dx ⩽ aC η−1∥[u]α∥2 +H1+ +aC +� +η−1∥[u]α∥2 +H1 + η +� +Ω +v4v2 +xxx dx +� +∥vxx∥2 +2. +Integrating this inequality in time, we have +� +Ω +v2 +xx dx + ηa +�� +QT +v4v2 +xxxx dxdt ⩽ +� +∥vηε,0xx∥2 +2+ +aC η−1 +T +� +0 +∥[u]α∥2 +H1e +−aC +t� +0 +� +η−1∥[u]α∥2 +H1+η +� +Ω +v4v2 +xxx dx +� +ds +dt +� +× +e +aC +t� +0 +� +η−1∥[u]α∥2 +H1+η +� +Ω +v4v2 +xxx dx +� +ds +. +(3.26) + +Travelling waves of a control-volume fibre coating model +19 +By (3.26) and (3.24), ∥v0α∥C +3 +2 , 3 +8 ( ¯QT ) is uniformly bounded with respect +to α. This uniform bound follows from v0α ∈ L∞(0, T; H2(Ω)) and +v0α,t ∈ L2(QT) (see [33, Lemma 7.19, p. 175]. +Taking α → 0, it +completes the proof. +For the given δ > 0, η > 0 and ε > 0, equation (3.17) is uniformly +parabolic with respect to u for any v is from Lemma 3.3. By using +Faedo-Galerkin approximation (see, e. g., [4]), the system (3.17)–(3.18) +with periodic boundary conditions has a local in time weak solution +(v, u) := (vδηε, uδηε). +Next, we establish a priori estimates which +guarantee the global in time solvability. +Lemma 3.4 (a priori estimates). For fixed and positive constants +δ > 0, η > 0, ε > 0, and T > 0, let (vδηε, uδηε) be the solution to +the problem (3.17)–(3.20) in the following sense +�� +QT +uvψt dxdt+ +� +Ω +uεη,0vεη,0ψ(x, 0) dx + a +�� +QT +(uv + ηv4vxxx)uψx dxdt+ +b +�� +QT +κvxψ dxdt + b +�� +QT +κvψx dxdt − c +�� +QT +vuxψx dxdt+ +�� +QT +� +v − +uv +g(h) +� +ψ dxdt − δ +�� +QT +uxxψxx dxdt − εa +�� +QT +p(v)ψx dxdt− +εa +�� +QT +vxxx(vxxψ + 2vxψx + vψxx) dxdt = 0, +(3.27) + +Travelling waves of a control-volume fibre coating model +20 +�� +QT +vφt dxdt + +� +Ω +vεη,0φ(x, 0) dx+ +a +�� +QT +uvφx dxdt + aη +�� +QT +v4vxxxφx dxdt = 0 +(3.28) +for all φ ∈ C∞ +c ( ¯QT) and ψ ∈ C∞ +c ( ¯QT) such that φ(x, T) = ψ(x, T) = 0. +Moreover, there exists a positive constant C > 0 depending only +on a, b, c, T, E(0), and Si(u0, v0) such that the following terms are +bounded by C in respective norms +√v ∈ L∞(0, T; H1(Ω)), √vu ∈ L∞(0, T; L2(Ω)), +(3.29) +− log(v), hΦ(hx) ∈ L∞(0, T; L1(Ω)), h−1f(hx) ∈ L1(QT), +(3.30) +� +hf 3(hx)hxx, √vux, +� +v +g(h)u ∈ L2(QT), +(3.31) +ε +1 +2v−1, ε +1 +2vxx ∈ L∞(0, T; L2(Ω)), +(3.32) +δ +1 +2uxx, (εη) +1 +2v2vxxxx, ε +1 +2vxxx, ε +1 +2(v−1)x ∈ L2(QT), +(3.33) +and +0 < ε ⩽ v(x, t) ⩽ C for all (x, t) ∈ QT. +(3.34) +Proof of Lemma 3.4. Let us denote by +Eε(t) := 1 +2 +� +Ω +(v u2 + 2b +a hΦ(hx) + ε +3v−2 + εv2 +xx) dx. +Multiplying (3.17) by u and integrating over Ω, we have +d +dtEε(t) + δ +� +Ω +u2 +xx dx + εηa +� +Ω +v4v2 +xxxx dx+ +εηa +� +Ω +v2 +xx dx + c +� +Ω +vu2 +x dx + +� +Ω +u2v +gε(h) dx = +� +Ω +uv dx− +4εηa +� +Ω +v3vxvxxxvxxxx dx − bη +� +Ω +(v4vxxx)xκ dx. +(3.35) + +Travelling waves of a control-volume fibre coating model +21 +Note that +� +Ω +uv dx ⩽ M +1 +2 +�� +Ω +vu2 dx +� 1 +2. +We will use the following estimates +∥v∥∞ ⩽ C ∥vxx∥2+ M +|Ω|, ∥vx∥∞ ⩽ C ∥vxx∥2, ∥vxxx∥2 ⩽ C ∥vxxxx∥ +1 +2 +2 ∥vxx∥ +1 +2 +2 , +∥v−1∥∞ ⩽ C +� +∥v−1∥ +3 +2 +2 (∥vxx∥2 +2 + ν) +1 +4 + (∥vxx∥2 +2 + ν)− 1 +2� +∀ ν ⩾ 0. (3.36) +For the proof of (3.36), see [6, Lemma 3.2, p.807]. From (3.36), we +deduce that +� +Ω +v3vxvxxxvxxxx dx ⩽ ∥v2vxxxx∥2∥vvxvxxx∥2 ⩽ +∥v2vxxxx∥2∥v∥∞∥vx∥∞∥vxxx∥2 ⩽ +C ∥v2vxxxx∥2∥v∥∞∥vx∥∞∥vxxxx∥ +1 +2 +2 ∥vxx∥ +1 +2 +2 ⩽ +C ∥v2vxxxx∥ +3 +2 +2 ∥v−1∥∞ +� M +|Ω| + ∥vxx∥2 +� +∥vxx∥ +3 +2 +2 ⩽ +C ∥v2vxxxx∥ +3 +2 +2 +� +∥v−1∥ +3 +2 +2 (∥vxx∥2 +2 + 1) +1 +4 + 1 +�� M +|Ω| + ∥vxx∥2 +� +∥vxx∥ +3 +2 +2 , +� +Ω +(v4vxxx)xκ dx = +� +Ω +v4vxxxxκ dx + 4 +� +Ω +v3vxvxxxκ dx ⩽ +ε +4 +� +Ω +v4v2 +xxxx dx + 2 +ε +� +Ω +v4κ2 dx + C +ε ∥v−1∥4 +∞ +� +Ω +v6v2 +xκ2 dx ⩽ +ε +4 +� +Ω +v4v2 +xxxx dx + C +ε +� +Ω +(v3 + v3v2 +xx) dx + C +ε ∥v−1∥4 +∞ +� +Ω +(v6 + v6v2 +xx) dx ⩽ +ε +4 +� +Ω +v4v2 +xxxx dx+ C +ε (1+∥vxx∥2 +2) +� +( M +|Ω|)3+∥vxx∥3 +2+∥v−1∥4 +∞(( M +|Ω|)6+∥vxx∥6 +2) +� +. + +Travelling waves of a control-volume fibre coating model +22 +By these estimates, from (3.35) we have +d +dtEε(t)+δ +� +Ω +u2 +xx dx+ εηa +2 +� +Ω +v4v2 +xxxx dx+εηa +� +Ω +v2 +xx dx+c +� +Ω +vu2 +x dx+ +� +Ω +u2v +gε(h) dx ⩽ C E +1 +2ε (t) + C bηε−9E8 +ε(t) + C aηε−8E9 +ε(t), +(3.37) +whence +Eε(T) + δ +�� +QT +u2 +xx dxdt + εηa +2 +�� +QT +v4v2 +xxx dxdt+ +εηa +�� +QT +v2 +xx dxdt + c +�� +QT +vu2 +x dxdt + +�� +QT +u2v +g(h) dxdt ⩽ C(T) +(3.38) +for all T ⩽ Tη := [16CE8 +ε(0)ηε−8]−1 → +∞ as η → 0. In particular, +from (3.38), due to (3.36), we arrive at +∥v−1∥∞ ⩽ C ε−1, +(3.39) +whence inf +Ω v ⩾ C ε. +Next, we consider the plug flow case g(h) = v only, and the laminar +flow case can be addressed in a similar manner. Multiplying (3.17) by + +Travelling waves of a control-volume fibre coating model +23 +vx +v and integrating over Ω, we obtain that +d +dtS1,ε(u, v) + 2bc +a +� +Ω +hf 3(hx)h2 +xx dx + +� +Ω +u2 dx + 2bc +a +� +Ω +h−1f(hx) dx+ +δ +� +Ω +u2 +xx dx + εa +� +Ω +v2 +xxx dx + εa +3 +� +Ω +v2 +x +v4 dx = +2bc +a +� +Ω +h−1Φ(hx) dx + +� +Ω +uv dx − δ +� +Ω +� vx +v +� +xxuxx dx+ +η c +� +Ω +vux(v3vxxx)x dx + η c2 +a +� +Ω +� vx +v +� +xv(v3vxxx)x dx+ +η c +a +� +Ω +(v4vxxx)x +g(h) +dx, +(3.40) +where +S1,ε(u, v) := 1 +2 +� +Ω +� +v(u + c +a +vx +v )2 + 4b +a hΦ(hx) + 2c +a2(v − log(v))+ +2εv−2 + εv2 +xx +� +dx. +Using the equality +� vx +v +� +xx = vxxx +v +− 3 vxvxx +v2 ++ 2 v3 +x +v3 , +(3.41) +and the following estimates +∥vx∥6 ⩽ C∥vxx∥ +1 +3 +2 ∥vx∥ +2 +3 +2 , ∥vxx∥∞ ⩽ C∥vxxx∥2, + +Travelling waves of a control-volume fibre coating model +24 +we find that +δ +� +Ω +� vx +v +� +xxuxx dx ⩽ δ +1 +2∥δ +1 +2uxx∥2 +� +∥vxxx +v ∥2 + 3∥vxvxx +v2 ∥2 + 2∥vx +v ∥3 +6 +� +⩽ +δ +1 +2∥δ +1 +2uxx∥2 +� +∥v−1∥∞∥vxxx∥2+3∥v−1∥2 +∞∥vx∥2∥vxx∥∞+2∥v−1∥3 +∞∥vx∥3 +6 +� +⩽ +C δ +1 +2∥δ +1 +2uxx∥2 +� +∥v−1∥∞∥vxxx∥2 + ∥v−1∥2 +∞∥vx∥2∥vxxx∥2+ +∥v−1∥3 +∞∥vx∥2 +2∥vxxx∥2 +� +⩽ C δ +1 +2ε−4∥δ +1 +2uxx∥2∥vxxx∥2 ⩽ +C δ +1 +2ε−4∥δ +1 +2uxx∥2∥vxxxx∥2 ⩽ C δ +1 +2ε− 13 +2 η− 1 +2∥δ +1 +2uxx∥2∥(εη) +1 +2v2vxxxx∥2. +As a result, due to (3.38), we find that +δ +�� +QT +� vx +v +� +xxuxx dxdt ⩽ C δ +1 +2η− 1 +2ε− 13 +2 . +(3.42) +Using (3.41) and the estimates +� +Ω +vv2 +xv2 +xx dx ⩽ 1 +3 +�� +Ω +v3v2 +xxx dx +� 1 +2�� +Ω +v6 +x +v dx +� 1 +2, ∥vx∥6 ⩽ C∥vxxx∥ +1 +6 +2 ∥vx∥ +5 +6 +2 , + +Travelling waves of a control-volume fibre coating model +25 +we find that +� +Ω +� vx +v +� +xv(v3vxxx)x dx = − +� +Ω +� vx +v +� +xxv4vxxx dx− +� +Ω +� vx +v +� +xv3vxvxxx dx = +− +� +Ω +v3v2 +xxx dx + 2 +� +Ω +v2vxvxxvxxx dx − +� +Ω +vv3 +xvxxx dx ⩽ +− +� +Ω +v3v2 +xxx dx + +2 +√ +3 +�� +Ω +v3v2 +xxx dx +� 3 +4�� +Ω +v6 +x +v dx +� 1 +4+ +�� +Ω +v3v2 +xxx dx +� 1 +2�� +Ω +v6 +x +v dx +� 1 +2 ⩽ − 1 +2 +� +Ω +v3v2 +xxx dx + C∥v−1∥∞∥vx∥6 +6 ⩽ +− 1 +2 +� +Ω +v3v2 +xxx dx + C∥v−1∥∞∥vx∥5 +2∥vxxx∥2 ⩽ +− 1 +2 +� +Ω +v3v2 +xxx dx + C∥v−1∥ +5 +2∞∥vx∥5 +2∥v +3 +2vxxx∥2 ⩽ +− 1 +4 +� +Ω +v3v2 +xxx dx + C∥v−1∥5 +∞∥vxx∥10 +2 ⩽ − 1 +4 +� +Ω +v3v2 +xxx dx + C ε−10. +As a result, due to (3.38), we arrive at +η +�� +QT +� vx +v +� +xv(v3vxxx)x dxdt ⩽ − η +4 +�� +QT +v3v2 +xxx dxdt + C η ε−10. +(3.43) +Using the estimate +� +Ω +v(v3vxxx)2 +x dx ⩽ C +� +Ω +v7v2 +xxxx dx + C +� +Ω +v5v2 +xv2 +xxx dx ⩽ +C∥v∥3 +∞ +� +Ω +v4v2 +xxxx dx + C∥v∥5 +∞∥vx∥2 +∞ +� +Ω +v2 +xxx dx ⩽ +C +� +∥v∥3 +∞ + ∥v∥5 +∞∥vx∥2 +∞∥v−1∥4 +∞ +� � +Ω +v4v2 +xxxx dx ⩽ C ε− 15 +2 +� +Ω +v4v2 +xxxx dx, + +Travelling waves of a control-volume fibre coating model +26 +we deduce that +η c +�� +QT +vux(v3vxxx)x dxdt ⩽ +η c +��� +QT +vu2 +x dxdt +� 1 +2��� +QT +v(v3vxxx)2 +x dxdt +� 1 +2 ⩽ +η c +��� +QT +vu2 +x dxdt +� 1 +2� +C ε− 15 +2 +�� +QT +v4v2 +xxxx dxdt +� 1 +2, +whence, due to (3.38), we have +η c +�� +QT +vux(v3vxxx)x dxdt ⩽ C η +1 +2ε− 17 +4 . +(3.44) +Using the estimate +� +Ω +(v4vxxx)x +v +dx = +� +Ω +v3vxxxx dx + 4 +� +Ω +v2vxvxxx dx ⩽ +∥v2vxxxx∥2(∥v∥2 + C ∥v−1∥2 +∞∥v∥2 +∞∥vx∥2) ⩽ +C ε− 7 +2∥v2vxxxx∥2 = C η− 1 +2ε−4∥(εη) +1 +2v2vxxxx∥2, +we get +η c +a +�� +QT +(v4vxxx)x +g(h) +dxdt ⩽ C η +1 +2ε−4. +(3.45) +Integrating (3.40) in time, taking into account (3.38) and (3.42)–(3.45), + +Travelling waves of a control-volume fibre coating model +27 +we obtain +S1,ε(u, v) + 2bc +a +�� +QT +hf 3(hx)h2 +xx dxdt + +�� +QT +u2 dxdt+ +2bc +a +�� +QT +h−1f(hx) dxdt + δ +�� +QT +u2 +xx dxdt + εa +�� +QT +v2 +xxx dxdt+ +2εa +3 +�� +QT +v2 +x +v4 dxdt + ηc2 +4a +�� +QT +v3v2 +xxx dxdt ⩽ S1,ε(uεη,0, vεη,0)+ +C(T) + C δ +1 +2η− 1 +2ε− 13 +2 + C η ε−10 + C η +1 +2ε− 17 +4 +(3.46) +for all T ⩽ Tη. +3.4. Compactness and limit processes +Passage to the limit δ → 0. Denote the corresponding solution to +the approximate problem (3.17)–(3.20) by (vδηε, uδηε). Let T ⩽ Tη. We +study the compactness properties of the sequence (vδηε, uδηε) by using +the estimates derived in Lemma 3.4. Due to (3.32) and (3.33), we have +{vδηε}δ>0 is bounded in L∞(0, T; H2(Ω)) and {vδηε,t}δ>0 is bounded in +L2(QT), whence, using [33, Lemma 7.19, p. 175], we arrive at {vδηε}δ>0 +is bounded in C +3 +2 , 3 +8( ¯QT). By the Arzela-Ascoli theorem, after possibly +extracting a subsequence, we obtain that +vδηε → +δ→0 vηε uniformly in C +3 +2 , 3 +8( ¯QT), +vδηε,t → +δ→0 vηε,t weakly in L2(QT), +whence +v−1 +δηε → +δ→0 v−1 +ηε uniformly in C +3 +2 , 3 +8( ¯QT). +Also, by (3.33) we obtain that +vδηε → +δ→0 vηε weakly in L2(0, T; H4(Ω)), + +Travelling waves of a control-volume fibre coating model +28 +vδηε → +δ→0 vηε strongly in L2(0, T; H3(Ω)). +We next turn to compactness properties of {uδηε}δ>0. By (3.29)–(3.34) +and the boundedness vδηε away from zero, we have {uδηε}δ>0 is bounded +in L∞(0, T; L2(Ω)) ∩ L2(0, T; H1(Ω)); {(vδηεuδηε)t}δ>0 and {uδηε,t}δ>0 +are bounded in L2(0, T; H−2(Ω)). So, +uδηε → +δ→0 uηε strongly in L2(QT), +uδηε,x → +δ→0 uηε,x weakly in L2(QT), +uδηε,t → +δ→0 uηε,t ∗ − weakly in L2(0, T; H−2(Ω)), +vδηεuδηε → +δ→0 vηεuηε strongly in L2(QT), +(vδηεuδηε)t → +δ→0(vηεuηε)t ∗ − weakly in L2(0, T; H−2(Ω)). +Moreover, by (3.33) +δ +��� +�� +QT +uxxψxx dxdt +��� ⩽ δ +1 +2∥δ +1 +2uxx∥L2(QT )∥ψ∥L2(0,T;H2(Ω)) ⩽ C δ +1 +2. +The obtained convergences allow to pass to the limit as δ → 0 in (3.27) +and (3.28). +Passage to the limit η → 0. +Since Tη → +∞ as η → 0 +then we can to take any T > 0. Now, we consider the compactness +properties of the sequence (vηε, uηε) by using the estimates derived in +Lemma 3.4. Due to (3.32) and (3.33), we have {vηε}η>0 is bounded +in L∞(0, T; H2(Ω)) and {vηε,t}η>0 is bounded in L2(0, T; H−1(Ω)), +whence, similar to [1, Lemma 2.1, p. 183], we arrive at {vηε}η>0 is +bounded in C +3 +2 , 1 +4( ¯QT). By the Arzela-Ascoli theorem, after possibly +extracting a subsequence, we obtain that +vηε → +η→0 vε uniformly in C +3 +2 , 1 +4( ¯QT), + +Travelling waves of a control-volume fibre coating model +29 +vηε,t → +η→0 vε,t ∗ − weakly in L2(0, T; H−1(Ω)), +whence +v−1 +ηε → +η→0 v−1 +ε +uniformly in C +3 +2 , 1 +4( ¯QT). +Also, by (3.33) we obtain that +vηε → +η→0 vε weakly in L2(0, T; H3(Ω)), +vηε → +η→0 vε strongly in L2(0, T; H2(Ω)). +Next, we turn to compactness properties of {uηε}η>0. By (3.29)–(3.34) +and the boundedness vηε away from zero, we have {uηε}η>0 is bounded +in L∞(0, T; L2(Ω))∩L2(0, T; H1(Ω)), whence, in particular, {vηεu2 +ηε}η>0 +is bounded in Lp(QT) for p ∈ (1, 3); {(vηεuηε)t}η>0 and {uηε,t}ε>0 are +bounded in L2(0, T; H−2(Ω)). So, +uηε → +η→0 uε strongly in L2(QT), +uηε,x → +η→0 uε,x weakly in L2(QT), +uηε,t → +η→0 uε,t ∗ − weakly in L2(0, T; H−2(Ω)), +vηεuηε → +η→0 vεuε strongly in L2(QT), +vηεu2 +ηε → +η→0 vεu2 +ε strongly in L2(QT), +(vηεuηε)t → +η→0(vεuε)t ∗ − weakly in L2(0, T; H−2(Ω)). +Moreover, by (3.29) and (3.33) we get +η +�� +QT +u v4vxxxψx dxdt ⩽ η +T +� +0 +∥√vu∥2∥v∥ +7 +2∞∥vxxx∥2∥ψx∥∞ dt ⩽ +C ηε− 1 +2∥ε +1 +2vxxx∥L2(QT )∥ψ∥L2(0,T;H2(Ω)) ⩽ C ηε− 1 +2, + +Travelling waves of a control-volume fibre coating model +30 +η +�� +QT +v4vxxxφx dxdt ⩽ ηε− 1 +2∥v∥L∞(QT )∥ε +1 +2vxxx∥L2(QT )∥ψx∥L2(QT ) ⩽ +C ηε− 1 +2∥ψ∥L2(0,T;H1(Ω)) ⩽ C ηε− 1 +2. +The obtained convergences allow to pass to the limit as η → 0 in (3.27) +and (3.28) with δ = 0. +Passage to the limit ε → 0. Next, we study the compactness +properties of the sequence (vε, uε) by using the estimates derived in +Lemma 3.4. Taking into account +(√v)t = −a(√vu)x + a +2 +√vux, +by (3.29) and (3.31), we deduce that {(√vε)t}ε>0 is uniformly +bounded in L2(0, T; H−1(Ω)), and {√vε}ε>0 is uniformly bounded in +L∞(0, T; H1(Ω)). So, due to the lemma of compactness embedding +from [30, Corollary 4, p. 85], we obtain that +√vε → +ε→0 +√v uniformly in C +1 +2 ,0( ¯QT), +(3.47) +whence it follows that +vε → +ε→0 v uniformly in C( ¯QT). +(3.48) +Also, by (3.29) and (3.47), {uεvε}ε>0 is uniformly bounded in +L2(QT), whence we find that {vε,t}ε>0 is uniformly bounded in +L2(0, T; H−1(Ω)). It implies +vε,t → +ε→0 vt ∗ − weakly in L2(0, T; H−1(Ω)). +From the boundedness of {hεΦ(hε,x)}ε>0 in L∞(0, T; L1(Ω)) we deduce +that +{vε}ε>0 is uniformly bounded in L∞(0, T; W 1 +1 (Ω)), +(3.49) + +Travelling waves of a control-volume fibre coating model +31 +whence, due to (3.48), it follows +vε, hε → +ε→0 v, h ∗ − weakly in L∞(0, T; W 1 +1 (Ω)), +and the set {|hx(., t)| = ∞} has measure zero for any t > 0. +By +the boundedness {log(vε)}ε>0 in L∞(0, T; L1(Ω)) and (3.48), the set +{v(., t) = 0} has measure zero for any t > 0, it follows that +p(vε) → +ε→0 p(v) +holds for almost all x and for any t > 0, whence we arrive at +ε +��� +�� +QT +p(vε)ψx dxdt +��� ⩽ sup +t∈[0,T] +� +ε +� +Ω +p(vε) dx +� +T +� +0 +∥ψx∥∞ dt ⩽ +T +1 +2 sup +t∈[0,T] +� +ε +� +Ω +p(vε) dx +� +∥ψ∥L2(0,T;H2(Ω)) → +ε→0 0. +Using the following estimates +|κε| = +��h−1 +ε f(hε,x)− +� +f3(hε,x) +hε +� +hεf 3(hε,x)hε,xx +�� ⩽ 1+ +� +hεf 3(hε,x)|hε,xx|, +|κεvε,x| = 2 +��hε,xf(hε,x) − hε,x +� +hεf 3(hε,x) +� +hεf 3(hε,x)hε,xx +�� ⩽ +2 + 2∥ +� +hε∥∞ +� +hεf 3(hε,x)|hε,xx|, +due to (3.48) and (3.31), we have {κε}ε>0 and {κεvε,x}ε>0 are uniformly +bounded in L2(QT). In particular, by (3.31), we get {χ{|hx|<∞}hε}ε>0 +is uniformly bounded in L2(0, T; H2(Ω)), whence +hε → +ε→0 χ{|hx|<∞}h weakly in L2(0, T; H2(Ω)). +(3.50) +By (3.48) and (3.50), we arrive at +�� +QT +κεvεψx dxdt → +ε→0 +�� +QT +χ{|hx|<∞}(h−1f(hx) − f 3(hx)hxx)vψx dxdt, + +Travelling waves of a control-volume fibre coating model +32 +�� +QT +κεvε,xψ dxdt → +ε→0 +�� +QT +χ{|hx|<∞}(h−1f(hx) − f 3(hx)hxx)vxψ dxdt. +By (3.33), (3.49), and +∥vxx∥2 ⩽ C∥vxxx∥ +3 +5 +2 ∥vx∥ +2 +5 +1 , +∥vx∥2 ⩽ C∥vxxx∥ +1 +5 +2 ∥vx∥ +4 +5 +1 , +we have +ε +��� +�� +QT +vxxx(vxxψ + 2vxψx + vψxx) dxdt +��� ⩽ ε +T +� +0 +∥vxxx∥2∥vxx∥2∥ψ∥∞ dt+ +2ε +T +� +0 +∥vxxx∥2∥vx∥2∥ψx∥∞ dt + ε +T +� +0 +∥vxxx∥2∥ψxx∥2∥v∥L∞(QT ) dt ⩽ +C ε +1 +5∥ε +1 +2vxxx∥ +8 +5 +L2(QT )∥vx∥ +2 +5 +L∞(0,T;L1(Ω))∥ψ∥L5(0,T;L∞(Ω))+ +C ε +2 +5∥ε +1 +2vxxx∥ +6 +5 +L2(QT )∥vx∥ +4 +5 +L∞(0,T;L1(Ω))∥ψx∥L +5 +2 (0,T;L∞(Ω))+ +ε +1 +2∥ε +1 +2vxxx∥L2(QT )∥ψ∥L2(0,T;H2(Ω)) ⩽ C ε +1 +5. +By (3.31), {uε}ε>0 is uniformly bounded in L2(QT), and {χ{v>0}uε}ε>0 +is uniformly bounded in L2(0, T; H1(Ω)), whence +uε → +ε→0 u weakly in L2(QT), +(3.51) +vεuε,x → +ε→0 χ{v>0}vux weakly in L2(QT), +�� +QT +vεu2 +εψx dxdt → +ε→0 +�� +QT +χ{v>0}vu2ψx dxdt. +By (3.48) and (3.51), we get +uεvε → +ε→0 uv weakly in L2(QT). +The obtained convergences allow to pass to the limit as ε → 0 in (3.27) +and (3.28) with δ = η = 0. As a result, we obtain a weak solution (v, u) +in the sense of Definition 2.1. + +Travelling waves of a control-volume fibre coating model +33 +4. Travelling wave solutions +Next, we focus on the travelling wave solutions to the control-volume +model. Specifically, we look for a solution to (2.5)– (2.6) in the form: +u(x, t) = U(ξ), +v(x, t) = V (ξ) = H2(ξ) − 1, where ξ = x − s t, +where s is the propagation speed. Substituting the ansatz into (2.5)– +(2.6), we obtain the system of travelling wave ODEs for (U(ξ), V (ξ)) +for 0 ≤ ξ ≤ L, +− s U ′ + aU U ′ + b κ′ = c (V U ′)′ +V ++ 1 − +U +g(H), +(4.1) +− s V ′ + a(U V )′ = 0 +(4.2) +subject to the L-periodic boundary conditions. We also impose the +following mass constraint +L +� +0 +H2(ξ) dξ = M > 0, +where M is related to the mass M defined in (2.7) by M = M + L. +To study the structure of travelling wave solutions, we first consider +the case when the film profile touches down to zero at the boundary. +That is, we assume that +V (0) = V (L) = 0, +or, equivalently, H(0) = H(L) = 1. +(4.3) +From (4.2) it follows that +− V (s − a U) = C0, +(4.4) +whence by (4.3) we obtain C0 = 0, and U(ξ) becomes a trivial solution +U ≡ Uc := s +a. +(4.5) + +Travelling waves of a control-volume fibre coating model +34 +Lemma 4.1. There exists s > 0 such that the problem (4.1)–(4.2) has +a periodic solution (H, U) such that +H(0) = H(L) = 1, H′(0) = H′(L) = 0, +where the average fluid film radius +¯M := M +L = +L� +0 +H2(y) +g(H(y)) dy +L� +0 +dy +g(H(y)) +. +(4.6) +Remark 4.1. If g(H) = H2−1 (the plug flow case), then (4.6) implies +¯M = s +a + 1. +Proof of Lemma 4.1. Since U is a trivial solution satisfying (4.5), the +ODE (4.1) reduces to +b κ′ = 1 − +Uc +g(H) ⇔ b +� +f(H′)H−1 − f 3(H′)H′′�′ = 1 − +Uc +g(H), +whence +f 3(H′)H′′ − f(H′)H−1 = F(ξ) := b−1 +ξ +� +0 +� +Uc +g(H(y)) − 1 +� +dy. +(4.7) +By periodicity, we find that +F(0) = F(L) = 0 ⇒ Uc = +� +1 +L +L +� +0 +dy +g(H(y)) +�−1 +. +Multiplying (4.7) by H H′ ̸= 0, we deduce that +[Hf(H′)]′ = −HH′F(ξ). +(4.8) +We will look for the first integral to (4.8) in the form: +f(H′) = A(ξ)H + B(ξ)H−1, +(4.9) + +Travelling waves of a control-volume fibre coating model +35 +where A(0) = A(L) and B(0) = B(L). Substituting (4.9) into (4.8), +we find that +A(ξ) = − 1 +2F(ξ), +B(ξ) = B(0) + 1 +2 +ξ +� +0 +F ′(y)H2(y) dy. +By B(0) = B(L), we get +M = Uc +L +� +0 +H2(y) +g(H(y))dy = +� +1 +L +L +� +0 +dy +g(H(y)) +�−1 +L +� +0 +H2(y) +g(H(y))dy ⇔ M +L = +L� +0 +H2(y) +g(H(y)) dy +L� +0 +dy +g(H(y)) +. +As a result, by (4.9) we arrive at +[H′]2 = 1 − [A(ξ)H + B(ξ)H−1]2 +[A(ξ)H + B(ξ)H−1]2 +. +Furthermore, if we have H(0) = H(L) = |B(0)| = 1, then we have +H′(0) = H′(L) = 0. +Next, we consider a general travelling wave solution that satisfies +the periodic boundary condition +V (0) = V (L). +(4.10) +In this case, the relation (4.4) implies that +U = Uc + C0 +a V +∀ C0 ∈ R1, where Uc := s +a. +(4.11) +Lemma 4.2. There exist s and C0 such that the problem (4.1)–(4.2) +has at least one periodic solution (H, U) satisfying +H(0) = H(L), H′(0) = H′(L) = 0. +Proof of Lemma 4.2. From (4.1) it follows that +� +b κ + a +2(U − Uc)2�′ = c (V U′)′ +V ++ 1 − +U +g(H), + +Travelling waves of a control-volume fibre coating model +36 +whence, due to (4.11), we obtain +� +b κ + C2 +0 +2a V −2�′ += − c C0 +a +1 +V +� V ′ +V +�′ + 1 − +Uc +g(H) − +C0 +a V g(H). +(4.12) +Let C0 ̸= 0. Then we have +� +Hf(H′) + C2 +0 +4abV −1�′ += −HH′G(ξ), +(4.13) +where +G(ξ) := b−1 +ξ +� +0 +Uc +g(H(y)) + +C0 +a V (y)g(H(y)) − 1 + c C0 +a +1 +V +� V ′ +V +�′ dy. +By imposing the periodicity G(0) = G(L) = 0, we obtain +Uc +L +� +0 +dy +g(H(y)) + C0 +a +L +� +0 +� +1 +V (y)g(H(y)) + c V −3(y)V ′(y)2� +dy = L. +(4.14) +Integrating (4.13), we deduce that +f(H′) = A(ξ)H + B(ξ)H−1 − C2 +0 +4abH−1V −1, +(4.15) +where +A(ξ) = − 1 +2G(ξ), +B(ξ) = B(0) + 1 +2 +ξ +� +0 +G′(y)H2(y) dy +such that A(0) = A(L) and B(0) = B(L). From B(0) = B(L), it +follows that +Uc +L +� +0 +H2(y)dy +g(H(y)) + C0 +a +L +� +0 +� +H2(y) +V (y)g(H(y)) + c V −3(y)V ′(y)2� +dy = M. + +Travelling waves of a control-volume fibre coating model +37 +So, we find that +Uc = 1 +∆ +� +L +L +� +0 +H2(y)dy +V (y)g(H(y))−M +L +� +0 +dy +V (y)g(H(y))+c(L−M) +L +� +0 +V −3(y)V ′(y)2 dy +� +, +C0 +a = − 1 +∆ +� +L +L +� +0 +H2(y)dy +g(H(y)) − M +L +� +0 +dy +g(H(y)) +� +, +where +∆ = +� L +� +0 +dy +g(H(y)) +�� L +� +0 +H2(y)dy +V (y)g(H(y)) +� +− +� L +� +0 +H2(y)dy +g(H(y)) +�� L +� +0 +dy +V (y)g(H(y)) +� +− +c +� L +� +0 +V (y)dy +g(H(y)) +�� L +� +0 +V −3(y)V ′(y)2 dy +� +. +As a result, by (4.15) we arrive at +H′(ξ)2 = +1 − +� +A(ξ)H + B(ξ)H−1 − C2 +0 +4abH−1V −1�2 +� +A(ξ)H + B(ξ)H−1 − C2 +0 +4abH−1V −1 +�2 +. +Furthermore, if H′(0) = H′(L) = 0 and H(0) = H(L) > 1, then we +have +B(0)H−1(0) − C2 +0 +4abH−1(0)V −1(0) = 1 ⇒ +(H2(0) − 1)(H(0) − B(0)) = − C2 +0 +4ab. +This equation with respect to H(0) has two solutions provided that +B(0) > 1 and +C2 +0 +4ab < +1 +27 +� +(B(0) + +� +B2(0) + 3)2 − 9 +� +(2B(0) − +� +B2(0) + 3), +and one solution if +C2 +0 +4ab = +1 +27 +� +(B(0) + +� +B2(0) + 3)2 − 9 +� +(2B(0) − +� +B2(0) + 3). + +Travelling waves of a control-volume fibre coating model +38 +5. Numerical studies +In this section, we numerically investigate the coupled PDE system +(2.1) – (2.4) to explore the fibre coating dynamics and verify the +analytical results in previous sections. Following the work of Ruan et +al. [23], we specify the form of the function g(h) based on two models +- the plug flow model and the laminar flow model. For the plug flow +model, we set g(h) based on the form in (1.4). For the laminar flow +model, the function g(h) takes the form in equation (1.5). +Firstly, we numerically investigate the travelling wave solutions +(H(ξ), U(ξ)) that satisfy the coupled ODE system (4.1) - (4.2) with the +mass constraint (2.7), +� L +0 V (ξ) dξ = +� L +0 (H2(ξ)−1) dξ = M. We apply +Newton’s method to solve this nonlinear eigenvalue problem, where the +speed s is treated as an unknown variable. The coupled differential +equations are discretized for 0 ≤ ξ ≤ L with periodic boundary +conditions on H and U by second-order center finite differences. An +additional constraint H(ξ∗) = H∗ for some 0 ≤ ξ∗ ≤ L is imposed to +guarantee the local uniqueness of the solution. +Figure 1 presents typical travelling wave solutions (H(ξ), U(ξ)) +corresponding to two cases for the plug flow model and two cases for +the laminar flow model: +(a) Plug flow: a = 0.2, b = 10, c = 1 with travelling speed s = 1.396 +(b) Plug flow: a = 0.4, b = 12, c = 3 with travelling speed s = 2.517; +(c) laminar flow a = 1.5, b = 13, c = 4 with travelling speed s = 1.482; +(d) laminar flow a = 0.1, b = 11, c = 4 with travelling speed s = 0.1. +A fixed domain size L = 20 and mass constraint M = 84.8 are set +for all cases. +The profiles are shifted so that the maximum of the +droplet peaks are located at ξ = L/2. This comparison shows that in +a fixed domain with equal volumes, the travelling waves for the plug +flow model have more prominent peaks and higher velocity magnitude +than those obtained from the laminar flow model. + +Travelling waves of a control-volume fibre coating model +39 +1 +2 +3 +4 +0 +5 +10 +15 +20 +H +ξ +(a) +(b) +(c) +(d) +0 +2 +4 +6 +8 +0 +5 +10 +15 +20 +U +ξ +(a) +(b) +(c) +(d) +Figure 1. Typical travelling wave profiles (left) H(ξ) and (right) +U(ξ) for two plug flow cases ((a) and (b)) and two laminar flow +cases ((c) and (d)). +Next, we study the transient PDE solutions of the governing model +(2.1) - (2.4) and verify the derived energy and entropy estimates in +previous sections. To numerically solve the coupled fourth-order PDEs, +we use the Keller box method [16] to decompose the model into a +system of first-order differential equations, +k = hx, +p = kx, +w = ux, +ut + a +�u2 +2 +� +x ++ b +� +f(k)h−1 − f 3(k)p +� +x = c[(h2 − 1)w]x +h2 − 1 ++ 1 − +u +g(h), +2hht + a[u(h2 − 1)]x = 0. +(5.1) +Starting from the initial fluid film radius and the initial velocity +h(x, 0) = h0 + 0.1 sin(2πx/L), +u(x, 0) = g(h(x, 0)), +(5.2) +we then solve the system (5.1) using the fully implicit second-order +centered finite differences over the domain 0 +≤ +x +≤ +L, with +periodic boundary conditions imposed on both u and h. For all PDE +simulations, we keep the domain size L = 20 and h0 = 2.29, so that +the mass M = 84.8. + +Travelling waves of a control-volume fibre coating model +40 +1 +2 +3 +4 +0 +5 +10 +15 +20 +h +x − xmax + L/2 +h(x, 0) +h(x, t) +H(ξ) +0 +2 +4 +6 +8 +0 +5 +10 +15 +20 +u +x − xmax + L/2 +u(x, 0) +u(x, t) +U(ξ) +103 +105 +107 +0 +20 +40 +60 +80 +t +E(t) + I(t) +C0(t) +10−1 +102 +105 +0 +20 +40 +60 +80 +t +� L +0 v2 +x/v dx +C3(t) +Figure 2. Dynamics of plug flow with (top left) h(x, t) and (top +right) u(x, t) starting from initial profiles (5.2) with h0 = 2.29, +showing that the PDE solution approaches a travelling wave +solution (H(ξ), U(ξ)) satisfying equations (4.1) - (4.2) with the +velocity s = 1.396. The solutions are shifted so that the maximums +are aligned. The corresponding energy (bottom left) satisfies the +estimate (3.1), E(t) + I(t) < C0(t), where I(t) = c +�� +Qt +vu2 +x dxdt + +�� +Qt +u2v +g(h) dxdt. The entropy (bottom right) satisfies the estimate +(3.7), +� L +0 v2 +x/v dx < C3(t). The system parameters are L = 20, +a = 0.2, b = 10, c = 1 with g(h) = h2 − 1. +The top two plots in Figure 2 show the dynamics of (h(x, t), u(x, t)) +for the plug flow case, where the PDE solution converges to a travelling +wave solution (H(ξ), U(ξ)) that satisfies the ODE system (4.1) - (4.2) +with the velocity s = 1.396. +The solution profiles are shifted by +x → x − xmax(t) + L/2, where xmax(t) is the location of the peaks + +Travelling waves of a control-volume fibre coating model +41 +of the travelling waves in time, so that the peaks are aligned as the +wave evolves and travels to the right. The system parameters are given +by a = 0.2, b = 10, c = 1 with g(h) = h2 − 1 in (1.4), and the traveling +wave corresponds to the case (a) presented in Fig. 1. +We also numerically verify the analytically derived energy and +entropy estimates. In Figure 2 (bottom left), we show that the energy +estimate (3.1) +E(t) + I(t) < C0(t) +is satisfied as the transient PDE solution approaches the travelling +wave profile in time, where I(t) = c +�� +Qt +vu2 +x dxdt + +�� +Qt +u2v +g(h) dxdt. Fig. 2 +(bottom right) presents the numerically evaluated integral +L� +0 +v2 +x +v dx and +the upper bound C3(t) defined in (3.8) for the dynamic PDE solution, +indicating that the entropy estimate (3.7), +L +� +0 +v2 +x +v dx < C3(t), +is also satisfied in time. Here, we set ϵ = 0.5 in the definition of C3(t) +in (3.8), and this estimate holds for any ϵ ∈ (0, 1). +Figure 3 shows a similar numerical study for the laminar flow case +with the system parameters a = 0.1, b = 11, and c = 4, and the +function g(h) is given by (1.5). In this case, the laminar flow fluid +radius h(x, t) and velocity u(x, t) starting from identical initial data +used in Fig. 2 converge to a slowly-moving travelling wave parametrized +by (H(ξ), U(ξ)) with the speed of propagation s = 0.1 (see Fig. 3 (top +panel)). +The obtained travelling wave corresponds to the case (d) +presented in Fig. 1. Similar to the plug-flow case shown in Fig. 2, the +laminar flow solution also satisfies the energy and entropy estimates, +as demonstrated in Fig. 3 (bottom panel). + +Travelling waves of a control-volume fibre coating model +42 +1 +2 +3 +0 +5 +10 +15 +20 +h +x − xmax + L/2 +h(x, 0) +h(x, t) +H(ξ) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +15 +20 +u +x − xmax + L/2 +u(x, 0) +u(x, t) +U(ξ) +10−1 +102 +105 +108 +0 +100 +200 +300 +400 +t +E(t) + I(t) +C0(t) +10−1 +102 +105 +0 +100 +200 +300 +400 +t +� L +0 v2 +x/v dx +C3(t) +Figure 3. Dynamics of laminar flow (top left) h(x, t) and (top +right) u(x, t) starting from initial profiles (5.2) with h0 = 2.29, +showing that the PDE solution approaches a travelling wave +solution (H(ξ), U(ξ)) satisfying equations (4.1) - (4.2) with the +velocity s = 0.1. The solutions are shifted so that the maximums +are aligned. Again, the corresponding energy plot (bottom left) +shows that the energy satisfies the estimate (3.1), E(t) + I(t) < +C0(t), where I(t) = c +�� +Qt +vu2 +x dxdt + +�� +Qt +u2v +g(h) dxdt. +The entropy +(bottom right) satisfies the estimate (3.7), +� L +0 v2 +x/v dx < C3(t). +The system parameters are L = 20, a = 0.1, b = 11, c = 4, +g(h) = I(h)/(h2 − 1). +6. Conclusions +The main contribution of this paper is the proof of the existence of +weak solutions to the coupled PDE system (2.1)–(2.2) for the control- +volume fibre coating model. +This result establishes the analytical + +REFERENCES +43 +foundation for the control-volume model in real-world applications. +The a priori energy-entropy functional estimates used in the proof also +provide a possible pathway for showing the regularity of solutions in +similar coupled PDE systems in other fibre coating models [13, 25]. In +contrast to the work of Bresch et al. [4] and Kitavtsev et al. [17], for +the proof of existence, we use another approximation of the continuity +equation by the family of thin film equations (see (3.18)). +This +new idea can be applied for the analysis of other systems with the +same structure. +Typical numerical simulations of the PDE model +are presented to support the analytical results, with a focus on the +travelling wave solutions. 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Desalination, 449:92–100, 2019. +[37] Z. Zeng, A. Sadeghpour, G. Warrier, and Y. S. Ju. Experimental +study of heat transfer between thin liquid films flowing down a +vertical string in the Rayleigh-Plateau instability regime and a +counterflowing gas stream. +International Journal of Heat and +Mass Transfer, 108:830–840, 2017. + diff --git a/WNE0T4oBgHgl3EQf3AKg/content/tmp_files/load_file.txt b/WNE0T4oBgHgl3EQf3AKg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad41330f07c24d1ac842ba5b11f114ba66c2c7df --- /dev/null +++ b/WNE0T4oBgHgl3EQf3AKg/content/tmp_files/load_file.txt @@ -0,0 +1,1211 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf,len=1210 +page_content='On travelling waves of a control-volume model for liquid films flowing down a fibre Roman M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Taranets1, Hangjie Ji2, and Marina Chugunova3 1 Institute of Applied Mathematics and Mechanics of the NASU, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Batyuka Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 19, 84100, Sloviansk, Ukraine 2 Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA 3 Claremont Graduate University, 150 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 10th Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=', Claremont, CA 91711, USA E-mail: taranets_r@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='com, hangjie_ji@ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='edu, and marina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='chugunova@cgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='edu 10 January 2023 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' This paper analytically investigates the solutions to a control-volume model for liquid films flowing down a vertical fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The dynamic evolution of the free surface is governed by a coupled degenerate nonlinear PDE system for the fluid film radius and the axial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We prove the existence of weak solutions to the coupled system based on the application of a priori estimates derived for energy-entropy functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Existence of travelling wave solutions to the system is also established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Numerical studies are presented to illustrate the derived analytical results for both the dynamic PDE solutions and the travelling wave structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Keywords: Thin films;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' travelling waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' fourth-order parabolic partial differential equations arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='02720v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='AP] 6 Jan 2023 Travelling waves of a control-volume fibre coating model 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Introduction Thin liquid films flowing down a vertical fibre have attracted many interests in the past decades due to their importance in a variety of engineering applications, including heat and mass exchangers, thermal desalination, and vapor and CO2 capturing [27, 28, 35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' These liquid films are fundamentally driven by Rayleigh-Plateau instability and gravity modulation, spontaneously exhibiting complex interfacial instability and pattern formation [20, 21, 24, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Previous experimental works have found that the downstream flow dynamics and pattern formation highly depend on the flow rate, fibre radius, liquid properties, and inlet geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Specifically, three typical flow regimes have been extensively studied [7, 10, 18, 25, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' At high flow rates, the convective instability dominates the system and irregular droplet coalescence occur frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For lower flow rates, the Rayleigh-Plateau regime emerges where stable travelling droplets move at a constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' If flow rates are further reduced, the isolated droplet regime occurs where widely-spaced droplets coexist with small amplitude wave patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Similar regime transitions can also be triggered by varying the nozzle diameters or imposing a gradient to the liquid property along the fibre [8, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' A good understanding of these dynamics is critical to the design and control of engineering systems that involve a stable train of travelling droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In the low Reynolds number limit, classical lubrication models have been developed for the dynamics of falling viscous liquid films along an axisymmetric cylindrical fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Under the long-wave approximation, the resultant evolution equations are a family of fourth-order degenerate parabolic PDEs for the fluid film thickness [5, 10, 11, 15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' These models incorporate gravity and both stabilizing and destabilizing roles of surface tension by characterizing the axial and azimuthal curvature of the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Numerical and analytical investigations for this type of models also revealed Travelling waves of a control-volume fibre coating model 3 the dependence of the droplet dynamics on the substrate effects and external physical fields [12, 14, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For higher flow rates and for fluid films near the nozzle where inertial effects are significant, systems of coupled equations for both the film thickness and the local flow rate have also been investigated [13, 25, 26, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' These models include inertia effects based on a weighted-residual integeral boundary layer approach by assuming a local velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' More recently, Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' [23] proposed a new framework for liquid films flowing down a fibre using a control-volume approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Their model expresses the conservation of mass and axial momentum via the coupled dimensionless equation for the fluid film radius h(x, t) and the mean axial velocity u(x, t), where the momentum equation is ut + a �u2 2 � x + b κx = c [(h2 − 1)ux]x h2 − 1 + 1 − u g(h) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) and the mass conservation equation is 2hht + a[u(h2 − 1)]x = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) where the dimensionless parameter a represents the square of the Froude number, b is the reciprocal of the Bond number, c represents the ratio of axial viscous to gravitational forces, and g(h) represents the axial velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The film thickness is given by h(x, t)−1, and κ represents the combined azimuthal and streamwise curvatures of the free surface, κ = 1 h(1 + h2 x)1/2 − � hx (1 + h2 x)1/2 � x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3) Furthermore, by taking different forms of g(h), the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) corresponds to different flow regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For the high Reynolds number regime, we have the plug flow model with g(h) = h2 − 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4) Travelling waves of a control-volume fibre coating model 4 that assumes a uniform velocity in the cross section with a viscous drag force on the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For the low Reynolds number case, the fully- developed laminar velocity profile is assumed, with g(h) = I(h) h2 − 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5) where I(h) = 1 16[4h4 ln(h) + (h2 − 1)(1 − 3h2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' While extensive modelling works have been carried out for falling liquid films, relatively less research [14] have focused on establishing analytical understanding of the developed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In this work, we will analytically investigate the coupled equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2), with a focus on the travelling wave solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Energy and entropy estimates will be constructed to establish the existence of weak solutions to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Similar analytical techniques were also applied to other models (for example, see [2–4, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In section 2, we formulate the problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In section 3, we show the existence of weak solutions to the problem via energy and entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Section 4 presents a detailed discussion on travelling wave solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Numerical studies are presented in section 5 for both the plug flow and the laminar flow cases, followed by concluding remarks in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Problem statement We study the following initial boundary value problem: ut + a �u2 2 � x + b κx = c [(h2 − 1)ux]x h2 − 1 + 1 − u g(h) in QT, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) 2hht + a[u(h2 − 1)]x = 0 in QT, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) u and h are |Ω| − periodic, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3) u(x, 0) = u0(x), h(x, 0) = h0(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4) Travelling waves of a control-volume fibre coating model 5 where Ω ⊂ R1 is an open interval, QT := Ω × (0, T), a, b, c are non- negative constants, and κ = f(hx)h−1 − f 3(hx)hxx, f(z) = (1 + z2)− 1 2, Φ(z) = 1 f(z), Φ′(z) = zf(z), Φ′′(z) = f 3(z), g(h) = h2 − 1 or g(h) = I(h) h2 − 1, where I(h) := 1 16[4h4 ln(h) + (h2 − 1)(1 − 3h2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Let v = h2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Then we can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) in the following form: ut + a �u2 2 � x + b κx = c (v ux)x v + 1 − u g(h) in QT, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5) vt + a(u v)x = 0 in QT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) Integrating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) in Qt, we find that v(x, t) satisfies the conservation of mass, � Ω v(x, t) dx = � Ω v0(x) dx := M > 0 ∀ t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7) Furthermore, we assume that the initial data (v0, u0) satisfy h0 ⩾ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' v0 := h2 0 − 1 ⩾ 0, for all x ∈ ¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' √v0 ∈ H1(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' h0Φ(h0,x), − log(v0), v0u2 0 ∈ L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' A pair (v, u) is a weak solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) with periodic boundary conditions and initial conditions (v0, u0) if 0 ⩽ v ∈ C( ¯QT) and u satisfy the regularity properties √v ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H1(Ω));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' − log(v), vu2 ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L1(Ω));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9) Travelling waves of a control-volume fibre coating model 6 hΦ(hx) ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L1(Ω));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' χ{|hx|<∞}h−1f(hx) ∈ L1(QT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='10) χ{|hx|<∞} � hf 3(hx)hxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' χ{v>0} √vux, � v g(h)u ∈ L2(QT), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='11) and the following holds �� QT vφt dxdt + � Ω v0φ(x, 0) dx + �� QT uvφx dxdt = 0, �� QT uvψt dxdt + � Ω u0v0ψ(x, 0) dx + a �� QT χ{v>0}vu2ψx dxdt+ b �� QT χ{|hx|<∞}κvxψ dxdt + b �� QT χ{|hx|<∞}κvψx dxdt− c �� QT χ{v>0}vuxψx dxdt + �� QT � v − uv g(h) � ψ dxdt = 0 for all φ ∈ C∞ c ( ¯QT) and ψ ∈ C∞ c ( ¯QT) such that φ(x, T) = ψ(x, T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Based on the definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1, we will establish the existence of weak solutions to the problem and prove the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Let the initial data (v0, u0) satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8), and T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Then there exists a weak solution (v, u) in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Moreover, the sets {v(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=', t) = 0} and {|hx(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=', t)| = ∞} have measure zero for any t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Existence of weak solutions In this section, we will introduce the energy and entropy functionals for the problem and show their estimates in subsections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The proof of key results in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 follows the work of Kitavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Energy estimate Let us denote the energy functional by E(t) := 1 2 � Ω (vu2 + 4b a hΦ(hx)) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 (Energy inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Let (v, u) be sufficiently smooth solution to the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) with periodic boundary conditions, then (v, u) satisfy the following inequality E(T) + c �� QT vu2 x dxdt + �� QT u2v g(h) dxdt ⩽ C0(T), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) where C0(T) = (E 1 2(0) + √ 2M 2 T)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5) by uv and integrating over Ω, we have � Ω uvut dx + a � Ω uv( u2 2 )x dx + b � Ω uvκx dx = c � Ω u(vux)x dx + � Ω uv(1 − u g(h)) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) Since the first two integrals on the left-hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) satisfy � Ω uvut dx + a � Ω uv( u2 2 )x dx = � Ω v( u2 2 )t dx − a � Ω (uv)x u2 2 dx = � Ω v( u2 2 )t dx + � Ω vt u2 2 dx = 1 2 d dt � Ω vu2 dx, Travelling waves of a control-volume fibre coating model 8 and the third integral on the left-hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) satisfies b � Ω uvκx dx = −b � Ω (uv)xκ dx = b a � Ω vtκ dx = 2b a � Ω hht(f(hx)h−1 − f 3(hx)hxx) dx = 2b a � Ω (htf(hx) − hhtΦ′′(hx)hxx) dx = 2b a � Ω (htf(hx) − hht(Φ′(hx))x) dx = 2b a � Ω (htf(hx) + (hht)xΦ′(hx)) dx = 2b a � Ω (htf(hx) + hthxΦ′(hx) + hhxtΦ′(hx)) dx = 2b a � Ω (htf(hx) + hth2 xf(hx) + h(Φ(hx))t) dx = 2b a � Ω (htΦ(hx) + h(Φ(hx))t) dx = 2b a d dt � Ω hΦ(hx) dx, then from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) it follows that d dtE(t) + c � Ω vu2 x dx + � Ω u2v g(h) dx = � Ω uv dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3) Taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7) and � Ω uv dx ⩽ M 1 2 �� Ω vu2 dx � 1 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4) and after integrating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3) in time, we obtain the energy estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Entropy estimate We define two entropy-like functionals for the plug flow and laminar flow models separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For the plug flow model with g(h) = v, we Travelling waves of a control-volume fibre coating model 9 denote the entropy functional by S1(u, h) := 1 2 � Ω � v(u + c a vx v )2 + 4b a hΦ(hx) + 2c a2(v − log(v)) � dx For the laminar flow model with g(h) = I(h) v , we define the entropy functional as S2(u, h) := 1 2 � Ω � v(u + c a+ϵ0 vx v )2 + c2ϵ0 a(a+ϵ0)2 v2 x v + 4b a hΦ(hx)+ 4c a(a+ϵ0)(8v − G(h)) � dx where G′(h) = h g(h) and ϵ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We will prove the following entropy estimates for S1(u, h) and S2(u, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 (Entropy inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Let (v, u) be sufficiently smooth solution to the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) with periodic boundary conditions, then (v, u) satisfy the following inequality S1(u, h) + 2bc a �� QT hf 3(hx)h2 xx dxdt + �� QT u2 dxdt+ 2bc a �� QT h−1f(hx) dxdt ⩽ C1(T) if g(h) = v, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5) S2(u, h) + 2bc a+ϵ0 �� QT hf 3(hx)h2 xx dxdt + �� QT u2v2 I(h) dxdt+ 2bc a+ϵ0 �� QT h−1f(hx) dxdt ⩽ C2(T) if g(h) = I(h) v , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) where C1(T) := S1(u0, h0) + T � 0 (c C0(t) + (2M) 1 2C 1 2 0 (t)) dt, Travelling waves of a control-volume fibre coating model 10 C2(T) := S2(u0, h0) + T � 0 ( a c a+ϵ0C0(t) + (2M) 1 2C 1 2 0 (t) + 16c ϵ0(a+ϵ0)M) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Using the estimate (a + b)2 ⩾ ϵa2 − ϵ 1−ϵb2 for any ϵ ∈ [0, 1), from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1), we deduce that for the plug flow model, � Ω v2 x v dx ⩽ ( a c)2� 1 1−ϵ � Ω v u2 dx + 2 ϵC1(T) � ⩽ C3(T), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7) where C3(T) := ( a c)2� 2 1−ϵC0(T) + 2 ϵC1(T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7) it follows that (v 1 2)x ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The same can be shown in the case of laminar flows with g(h) = I(h) v as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5) by vx and integrating over Ω, we have � Ω (ut + auux)vx dx + b � Ω vxκx dx = − c � Ω vux( vx v )x dx − � Ω u vx g(h) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9) For the first integral on the left-hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9), we have � Ω (ut + auux)vx dx = � Ω (utvx + a(uv)xux − a vu2 x) dx = � Ω (utvx − vtux − a vu2 x) dx = � Ω (utvx + uvxt − a vu2 x) dx = d dt � Ω uvx dx − a � Ω vu2 x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 11 To handle the first integral on the right-hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9), we deduce d dt � Ω v2 x v dx = 2 � Ω vxvxt v dx − � Ω v2 xvt v2 dx = −2 � Ω ( vx v )xvt dx− � Ω v2 x v2 vt dx = 2a � Ω ( vx v )x(uv)xdx + a � Ω v2 x v2 (uv)x dx = 2a � Ω ( vx v )x(uv)xdx − 2a � Ω ( vx v )xuvxdx = 2a � Ω vux( vx v )xdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For the second integral on the left-hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9), we have b � Ω vxκx dx = −b � Ω vxx(f(hx)h−1 − f 3(hx)hxx) dx = − 2b � Ω (h2 x + hhxx)(f(hx)h−1 − f 3(hx)hxx) dx = − 2b � Ω h−1h2 xf(hx) dx − 2b � Ω f(hx)(1 − f 2(hx)h2 x)hxx dx+ + 2b � Ω hf 3(hx)h2 xx dx = −2b � Ω h−1h2 xf(hx) dx − 2b � Ω f 3(hx)hxx dx+ 2b � Ω hf 3(hx)h2 xx dx = −2b � Ω h−1h2 xf(hx) dx + 2b � Ω hf 3(hx)h2 xx dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Then from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9), it follows that d dt � Ω (uvx + c 2a v2 x v ) dx + 2b � Ω hf 3(hx)h2 xx dx = a � Ω vu2 x dx + 2b � Ω h−1h2 xf(hx) dx − � Ω u vx g(h) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='10) Travelling waves of a control-volume fibre coating model 12 Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='10) by c a and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3), we arrive at d dt � Ω ( c auvx + c2 2a2 v2 x v ) dx + d dtE(t) + 2bc a � Ω hf 3(hx)h2 xx dx+ � Ω u2v g(h) dx = 2bc a � Ω h−1h2 xf(hx) dx − c a � Ω u vx g(h) dx + � Ω uv dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='11) Note that � Ω ( c auvx + c2 2a2 v2 x v ) dx + 1 2 � Ω vu2 dx = 1 2 � Ω v(u + c a vx v )2 dx, c a � Ω u vx g(h) dx = c a � Ω (uv)x g(h) dx − c a � Ω v g(h)ux dx = − c a2 � Ω vt g(h) dx − c a � Ω v g(h)ux dx = − c a2 d dt � Ω log(v) dx if g(h) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Therefore, for the plug flow model with g(h) = v, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='11) has the form 1 2 d dt � Ω � v(u + c a vx v )2 + 4b a hΦ(hx) − 2c a2 log(v) � dx+ 2bc a � Ω hf 3(hx)h2 xx dx + � Ω u2 dx + 2bc a � Ω h−1f(hx) dx = 2bc a � Ω h−1Φ(hx) dx + � Ω uv dx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='12) If we set g(h) = I(h) v for the laminar flow model, then by using � Ω u vx g(h) dx = − 2 a � Ω hht g(h) dx − � Ω v2 I(h)ux dx = − 2 a d dt � Ω G(h) dx − � Ω v2 I(h)ux dx, where G′(h) = h g(h), Travelling waves of a control-volume fibre coating model 13 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='10) we get d dt � Ω (uvx + c 2a v2 x v ) dx + 2b � Ω hf 3(hx)h2 xx dx ⩽ (a + ϵ0) � Ω vu2 x dx+ 2b � Ω h−1h2 xf(hx) dx + 2 a d dt � Ω G(h) dx + 1 4ϵ0 � Ω v3 I2(h) dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='13) where ϵ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='13) by c a+ϵ0 and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3), we arrive at 1 2 d dt � Ω � v(u + c a+ϵ0 vx v )2 + c2ϵ0 a(a+ϵ0)2 v2 x v + 4b a hΦ(hx) − 4c a(a+ϵ0)G(h) � dx+ 2bc a+ϵ0 � Ω hf 3(hx)h2 xx dx + � Ω u2v2 I(h) dx + 2bc a+ϵ0 � Ω h−1f(hx) dx ⩽ 2bc a+ϵ0 � Ω h−1Φ(hx) dx + � Ω uv dx + c 4ϵ0(a+ϵ0) � Ω v3 I2(h) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='14) Note that v − log(v) ⩾ 1 and 8v − G(h) ⩾ 16 for all v ⩾ 0 (as v I(h) → 8 when h → 1 and v I(h) → 0 when h → +∞ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Then, due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='14) can be rewritten in the form d dtS1(u, h) + 2bc a � Ω hf 3(hx)h2 xx dx + � Ω u2 dx + 2bc a � Ω h−1f(hx) dx = 2bc a � Ω h−1Φ(hx) dx + � Ω uv dx if g(h) = v, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='15) d dtS2(u, h) + 2bc a+ϵ0 � Ω hf 3(hx)h2 xx dx + � Ω u2v2 I(h) dx+ 2bc a+ϵ0 � Ω h−1f(hx) dx ⩽ 2bc a+ϵ0 � Ω h−1Φ(hx) dx + � Ω uv dx+ c 4ϵ0(a+ϵ0) � Ω v3 I2(h) dx if g(h) = I(h) v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='16) Travelling waves of a control-volume fibre coating model 14 Taking into account (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4), � Ω h−1Φ(hx) dx ⩽ � Ω hΦ(hx) dx for v ⩾ 0, � Ω v3 I2(h) dx ⩽ 64M for v ⩾ 0 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3), from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='16) we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Approximate problem For given δ > 0, η > 0 and ε > 0, we consider the following approximate system (v u)t + a [(uv + ηv4vxxx)u]x + b v κx = c (v ux)x + v � 1 − u g(h) � − δuxxxx + ε a[p(v)]x − ε a vvxxxxx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='17) vt + a(u v)x = −a η � v4vxxx � x , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='18) u and v are |Ω| − periodic, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='19) u(x, 0) = uεη,0(x), v(x, 0) = vεη,0(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='20) where uεη,0 ∈ H1(Ω) and 0 < vεη,0 ∈ H2(Ω) such that uε,0(x) → u0(x) strongly in L2(Ω), vε,0(x) ⩾ v0(x) + εθ, θ ∈ (0, 1 2), vε,0(x) → v0(x) strongly in W 1 1 (Ω) ∩ C(¯Ω), ε 1 2vε,0xx(x) → 0 strongly in L2(Ω), and p(z) = 1 2z−2, g(h) = v or g(h) = |I(h)| v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Let u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)) be a periodic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For any 0 < vηε,0(x) ∈ H2(Ω), the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='18)—(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='19) has a unique weak positive solution v ∈ C 3 2 , 3 8( ¯QT) such that vt ∈ L2(QT), v ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)), v ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H4(Ω)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='21) � Ω v dx = � Ω vηε,0 dx =: Mηε > 0, and v satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='18) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' in QT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Moreover, there exists a constant C > 1 depending on η such that 1 C ⩽ v ⩽ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The main line of proof follows the approach in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We omit details and restrict the discussion only to the key elements of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' First of all, we approximate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='18) by vt + a([u]αv)x = −a η((v4 + β)vxxx)x, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='22) where β > 0 and [u]α denotes a smooth approximation of u such that [u]α → u strongly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)) as α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We also approximate vηε,0 in the H2-norm by C4+γ functions vβηε,0, satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='18), and replace (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='20) by v(x, 0) = vβεη,0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='23) Using the parabolic Schauder estimates from [31], one can generalise [9, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 302] and show that the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='22)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='23) has a unique classical solution vβα ∈ C 4+γ,1+ γ 4 x,t (Ω×[0, τβα]) for some τβα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Next, for simplicity, we will write v instead vβα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='22) Travelling waves of a control-volume fibre coating model 16 by −vxx and integrating by parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' we have 1 2 d dt � Ω v2 x dx + ηa � Ω (v4 + β)v2 xxx dx = a � Ω ([u]αv)xvxx dx = −a � Ω ([u]α)xxvvx dx − 3a 2 � Ω ([u]α)xv2 x dx ⩽ a sup Ω |v| �� Ω ([u]α)2 xx dx � 1 2�� Ω v2 x dx � 1 2 + 3a 2 sup Ω |([u]α)x| � Ω v2 x dx ⩽ �� Ω ([u]α)2 xx dx � 1 2� 5a|Ω| 1 2 2 � Ω v2 x dx + Mβηε |Ω| �� Ω v2 x dx � 1 2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' whence � Ω v2 x dx + ηa �� QT (v4 + β)v2 xxx dxdt ⩽ � ∥vβηε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='0x∥2+ Mβηε |Ω| T � 0 ∥([u]α)xx∥2e − 5a|Ω| 1 2 2 t� 0 ∥([u]α)xx∥2 ds dt �2 e 5a|Ω| 1 2 T� 0 ∥([u]α)xx∥2 dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='24) Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='24), we deduce that ∥vβ∥C 1 2 , 1 8 ( ¯QT ) is uniformly bounded with respect to α, β and τβα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For any fixed values of β and α, by [9, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 316], we can to extend the solution vβ step-by-step to all of QT for any T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Let us denote by Gβ(z) ⩾ 0 such that Gβ(z) = z � 1 y � 1 dsdy s4+β, G′′ β(z) = 1 z4+β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 17 Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='18) by G′ β(v) and integrating by parts, we arrive at d dt � Ω Gβ(v) dx + ηa � Ω v2 xx dx = −a � Ω ([u]α)x(vG′ β(v) − Gβ(v)) dx ⩽ aC sup Ω |([u]α)x| � Ω Gβ(v) dx, whence � Ω Gβ(v) dx + ηa �� QT v2 xx dxdt ⩽ e aC T� 0 sup Ω |([u]α)x| dt � Ω Gβ(vβηε,0) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='25) Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='24) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='25), similar to the proof of [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='190], after taking β → 0, we obtain the global existence of a unique positivity classical solution v0α for any α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Moreover, 1 C ⩽ v0α ⩽ C < ∞, where C > 1 is independent of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For the limit process α → 0, we need the following a priori estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='22) with β = 0 by vxxxx and integrating by parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' we ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Travelling waves of a control-volume fibre coating model ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xv2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx ⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='ηa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxxx dx + 2a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∥v−1∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥([u]α)x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 + ∥v−1∥4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥vx∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥[u]α∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='16aη∥v−1∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥vx∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v4v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' whence d dt∥vxx∥2 2 + ηa � Ω v4v2 xxxx dx ⩽ aC η−1∥[u]α∥2 H1+ aC � η−1∥[u]α∥2 H1 + η � Ω v4v2 xxx dx � ∥vxx∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Integrating this inequality in time, we have � Ω v2 xx dx + ηa �� QT v4v2 xxxx dxdt ⩽ � ∥vηε,0xx∥2 2+ aC η−1 T � 0 ∥[u]α∥2 H1e −aC t� 0 � η−1∥[u]α∥2 H1+η � Ω v4v2 xxx dx � ds dt � × e aC t� 0 � η−1∥[u]α∥2 H1+η � Ω v4v2 xxx dx � ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='26) Travelling waves of a control-volume fibre coating model 19 By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='24), ∥v0α∥C 3 2 , 3 8 ( ¯QT ) is uniformly bounded with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' This uniform bound follows from v0α ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)) and v0α,t ∈ L2(QT) (see [33, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='19, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Taking α → 0, it completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For the given δ > 0, η > 0 and ε > 0, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='17) is uniformly parabolic with respect to u for any v is from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By using Faedo-Galerkin approximation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=', [4]), the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='17)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='18) with periodic boundary conditions has a local in time weak solution (v, u) := (vδηε, uδηε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Next, we establish a priori estimates which guarantee the global in time solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4 (a priori estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For fixed and positive constants δ > 0, η > 0, ε > 0, and T > 0, let (vδηε, uδηε) be the solution to the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='17)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='20) in the following sense �� QT uvψt dxdt+ � Ω uεη,0vεη,0ψ(x, 0) dx + a �� QT (uv + ηv4vxxx)uψx dxdt+ b �� QT κvxψ dxdt + b �� QT κvψx dxdt − c �� QT vuxψx dxdt+ �� QT � v − uv g(h) � ψ dxdt − δ �� QT uxxψxx dxdt − εa �� QT p(v)ψx dxdt− εa �� QT vxxx(vxxψ + 2vxψx + vψxx) dxdt = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='27) Travelling waves of a control-volume fibre coating model 20 �� QT vφt dxdt + � Ω vεη,0φ(x, 0) dx+ a �� QT uvφx dxdt + aη �� QT v4vxxxφx dxdt = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='28) for all φ ∈ C∞ c ( ¯QT) and ψ ∈ C∞ c ( ¯QT) such that φ(x, T) = ψ(x, T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Moreover, there exists a positive constant C > 0 depending only on a, b, c, T, E(0), and Si(u0, v0) such that the following terms are bounded by C in respective norms √v ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H1(Ω)), √vu ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L2(Ω)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29) − log(v), hΦ(hx) ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L1(Ω)), h−1f(hx) ∈ L1(QT), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='30) � hf 3(hx)hxx, √vux, � v g(h)u ∈ L2(QT), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='31) ε 1 2v−1, ε 1 2vxx ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L2(Ω)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='32) δ 1 2uxx, (εη) 1 2v2vxxxx, ε 1 2vxxx, ε 1 2(v−1)x ∈ L2(QT), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='33) and 0 < ε ⩽ v(x, t) ⩽ C for all (x, t) ∈ QT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='34) Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Let us denote by Eε(t) := 1 2 � Ω (v u2 + 2b a hΦ(hx) + ε 3v−2 + εv2 xx) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='17) by u and integrating over Ω, we have d dtEε(t) + δ � Ω u2 xx dx + εηa � Ω v4v2 xxxx dx+ εηa � Ω v2 xx dx + c � Ω vu2 x dx + � Ω u2v gε(h) dx = � Ω uv dx− 4εηa � Ω v3vxvxxxvxxxx dx − bη � Ω (v4vxxx)xκ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='35) Travelling waves of a control-volume fibre coating model 21 Note that � Ω uv dx ⩽ M 1 2 �� Ω vu2 dx � 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We will use the following estimates ∥v∥∞ ⩽ C ∥vxx∥2+ M |Ω|, ∥vx∥∞ ⩽ C ∥vxx∥2, ∥vxxx∥2 ⩽ C ∥vxxxx∥ 1 2 2 ∥vxx∥ 1 2 2 , ∥v−1∥∞ ⩽ C � ∥v−1∥ 3 2 2 (∥vxx∥2 2 + ν) 1 4 + (∥vxx∥2 2 + ν)− 1 2� ∀ ν ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='36) For the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='36), see [6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='807].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='36),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' we deduce that � Ω v3vxvxxxvxxxx dx ⩽ ∥v2vxxxx∥2∥vvxvxxx∥2 ⩽ ∥v2vxxxx∥2∥v∥∞∥vx∥∞∥vxxx∥2 ⩽ C ∥v2vxxxx∥2∥v∥∞∥vx∥∞∥vxxxx∥ 1 2 2 ∥vxx∥ 1 2 2 ⩽ C ∥v2vxxxx∥ 3 2 2 ∥v−1∥∞ � M |Ω| + ∥vxx∥2 � ∥vxx∥ 3 2 2 ⩽ C ∥v2vxxxx∥ 3 2 2 � ∥v−1∥ 3 2 2 (∥vxx∥2 2 + 1) 1 4 + 1 �� M |Ω| + ∥vxx∥2 � ∥vxx∥ 3 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' � Ω (v4vxxx)xκ dx = � Ω v4vxxxxκ dx + 4 � Ω v3vxvxxxκ dx ⩽ ε 4 � Ω v4v2 xxxx dx + 2 ε � Ω v4κ2 dx + C ε ∥v−1∥4 ∞ � Ω v6v2 xκ2 dx ⩽ ε 4 � Ω v4v2 xxxx dx + C ε � Ω (v3 + v3v2 xx) dx + C ε ∥v−1∥4 ∞ � Ω (v6 + v6v2 xx) dx ⩽ ε 4 � Ω v4v2 xxxx dx+ C ε (1+∥vxx∥2 2) � ( M |Ω|)3+∥vxx∥3 2+∥v−1∥4 ∞(( M |Ω|)6+∥vxx∥6 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 22 By these estimates, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='35) we have d dtEε(t)+δ � Ω u2 xx dx+ εηa 2 � Ω v4v2 xxxx dx+εηa � Ω v2 xx dx+c � Ω vu2 x dx+ � Ω u2v gε(h) dx ⩽ C E 1 2ε (t) + C bηε−9E8 ε(t) + C aηε−8E9 ε(t), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='37) whence Eε(T) + δ �� QT u2 xx dxdt + εηa 2 �� QT v4v2 xxx dxdt+ εηa �� QT v2 xx dxdt + c �� QT vu2 x dxdt + �� QT u2v g(h) dxdt ⩽ C(T) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='38) for all T ⩽ Tη := [16CE8 ε(0)ηε−8]−1 → +∞ as η → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In particular, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='38), due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='36), we arrive at ∥v−1∥∞ ⩽ C ε−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='39) whence inf Ω v ⩾ C ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Next, we consider the plug flow case g(h) = v only, and the laminar flow case can be addressed in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='17) by Travelling waves of a control-volume fibre coating model 23 vx v and integrating over Ω, we obtain that d dtS1,ε(u, v) + 2bc a � Ω hf 3(hx)h2 xx dx + � Ω u2 dx + 2bc a � Ω h−1f(hx) dx+ δ � Ω u2 xx dx + εa � Ω v2 xxx dx + εa 3 � Ω v2 x v4 dx = 2bc a � Ω h−1Φ(hx) dx + � Ω uv dx − δ � Ω � vx v � xxuxx dx+ η c � Ω vux(v3vxxx)x dx + η c2 a � Ω � vx v � xv(v3vxxx)x dx+ η c a � Ω (v4vxxx)x g(h) dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='40) where S1,ε(u, v) := 1 2 � Ω � v(u + c a vx v )2 + 4b a hΦ(hx) + 2c a2(v − log(v))+ 2εv−2 + εv2 xx � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Using the equality � vx v � xx = vxxx v − 3 vxvxx v2 + 2 v3 x v3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='41) and the following estimates ∥vx∥6 ⩽ C∥vxx∥ 1 3 2 ∥vx∥ 2 3 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' ∥vxx∥∞ ⩽ C∥vxxx∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Travelling waves of a control-volume fibre coating model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='we find that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� vx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxuxx dx ⩽ δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2∥δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2uxx∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∥vxxx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v ∥2 + 3∥vxvxx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v2 ∥2 + 2∥vx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v ∥3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2∥δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2uxx∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∥v−1∥∞∥vxxx∥2+3∥v−1∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥vx∥2∥vxx∥∞+2∥v−1∥3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥vx∥3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='C δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2∥δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2uxx∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∥v−1∥∞∥vxxx∥2 + ∥v−1∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥vx∥2∥vxxx∥2+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∥v−1∥3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥vx∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2∥vxxx∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='⩽ C δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2ε−4∥δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2uxx∥2∥vxxx∥2 ⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='C δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2ε−4∥δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2uxx∥2∥vxxxx∥2 ⩽ C δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2ε− 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 η− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2∥δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2uxx∥2∥(εη) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2v2vxxxx∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' As a result, due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='38), we find that δ �� QT � vx v � xxuxx dxdt ⩽ C δ 1 2η− 1 2ε− 13 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='42) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='41) and the estimates � Ω vv2 xv2 xx dx ⩽ 1 3 �� Ω v3v2 xxx dx � 1 2�� Ω v6 x v dx � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' ∥vx∥6 ⩽ C∥vxxx∥ 1 6 2 ∥vx∥ 5 6 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Travelling waves of a control-volume fibre coating model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='we find that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� vx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xv(v3vxxx)x dx = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� vx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxv4vxxx dx− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� vx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xv3vxvxxx dx = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v2vxvxxvxxx dx − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='vv3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xvxxx dx ⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 ⩽ − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx + C∥v−1∥∞∥vx∥6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6 ⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx + C∥v−1∥∞∥vx∥5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2∥vxxx∥2 ⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx + C∥v−1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2∞∥vx∥5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2∥v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2vxxx∥2 ⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx + C∥v−1∥5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='∞∥vxx∥10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 ⩽ − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='v3v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='xxx dx + C ε−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' As a result, due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='38), we arrive at η �� QT � vx v � xv(v3vxxx)x dxdt ⩽ − η 4 �� QT v3v2 xxx dxdt + C η ε−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='43) Using the estimate � Ω v(v3vxxx)2 x dx ⩽ C � Ω v7v2 xxxx dx + C � Ω v5v2 xv2 xxx dx ⩽ C∥v∥3 ∞ � Ω v4v2 xxxx dx + C∥v∥5 ∞∥vx∥2 ∞ � Ω v2 xxx dx ⩽ C � ∥v∥3 ∞ + ∥v∥5 ∞∥vx∥2 ∞∥v−1∥4 ∞ � � Ω v4v2 xxxx dx ⩽ C ε− 15 2 � Ω v4v2 xxxx dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 26 we deduce that η c �� QT vux(v3vxxx)x dxdt ⩽ η c ��� QT vu2 x dxdt � 1 2��� QT v(v3vxxx)2 x dxdt � 1 2 ⩽ η c ��� QT vu2 x dxdt � 1 2� C ε− 15 2 �� QT v4v2 xxxx dxdt � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' whence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='38), we have η c �� QT vux(v3vxxx)x dxdt ⩽ C η 1 2ε− 17 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='44) Using the estimate � Ω (v4vxxx)x v dx = � Ω v3vxxxx dx + 4 � Ω v2vxvxxx dx ⩽ ∥v2vxxxx∥2(∥v∥2 + C ∥v−1∥2 ∞∥v∥2 ∞∥vx∥2) ⩽ C ε− 7 2∥v2vxxxx∥2 = C η− 1 2ε−4∥(εη) 1 2v2vxxxx∥2, we get η c a �� QT (v4vxxx)x g(h) dxdt ⩽ C η 1 2ε−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='45) Integrating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='40) in time, taking into account (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='38) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='42)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='45), Travelling waves of a control-volume fibre coating model 27 we obtain S1,ε(u, v) + 2bc a �� QT hf 3(hx)h2 xx dxdt + �� QT u2 dxdt+ 2bc a �� QT h−1f(hx) dxdt + δ �� QT u2 xx dxdt + εa �� QT v2 xxx dxdt+ 2εa 3 �� QT v2 x v4 dxdt + ηc2 4a �� QT v3v2 xxx dxdt ⩽ S1,ε(uεη,0, vεη,0)+ C(T) + C δ 1 2η− 1 2ε− 13 2 + C η ε−10 + C η 1 2ε− 17 4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='46) for all T ⩽ Tη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Compactness and limit processes Passage to the limit δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Denote the corresponding solution to the approximate problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='17)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='20) by (vδηε, uδηε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Let T ⩽ Tη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We study the compactness properties of the sequence (vδηε, uδηε) by using the estimates derived in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='32) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='33), we have {vδηε}δ>0 is bounded in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)) and {vδηε,t}δ>0 is bounded in L2(QT), whence, using [33, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='19, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 175], we arrive at {vδηε}δ>0 is bounded in C 3 2 , 3 8( ¯QT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By the Arzela-Ascoli theorem, after possibly extracting a subsequence, we obtain that vδηε → δ→0 vηε uniformly in C 3 2 , 3 8( ¯QT), vδηε,t → δ→0 vηε,t weakly in L2(QT), whence v−1 δηε → δ→0 v−1 ηε uniformly in C 3 2 , 3 8( ¯QT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Also, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='33) we obtain that vδηε → δ→0 vηε weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H4(Ω)), Travelling waves of a control-volume fibre coating model 28 vδηε → δ→0 vηε strongly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H3(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We next turn to compactness properties of {uδηε}δ>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='34) and the boundedness vδηε away from zero, we have {uδηε}δ>0 is bounded in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L2(Ω)) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H1(Ω));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' {(vδηεuδηε)t}δ>0 and {uδηε,t}δ>0 are bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' So, uδηε → δ→0 uηε strongly in L2(QT), uδηε,x → δ→0 uηε,x weakly in L2(QT), uδηε,t → δ→0 uηε,t ∗ − weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−2(Ω)), vδηεuδηε → δ→0 vηεuηε strongly in L2(QT), (vδηεuδηε)t → δ→0(vηεuηε)t ∗ − weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='33) δ ��� �� QT uxxψxx dxdt ��� ⩽ δ 1 2∥δ 1 2uxx∥L2(QT )∥ψ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='H2(Ω)) ⩽ C δ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The obtained convergences allow to pass to the limit as δ → 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Passage to the limit η → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Since Tη → +∞ as η → 0 then we can to take any T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Now, we consider the compactness properties of the sequence (vηε, uηε) by using the estimates derived in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='32) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='33), we have {vηε}η>0 is bounded in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)) and {vηε,t}η>0 is bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−1(Ω)), whence, similar to [1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 183], we arrive at {vηε}η>0 is bounded in C 3 2 , 1 4( ¯QT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By the Arzela-Ascoli theorem, after possibly extracting a subsequence, we obtain that vηε → η→0 vε uniformly in C 3 2 , 1 4( ¯QT), Travelling waves of a control-volume fibre coating model 29 vηε,t → η→0 vε,t ∗ − weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−1(Ω)), whence v−1 ηε → η→0 v−1 ε uniformly in C 3 2 , 1 4( ¯QT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Also, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='33) we obtain that vηε → η→0 vε weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H3(Ω)), vηε → η→0 vε strongly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Next, we turn to compactness properties of {uηε}η>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='34) and the boundedness vηε away from zero, we have {uηε}η>0 is bounded in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L2(Ω))∩L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H1(Ω)), whence, in particular, {vηεu2 ηε}η>0 is bounded in Lp(QT) for p ∈ (1, 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' {(vηεuηε)t}η>0 and {uηε,t}ε>0 are bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' So, uηε → η→0 uε strongly in L2(QT), uηε,x → η→0 uε,x weakly in L2(QT), uηε,t → η→0 uε,t ∗ − weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−2(Ω)), vηεuηε → η→0 vεuε strongly in L2(QT), vηεu2 ηε → η→0 vεu2 ε strongly in L2(QT), (vηεuηε)t → η→0(vεuε)t ∗ − weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='33) we get η �� QT u v4vxxxψx dxdt ⩽ η T � 0 ∥√vu∥2∥v∥ 7 2∞∥vxxx∥2∥ψx∥∞ dt ⩽ C ηε− 1 2∥ε 1 2vxxx∥L2(QT )∥ψ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='H2(Ω)) ⩽ C ηε− 1 2, Travelling waves of a control-volume fibre coating model 30 η �� QT v4vxxxφx dxdt ⩽ ηε− 1 2∥v∥L∞(QT )∥ε 1 2vxxx∥L2(QT )∥ψx∥L2(QT ) ⩽ C ηε− 1 2∥ψ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='H1(Ω)) ⩽ C ηε− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The obtained convergences allow to pass to the limit as η → 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='28) with δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Passage to the limit ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Next, we study the compactness properties of the sequence (vε, uε) by using the estimates derived in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Taking into account (√v)t = −a(√vu)x + a 2 √vux, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='31), we deduce that {(√vε)t}ε>0 is uniformly bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−1(Ω)), and {√vε}ε>0 is uniformly bounded in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' So, due to the lemma of compactness embedding from [30, Corollary 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 85], we obtain that √vε → ε→0 √v uniformly in C 1 2 ,0( ¯QT), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='47) whence it follows that vε → ε→0 v uniformly in C( ¯QT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='48) Also, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='47), {uεvε}ε>0 is uniformly bounded in L2(QT), whence we find that {vε,t}ε>0 is uniformly bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' It implies vε,t → ε→0 vt ∗ − weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H−1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' From the boundedness of {hεΦ(hε,x)}ε>0 in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L1(Ω)) we deduce that {vε}ε>0 is uniformly bounded in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' W 1 1 (Ω)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='49) Travelling waves of a control-volume fibre coating model 31 whence, due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='48), it follows vε, hε → ε→0 v, h ∗ − weakly in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' W 1 1 (Ω)), and the set {|hx(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=', t)| = ∞} has measure zero for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By the boundedness {log(vε)}ε>0 in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' L1(Ω)) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='48), the set {v(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=', t) = 0} has measure zero for any t > 0, it follows that p(vε) → ε→0 p(v) holds for almost all x and for any t > 0, whence we arrive at ε ��� �� QT p(vε)ψx dxdt ��� ⩽ sup t∈[0,T] � ε � Ω p(vε) dx � T � 0 ∥ψx∥∞ dt ⩽ T 1 2 sup t∈[0,T] � ε � Ω p(vε) dx � ∥ψ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='H2(Ω)) → ε→0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Using the following estimates |κε| = ��h−1 ε f(hε,x)− � f3(hε,x) hε � hεf 3(hε,x)hε,xx �� ⩽ 1+ � hεf 3(hε,x)|hε,xx|, |κεvε,x| = 2 ��hε,xf(hε,x) − hε,x � hεf 3(hε,x) � hεf 3(hε,x)hε,xx �� ⩽ 2 + 2∥ � hε∥∞ � hεf 3(hε,x)|hε,xx|, due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='48) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='31), we have {κε}ε>0 and {κεvε,x}ε>0 are uniformly bounded in L2(QT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In particular, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='31), we get {χ{|hx|<∞}hε}ε>0 is uniformly bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)), whence hε → ε→0 χ{|hx|<∞}h weakly in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='50) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='48) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='50), we arrive at �� QT κεvεψx dxdt → ε→0 �� QT χ{|hx|<∞}(h−1f(hx) − f 3(hx)hxx)vψx dxdt, Travelling waves of a control-volume fibre coating model 32 �� QT κεvε,xψ dxdt → ε→0 �� QT χ{|hx|<∞}(h−1f(hx) − f 3(hx)hxx)vxψ dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='33), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='49), and ∥vxx∥2 ⩽ C∥vxxx∥ 3 5 2 ∥vx∥ 2 5 1 , ∥vx∥2 ⩽ C∥vxxx∥ 1 5 2 ∥vx∥ 4 5 1 , we have ε ��� �� QT vxxx(vxxψ + 2vxψx + vψxx) dxdt ��� ⩽ ε T � 0 ∥vxxx∥2∥vxx∥2∥ψ∥∞ dt+ 2ε T � 0 ∥vxxx∥2∥vx∥2∥ψx∥∞ dt + ε T � 0 ∥vxxx∥2∥ψxx∥2∥v∥L∞(QT ) dt ⩽ C ε 1 5∥ε 1 2vxxx∥ 8 5 L2(QT )∥vx∥ 2 5 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='L1(Ω))∥ψ∥L5(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='L∞(Ω))+ C ε 2 5∥ε 1 2vxxx∥ 6 5 L2(QT )∥vx∥ 4 5 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='L1(Ω))∥ψx∥L 5 2 (0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='L∞(Ω))+ ε 1 2∥ε 1 2vxxx∥L2(QT )∥ψ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='H2(Ω)) ⩽ C ε 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='31), {uε}ε>0 is uniformly bounded in L2(QT), and {χ{v>0}uε}ε>0 is uniformly bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' H1(Ω)), whence uε → ε→0 u weakly in L2(QT), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='51) vεuε,x → ε→0 χ{v>0}vux weakly in L2(QT), �� QT vεu2 εψx dxdt → ε→0 �� QT χ{v>0}vu2ψx dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='48) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='51), we get uεvε → ε→0 uv weakly in L2(QT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The obtained convergences allow to pass to the limit as ε → 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='28) with δ = η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' As a result, we obtain a weak solution (v, u) in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling wave solutions Next, we focus on the travelling wave solutions to the control-volume model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Specifically, we look for a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5)– (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) in the form: u(x, t) = U(ξ), v(x, t) = V (ξ) = H2(ξ) − 1, where ξ = x − s t, where s is the propagation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Substituting the ansatz into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5)– (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6), we obtain the system of travelling wave ODEs for (U(ξ), V (ξ)) for 0 ≤ ξ ≤ L, − s U ′ + aU U ′ + b κ′ = c (V U ′)′ V + 1 − U g(H), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) − s V ′ + a(U V )′ = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) subject to the L-periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We also impose the following mass constraint L � 0 H2(ξ) dξ = M > 0, where M is related to the mass M defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7) by M = M + L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' To study the structure of travelling wave solutions, we first consider the case when the film profile touches down to zero at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' That is, we assume that V (0) = V (L) = 0, or, equivalently, H(0) = H(L) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3) From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) it follows that − V (s − a U) = C0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4) whence by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='3) we obtain C0 = 0, and U(ξ) becomes a trivial solution U ≡ Uc := s a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5) Travelling waves of a control-volume fibre coating model 34 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' There exists s > 0 such that the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) has a periodic solution (H, U) such that H(0) = H(L) = 1, H′(0) = H′(L) = 0, where the average fluid film radius ¯M := M L = L� 0 H2(y) g(H(y)) dy L� 0 dy g(H(y)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' If g(H) = H2−1 (the plug flow case), then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6) implies ¯M = s a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Since U is a trivial solution satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5), the ODE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) reduces to b κ′ = 1 − Uc g(H) ⇔ b � f(H′)H−1 − f 3(H′)H′′�′ = 1 − Uc g(H), whence f 3(H′)H′′ − f(H′)H−1 = F(ξ) := b−1 ξ � 0 � Uc g(H(y)) − 1 � dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7) By periodicity, we find that F(0) = F(L) = 0 ⇒ Uc = � 1 L L � 0 dy g(H(y)) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7) by H H′ ̸= 0, we deduce that [Hf(H′)]′ = −HH′F(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8) We will look for the first integral to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8) in the form: f(H′) = A(ξ)H + B(ξ)H−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9) Travelling waves of a control-volume fibre coating model 35 where A(0) = A(L) and B(0) = B(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8), we find that A(ξ) = − 1 2F(ξ), B(ξ) = B(0) + 1 2 ξ � 0 F ′(y)H2(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By B(0) = B(L), we get M = Uc L � 0 H2(y) g(H(y))dy = � 1 L L � 0 dy g(H(y)) �−1 L � 0 H2(y) g(H(y))dy ⇔ M L = L� 0 H2(y) g(H(y)) dy L� 0 dy g(H(y)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' As a result, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='9) we arrive at [H′]2 = 1 − [A(ξ)H + B(ξ)H−1]2 [A(ξ)H + B(ξ)H−1]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Furthermore, if we have H(0) = H(L) = |B(0)| = 1, then we have H′(0) = H′(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Next, we consider a general travelling wave solution that satisfies the periodic boundary condition V (0) = V (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='10) In this case, the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4) implies that U = Uc + C0 a V ∀ C0 ∈ R1, where Uc := s a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='11) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' There exist s and C0 such that the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) has at least one periodic solution (H, U) satisfying H(0) = H(L), H′(0) = H′(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) it follows that � b κ + a 2(U − Uc)2�′ = c (V U′)′ V + 1 − U g(H), Travelling waves of a control-volume fibre coating model 36 whence, due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='11), we obtain � b κ + C2 0 2a V −2�′ = − c C0 a 1 V � V ′ V �′ + 1 − Uc g(H) − C0 a V g(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='12) Let C0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Then we have � Hf(H′) + C2 0 4abV −1�′ = −HH′G(ξ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='13) where G(ξ) := b−1 ξ � 0 Uc g(H(y)) + C0 a V (y)g(H(y)) − 1 + c C0 a 1 V � V ′ V �′ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' By imposing the periodicity G(0) = G(L) = 0, we obtain Uc L � 0 dy g(H(y)) + C0 a L � 0 � 1 V (y)g(H(y)) + c V −3(y)V ′(y)2� dy = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='14) Integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='13), we deduce that f(H′) = A(ξ)H + B(ξ)H−1 − C2 0 4abH−1V −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='15) where A(ξ) = − 1 2G(ξ), B(ξ) = B(0) + 1 2 ξ � 0 G′(y)H2(y) dy such that A(0) = A(L) and B(0) = B(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' From B(0) = B(L), it follows that Uc L � 0 H2(y)dy g(H(y)) + C0 a L � 0 � H2(y) V (y)g(H(y)) + c V −3(y)V ′(y)2� dy = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 37 So, we find that Uc = 1 ∆ � L L � 0 H2(y)dy V (y)g(H(y))−M L � 0 dy V (y)g(H(y))+c(L−M) L � 0 V −3(y)V ′(y)2 dy � , C0 a = − 1 ∆ � L L � 0 H2(y)dy g(H(y)) − M L � 0 dy g(H(y)) � , where ∆ = � L � 0 dy g(H(y)) �� L � 0 H2(y)dy V (y)g(H(y)) � − � L � 0 H2(y)dy g(H(y)) �� L � 0 dy V (y)g(H(y)) � − c � L � 0 V (y)dy g(H(y)) �� L � 0 V −3(y)V ′(y)2 dy � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' As a result, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='15) we arrive at H′(ξ)2 = 1 − � A(ξ)H + B(ξ)H−1 − C2 0 4abH−1V −1�2 � A(ξ)H + B(ξ)H−1 − C2 0 4abH−1V −1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Furthermore, if H′(0) = H′(L) = 0 and H(0) = H(L) > 1, then we have B(0)H−1(0) − C2 0 4abH−1(0)V −1(0) = 1 ⇒ (H2(0) − 1)(H(0) − B(0)) = − C2 0 4ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' This equation with respect to H(0) has two solutions provided that B(0) > 1 and C2 0 4ab < 1 27 � (B(0) + � B2(0) + 3)2 − 9 � (2B(0) − � B2(0) + 3), and one solution if C2 0 4ab = 1 27 � (B(0) + � B2(0) + 3)2 − 9 � (2B(0) − � B2(0) + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Numerical studies In this section, we numerically investigate the coupled PDE system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) – (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4) to explore the fibre coating dynamics and verify the analytical results in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Following the work of Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' [23], we specify the form of the function g(h) based on two models the plug flow model and the laminar flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For the plug flow model, we set g(h) based on the form in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For the laminar flow model, the function g(h) takes the form in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Firstly, we numerically investigate the travelling wave solutions (H(ξ), U(ξ)) that satisfy the coupled ODE system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) - (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) with the mass constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7), � L 0 V (ξ) dξ = � L 0 (H2(ξ)−1) dξ = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We apply Newton’s method to solve this nonlinear eigenvalue problem, where the speed s is treated as an unknown variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The coupled differential equations are discretized for 0 ≤ ξ ≤ L with periodic boundary conditions on H and U by second-order center finite differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' An additional constraint H(ξ∗) = H∗ for some 0 ≤ ξ∗ ≤ L is imposed to guarantee the local uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Figure 1 presents typical travelling wave solutions (H(ξ), U(ξ)) corresponding to two cases for the plug flow model and two cases for the laminar flow model: (a) Plug flow: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2, b = 10, c = 1 with travelling speed s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='396 (b) Plug flow: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4, b = 12, c = 3 with travelling speed s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='517;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (c) laminar flow a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5, b = 13, c = 4 with travelling speed s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='482;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (d) laminar flow a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1, b = 11, c = 4 with travelling speed s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' A fixed domain size L = 20 and mass constraint M = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8 are set for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The profiles are shifted so that the maximum of the droplet peaks are located at ξ = L/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' This comparison shows that in a fixed domain with equal volumes, the travelling waves for the plug flow model have more prominent peaks and higher velocity magnitude than those obtained from the laminar flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 39 1 2 3 4 0 5 10 15 20 H ξ (a) (b) (c) (d) 0 2 4 6 8 0 5 10 15 20 U ξ (a) (b) (c) (d) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Typical travelling wave profiles (left) H(ξ) and (right) U(ξ) for two plug flow cases ((a) and (b)) and two laminar flow cases ((c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Next, we study the transient PDE solutions of the governing model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) - (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4) and verify the derived energy and entropy estimates in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' To numerically solve the coupled fourth-order PDEs, we use the Keller box method [16] to decompose the model into a system of first-order differential equations, k = hx, p = kx, w = ux, ut + a �u2 2 � x + b � f(k)h−1 − f 3(k)p � x = c[(h2 − 1)w]x h2 − 1 + 1 − u g(h), 2hht + a[u(h2 − 1)]x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) Starting from the initial fluid film radius and the initial velocity h(x, 0) = h0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 sin(2πx/L), u(x, 0) = g(h(x, 0)), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) we then solve the system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) using the fully implicit second-order centered finite differences over the domain 0 ≤ x ≤ L, with periodic boundary conditions imposed on both u and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For all PDE simulations, we keep the domain size L = 20 and h0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29, so that the mass M = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 40 1 2 3 4 0 5 10 15 20 h x − xmax + L/2 h(x, 0) h(x, t) H(ξ) 0 2 4 6 8 0 5 10 15 20 u x − xmax + L/2 u(x, 0) u(x, t) U(ξ) 103 105 107 0 20 40 60 80 t E(t) + I(t) C0(t) 10−1 102 105 0 20 40 60 80 t � L 0 v2 x/v dx C3(t) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Dynamics of plug flow with (top left) h(x, t) and (top right) u(x, t) starting from initial profiles (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) with h0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29, showing that the PDE solution approaches a travelling wave solution (H(ξ), U(ξ)) satisfying equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) - (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) with the velocity s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The solutions are shifted so that the maximums are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The corresponding energy (bottom left) satisfies the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1), E(t) + I(t) < C0(t), where I(t) = c �� Qt vu2 x dxdt + �� Qt u2v g(h) dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The entropy (bottom right) satisfies the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7), � L 0 v2 x/v dx < C3(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The system parameters are L = 20, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2, b = 10, c = 1 with g(h) = h2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The top two plots in Figure 2 show the dynamics of (h(x, t), u(x, t)) for the plug flow case, where the PDE solution converges to a travelling wave solution (H(ξ), U(ξ)) that satisfies the ODE system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) - (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) with the velocity s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The solution profiles are shifted by x → x − xmax(t) + L/2, where xmax(t) is the location of the peaks Travelling waves of a control-volume fibre coating model 41 of the travelling waves in time, so that the peaks are aligned as the wave evolves and travels to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The system parameters are given by a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2, b = 10, c = 1 with g(h) = h2 − 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4), and the traveling wave corresponds to the case (a) presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' We also numerically verify the analytically derived energy and entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In Figure 2 (bottom left), we show that the energy estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) E(t) + I(t) < C0(t) is satisfied as the transient PDE solution approaches the travelling wave profile in time, where I(t) = c �� Qt vu2 x dxdt + �� Qt u2v g(h) dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 2 (bottom right) presents the numerically evaluated integral L� 0 v2 x v dx and the upper bound C3(t) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8) for the dynamic PDE solution, indicating that the entropy estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7), L � 0 v2 x v dx < C3(t), is also satisfied in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Here, we set ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5 in the definition of C3(t) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8), and this estimate holds for any ϵ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Figure 3 shows a similar numerical study for the laminar flow case with the system parameters a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1, b = 11, and c = 4, and the function g(h) is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In this case, the laminar flow fluid radius h(x, t) and velocity u(x, t) starting from identical initial data used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 2 converge to a slowly-moving travelling wave parametrized by (H(ξ), U(ξ)) with the speed of propagation s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 3 (top panel)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The obtained travelling wave corresponds to the case (d) presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Similar to the plug-flow case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 2, the laminar flow solution also satisfies the energy and entropy estimates, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 3 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Travelling waves of a control-volume fibre coating model 42 1 2 3 0 5 10 15 20 h x − xmax + L/2 h(x, 0) h(x, t) H(ξ) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='8 1 0 5 10 15 20 u x − xmax + L/2 u(x, 0) u(x, t) U(ξ) 10−1 102 105 108 0 100 200 300 400 t E(t) + I(t) C0(t) 10−1 102 105 0 100 200 300 400 t � L 0 v2 x/v dx C3(t) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Dynamics of laminar flow (top left) h(x, t) and (top right) u(x, t) starting from initial profiles (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) with h0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='29, showing that the PDE solution approaches a travelling wave solution (H(ξ), U(ξ)) satisfying equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1) - (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) with the velocity s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The solutions are shifted so that the maximums are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Again, the corresponding energy plot (bottom left) shows that the energy satisfies the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1), E(t) + I(t) < C0(t), where I(t) = c �� Qt vu2 x dxdt + �� Qt u2v g(h) dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The entropy (bottom right) satisfies the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='7), � L 0 v2 x/v dx < C3(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The system parameters are L = 20, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1, b = 11, c = 4, g(h) = I(h)/(h2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Conclusions The main contribution of this paper is the proof of the existence of weak solutions to the coupled PDE system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='2) for the control- volume fibre coating model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' This result establishes the analytical REFERENCES 43 foundation for the control-volume model in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' The a priori energy-entropy functional estimates used in the proof also provide a possible pathway for showing the regularity of solutions in similar coupled PDE systems in other fibre coating models [13, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' In contrast to the work of Bresch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' [4] and Kitavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' [17], for the proof of existence, we use another approximation of the continuity equation by the family of thin film equations (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content='18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' This new idea can be applied for the analysis of other systems with the same structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Typical numerical simulations of the PDE model are presented to support the analytical results, with a focus on the travelling wave solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' For future studies, it would be interesting to further investigate the convergence criteria of PDE solutions to travelling wave solutions and other coherent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Acknowledgements H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Ji was supported by NC State University Faculty Research and Professional Development Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Bernis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Friedman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Higher order nonlinear degenerate parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Journal of differential equations, 83(1):179– 206, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Bresch and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Desjardins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Communications in mathematical physics, 238(1):211–223, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Bresch and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' Desjardins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids.' metadata={'source': 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transfer between thin liquid films flowing down a vertical string in the Rayleigh-Plateau instability regime and a counterflowing gas stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} +page_content=' International Journal of Heat and Mass Transfer, 108:830–840, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQf3AKg/content/2301.02720v1.pdf'} diff --git a/WdAyT4oBgHgl3EQf8_qr/content/tmp_files/2301.00867v1.pdf.txt b/WdAyT4oBgHgl3EQf8_qr/content/tmp_files/2301.00867v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0d23943419a04494aac265a7d512b428e78b6ea --- /dev/null +++ b/WdAyT4oBgHgl3EQf8_qr/content/tmp_files/2301.00867v1.pdf.txt @@ -0,0 +1,1951 @@ +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary +in Chronological Order +XIUYING CHEN∗, Computational Bioscience Reseach Center, King Abdullah University of Science and Technology +MINGZHE LI∗, Wangxuan Institute of Computer Technology, Peking University +SHEN GAO, Wangxuan Institute of Computer Technology, Peking University +ZHANGMING CHAN, Wangxuan Institute of Computer Technology, Peking University +DONGYAN ZHAO, Wangxuan Institute of Computer Technology, Peking University +XIN GAO, Computational Bioscience Reseach Center, King Abdullah University of Science and Technology +XIANGLIANG ZHANG, 1 University of Notre Dame; 2 King Abdullah University of Science and Technology +RUI YAN†, Gaoling School of Artificial Intelligence, Renmin University of China +Nowadays, time-stamped web documents related to a general news query floods spread throughout the Internet, and timeline +summarization targets concisely summarizing the evolution trajectory of events along the timeline. Unlike traditional document +summarization, timeline summarization needs to model the time series information of the input events and summarize important +events in chronological order. To tackle this challenge, in this paper, we propose a Unified Timeline Summarizer (UTS) that can generate +abstractive and extractive timeline summaries in time order. Concretely, in the encoder part, we propose a graph-based event encoder +that relates multiple events according to their content dependency and learns a global representation of each event. In the decoder +part, to ensure the chronological order of the abstractive summary, we propose to extract the feature of event-level attention in +its generation process with sequential information remained and use it to simulate the evolutionary attention of the ground truth +summary. The event-level attention can also be used to assist in extracting summary, where the extracted summary also comes in +time sequence. We augment the previous Chinese large-scale timeline summarization dataset and collect a new English timeline +dataset. Extensive experiments conducted on these datasets and on the out-of-domain Timeline 17 dataset show that UTS achieves +state-of-the-art performance in terms of both automatic and human evaluations1. +CCS Concepts: • Information retrieval → Summarization. +Additional Key Words and Phrases: Timeline Summarization, Extractive Summarization, Abstractive Summarization +∗Equal contribution. Ordering is decided by a coin flip. +†Corresponding Author: Rui Yan (ruiyan@ruc.edu.cn) +1https://github.com/iriscxy/Unified-Timeline-Summarizer +Authors’ addresses: Xiuying Chen, Computational Bioscience Reseach Center, King Abdullah University of Science and Technology, xiuying.chen@kaust. +edu.sa; Mingzhe Li, Wangxuan Institute of Computer Technology, Peking University, li_mingzhe@pku.edu.cn; Shen Gao, Wangxuan Institute of Computer +Technology, Peking University, shengao@pku.edu.cn; Zhangming Chan, Wangxuan Institute of Computer Technology, Peking University, zhangming. +chan@pku.edu.cn; Dongyan Zhao, Wangxuan Institute of Computer Technology, Peking University, zhaody@pku.edu.cn; Xin Gao, Computational +Bioscience Reseach Center, King Abdullah University of Science and Technology, xin.gao@kaust.edu.sa; Xiangliang Zhang, 1 University of Notre Dame; 2 +King Abdullah University of Science and Technology, xzhang33@nd.edu; Rui Yan, Gaoling School of Artificial Intelligence, Renmin University of China, +ruiyan@ruc.edu.cn. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2022 Association for Computing Machinery. +Manuscript submitted to ACM +Manuscript submitted to ACM +1 +arXiv:2301.00867v1 [cs.CL] 2 Jan 2023 + +2 +Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +ACM Reference Format: +Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan. 2022. Follow the +Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order. ACM Transactions on Information Systems +1, 1, Article 1 (January 2022), 30 pages. https://doi.org/10.1145/3517221 +1 +INTRODUCTION +The rapid growth of World Wide Web means that time-stamped document floods spread throughout the Internet. +General search engines simply return web pages ranked by query relevance, but they are not quite capable of handling +ambiguous intentioned queries, such as a query about evolving news “COVID-19”. People may have a myriad of general +interests about the beginning, the evolution, or the most up-to-date situation, while simply ranking the returned +webpages according to their relevance is insufficient. In many cases, readers are tired of navigating every document +in the overwhelming collection: they want to monitor the evolution trajectory of hot topics by simply browsing. +Summarization is an ideal solution to provide a condensed, informative document reorganization for a faster and +better representation of news evolution. Timeline summary temporally summarizes evolutionary news as a series of +individual but correlated component summaries and hence offers an option to understand the big picture of a developing +situation [75]. +Existing timeline summarization approaches such as [34, 53, 75] are all based on extraction methods. However, these +methods rely on human-engineered features and sophisticated abilities that are crucial to high-quality summarization, +such as paraphrasing, generalization, or the incorporation of real-world knowledge, which are possible only in an +abstractive framework. Recently, with the emergence of strong generative neural models for text [4], abstractive +techniques are also becoming increasingly popular. Hence, we propose the abstractive timeline summarization task in +our early work [10], which aims to concisely paraphrase the event information in the input article. An example case +is shown in Table 1, where the article consists of events of a great entertainer in different periods, and the summary +correctly summarizes the important events from the input article in order. +Abstractive summarization approaches including [24, 28, 56, 80] have been proven to be useful in traditional +summarization task. However, unlike traditional document summarization, the timeline summarization dataset consists +of a series of time-stamped events, and it is crucial for the timeline summarization model to capture this time series +information to better guide the chronological summary generation process. Besides, the fidelity problem is also of vital +importance for timeline summarization, where mixing the information of different events leads to a bad summary. Take +the example in Table 1 for example, the bad summary confuses the birthplace and the residence, the first album, and +the best-selling album of the celebrity. Herein, the good summary is the ground truth summary from our dataset, and +the bad summary is a wrong summary with typical errors we found in a preliminary experiment. As we found in the +experiment, such infidelity phenomena is a commonly-faced problem in summarization tasks. +To tackle the above challenges, in our previous work [10], we come up with a Memory-based Timeline Summarization +(MTS) model. Specifically, we first use an event embedding module with selective reading units to embed all events. +Then, we propose a key-value memory module storing time-series information to guide the summary generation +process. Concretely speaking, the key in the memory module is the time position embedding that represents the time +series information, and the values are the corresponding event representations. The value item includes local and global +representation, where local value is the output from the event embedding module, and global value is taken from the +average local representation. Keys together form a timeline and we use the time position of events on the timeline to +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +3 +Events +Michael Jackson (dubbed as “King of Pop”) was born on August 29, 1958 in Gary, Indiana. He is +the seventh child in his family. +In 1971, Jackson released his first solo “got to be there”, marking the beginning of his solo +career. +In late 1982, Jackson’s sixth album, “Thriller”, was released, where videos "Beat It", "Billie Jean" +in it are credited with breaking racial barriers and transforming the medium into an art form +and promotional tool. +In March 1988, Jackson built a new home named Neverland Ranch in California, where more +than 100 arcade machines were stored here. +In 2000, Guinness World Records recognized him for supporting 39 charities and donated more +than 300 million dollars to charities in his own name, more than any other entertainer. +Bad summary +Michael Jackson was born on August 29, 1958 in Gary, California. In 1971, his first album +“Thriller” was released. In 2000, Guinness World Records recognized him for supporting 39 +charities. +Good summary +Michael Jackson was born on August 29, 1958 in Gary, Indiana. His sixth album “Thriller” was +released in 1982. In 2000, Guinness World Records recognized him for supporting 39 charities. +Table 1. Example of timeline summarization. The text in pink demonstrates time stamp, and text in blue demonstrates wrong event +description. Events are split by lines. +guide the generation process. Finally, in each decoding step, we introduce event-level attention and use it to determine +word-level attention to avoid confusion between events. +In MTS, the time information is captured in an implicit and indirect way. MTS stores the time position embedded in +the memory and hopes the decoder will learn to attend to the correct time position in the training process. However, that +strategy is rather weak supervision, where it is hard to verify and ensure the decoder indeed captures the time-sequential +information. In this work, we take one step further and improve our previously proposed MTS framework with explicit +timeline guidance modeling. In other words, we carefully design a strategy that lets the time information be a clear +guidance signal for the summarization process. +Overall, in this paper, we propose a novel Unified Timeline Summarizer (UTS) that can generate abstractive and +extractive timeline summaries in time order. For the abstractive part, concretely, in the encoder part, we first propose a +graph-based event encoder that relates multiple events according to their content dependency and learns a representation +of each event. The motivation is that the importance of each event and whether it should be included in the summary +does not only depend on itself but also is related to other events. Take Table 1 for example, Jackson releases his first +solo album might be an important event, but its importance is weakened by his “Thriller” album that breaks the racial +barriers. Hence, the representation from the graph encoder incorporates global information from other events, thus +is used to replace the old global representation in the memory. In the decoder part, to avoid the situation in the bad +summary in Table 1, where it confuses the birthplace and the residence because the model is not sensitive to the timeline, +we propose a summary decoder that emphasizes the time information. Concretely, to ensure the chronological order +of the abstractive summary, we propose to extract the feature of event-level attention in its generation process with +sequential information remained and use it to simulate the evolutionary attention of the ground truth summary. +Manuscript submitted to ACM + +4 +Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +In terms of the extractive part, we present a sentence embedding module to encode each sentence. Next, a sentence +extractor sequential selects important sentences to be included in the summary. The event-level attention can also +be used to assist in extracting summary in this process, where we devise a time-aware inconsistency loss function to +penalize the inconsistency between abstractive attention and extractive attention. Note that the extractive summary is +extracted one by one, thus the extracted summary also comes in time sequence. +We empirically compare MTS and UTS on the public dataset2 proposed by our early work [10]. This is a large-scale +real-world timeline summarization dataset, which consists of a series of time-stamped events and the corresponding +summary. Moreover, since this previous dataset only includes a timeline corpus about celebrities, we augment the +dataset with cases about social events. We also collect an English timeline summarization dataset. Experimental results +on these datasets and on out-of-domain Timeline17 dataset show that our newly proposed UTS model can significantly +outperform the existing methods. Particularly, UTS-abs yields 4.47% and 5.90% percentage point improvement in terms of +ROUGE-1 on celebrity and event timeline datasets compared with our early work MTS. In addition to the comprehensive +evaluation, we also evaluate our proposed graph encoder and attention mechanism by a fine-grained analysis. The +analysis reveals how the model leverages the explicit timeline information to guide the abstractive and extractive +summarization process and provides us insights on why they can achieve big improvement over state-of-the-art +methods. +Overall, our contributions can be summarized as follows: +• We propose a unified abstractive and extractive timeline summarization framework, where a time-aware inconsis- +tency loss function is proposed to unify these two processes. +• We propose a graph-based encoder that relates multiple events according to their content dependency and learns +the global representation of each event. +• We propose to use the evolutionary attention of the ground truth summary to guide both the abstractive and +extractive summary generation process, to ensure that the generated summaries follow strict time order. +• We also augment the first real-world large-scale timeline summarization dataset with social event corpus and corpus +in English3. Experiments conducted on the three datasets and the out-of-domain benchmark Timeline 17 dataset show +that our model outperforms all baselines, including state-of-the-art models. Experiments also verify the effectiveness of +each module in UTS as well as its interpretability. +The rest of the paper is organized as follows: We summarize related work in §2. We then formulate our research +problem in §3 and elaborate our approach in §4. §5 gives the details of our experimental setup and §6 presents the +experimental results. Finally, §7 concludes the paper. +2 +RELATED WORK +We detail related work on text generation methods, timeline summarization, extractive summarization, abstractive +summarization, unified summarization, and memory network. +2.1 +Text Generation Methods +In recent years, sequence-to-sequence (seq2seq) [61] based neural networks have been proved effective in generating a +fluent sentence. The seq2seq model is originally proposed for machine translation and later adapted to various natural +language generation tasks, such as text summarization [25, 38, 50, 65, 67] and dialogue generation [6, 63, 77, 79, 81]. +2https://github.com/yingtaomj/Learning-towards-Abstractive-Timeline-Summarization +3Data will be released in camera-ready version. +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +5 +Rush et al. [54] apply the seq2seq mechanism with attention model to the text summarization field. Then See et al. [56] +add copy mechanism and coverage loss to generate summarization without out-of-vocabulary and redundancy words. +The seq2seq architecture has also been broadly used in a dialogue system. Tao et al. [62] propose a multi-head attention +mechanism to capture multiple semantic aspects of the query and generate a more informative response. Yao et al. [77] +propose to use the content introducing method to solve the problem of generating a meaningless response. Wang et al. +[68] use three channels for widening and deepening the topics of interest and try to make the conversational model +chat more turns. +2.2 +Timeline Summarization +The timeline summarization task is firstly proposed by Allan et al. [2], where they define temporal summaries of news +stories as extracting a single sentence from each event within a news topic. Later, a series of works [72, 74, 75, 82] +further investigate timeline summarization task. Yan et al. [75] formally formulate the task as an optimization problem +via iterative substitution from a set of sentences to a subset of sentences that satisfies the above requirements, balancing +coherence/diversity measurement and local/global summary quality. In follow-up work, Yan et al. [72] propose to +model trans-temporal correlations among component summaries for timelines, using inter-date and intra-date sentence +dependencies, and present a novel combination. There are also works focusing on tweets summarization that is related +to timeline summarization. For example, [53] focus on the problem of selecting meaningful tweets given a user’s +interests; the dynamic nature of user interests, the sheer volume, and the sparseness of individual messages make +this a challenging problem. Specifically, they consider the task of time-aware tweets summarization, based on a user’s +history and collaborative social influences from “social circles”. Ghalandari and Ifrim [26] compare different timeline +summarization strategies using appropriate evaluation frameworks. For a more robust evaluation, they also present a +new timeline summarization dataset, which spans longer time periods than previous datasets. However, all the above +works are based on extractive methods, which are not as flexible as abstractive approaches. +The most similar work to ours is proposed by [58], where they construct a word-adjacency graph, and then generate +new sentences from this graph by finding paths from the sentence start node to the sentence end node. This is very +different from our neural-based approach, and we demonstrate the superiority of our model in the experiment. +2.3 +Extractive Summarization +Despite the focus on abstractive summarization, extractive summarization remains an attractive method. In extractive +summarization, Kobayashi et al. [30] propose a summarization method using document-level similarity based on word +embeddings. Meanwhile, Filippova et al. [20] use an RNN to delete words in a document for sentence compression. Yan +and Wan [76] propose more meaningful and informative units named frequent deep dependency sub-structure and +a topic-sensitive multi-task learning model for multi-doc summarization. Cheng and Lapata [13] propose a general +framework for single-document text summarization using a hierarchical article encoder composed with an attention- +based extractor. Following this, Nallapati et al. [47] propose a simple RNN-based sequence classifier that outperforms or +matches the state-of-art models at the time. Chen et al. [11] introduce a model which iteratively polishes the document +representation on many passes through the document, so as to produce a better summary. In another approach, Narayan +et al. [49] use a reinforcement learning method to optimize the ROUGE evaluation metric for text summarization. Ren +et al. [52] study the use of sentence relations, e.g., contextual sentence relations, title sentence relations, and query +sentence relations, so as to improve the performance of extractive summarization. +Manuscript submitted to ACM + +6 +Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +Recently, pre-trained language models are also applied in summarization for contextual word representations [39, 83]. +Another intuitive structure for extractive summarization is the graph, which can better utilize the statistical or linguistic +information between sentences. Early works focus on document graphs constructed with the content similarity among +sentences, like LexRank [19] and TextRank [43]. Some recent works aim to incorporate a relational prior into the +encoder by graph neural networks (GNNs) [78]. +2.4 +Abstractive Summarization +Recently, with the emergence of strong generative neural models for text [3], abstractive summarization is also becoming +increasingly popular [47, 56]. These models typically take the form of convolutional neural networks (CNN) or recurrent +neural networks (RNN). For example, Rush et al. [54] propose an encoder-decoder model which uses a local attention +mechanism to generate summaries. Nallapati et al. [48] further develop this work by addressing problems that had not +been adequately solved by the basic architecture, such as keyword modeling and capturing the hierarchy of sentence-to- +word structures. In follow-up work, Nallapati et al. [46] propose a new summarization model which generates summaries +by sampling a topic one sentence at a time, then producing words using an RNN decoder conditioned on the sentence +topic. Zhu et al. [85] tackles the cross-lingual summarization task, which aims at summarizing a document in one +language into another language. They propose a method inspired by the translation pattern in the process of obtaining +a cross-lingual summary. A series of works relies on prototype text to assist in summarization. Cao et al. [7] chose the +template with the highest similarity to the input sentence as a soft template to generate summaries. Following this, Gao +et al. [22] proposed to generate the summary with pattern based on prototype editing. Summarization techniques have +also been used in other tasks such as related work generation [9] and headline generation [36]. +2.5 +Unified Summarization +Unified summarization here means unifying extractive and abstractive summarization tasks together. It is a common +way to propose a multi-task framework that utilizes the benefits from one task to augment the performance of the +other task. For example, Hsu et al. [28] proposed a unified framework that takes advantage of both extractive and +abstractive summarization using an attention mechanism, which is a combination of the sentence-level attention. Chen +and Bansal [12] introduced a multi-step procedure, namely compression paraphrase, for abstractive summarization, +which first extracts salient sentences from documents and then rewrites them in order to get final summaries. Li et al. +[33] introduced a guiding generation model, where the keywords in source texts are first retrieved with an extractive +model. The most similar work to ours is [28], where they use sentence-level attention to modulate the word-level +attention such that words in less attended sentences are less likely to be generated. Their sentence-level attention is +static during the generation process, while in our model, the high-level attention changes in each decode step depending +on the current generated word which is more reasonable. +2.6 +Memory Network +The memory network proposed by Sukhbaatar et al. [59] generally consists of two components. The first one is a +memory matrix to save information (i.e., memory slots) and the second one is a neural network to read/write the +memory slots. The memory network has shown better performance than traditional long-short term memory network +in several tasks, such as question answering [21, 40, 51, 59], machine translation [42], text summarization [10, 29], +dialog system [16, 69], job-resume matching [73] and recommendation [18, 66, 84]. The reason is that the memory +network can store the information in a long time range and has more memory storage units than LSTM which has a +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +7 +Symbol +Description +𝑋 +a document consists of multiple events +𝑌 +ground truth timeline summary +ˆ𝑌 +generated timeline summary +𝑥𝑖 +𝑖-th event in input document +𝑤𝑖 +𝑗 +𝑗-th word in 𝑖-th event +𝑇𝑒 +number of input events +𝑇𝑖𝑤 +number of words in 𝑖-th event +𝑇𝑦 +number of words in ground truth summary +𝑇𝑦𝑠 +number of sentences in the ground truth timeline summary +𝑙𝑖 +extract label for 𝑖-th sentence in the summary +Table 2. Glossary. +single hidden state. Following memory network, there are many variations of memory network have been proposed, +i.e., key-value memory network [45] and dynamic memory network [31, 70]. Representative works include [23], where +they generate more meaningful answers in E-commerce question-answering by a read-and-write memory consisting of +selective writing units to conduct reasoning among these reviews. +In our work, we apply the key-value memory network on the timeline summarization task and fuse it into the +generation process. +3 +PROBLEM FORMULATION +Before detailing our answer generation model, we first introduce our notations listed in Table 2. +UTS takes a list of events 𝑋 = (𝑥1, ...,𝑥𝑇𝑒 ) as inputs, where𝑇𝑒 is the number of events. Each event 𝑥𝑖 is a list of words: +𝑥𝑖 = (𝑤𝑖 +1,𝑤𝑖 +2, ...,𝑤𝑖 +𝑇 𝑖𝑤), where 𝑤𝑖 +𝑗 is the 𝑗-th word in 𝑖-th event, and 𝑇𝑖𝑤 is the word number of event 𝑥𝑖. +In the abstractive part, UTS-abs aims to generate a summary ˆ𝑌 = ( ˆ𝑦1, ..., ˆ𝑦𝑇𝑦) that is not only grammatically correct +but also consistent with the event information such as occurrence place and time. Essentially, UTS-abs tries to optimize +the parameters to maximize the probability 𝑃(𝑌 |𝑋) = �𝑇𝑦 +𝑡=1 𝑃(𝑦𝑡 |𝑋), where 𝑌 = (𝑦1, ...,𝑦𝑇𝑦) is the ground truth +summary. +For the extractive part, UTS-ext targets at generating a score vector ˆ𝐿 = {ˆ𝑙1, . . . , ˆ𝑙𝑇𝑦𝑠 } for each sentence, where each +score denotes the sentence’s extracting probability. We convert the human-written summaries to gold label vector +𝐿 = {𝑙1, ...,𝑙𝑇𝑦𝑠 }, where 𝑙𝑖 ∈ {0, 1} denotes whether the 𝑖-th sentence is selected (1) or not (0). During the training +process, the cross-entropy loss is calculated between 𝐿 and ˆ𝐿, which is minimized to optimize ˆ𝐿. +4 +MODEL +4.1 +Overview +In this section, we introduce our Unified Timeline Summarizer (UTS) in detail. The overview of UTS is shown in Figure 1 +and can be split into two parts, one aims to generate an abstractive summary, and one targets selecting important +sentences as a summary. +Abstractive part includes: (1) Event Embedding Module (See § 4.2): To obtain the vector representations for each event, +we employ a recurrent network with Selective Reading Units (SRU) to learn the local representations. (2) Graph-based +Encoder (See § 4.3): The representations learned in the last module do not incorporate interaction between events. +Manuscript submitted to ACM + +8 +Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +Table 3. Comparision between MTS and UTS. +MTS +UTS +Event Embedding Module +SRU +SRU +Graph-based Encoder +- +Transformer +Time-Event Memory +Key-Value Memory +Key-Value Memory +Summary Generator +Editing Gate +Editing Gate +Sentence Embedding Module +- +SRU +Sentence Extractor +- +RNN +Unifier +- +Inconsistency loss +Hence, we propose a graph-based encoder to learn the global representation of each event incorporating the information +from other events and the relationship between them. (3) Time-Event Memory (See § 4.4): we propose a time-event +memory, which stores the local and global event representation, with time position keys together forming a timeline. (4) +Summary Generator (See § 4.5): eventually, we use an RNN-based decoder to generate the summary under the guidance +of event-level attention and word-level attention. +Extractive part includes: (5) Sentence Embedding Module (See § 4.6): this module embeds the sentence to a vector +representation in a similar way to the event embedding module. (6) Sentence Extractor (See § 4.7): the sentence extractor +selects the salient sentences as the summary following the sequential time order. +Additionally, we propose (7) Chronological-Attention Unifier (See § 4.8), to let the two parts complement each other +by unifying the attention distributions of abstractive parts and extractive parts. Concretely, we propose a time-aware +inconsistency loss to penalize the inconsistency between these two tasks. +Although some encoder and decoder modules in MTS are similar to UTS, there are three significant differences in +our UTS model compared with MTS: +(1) MTS encodes each event independently, without considering the information interaction between events. While +in UTS, we propose a graph encoder, which learns global representations for input events. +(2) We propose a unified timeline framework that can not only generate an abstractive summary, but also an +extractive summary. That is, only UTS includes the extractive part. +(3) We propose to unify the abstractive and extractive parts together, where the two tasks can benefit each other. +Specifically, we show the comparison between MTS and UTS in Table 3. +4.2 +Event Embedding Module +We first propose an event embedding module to obtain the word-level and event-level vector representations. To begin +with, we use an embedding matrix 𝑒 to map a one-hot representation of each word in 𝑥𝑖 into a high-dimensional vector +space. We denote 𝑒(𝑤𝑖 +𝑡) as the embedding representation of word 𝑤𝑖 +𝑡. We then employ a bi-directional recurrent neural +network (Bi-RNN) to model the temporal interactions between words: +←− +ℎ𝑖 +𝑡 = LSTMenc([𝑒(𝑤𝑖 +𝑡);𝑝𝑖], +←−−− +ℎ𝑖 +𝑡−1), +(1) +−→ +ℎ𝑖 +𝑡 = LSTMenc([𝑒(𝑤𝑖 +𝑡);𝑝𝑖], +−−−→ +ℎ𝑖 +𝑡−1), +(2) +ℎ𝑖 +𝑡 = +−→ +ℎ𝑖 +𝑡 + +←− +ℎ𝑖 +𝑡, +(3) +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +9 +Chronological +Event-level Attention +Chronological +Sentence-level Attention +... +Event1 +(1) Event +Embedding Module +(1) Event +Embedding Module +(1) Event +Embedding Module +Key +Local Value Global Value +Time1 +Time2 +Time3 +(4) Summary +Generator +(3) Time-Event Memory +Abstractive +Timeline Summary + (2) Graph-based Encoder +Sentences +(5) Sentence  +Embedding Module +(5) Sentence  +Embedding Module +(5) Sentence  +Embedding Module +(6) Summary  +Extractor +Extractive +Timeline Summary +(7) Chronological-Attention Unifier +... +... +... +Event2 +... +Event3 +Fig. 1. Overview of UTS. We divide our model into abstractive summarization part and extractive summarization part. Abstractive +part includes: (1) Event Embedding Module, (2) Graph-based encoder, (3) Time-Event Memory, and (4) Summary Generator. Extractive +part includes: (5) Sentence Embedding Module and (6) Sentence Extractor. Additionally, there is a (7) Chronological Attention Unifier +that unifies the two tasks. +where “;” denotes the concatenation between vectors, and ℎ𝑖 +𝑡 denotes the hidden state of 𝑡-th word in Bi-RNN for event +𝑥𝑖. To capture the sequential information of events, we randomly initialize a time position encoding vector 𝑝𝑖 of 𝑖-th +event to be included in the Bi-RNN input. +Apart from obtaining word representation ℎ𝑖 +𝑡, we also need to gain event representation. Simply taking the final +state of Bi-RNN ℎ𝑖 +𝑇 𝑖𝑤 as the representation of the whole event cannot fully capture the feature of the whole event. Thus, +we employ the selective reading module consisted of SRU proposed in [11] to gain new event representation 𝑎𝑖: +𝑠𝑖 +𝑡 = SRU(𝑠𝑖 +𝑡−1, [ℎ𝑖 +𝑡,ℎ𝑖 +𝑇 𝑖𝑤]), +(4) +𝑎𝑖 = 𝑠𝑖 +𝑇 𝑖𝑤, +(5) +where 𝑠𝑖 +𝑡 is the hidden state of 𝑡-th SRU cell in 𝑖-th event. At the high level, SRU is a modified version of GRU, which +replaces the update gate in original GRU [15] with a new gate taking each input ℎ𝑖 +𝑡 and coarse event representation ℎ𝑖 +𝑇𝑤 +into consideration. We omit the details here due to limited space and readers can refer to [11] for details. So far, we +obtain the representation of 𝑖-th event 𝑎𝑖 and 𝑡-th word in 𝑎𝑖, i.e., ℎ𝑖 +𝑡. +4.3 +Graph-based Encoder +The event representation 𝑎𝑖 in the previous section is calculated independently, without considering the information +flow between different events. However, the importance of each event and whether it should be included in the summary +does not only depend on itself but also is related to other events. For example, in Table 1, Jackson releases his first +solo album might be an important event, but its importance is weakened by his “Thriller” album that breaks the racial +Manuscript submitted to ACM + +10 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +barriers. Hence, we propose a graph-based encoder to learn the relationship between events and obtain a global event +representation that incorporates such information. +As shown in Figure 1, to embed relationship information, we set up the relation edges in our document modeling +graph. The relation edge in our graph is firstly initialized by the event representation: +𝑟𝑖,𝑗 = MLP𝑎([𝑎𝑖;𝑎𝑗]), +(6) +where MLP is a multi-layer perceptron. +Next, during the relation-aware encoding process, we incorporate the relation edge 𝑟𝑖,𝑗 into the final event represen- +tation by self attention operation: +𝑏𝑖 = RE(𝑎𝑖,𝑎∗,𝑟𝑖,∗), +(7) +where ∗ denotes all indexes between 1 and𝑇𝑒. This module is based on Transformer. Thus, we first introduce Transformer: +𝑏𝑖′ = Transformer(𝑎𝑖,𝑎∗). +(8) +Concretely, the first input is for query and the second input is for keys and values. Each output element, 𝑏𝑖′, is computed +as weighted sum of a linearly transformed input values: +𝑏𝑖′ = +𝑇𝑒 +∑︁ +𝑗=1 +𝛼𝑖,𝑗 +𝑔 +� +𝑎𝑗𝑊 𝑉 � +. +(9) +Each weight coefficient, 𝛼𝑖,𝑗 +𝑔 , is computed using a softmax function: +𝛼𝑖,𝑗 +𝑔 += +exp +� +𝛽𝑖,𝑗 +𝑔 +� +�𝑇𝑒 +𝑘=1 exp +� +𝛽𝑖,𝑘 +𝑔 +� . +(10) +𝛽𝑖,𝑗 +𝑔 +is computed using a compatibility function that compares two input elements: +𝛽𝑖,𝑗 +𝑔 += +� +𝑎𝑖𝑊 𝑄� � +𝑎𝑗𝑊 𝐾 �𝑇 +√ +𝑑 +, +(11) +where 𝑑 is the hidden dimension, and 𝑊 𝑄,𝑊 𝐾,𝑊 𝑉 ∈ R𝑑×𝑑 are parameter matrices. +RE is similar to Transformer, with two changes in Equation 9 and 11. Specifically, we modify Equation 9 to propagate +edge information to the sub-layer output: +𝑏𝑖 = +𝑇𝑒 +∑︁ +𝑗=1 +𝛼𝑖,𝑗 +𝑔 +� +𝑎𝑗𝑊 𝑉 +𝑟 + 𝑟𝑖,𝑗 � +. +(12) +In this way, the representation of each event is more comprehensive, consisting of its relation dependency information +with other events. In the meantime, when deciding the weight of each edge, i.e., 𝛽𝑖,𝑗 +𝑔 , we also incorporate relation edge +information, since close relationships can have a great impact on edge weight. Concretely, Equation 11 is changed to: +𝛽𝑖,𝑗 +𝑔 += +� +𝑎𝑖𝑊 𝑄 +𝑟 +� � +𝑎𝑗𝑊 𝐾 +𝑟 + 𝑟𝑖,𝑗 �𝑇 +√ +𝑑 +. +(13) +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +11 +Word-attention +MJ +was +born +in +... +Event1 +Event2 +Event3 +Chronological event-attention +Initial State +Key +Time1 +Time2 +Time3 +Local  +Value +Global +Value +Update +Fig. 2. An overview of the summary generator in the abstractive part. The summary generator generates the next word based on +word-level and event-level attention, as well as the key-value memory. +The intuition for Transformer architecture is that each input is not isolated, and its representation depends on other +inputs as well. In our augmented Transformer, i.e., graph-based encoder, the polished event representation 𝑏𝑖 follows +the same idea and expands the dependency between input documents. 𝑏𝑖 does not only depend on its corresponding +content but also depends on other inputs, as well as the relationships with others. +4.4 +Time-Event Memory +As stated in the Introduction, in the timeline dataset, the generated summary should capture the time-series information +to guide the chronological generation process. Hence, we propose a key-value memory module where keys together +form a timeline, and this time series information is used to guide the generation process as shown in Figure 2. +The key in this memory is the time position encoding 𝑝𝑖 introduced in § 4.2. We will use this key as time guidance +to extract information from the value part in the memory, which will be introduced in detail in § 4.5. The value part +stores event information of local aspect in local value and global aspect in global value. Local value simply stores the +event representation 𝑎𝑖, which means that only captures information from the current event. On the other hand, the +global value is responsible for learning the event feature from a global perspective, not only based on itself but also its +relationship with other events. Hence, it stores the graph-based encoder output, 𝑏𝑖. +4.5 +Summary Generator +To generate a consistent and informative summary, so as to avoid mixing information from different time stamps due to +unawareness of correct timeline, we propose an RNN-based decoder that incorporates outputs of time-event memory +module and event representation as illustrated in Figure 2. +Manuscript submitted to ACM + +12 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +Following [35], we randomly initialize an LSTM cell taking the concatenation of all event representations as input, +and use the output as decoder initial state: +ℎ′ +0 = LSTMini � +ℎ𝑐, [𝑎1; ...;𝑎𝑇𝑒 ] +� +, +(14) +where ℎ𝑐 is a random variable. +4.5.1 +Word-level attention. Next, following traditional attention mechanism in [4], we summarize the input document +into context vector 𝑐′ +𝑡−1 dynamically, and the 𝑡-th decoding step is calculated as: +ℎ′ +𝑡 = LSTMdec �ℎ′ +𝑡−1, [𝑐′ +𝑡−1;𝑒(𝑦𝑡−1)]� , +(15) +where ℎ′ +𝑡 is the hidden state of 𝑡-th decoding step. Context vector 𝑐′ +𝑡−1 is calculated as: +𝛼𝑡′ +𝑖,𝑗 = 𝑊 ⊺ +𝑎 tanh +� +𝑊𝑏ℎ′ +𝑡−1 +𝑊ℎℎ𝑖 +𝑗 +� +, +(16) +𝛼𝑡 +𝑖,𝑗 = exp +� +𝛼𝑡′ +𝑖,𝑗 +� +/ +𝑇𝑒 +∑︁ +𝑘=1 +�� +� +𝑇 𝑖 +𝑤 +∑︁ +𝑗=1 +exp +� +𝛼𝑡′ +𝑘,𝑗 +��� +� +, +(17) +𝑐′ +𝑡−1 = +𝑇𝑒 +∑︁ +𝑖=1 +�� +� +𝑇 𝑖 +𝑤 +∑︁ +𝑗=1 +𝛼𝑡 +𝑖,𝑗ℎ𝑖 +𝑗 +�� +� +, +(18) +where we first use the decoder state ℎ′ +𝑡−1 to attend to each states ℎ𝑖 +𝑗 which results in the attention distribution 𝛼𝑡 +𝑖,𝑗, +shown in Equation 17. ℎ𝑖 +𝑗 denotes the representation of 𝑗-th word in event 𝑥𝑖. Then we use the attention distribution +𝛼𝑡 +𝑖,𝑗 to obtain the weighted sum of document states as the context vector 𝑐′ +𝑡−1. +Context vector 𝑐′ +𝑡−1 here only takes the word-level attention into consideration without considering event-level +information. However, in timeline summarization, it is important for the model to be aware of which event it is currently +describing, or it may confuse information from different events and result in an unfaithful summary. Hence, we introduce +an event-level attention 𝛽 similar to the calculation of word-level attention and use it to adjust word-level attention: +𝛽𝑡′ +𝑖 = 𝑊 ⊺ +𝑐 tanh +� +𝑊𝑑ℎ′ +𝑡−1 +𝑊𝑒𝑎𝑖� +, +(19) +𝛽𝑡 +𝑖 = exp +� +𝛽𝑡′ +𝑖 +� +/ +𝑇𝑒 +∑︁ +𝑗=1 +exp +� +𝛽𝑡′ +𝑗 +� +, +(20) +𝛾𝑡 +𝑖,𝑗 = 𝛼𝑡 +𝑖,𝑗𝛽𝑡 +𝑖 . +(21) +The new context vector 𝑐𝑡 (replacing 𝑐′ +𝑡 in Equation 15) is now calculated as: +𝑐𝑡 = +𝑇𝑒 +∑︁ +𝑖=1 +�� +� +𝑇 𝑖 +𝑤 +∑︁ +𝑗=1 +𝛾𝑡 +𝑖,𝑗ℎ𝑖 +𝑗 +�� +� +. +(22) +4.5.2 +Event-level attention. Apart from using event-level attention to directly guide word-level attention, we also use it +to obtain the weighted sum of event representation to be concatenated in the projection layer in Equation 31: +𝑒𝑡 = +𝑇𝑒 +∑︁ +𝑖=1 +𝛽𝑡 +𝑖 𝑎𝑖. +(23) +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +13 +4.5.3 +Memory guidance. So far, we have finished the calculation of the context vectors. Next, we introduce how to +incorporate the guidance from memory. We first use hidden state ℎ′ +𝑡 to attend to each key in memory. As stated in § 4.4, +keys, i.e., time position embeddings, conform the timeline that represents the time series information. Thus, we let +the model take advantage of this sequential information, and calculate the relevance between position encoding and +current state as time-attention 𝜋(𝑝𝑖,ℎ′ +𝑡): +𝜋(𝑝𝑖,ℎ′ +𝑡) = exp(ℎ′ +𝑡𝑊𝑒𝑝𝑖)/ +𝑇𝑒 +∑︁ +𝑗=1 +exp(ℎ′ +𝑡𝑊𝑒𝑝 𝑗). +(24) +Time-attention is then used to gain the weighted sum of local value 𝑣1 and global value 𝑣2 in the memory: +𝑚1′ +𝑡 = +𝑇𝑒 +∑︁ +𝑖=1 +𝜋(𝑝𝑖,ℎ′ +𝑡)𝑣𝑖 +1, +(25) +𝑚2′ +𝑡 = +𝑇𝑒 +∑︁ +𝑖=1 +𝜋(𝑝𝑖,ℎ′ +𝑡)𝑣𝑖 +2. +(26) +𝑚1′ +𝑡 and 𝑚2′ +𝑡 stores information from different level, thus should play different roles in generator. +By a fusion gate, local value 𝑚1′ +𝑡 is changed to 𝑚1 +𝑡 and will be incorporated into the projection layer in Euqation 31. +𝑔1 +𝑡 = 𝑊𝑜 ([ℎ′ +𝑡;𝑐𝑡;𝑚1′ +𝑡 ]), +(27) +𝑚1 +𝑡 = 𝑔1 +𝑡 · 𝑚1′ +𝑡 . +(28) +We place the local value in the projection layer since 𝑚1 +𝑡 stores the detailed information rather than the global feature +in the input, thus should play an important part when generating each word. +As for the global value 𝑚2′ +𝑡 , it stores the global feature of the event in a different position, thus should influence the +whole generation process. Concretely, information from 𝑚2′ +𝑡 is fusioned into the decoding state ℎ′ +𝑡 by a gate: +𝑔2 +𝑡 = 𝑊𝑛([ℎ′ +𝑡;𝑐𝑡;𝑚2′ +𝑡 ]), +(29) +ℎ′ +𝑡 = 𝑔2 +𝑡 · ℎ′ +𝑡 + (1 − 𝑔2 +𝑡 ) · 𝑚2′ +𝑡 . +(30) +Finally, an output projection layer is applied to get the final generating distribution 𝑃𝑣 over vocabulary: +𝑃𝑣 = softmax +� +𝑊𝑣 [𝑚1 +𝑡 ;ℎ′ +𝑡;𝑐𝑡;𝑒𝑡] + 𝑏𝑣 +� +. +(31) +We concatenate the output of decoder LSTM ℎ′ +𝑡, the word context vector 𝑐𝑡, the event context vector 𝑒𝑡, and memory +vector 𝑚1 +𝑡 as the input of the output projection layer. +In order to handle the out-of-vocabulary (OOV) problem, we equip the pointer network [27, 56] with our decoder, +which enables the decoder capable of copying words from the source text. The design of the pointer network is the +same as the model used in [56], thus we omit this procedure due to limited space. +Our objective function in the abstractive part is the negative log likelihood of the target word𝑦𝑡, shown in Equation 32: +Labs = − +𝑇𝑦 +∑︁ +𝑡=1 +log 𝑃𝑣(𝑦𝑡). +(32) +The gradient descent method is employed to update the parameters in the abstractive part to minimize this loss function. +Manuscript submitted to ACM + +14 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +Word-level +Encoding +SRU +Sentence-level +Encoding +0 +SRU +SRU +Sentence Embedding Module +Summary Extractor +xN +Fig. 3. An overview of the sentence extractor in the extractive part. In each decoding step, a sentence is to be included in the summary +in sequence. +4.6 +Sentence Embedding Module +So far, we introduce the abstractive timeline summarization part. Next, we will introduce the extractive summarization +part in UTS, and how to unify these two tasks. +Our sentence embedding module takes inspiration from [11], where the embedding module also takes the form of a +hierarchical structure and consists of an iterative polishing process to better encoder the input document. Concretely, +we employ a new Bi-RNN to process each sentence and obtain the representation ˆℎ𝑖 +𝑡, denoting the 𝑡-th word in 𝑖-th +sentence. We use the last hidden state to represent the overall sentence representation, denoted as ˆℎ𝑖 +𝑇𝑤. The document +representation is initialized as the average of all sentence representations: +𝐷1 = tanh �� +� +𝑊 1 +𝑇𝑦𝑠 +𝑇𝑦𝑠 +∑︁ +𝑖=1 +� +ˆℎ𝑖 +𝑇𝑤 +� ++ 𝑏�� +� +. +(33) +Next, to model the sequential relationship between sentences and obtain a more comprehensive sentence representation, +we iteratively polish the sentence and document representations. For brevity, we take the first iteration as an example +to illustrate the process. Concretely, there is an RNN based on SRU (introduced in Equation 4) in the iteration: +ˆ𝑎𝑖 +1 = SRU( ˆ𝑎𝑖−1 +0 +, [ ˆℎ𝑖−1 +𝑇𝑤 , 𝐷1]), +(34) +𝐷2 = GRUiter( ˆ𝑎𝑇𝑦𝑠 +1 +, 𝐷1), +(35) +where ˆ𝑎𝑖 +1 is the hidden state of 𝑖-th SRU cell in the first iteration. In this way, we iteratively polish the sentence and +document representation. We use 𝐼 to denote the iteration number, thus the final representation for 𝑖-th sentence is ˆ𝑎𝑖 +𝐼 . +4.7 +Sentence Extractor +Different from previous work that builds a classifier to assign importance score to each sentence, we use an RNN +consisting of LSTM cells to select sentences, wherein each step a sentence is selected as illustrated in Figure 3. Following +traditional attention mechanism in [4], we summarize the input document sentences into context vector𝑐ext dynamically, +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +15 +Selected +Sentence +Number +Sentence +Number +Tile +Time-aware +Inconsistency  +Loss +Selected +Sentence +Number +Selected +Sentence +Number +Event +Number +Decode Step Number +Convolution +Pooling +Event-level Attention Map +in Summary Generator +Sentence-level Attention +Map in Sentence Extractor +Fig. 4. The illustration of the Chronological-Attention Unifier. After a convolution and a tile operation, the event-level attention in the +summary generation is compared with the sentence-level attention in sentence extractor, where a novel inconsistency loss function is +introduced to penalize the inconsistency between these two levels of attentions. +and the 𝑡-th decoding step is calculated as: +˜ℎ𝑡+1 = LSTMext( ˜ℎ𝑡, [𝑐ext +𝑡 +; ˆ𝑎𝑜𝑡 +𝐼 ]), +(36) +𝑐ext +𝑡 += +𝑇𝑦𝑠 +∑︁ +𝑖=1 +ˆ𝛽𝑖 +𝑡 ˆ𝑎𝑖 +𝐼, +(37) +𝑜𝑡 = argmax(MLP( ˜ℎ𝑡)), +(38) +where 𝑜𝑡 is the index of the selected in 𝑡 step, and ˆ𝑎𝑜𝑡 +𝐼 +is the hidden state of the previously selected sentence. ˆ𝛽𝑖 +𝑡 is the +attention weight on 𝑖-th sentence in 𝑡-th step, and is computed in a similar way to §4.5.1. Thus, the details are omitted +here due to limited space. +In this way, our extracted summary is generated in sequence, so as to better capture the sequential information in +the input: +Lext = − +𝑇𝑦𝑠 +∑︁ +𝑖=1 +log 𝑃𝑠 (𝑙𝑖) . +(39) +4.8 +Chronological-Attention Unifier +In both abstractive and extractive timeline summarization tasks, the attention on the input document should both +follow the time sequential order. Hence, it is intuitive to encourage these two levels of attention to be mostly consistent +with each other during training as an intrinsic learning target for free (i.e., without additional human annotation). +In §4.5.1, we propose the event-level attention 𝛽 in abstractive part, while in §4.7, the extractor pays sentence-level +attention ˆ𝛽 on the input. The event-level attention evolves each time a new word is predicted in the summary generator, +while the sentence-level attention evolves when a new sentence is selected. Hence, we first use a convolutional neural +network (CNN) to extract the evolving event attention feature from the generator. Concretely, as shown in Figure 4, a +Manuscript submitted to ACM + +16 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +convolution along the decode step number axis is conducted on the event-level attention map, and the new attention +matrix with the sentence-numbered axis is obtained. Then, for each sentence-select step, we duplicate the event-level +attention 𝛽𝑖 +𝑡 multiple times, where the duplicate number is the sentence number in the 𝑖-th event. +Finally, we would like the event-level attention to be high when the sentence-level attention is high. Hence, we +design the following time-aware inconsistency loss: +Linc = − 1 +𝑇𝑦 +𝑇𝑦𝑠 +∑︁ +𝑡=1 +log +� +1 +|K| +∑︁ +𝑡 ∈K +ˆ𝛽𝑡 × 𝛽𝑡 +� +, +(40) +where K is the set of top K attended sentences. This implicitly encourages the distribution of the sentence-level +attentions to be sharp and event-level attention to be high. To avoid the degenerated solution for the distribution of +sentence attention to be one-hot and event attention to be high, we include the original loss functions for training +the extractor (Lext in §4.7) and abstracter (Labs in §4.5). Note that this module is the only part that the extractor is +interacting with the abstracter. Our time-aware inconsistency loss facilitates our end-to-end trained unified model to be +mutually beneficial to both the extractor and abstracter. +5 +EXPERIMENTAL SETUP +5.1 +Research Questions +We list seven research questions that guide the experiments: RQ1 (See § 6.1): What is the overall performance of UTS? +Does it outperform other baselines on multilingual datasets? RQ2 (See § 6.2): What is the performance of our model on +out-of-domain classic timeline summarization dataset? RQ3 (See § 6.3): What is the effect of each module in UTS? Does +our multi-task framework help better summarization performance? RQ4 (See § 6.4): Is the time position embedding +useful so that the summary generator can attend to correct information in the time-event memory? RQ5 (See § 6.5): +Can event-level attention correctly guide word-level attention in decoding process in the abstractive part? RQ6 (See +§ 6.6): Are the chronological attentions successfuly unified in the abstractive and extractive summarization tasks? RQ7 +(See § 6.7): What is the influence of the parameter settings? +5.2 +Dataset +To our best knowledge, there are no large-scale summarization datasets for timeline. Hence, in our previous work [10], +we collect a large-scale timeline dataset from the world’s largest Chinese encyclopedia4. The character subsection of +this website consists of celebrities at all times and in all countries or lands. On each website page, there is a timeline +summary for each character, and in the character experience section of this page, each event is set as a paragraph with +explanation and details, which is selected as an input article. In the previous timeline works [72], they did not pre-select +important sentences from the event news articles as a way to test the summarization ability of the proposed model. +Hence, in our work, we did not preprocess the event paragraph as well, since these event paragraphs are similar to +news articles in content and in style. We filter out irrelevant content such as cited sources and figures. We did not +specifically extract time information from the input, because our model learns the time information in an implicit way, +instead of particularly encoding it. In total, the training dataset amounts to 169,423 samples with 5,000 evaluation and +5,000 test samples. On average, there are 353.79 words and 61.19 words in the article and summary respectively. +4https://baike.baidu.com/ +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +17 +Datasets +# docs (train/val/test) +avg. document length +avg. summary length +vocabulary size +words +sentences +words +sentences +document +summary +TL17 +4,650 +1,252.33 +63.76 +43.66 +2.73 +102,099 +6,725 +Celebrity TS +169,423/5,000/5,000 +353.79 +12.76 +61.19 +3.97 +444,725 +191,334 +Event TS +83,188/3,000/3,000 +495.19 +18.73 +141.62 +6.05 +1,083,249 +368,619 +Wiki TS +140,000/5,000/5,000 +606.65 +27.79 +79.19 +7.89 +1,029,617 +438,011 +Table 4. Comparison of summarization datasets with respect to overall corpus size, size of training, validation, and test set, average +document (source) and summary (target) length (in terms of words and sentences), and vocabulary size on both on source and target. +TS denotes Timeline Summarization. +Furthermore, in this work, we first augment the previous dataset with event timeline summarization cases. On +the Chinese encyclopedia, there is also a social event subsection that includes the developments of related events +over time. Concretely, in the development history section of each page, there are event paragraphs that describe the +development of the corresponding event, and there is also a corresponding timeline summary for these events. After +the same cleaning operation, we have 83,188 training cases, 3,000 validation, and 3,000 test samples. On average, there +are 495.19 words and 141.62 words in the article and summary respectively. +Note that the above two datasets are both in the Chinese language. To test the performance of our model on multi- +lingual datasets, we collect an English timeline summarization dataset from Wikipedia websites. Since there are no +character or event subsections in Wikipedia, we filter timeline pages by checking if there are multiple timestamps in +the summary and document on each website. Other preprocesses are similar to the Chinese encyclopedia. A human +evaluation on 200 sampled cases from the collected corpus shows that 196 cases are timeline document-summary pairs, +145 of which are about characters, and 51 are about events. +Since we have large-scale English summarization datasets, we can test the generalization ability of our model on +classic timeline summarization datasets, which are small-scale. Concretely, we report the performance of UTS on +out-of-domain dataset Timeline 17 (TL17) [64]. TL17 contains human-written timelines about topics such as civil wars +or the British Petroleum oil disaster, collected from major news outlets. Each topic also has a set of related news articles +scraped from the web. +The statistics of the four datasets are listed in Table 4. We also give timeline statistic information in Table 5. It +can be seen that compared with TL17 dataset, our three datasets are significantly larger. This again demonstrates the +necessity of our datasets, which are large enough to train a neural-based model. In terms of timeline-related attributes, +the summaries in our datasets have more date stamps in each sentence on average, which requires the summarization +model to be more time-aware. The average date number in the document input is smaller in our datasets, this is because +that our input document is shorter than TL17. However, the average sentences/dates ratio of our datasets is comparable +to TL17, proving the time attribute of our datasets. +5.3 +Comparison Methods +We first conduct an ablation study to prove the effectiveness of each module in UTS. Then, to evaluate the performance +of our proposed dataset and model, we compare it with the following baselines: +Abstractive baselines: +(1) Pointer-Gen [56] is an RNN based model with an attention mechanism and allows the system to copy words from +the source text via pointing for abstractive summarization. +Manuscript submitted to ACM + +18 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +Datasets +Document +Summary length +Compression +Avg dates +Avg sents/dates +Avg dates +Avg sents/dates +Sent +Date +TL17 +77.55 +1.22 +1.61 +1.14 +34.70 +32.46 +Celebrity TS +11.03 +1.18 +3.15 +1.10 +3.76 +3.50 +Event TS +13.09 +1.54 +5.55 +1.19 +3.03 +2.36 +Wiki TS +6.81 +3.99 +2.49 +3.17 +3.44 +2.74 +Table 5. Timeline-specific statistic attributes of our datasets and TL17 dataset. +(2) FTSum leverages open information extraction and dependency parse technologies to extract actual fact descriptions +from the source text [8]. Since there is no open information extraction tool in Chinese, we use POS tagging to extract +entities and verbs to replace them. +(3) Unified is a unified model combining the strength of extractive and abstractive summarization proposed in [28], +where sentence-level attention is used to modulate the word-level attention such that words in less attended sentences +are less likely to be generated. +(4) GPG is a model proposed by Shen et al. [57] which generates summaries by “editing” pointed tokens instead of hard +copying. The editing is performed by transforming the pointed word vector into a target space with a learned relation +embedding. +(5) SAGCopy is an augmented Transformer with a self-attention guided copy mechanism, which was proposed by Xu +et al. [71]. Specifically, they first identify the importance of each source word based on the degree centrality with a +directed graph built by the self-attention layer in the Transformer. They then use the centrality of each source word to +guide the copy process explicitly. +(6) MTS is the first abstractive timeline summarization framework proposed in our previous work [10]. This method +achieves state-of-the-art performance on the celebrity timeline summarization dataset. +Extractive baselines: +(1) Lead3 is an extractive baseline that concatenates the first-3 sentences of each source document as a summary. +(2) TextRank [44] is an unsupervised algorithm while sentence importance scores are computed based on eigenvector +centrality within weighted-graphs for extractive sentence summarization. +(3) ITS One of state-of-the-art extractive summarization models proposed in [11]. ITS iteratively polishes the document +representation on many passes through the document, so as to extract better summaries. +For testing our models on out-of-domain dataset, we compare with a number of traditional timeline summarization +baselines: +(1) Chieu [14] is an unsupervised baseline based on direct summarization. +(2) Martschat [41] greedily selects a combination of sentences from the entire collection, which maximizes submodular +functions for content coverage, textual and temporal diversity and a high count of date references. +(3) Tran[5] is an original date-wise timeline summarization approach, using regression for both date selection and +summarization, and using all sentences of a date as candidate sentences. +(4) Pubcount [26] is a simple date-wise baseline that uses the publication count to rank dates, and all sentences +published on a date for candidate selection. +(5) Datawise [26] uses supervised date selection, PM-MEAN for candidate selection and CENTROID-OPT for summa- +rization. +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +19 +(6) Clust [26] uses DATEMENTIONCOUNT to rank clusters, and CENTROID-OPT for summarization. +The performance of these baselines are consistent with the result from [26]. +5.4 +Evaluation Metrics +For evaluation metrics, we adopt ROUGE F1 score in [37] which is widely applied for summarization evaluation [11, 60]. +The ROUGE metrics compare the generated summary with the reference summary by computing overlapping lexical +units, including ROUGE-1 (unigram), ROUGE-2 (bi-gram), and ROUGE-L (longest common subsequence). +For the out-of-domain test dataset, we follow [26], and use the specific timeline evaluation metric, i.e., Alignment- +based ROUGE F1-score, and Date F1-score. Alignment-based ROUGE F1-score compares the textual overlap between a +system and a ground-truth timeline, while also considering the assignments of dates to texts. Date F1-score compares +only the dates of a system and a ground-truth timeline. +[55] notes that only using the ROUGE metric to evaluate summarization quality can be misleading. Therefore, we also +evaluate our model by human evaluation. Three highly educated participants are asked to score 100 randomly sampled +summaries generated by GPG, SAGCopy, MTS, and UTS-abs. Statistical significance of observed differences between the +performance of two runs are tested using a two-tailed paired t-test and is denoted using ▲(or ▼) for strong significance +for 𝛼 = 0.01. +5.5 +Implementation Details +We implement our experiments in TensorFlow [1] on NVIDIA GTX 1080 Ti GPU. For all experiments, our model has +256-dimensional hidden states and 128-dimensional word embeddings. Following See et al. [56], we do not pretrain +the word embeddings, instead, they are learned from scratch during training. We use a vocabulary of 50k words for +both source and target. For time-event memory, the dimension of the key, global value, and local value are 128, 512, +and 256 respectively. We initialize all of the parameters randomly using a uniform distribution in [-0.02, 0.02]. The +batch size is set to 16, and the event number is set to 8. For the abstractive part, during training and at test time we +truncate the article to 400 tokens and limit the length of the summary to 70 tokens. For the extractive part, we used +a greedy algorithm similar to [47] to obtain an oracle summary for each document to train extractive models. The +algorithm generates an oracle consisting of multiple sentences which maximize the ROUGE-2 score against the gold +summary. We limit the input sentence number to 24, the length of each input sentence to 20, and the number of selected +sentences to 4. For the chronological-attention unifier, we set 𝐾 to 3 for computing Linc. We use Adagrad optimizer [17] +as our optimizing algorithm and the learning rate is 0.15. (This was found to work best of Stochastic Gradient Descent, +Adadelta, Momentum, Adam, and RMSProp). We use gradient clipping with a maximum gradient norm of 2, but do +not use any form of regularization. We use loss on the validation set to implement early stopping. In decoding, we +employ a beam search with beam size 4 to generate a more fluent summary sentence. When testing our model on the +out-of-domain Timeline 17 dataset, for each example with S source input documents, we take the first 400/S tokens +from each source document. +For the training efficiency, it takes about 9.7 hours to train an epoch, and our model reaches the best performance after +only 3 epochs. While for baseline Pointer-Gen, it takes 7 hours to train an epoch, but it reaches the best performance +after 7 epochs. In particular, our model makes much quicker progress in the early phases of training. This demonstrates +the effectiveness of our unified model. In terms of testing, it takes 1.06 hours to generate summaries for all the cases in +the test dataset. We selected the top-3 checkpoints based on the evaluation loss on the validation set, and report the +averaged results on the test set. +Manuscript submitted to ACM + +20 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +Models +Celebrity Timeline Dataset +Event Timeline Dataset +Wiki Timeline Dataset +RG-1 +RG-2 +RG-L +RG-1 +RG-2 +RG-L +RG-1 +RG-2 +RG-L +Sentence extraction methods +Lead3 +32.36 +17.96 +30.99 +21.47 +9.26 +15.73 +25.35 +5.94 +20.56 +TextRank +32.27 +15.34 +30.86 +23.89 +10.43 +16.66 +24.98 +5.47 +22.40 +ITS +34.03 +18.20 +31.24 +27.94 +14.28 +20.39 +27.82 +5.91 +25.37 +Unified-ext +34.18 +18.29 +31.16 +28.06 +14.39 +20.47 +26.48 +5.82 +24.28 +UTS-ext +34.81 +22.26 +32.03 +29.12 +16.01 +23.06 +29.00 +6.64 +25.81 +Abstractive methods +Pointer-Gen +36.61 +21.35 +34.51 +22.56 +7.84 +21.00 +23.12 +5.07 +19.65 +FTSum +37.84 +21.47 +35.37 +23.41 +6.95 +21.66 +24.08 +5.80 +20.05 +Unified-abs +38.24 +21.95 +36.42 +23.58 +7.93 +21.95 +24.34 +5.84 +20.37 +GPG +38.43 +21.59 +36.38 +22.38 +7.77 +20.81 +24.71 +5.80 +20.85 +SAGCopy +38.64 +20.84 +36.41 +23.40 +7.95 +21.72 +26.00 +5.84 +22.01 +MTS +39.78 +22.24 +37.69 +23.89 +8.38 +21.97 +26.68 +5.88 +23.18 +UTS-abs +41.56 +23.95 +39.18 +25.30 +9.63 +23.28 +27.71 +5.92 +24.62 +Table 6. RQ1: ROUGE scores comparison between baselines. Models and baselines in the top half are extractive, while those in the +bottom half are abstractive. All our ROUGE scores have a 95% confidence interval of at most ±0.24 as reported by the official ROUGE +script. +6 +EXPERIMENTAL RESULTS +6.1 +Overall Performance +For research question RQ1, we examine the performance of our model and baselines in terms of ROUGE as shown +in table 6. Firstly, on the celebrity timeline dataset, abstractive models outperform extractive models by a substantial +margin on our datasets. We attribute this result to the observation that the gold summary of this dataset tends to use +new expressions to summarize the original input documents. This demonstrates the necessity of abstractive timeline +summarization approaches. Secondly, we compare our previous model MTS with recently-proposed baselines including +SAGCopy and GPG. These two baselines obtain lower ROUGE scores on our datasets than MTS, which demonstrates the +effectiveness of our previous model. Finally, based on MTS, our augmented model UTS-ext and UTS-abs achieves even +better performance. +Concretely, for the abstractive part, UTS-abs outperforms SAGCopy by 7.56%, 14.92% and 7.61%, and outperforms MTS +by 4.47%, 7.68% and 3.95% in terms of ROUGE-1, ROUGE-2 and ROUGE-L respectively on celebrity timeline dataset. +For the extractive part, our extractive method achieves about 2.23% points improvement on ROUGE-2 compared with +ITS on the celebrity timeline dataset. We attribute the improvement to two aspects: Firstly, the abstractive objective +can promote the recognition of important sentences for the extractive model with the chronological attention unifier +network. Besides, while extractive gold label sequences are obtained by greedily optimizing ROUGE-2 on the gold- +standard summary, gold labels may not be accurate. Joint learning of two objectives may correct some biases for the +extractive model due to the inaccurate labels. The above results prove the superiority of our model. Note that we mainly +compare our model with ITS, because our extractive part is mostly based on ITS. Our framework can be applied to +other extractive models, and theoretically, will bring benefits for both tasks. We leave it as future work. +Our human evaluation study assessed the overall quality of the summaries on the celebrity timeline dataset by +asking three highly educated participants to rank them taking into account the following criteria: Fluency (is the +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +21 +Fluency +Informativeness +Fidelity +GPG +2.59 +2.53 +2.39 +SAGCopy +2.64 +2.57 +2.43 +MTS +2.71 +2.58 +2.61 +UTS-abs +2.77▲ +2.62▲ +2.65▲ +Table 7. RQ1: Human evaluation comparison with main baselines on celebrity timeline dataset. +summary fluent and grammatical?), Informativeness (does the summary convey important facts about the topic in +question?), and Fidelity (is the summary faithful to the input?). We pick SAGCopy and GPG as baselines since their +performance is relatively high compared to other baselines. The rating score ranges from 1 to 3 and 3 is the best. The +results are presented in Table 7. We can see that our model performs much better than all baselines. In the fluency +indicator, our model achieves a high score of 2.77, which is higher than 2.59 of GPG and 2.64 of SAGCopy, indicating +that our model can reduce the grammatical errors and improve the readability of the summary. In the informativeness +indicator, our model is 0.05 better than SAGCopy. It indicates that our model can effectively capture salient information. +In the fidelity indicator, UTS-abs outperforms all baselines by a large margin, which indicates the multi-granularity +semantic information and joint learning with extractive summarization does help to avoid the unfaithful information +of the generated summary. It is worth noticing that the infidelity problem is a serious problem existing in timeline +summarization, and MTS and UTS-abs greatly alleviates such problem. We also conduct the paired student t-test between +our model and SAGCopy (the row with shaded background), and the result demonstrates the significance of the above +results. The kappa statistics is 0.46 and 0.49 respectively, which indicates moderate agreement between annotators5. To +prove the significance of these results, we also conduct the paired student t-test between our model and SAGCopy. We +obtain a p-value of 3 × 10−8, 8 × 10−12, and 9 × 10−11 for fluency, informativeness, and fidelity, respectively. +We also show a case study in Table 9 with translated version in Table 8 selected from celebrity timelime dataset. The +case is about James Cameron’s career as a director. We omit unimportant information in the input document due to +limited space. The input document includes most of his works, and the detailed information of each event, while the +summary reference only introduces the main event of his experience, omitting those details and unimportant events. +It can be seen that the summary generated by UTS-abs successfully captures the important events, and introduces +them in the correct order. The output of our UTS-ext has a high overlap with the ground truth. As for baseline GPG, it +fails to capture the most important events, but includes irrelevant information such as details in filming “Piranha II”. +For baseline SAGCopy, it also generates unimportant descriptions including information “The dark angel of the last +world”. Moreover, our extractive and abstractive summary show consistent behavior with the high overlap, which +further indicates that the two methods can jointly promote the recognition of important information. Compared with +the extracted summary, the generated summary is more concise and coherent. +6.2 +Out of Domain Test +Next, we address research question RQ2. In Table 10, we present the performance of UTS on the classic timeline +summarization TL17 dataset as an out-of-domain test. It can be seen that both of our models outperform existing +baselines. Specifically, UTS-ext outperforms the best baseline Datawise by 19.4% on AR1-F score, demonstrating the +5[32] characterize kappa values < 0 as no agreement, 0-0.20 as slight, 0.21-0.40 as fair, 0.41-0.60 as moderate, 0.61-0.80 as substantial, and 0.81-1 as almost +perfect agreement. +Manuscript submitted to ACM + +22 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +In 1981, James Cameron directed his first film, “Piranha II”, which was shot entirely in Italy. Cameron didn’t +get along well with an Italian speaking staff, and the producers didn’t let him participate in the final editing of +the film... In 1984, Cameron released his first self-made and self-directed film “Terminator”, which costs only 6.5 +million dollars... In 1986, James Cameron’s second self-made work, “Alien 2”, was published... In 1987, “Alien 2” +won seven Academy Award nominations... James Cameron won the best director award at the 14th Saturn Awards +for this film. In 1989, Cameron wrote and directed his third film, “The Abyss”... In 1991, his film “Terminator 2” +made 200 million dollars at the box office in the United States, and he also won the 18th Saturn Awards for best +director and best screenwriter for this film. In 1997, Cameron directed the film “Titanic”, which wins 1.84 billion +at the box office, and starred Leonardo DiCaprio and Kate Winslet. In 1998, the film won 14 nominations and 11 +awards at the 70th Academy Awards. In 2000, he directed and supervised the TV series “The dark angel of the last +world” with the theme of gene therapy. +reference +In 1981, he directed the first film “Piranha II”. In 1984, he became famous for his science fiction film +“Terminator”. In 1986, he wrote and directed the film “Alien 2”. In 1991, he won the best director +award and best screenwriter award at the 18th Saturn awards for his film “Terminator 2”. In 1997, +his film “Titanic” won 1.84 billion dollars at the box office, breaking the global box office record; it +won 11 awards including best picture at the 70th Academy Awards, and James Cameron won the +best director award at the Oscars. +GPG +In 1981, James Cameron directed the film “Piranha II”. Cameron Cameron didn’t get along well with +an Italian speaking staff, and the producers didn’t let him participate in the final editing of the film. +In 1984, he became famous for his science fiction film “Terminator”. In 1986, he wrote and directed +the film “Alien 2”. In 1986, he wrote and directed the film “Alien 2”. In 1991, he won the best director +award and best screenwriter award at the 18th Saturn Awards for his film “Terminator 2”. In 1997, +his film Titanic won 1.84 billion US dollars at the box office. +SAGCopy +In 1981, James Cameron directed his first work, “Piranha II”. In 1984, Cameron released his first film, +“Terminator”. In 1986, James Cameron wrote and directed his second work, “Alien 2”. In 1997, James +Cameron directed the film “Titanic”. In 2000, he directed and supervised the TV series “The dark +angel of the last world” with the theme of gene therapy. +UTS-ext +In 1981, James Cameron directed his first film, “Piranha II”, which was shot entirely in Italy. In 1984, +Cameron released his first self-made and self-directed film “Terminator”, which cost only 6.5 million +dollars. In 1986, James Cameron’s second self-made work, “Alien 2”, was published. in 1997, Cameron +directed the film “Titanic”, which wins 1.84 billion at the box office, and starred Leonardo DiCaprio +and Kate Winslet. +UTS-abs +In 1981, Cameron directed his first work, “Piranha II”. In 1984, he released his first film “Terminator”. +In 1986, his second film, “Alien 2”, was published. In 1987, “Alien 2” won the 14th Saturn Award for +best director. In 1991, his film “Terminator 2” made 200 million dollars in the United States. In 1997, +he directed the film “Titanic”. In 1998, the film won 14 Academy Awards nominations and 11 of them +at the 70th Academy Awards. +Table 8. RQ1: Examples of the generated answers by UTS-abs, UTS-ext and baselines (translated version). +effectiveness of the neural network in the traditional extractive style. UTS-abs performs similar to UTS-ext, improving +the AR1-F score of Pubcount by 3.79. This demonstrates that the abstractive methods can be adapted to out-of-domain +small-scale datasets. Specifically, since our original WikiTS dataset is in encyclopedia style, while Timeline 17 is a news +dataset, this demonstrates that our model can be applied to datasets of different language styles. +6.3 +Ablation Study +Next, we turn to research question RQ3, where we perform an ablation study on the test set to investigate the influence +of different modules in our proposed UTS model. Modules are tested in four ways: (1) we remove the sentence extractor +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +23 +1981年，詹姆斯·卡梅隆执导了第一部作品《食人鱼2》，影片完全在意大利拍摄。卡梅隆和一口意大利 +语的工作人员相处得并不愉快，而拍摄完毕后制片方不让他参与影片的最终剪辑. 1984年，卡梅隆推出了 +他第一部自编自导的影片《终结者》，这部影片的拍摄只花了650万美元... 1986年，詹姆斯·卡梅隆自编 +自导的第二部作品《异形2》问世... 1987年，《异形2》获得了七项奥斯卡奖提名...詹姆斯·卡梅隆凭借此 +片获得了第14届土星奖最佳导演奖... 1989年，卡梅隆自编自导了第三部电影《深渊》... 1991年，他执导的 +电影《终结者2》在美国上映后取得了2亿美元的票房，他也凭借该片获得了第18届土星奖最佳导演奖以 +及最佳编剧奖... 1997年，詹姆斯·卡梅隆执导了电影《泰坦尼克号》，该片获得18.4亿美元的票房，由莱 +昂纳多·迪卡普里奥、凯特·温斯莱特等主演. 1998年，在第70届奥斯卡金像奖上这部影片获得了14个奥 +斯卡奖的提名并获得了其中的11个奖项... 2000年，他执导并监制了以基因治疗为题材的电视剧《末世黑天 +使》... +reference +1981年，詹姆斯卡梅隆执导首部电影《食人鱼2》。1984年，因自编自导科幻电影《终结者》 +成名。1986年，自编自导电影《异形2》。1991年，凭借电影《终结者2》获得第18届土星奖最 +佳导演奖以及最佳编剧奖。1997年，他执导的电影《泰坦尼克号》取得了18.4亿美元的票房， +打破全球影史票房纪录;该片在第70届奥斯卡金像奖上获得了包括最佳影片在内的11个奖项，詹 +姆斯·卡梅隆凭借该片获得了奥斯卡奖最佳导演奖。 +GPG +1981年，詹姆斯·卡梅隆执导了部作品《食人鱼2》，卡梅隆卡梅隆和一口意大利语的工作 +人员相处得并不愉快，而拍摄完毕后制片方不让他参与影片的最终剪辑. 1984年，卡梅隆凭 +借科幻电影《终结者》出名。1986年，他自编自导了电影《异形2》。1986年，他自编自导 +了电影《异形2》。1991年，他凭借《终结者2》获得了第18届土星奖最佳导演奖和最佳编剧 +奖。1997年，他的电影《泰坦尼克号》在美国获得了18.4亿票房。 +SAGCopy +1981年，詹姆斯·卡梅隆执导了第一部作品《食人鱼2》。1984年，卡梅隆推出了他第一 +部自编自导的影片《终结者》。1986年，詹姆斯·卡梅隆自编自导的第二部作品《异 +形2》。1997年，詹姆斯·卡梅隆执导了电影《泰坦尼克号》。2000年，他执导并监制了以 +基因治疗为题材的电视剧《末世黑天使》。 +UTS-ext +1981年，詹姆斯·卡梅隆执导了第一部作品《食人鱼2》，影片完全在意大利拍摄。1984年， +卡 梅 隆 推 出 了 他 第 一 部 自 编 自 导 的 影 片 《 终 结 者 》.这 部 影 片 的 拍 摄 只 花 了650万 美 +元。1986年，詹姆斯·卡梅隆自编自导的第二部作品《异形2》问世。1997年，詹姆斯·卡 +梅隆执导了电影《泰坦尼克号》，该片获得18.4亿美元的票房，由莱昂纳多·迪卡普里奥、凯 +特·温斯莱特等主演。 +UTS-abs +1981年，卡梅隆执导了第一部作品《食人鱼2》。1984年，卡梅隆推出了他第一部自编自导的 +影片《终结者》。1986年，自编自导的第二部作品《异形2》问世。1987年，《异形2》获得了 +第14届土星奖最佳导演奖。1991年，他执导的电影《终结者2》在美国上映后取得了2亿美元的 +票房。1997年，执导了电影《泰坦尼克号》。1998年，在第70届奥斯卡金像奖上这部影片获得 +了14个奥斯卡奖的提名并获得了其中的11个奖项。 +Table 9. RQ1: Examples of the generated answers by UTS-abs, UTS-ext and baselines. +Fig. 5. RQ3: Visualizations of time-attention. The figure in the left part is the attention map in the first decoding step, and the figure +in the right part is in the final decoding step. +and only train the generator to verify the effectiveness of joint learning on the abstractive summarization; (2) we +remove the summary generator part and only train the sentence extractor to verify the effectiveness of joint learning +on the extractive summarization; (3) we remove the graph-based encoder and only stores the local representation in the +Manuscript submitted to ACM + +24 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +AR1-F +AR2-F +Date-F1 +Chieu +6.66 +1.9 +25.1 +Martschat +10.5 +3.0 +54.4 +Tran +9.4 +2.2 +51.7 +Pubcount +10.5 +2.7 +48.1 +Datewise +12.0 +3.5 +54.4 +Clust +8.2 +2.0 +40.7 +UTS-ext +16.73 +4.08 +54.9 +UTS-abs +14.29 +3.51 +54.6 +Table 10. RQ2: ROUGE scores on out-of-domain TL17 summarization dataset. +ROUGE-1 +ROUGE-2 +ROUGE-L +UTS-abs +41.56 +23.95 +39.18 +without multitask +39.58 +22.54 +37.55 +without global +38.66 +22.87 +36.76 +without local +39.14 +23.15 +36.00 +UTS-ext +34.81 +22.26 +32.03 +without multitask +33.78 +21.09 +29.03 +without global +33.00 +18.89 +27.07 +without local +33.28 +20.98 +29.69 +Table 11. RQ3: ROUGE scores of different ablation models. +memory to verify the effectiveness of global representation; (4) we remove the time-event memory entirely to verify +the importance of global and local representation further. +Table 11 presents the results. We find that the ROUGE-2 score of extractive summarization drops by 5.26% after +the summary generator is removed. This indicates that the joint learning method helps extractive summarization +to benefit from abstractive summarization. ROUGE-2 score of abstractive summarization drops by 5.54% after the +sentence extractor is removed. This indicates that extractive summarization does help abstractive summarization +identify important sentences during the interactive decoding phrase. ROUGE-2 score of extractive summarization drops +by 4.72%, while the ROUGE-2 score of abstractive summarization drops by 6.25% after the global representation is +removed. It indicates establishing the graph-based encoder to simulate the relationships between events is necessary to +improve the performance of both extractive and abstractive summarization. ROUGE-2 score drops by 4.72% and 3.45% +compared with UTS-abs after removing the global representation and the local representation. It indicates the semantic +information of the time-event memory is of great importance to encode multiple events. +6.4 +Analysis of Time Position Embedding +We then address RQ4. The usefulness of time position embedding is reflected by time-attention in the memory, denoted +as 𝜋 as introduced in Equation 24. If the time position embedding successfully encodes the time information, then the +time-attention should obey the development of the input document. We visualize the attention map of two randomly +sampled examples as shown in Figure 5 from the celebrity timeline dataset. The figure on the left is the attention map +in the first decoding step, and the figure on the right is in the final decoding step. The darker the color is, the higher +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +25 +Fig. 6. RQ5: Visualizations of two level attentions. The figure above is the event-level attention and the three figures below are the +word-level attentions of first lead three events. +the attention is. Due to limited space, we omit the corresponding event descriptions. When decoding starts, UTS-abs +learns to pay attention to the first two events, which always consist of parallel information such as the birthplace and +birth date of the character. The attentions on the last several events are low since it does not need this information in +advance. When decoding ends, UTS-abs focuses more on the last several events. However, it also pays attention to the +first few events, since timeline summarization is a process of information accumulation, and later sentences should +consider previous information. The above example demonstrates the effectiveness of time position embedding. +6.5 +Analysis of Event-level Attention +We now turn to RQ5, whether event-level attention can guide word-level attention in the abstractive part. We first +conduct a case study to visualize the two-level attention, as shown in Figure 6. The figure above is the event-level +attention, and the three figures below are word-level attention corresponding to the first three events. We only show +the first 11 words in an event. The result shows that the third event is the most important event in this decoding +step, and the weights of the words in this event are also greater than other words on average. The above observation +demonstrates that event-level attention gives the correct guidance for word-level attention. +Apart from the visualization, we also conduct a quantitative analysis to measure how greatly the word-level attention +is influenced by event-level information, which is reflected by inconsistency loss. We adjust the inconsistency loss +proposed in §4.8 to evaluate the inconsistency between event attention and word attention. The new consistency loss +at 𝑡-th decoding step is the negative log-likelihood of the product of attention value of most attended words and their +corresponding event-level attention. The intuition is to verify whether the event-level attention is high too when +word-level attention is high. When training starts, the inconsistency loss is around 5.3, and when training ends, the loss +drops to 2.1. This means that event-level information greatly influences the word-level attention and the model learns +to unify these two attentions. We did not directly add inconsistency loss to training because we found that made UTS +perform worse. Instead, we let the model learn by itself to unify these two attentions. +6.6 +Analysis of the Unified Chronological Attentions +We then address RQ6, examining whether the chronological attentions in the abstractive and extractive parts are indeed +unified. Remember that we come up with a time-aware inconsistency loss to unify the two attentions, thus, by looking +at the loss curve, we can examine the effectiveness of this part. +Manuscript submitted to ACM + +26 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan +0k +2k +4k +6k +8k +10k +Step +2.5 +3.0 +3.5 +4.0 +Time-aware +Inconsistency Loss +Fig. 7. RQ6: Time-aware inconsistency loss curve. +The loss curve of the inconsistency is shown in Figure 7. We can see that when the training begins, the inconsistency +loss fluctuates from time to time, probably because the model aims to train the extractor and generator separately at +the beginning of the process. However, the average of the inconsistency loss presents a falling tendency, which means +that the extractor and generator unify during the whole training procedure. In the end, the time-aware inconsistency +loss drops from 4.0 to 2.5. +6.7 +Robustness of Parameter Setting +64 +128 +256 +448 +512 +Hidden Size +0 +5 +10 +15 +20 +25 +30 +35 +40 +ROUGE score +ROUGE-1 +ROUGE-2 +ROUGE-L +Fig. 8. Performance of UTS-abs with different parameter settings. +Manuscript submitted to ACM + +Follow the Timeline! Generating Abstractive and Extractive Timeline Summary in Chronological Order +27 +Finally, we turn to address RQ7 to investigate the robustness of parameter setting. We train our model in different +parameter settings as shown in Figure 8. The hidden size of the RNN is tuned from 64 to 512, and we use the ROUGE 𝐹1 +score to evaluate each model. As the hidden size grows larger from 64 to 256, the performance rises along with. The +increment of hidden size improves the ROUGE-1 and ROUGE-L scores by 0.54 and 0.77 score. When the hidden size +continuously goes larger from 256 to 512, the performance is declined slightly. The increment of hidden size leads to a +1.15% and 1.25% drop in terms of ROUGE-1 and ROUGE-L respectively. Nonetheless, we can find that each metric is +maintained at a stable interval, which demonstrates that our UTS is robust in terms of different parameter sizes. +7 +CONCLUSION AND FUTURE WORK +In our previous work, we propose a framework named MTS which aims to generate summaries that concisely summarize +the evolution trajectory along the timeline. However, in this method, the time information is captured in an implicit +and indirect way, where it is hard to verify and ensure the decoder indeed captures the time-sequential information. +Hence, in this work, we propose a novel Unified Timeline Summarizer (UTS) that can generate abstractive and extractive +timeline summaries in time order. Specifically, in the encoder part, we propose a graph-based event encoder that relates +multiple events according to their content dependency and learns a representation of each event. In the decoder part, to +ensure the chronological order of the abstractive summary, we propose to extract the feature of event-level attention in +its generation process with sequential information remained and use it to simulate the evolutionary attention of the +ground truth summary. The event-level attention can also be used to assist in extracting summary, where we devise +a time-aware inconsistency loss function to penalize the inconsistency between abstractive attention and extractive +attention. Note that the extractive summary is generated one by one, thus the extracted summary also comes in time +sequence. We augment the character timeline summarization dataset proposed in our previous work with the event +timeline summarization corpus and English corpus. Experimental results on these datasets and on out-of-domain +Timeline 17 dataset show that our UTS model can significantly outperform the existing methods. 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In ACL. +Manuscript submitted to ACM + diff --git a/WdAyT4oBgHgl3EQf8_qr/content/tmp_files/load_file.txt b/WdAyT4oBgHgl3EQf8_qr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3dd3652e8f456e3c70f9e8a4be2e29b636c5750d --- /dev/null +++ b/WdAyT4oBgHgl3EQf8_qr/content/tmp_files/load_file.txt @@ -0,0 +1,1559 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf,len=1558 +page_content='Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order XIUYING CHEN∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Computational Bioscience Reseach Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' King Abdullah University of Science and Technology MINGZHE LI∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Wangxuan Institute of Computer Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Peking University SHEN GAO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Wangxuan Institute of Computer Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Peking University ZHANGMING CHAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Wangxuan Institute of Computer Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Peking University DONGYAN ZHAO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Wangxuan Institute of Computer Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Peking University XIN GAO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Computational Bioscience Reseach Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' King Abdullah University of Science and Technology XIANGLIANG ZHANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1 University of Notre Dame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2 King Abdullah University of Science and Technology RUI YAN†, Gaoling School of Artificial Intelligence, Renmin University of China Nowadays, time-stamped web documents related to a general news query floods spread throughout the Internet, and timeline summarization targets concisely summarizing the evolution trajectory of events along the timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Unlike traditional document summarization, timeline summarization needs to model the time series information of the input events and summarize important events in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' To tackle this challenge, in this paper, we propose a Unified Timeline Summarizer (UTS) that can generate abstractive and extractive timeline summaries in time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, in the encoder part, we propose a graph-based event encoder that relates multiple events according to their content dependency and learns a global representation of each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the decoder part, to ensure the chronological order of the abstractive summary, we propose to extract the feature of event-level attention in its generation process with sequential information remained and use it to simulate the evolutionary attention of the ground truth summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The event-level attention can also be used to assist in extracting summary, where the extracted summary also comes in time sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We augment the previous Chinese large-scale timeline summarization dataset and collect a new English timeline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Extensive experiments conducted on these datasets and on the out-of-domain Timeline 17 dataset show that UTS achieves state-of-the-art performance in terms of both automatic and human evaluations1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' CCS Concepts: • Information retrieval → Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Additional Key Words and Phrases: Timeline Summarization, Extractive Summarization, Abstractive Summarization ∗Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Ordering is decided by a coin flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' †Corresponding Author: Rui Yan (ruiyan@ruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='cn) 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='com/iriscxy/Unified-Timeline-Summarizer Authors’ addresses: Xiuying Chen, Computational Bioscience Reseach Center, King Abdullah University of Science and Technology, xiuying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='chen@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='sa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Mingzhe Li, Wangxuan Institute of Computer Technology, Peking University, li_mingzhe@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Shen Gao, Wangxuan Institute of Computer Technology, Peking University, shengao@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Zhangming Chan, Wangxuan Institute of Computer Technology, Peking University, zhangming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' chan@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Dongyan Zhao, Wangxuan Institute of Computer Technology, Peking University, zhaody@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Xin Gao, Computational Bioscience Reseach Center, King Abdullah University of Science and Technology, xin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='gao@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='sa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Xiangliang Zhang, 1 University of Notre Dame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2 King Abdullah University of Science and Technology, xzhang33@nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Rui Yan, Gaoling School of Artificial Intelligence, Renmin University of China, ruiyan@ruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial 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Xiangliang Zhang, and Rui Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ACM Transactions on Information Systems 1, 1, Article 1 (January 2022), 30 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1145/3517221 1 INTRODUCTION The rapid growth of World Wide Web means that time-stamped document floods spread throughout the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' General search engines simply return web pages ranked by query relevance, but they are not quite capable of handling ambiguous intentioned queries, such as a query about evolving news “COVID-19”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' People may have a myriad of general interests about the beginning, the evolution, or the most up-to-date situation, while simply ranking the returned webpages according to their relevance is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In many cases, readers are tired of navigating every document in the overwhelming collection: they want to monitor the evolution trajectory of hot topics by simply browsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Summarization is an ideal solution to provide a condensed, informative document reorganization for a faster and better representation of news evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Timeline summary temporally summarizes evolutionary news as a series of individual but correlated component summaries and hence offers an option to understand the big picture of a developing situation [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Existing timeline summarization approaches such as [34, 53, 75] are all based on extraction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, these methods rely on human-engineered features and sophisticated abilities that are crucial to high-quality summarization, such as paraphrasing, generalization, or the incorporation of real-world knowledge, which are possible only in an abstractive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Recently, with the emergence of strong generative neural models for text [4], abstractive techniques are also becoming increasingly popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, we propose the abstractive timeline summarization task in our early work [10], which aims to concisely paraphrase the event information in the input article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' An example case is shown in Table 1, where the article consists of events of a great entertainer in different periods, and the summary correctly summarizes the important events from the input article in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Abstractive summarization approaches including [24, 28, 56, 80] have been proven to be useful in traditional summarization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, unlike traditional document summarization, the timeline summarization dataset consists of a series of time-stamped events, and it is crucial for the timeline summarization model to capture this time series information to better guide the chronological summary generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Besides, the fidelity problem is also of vital importance for timeline summarization, where mixing the information of different events leads to a bad summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Take the example in Table 1 for example, the bad summary confuses the birthplace and the residence, the first album, and the best-selling album of the celebrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Herein, the good summary is the ground truth summary from our dataset, and the bad summary is a wrong summary with typical errors we found in a preliminary experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' As we found in the experiment, such infidelity phenomena is a commonly-faced problem in summarization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' To tackle the above challenges, in our previous work [10], we come up with a Memory-based Timeline Summarization (MTS) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Specifically, we first use an event embedding module with selective reading units to embed all events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Then, we propose a key-value memory module storing time-series information to guide the summary generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely speaking, the key in the memory module is the time position embedding that represents the time series information, and the values are the corresponding event representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The value item includes local and global representation, where local value is the output from the event embedding module, and global value is taken from the average local representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Keys together form a timeline and we use the time position of events on the timeline to Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 3 Events Michael Jackson (dubbed as “King of Pop”) was born on August 29, 1958 in Gary, Indiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' He is the seventh child in his family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1971, Jackson released his first solo “got to be there”, marking the beginning of his solo career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In late 1982, Jackson’s sixth album, “Thriller”, was released, where videos "Beat It", "Billie Jean" in it are credited with breaking racial barriers and transforming the medium into an art form and promotional tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In March 1988, Jackson built a new home named Neverland Ranch in California, where more than 100 arcade machines were stored here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 2000, Guinness World Records recognized him for supporting 39 charities and donated more than 300 million dollars to charities in his own name, more than any other entertainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Bad summary Michael Jackson was born on August 29, 1958 in Gary, California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1971, his first album “Thriller” was released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 2000, Guinness World Records recognized him for supporting 39 charities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Good summary Michael Jackson was born on August 29, 1958 in Gary, Indiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' His sixth album “Thriller” was released in 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 2000, Guinness World Records recognized him for supporting 39 charities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Example of timeline summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The text in pink demonstrates time stamp, and text in blue demonstrates wrong event description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Events are split by lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' guide the generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Finally, in each decoding step, we introduce event-level attention and use it to determine word-level attention to avoid confusion between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In MTS, the time information is captured in an implicit and indirect way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' MTS stores the time position embedded in the memory and hopes the decoder will learn to attend to the correct time position in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, that strategy is rather weak supervision, where it is hard to verify and ensure the decoder indeed captures the time-sequential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In this work, we take one step further and improve our previously proposed MTS framework with explicit timeline guidance modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In other words, we carefully design a strategy that lets the time information be a clear guidance signal for the summarization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Overall, in this paper, we propose a novel Unified Timeline Summarizer (UTS) that can generate abstractive and extractive timeline summaries in time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For the abstractive part, concretely, in the encoder part, we first propose a graph-based event encoder that relates multiple events according to their content dependency and learns a representation of each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The motivation is that the importance of each event and whether it should be included in the summary does not only depend on itself but also is related to other events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Take Table 1 for example, Jackson releases his first solo album might be an important event, but its importance is weakened by his “Thriller” album that breaks the racial barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, the representation from the graph encoder incorporates global information from other events, thus is used to replace the old global representation in the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the decoder part, to avoid the situation in the bad summary in Table 1, where it confuses the birthplace and the residence because the model is not sensitive to the timeline, we propose a summary decoder that emphasizes the time information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, to ensure the chronological order of the abstractive summary, we propose to extract the feature of event-level attention in its generation process with sequential information remained and use it to simulate the evolutionary attention of the ground truth summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 4 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan In terms of the extractive part, we present a sentence embedding module to encode each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Next, a sentence extractor sequential selects important sentences to be included in the summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The event-level attention can also be used to assist in extracting summary in this process, where we devise a time-aware inconsistency loss function to penalize the inconsistency between abstractive attention and extractive attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Note that the extractive summary is extracted one by one, thus the extracted summary also comes in time sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We empirically compare MTS and UTS on the public dataset2 proposed by our early work [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This is a large-scale real-world timeline summarization dataset, which consists of a series of time-stamped events and the corresponding summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Moreover, since this previous dataset only includes a timeline corpus about celebrities, we augment the dataset with cases about social events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We also collect an English timeline summarization dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Experimental results on these datasets and on out-of-domain Timeline17 dataset show that our newly proposed UTS model can significantly outperform the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Particularly, UTS-abs yields 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='47% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='90% percentage point improvement in terms of ROUGE-1 on celebrity and event timeline datasets compared with our early work MTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In addition to the comprehensive evaluation, we also evaluate our proposed graph encoder and attention mechanism by a fine-grained analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The analysis reveals how the model leverages the explicit timeline information to guide the abstractive and extractive summarization process and provides us insights on why they can achieve big improvement over state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Overall, our contributions can be summarized as follows: We propose a unified abstractive and extractive timeline summarization framework, where a time-aware inconsis- tency loss function is proposed to unify these two processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We propose a graph-based encoder that relates multiple events according to their content dependency and learns the global representation of each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We propose to use the evolutionary attention of the ground truth summary to guide both the abstractive and extractive summary generation process, to ensure that the generated summaries follow strict time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We also augment the first real-world large-scale timeline summarization dataset with social event corpus and corpus in English3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Experiments conducted on the three datasets and the out-of-domain benchmark Timeline 17 dataset show that our model outperforms all baselines, including state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Experiments also verify the effectiveness of each module in UTS as well as its interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The rest of the paper is organized as follows: We summarize related work in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We then formulate our research problem in §3 and elaborate our approach in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' §5 gives the details of our experimental setup and §6 presents the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Finally, §7 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2 RELATED WORK We detail related work on text generation methods, timeline summarization, extractive summarization, abstractive summarization, unified summarization, and memory network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1 Text Generation Methods In recent years, sequence-to-sequence (seq2seq) [61] based neural networks have been proved effective in generating a fluent sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The seq2seq model is originally proposed for machine translation and later adapted to various natural language generation tasks, such as text summarization [25, 38, 50, 65, 67] and dialogue generation [6, 63, 77, 79, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='com/yingtaomj/Learning-towards-Abstractive-Timeline-Summarization 3Data will be released in camera-ready version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 5 Rush et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [54] apply the seq2seq mechanism with attention model to the text summarization field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Then See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [56] add copy mechanism and coverage loss to generate summarization without out-of-vocabulary and redundancy words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The seq2seq architecture has also been broadly used in a dialogue system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [62] propose a multi-head attention mechanism to capture multiple semantic aspects of the query and generate a more informative response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [77] propose to use the content introducing method to solve the problem of generating a meaningless response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [68] use three channels for widening and deepening the topics of interest and try to make the conversational model chat more turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2 Timeline Summarization The timeline summarization task is firstly proposed by Allan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [2], where they define temporal summaries of news stories as extracting a single sentence from each event within a news topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Later, a series of works [72, 74, 75, 82] further investigate timeline summarization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [75] formally formulate the task as an optimization problem via iterative substitution from a set of sentences to a subset of sentences that satisfies the above requirements, balancing coherence/diversity measurement and local/global summary quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In follow-up work, Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [72] propose to model trans-temporal correlations among component summaries for timelines, using inter-date and intra-date sentence dependencies, and present a novel combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' There are also works focusing on tweets summarization that is related to timeline summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For example, [53] focus on the problem of selecting meaningful tweets given a user’s interests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' the dynamic nature of user interests, the sheer volume, and the sparseness of individual messages make this a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Specifically, they consider the task of time-aware tweets summarization, based on a user’s history and collaborative social influences from “social circles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Ghalandari and Ifrim [26] compare different timeline summarization strategies using appropriate evaluation frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For a more robust evaluation, they also present a new timeline summarization dataset, which spans longer time periods than previous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, all the above works are based on extractive methods, which are not as flexible as abstractive approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The most similar work to ours is proposed by [58], where they construct a word-adjacency graph, and then generate new sentences from this graph by finding paths from the sentence start node to the sentence end node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This is very different from our neural-based approach, and we demonstrate the superiority of our model in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='3 Extractive Summarization Despite the focus on abstractive summarization, extractive summarization remains an attractive method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In extractive summarization, Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [30] propose a summarization method using document-level similarity based on word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Meanwhile, Filippova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [20] use an RNN to delete words in a document for sentence compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Yan and Wan [76] propose more meaningful and informative units named frequent deep dependency sub-structure and a topic-sensitive multi-task learning model for multi-doc summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Cheng and Lapata [13] propose a general framework for single-document text summarization using a hierarchical article encoder composed with an attention- based extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Following this, Nallapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [47] propose a simple RNN-based sequence classifier that outperforms or matches the state-of-art models at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [11] introduce a model which iteratively polishes the document representation on many passes through the document, so as to produce a better summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In another approach, Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [49] use a reinforcement learning method to optimize the ROUGE evaluation metric for text summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [52] study the use of sentence relations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', contextual sentence relations, title sentence relations, and query sentence relations, so as to improve the performance of extractive summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 6 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan Recently, pre-trained language models are also applied in summarization for contextual word representations [39, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Another intuitive structure for extractive summarization is the graph, which can better utilize the statistical or linguistic information between sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Early works focus on document graphs constructed with the content similarity among sentences, like LexRank [19] and TextRank [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Some recent works aim to incorporate a relational prior into the encoder by graph neural networks (GNNs) [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4 Abstractive Summarization Recently, with the emergence of strong generative neural models for text [3], abstractive summarization is also becoming increasingly popular [47, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' These models typically take the form of convolutional neural networks (CNN) or recurrent neural networks (RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For example, Rush et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [54] propose an encoder-decoder model which uses a local attention mechanism to generate summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Nallapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [48] further develop this work by addressing problems that had not been adequately solved by the basic architecture, such as keyword modeling and capturing the hierarchy of sentence-to- word structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In follow-up work, Nallapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [46] propose a new summarization model which generates summaries by sampling a topic one sentence at a time, then producing words using an RNN decoder conditioned on the sentence topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [85] tackles the cross-lingual summarization task, which aims at summarizing a document in one language into another language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' They propose a method inspired by the translation pattern in the process of obtaining a cross-lingual summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' A series of works relies on prototype text to assist in summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [7] chose the template with the highest similarity to the input sentence as a soft template to generate summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Following this, Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [22] proposed to generate the summary with pattern based on prototype editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Summarization techniques have also been used in other tasks such as related work generation [9] and headline generation [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 Unified Summarization Unified summarization here means unifying extractive and abstractive summarization tasks together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' It is a common way to propose a multi-task framework that utilizes the benefits from one task to augment the performance of the other task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For example, Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [28] proposed a unified framework that takes advantage of both extractive and abstractive summarization using an attention mechanism, which is a combination of the sentence-level attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Chen and Bansal [12] introduced a multi-step procedure, namely compression paraphrase, for abstractive summarization, which first extracts salient sentences from documents and then rewrites them in order to get final summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [33] introduced a guiding generation model, where the keywords in source texts are first retrieved with an extractive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The most similar work to ours is [28], where they use sentence-level attention to modulate the word-level attention such that words in less attended sentences are less likely to be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Their sentence-level attention is static during the generation process, while in our model, the high-level attention changes in each decode step depending on the current generated word which is more reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='6 Memory Network The memory network proposed by Sukhbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [59] generally consists of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The first one is a memory matrix to save information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', memory slots) and the second one is a neural network to read/write the memory slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The memory network has shown better performance than traditional long-short term memory network in several tasks, such as question answering [21, 40, 51, 59], machine translation [42], text summarization [10, 29], dialog system [16, 69], job-resume matching [73] and recommendation [18, 66, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The reason is that the memory network can store the information in a long time range and has more memory storage units than LSTM which has a Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='Symbol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑋 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='a document consists of multiple events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑌 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='ground truth timeline summary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='ˆ𝑌 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='generated timeline summary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑖-th event in input document ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑤𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑗-th word in 𝑖-th event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑇𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='number of input events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑇𝑖𝑤 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='number of words in 𝑖-th event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑇𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='number of words in ground truth summary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑇𝑦𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='number of sentences in the ground truth timeline summary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑙𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='extract label for 𝑖-th sentence in the summary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Glossary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' single hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Following memory network, there are many variations of memory network have been proposed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', key-value memory network [45] and dynamic memory network [31, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Representative works include [23], where they generate more meaningful answers in E-commerce question-answering by a read-and-write memory consisting of selective writing units to conduct reasoning among these reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In our work, we apply the key-value memory network on the timeline summarization task and fuse it into the generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 3 PROBLEM FORMULATION Before detailing our answer generation model, we first introduce our notations listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' UTS takes a list of events 𝑋 = (𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=',𝑥𝑇𝑒 ) as inputs, where𝑇𝑒 is the number of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Each event 𝑥𝑖 is a list of words: 𝑥𝑖 = (𝑤𝑖 1,𝑤𝑖 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=',𝑤𝑖 𝑇 𝑖𝑤), where 𝑤𝑖 𝑗 is the 𝑗-th word in 𝑖-th event, and 𝑇𝑖𝑤 is the word number of event 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the abstractive part, UTS-abs aims to generate a summary ˆ𝑌 = ( ˆ𝑦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', ˆ𝑦𝑇𝑦) that is not only grammatically correct but also consistent with the event information such as occurrence place and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Essentially, UTS-abs tries to optimize the parameters to maximize the probability 𝑃(𝑌 |𝑋) = �𝑇𝑦 𝑡=1 𝑃(𝑦𝑡 |𝑋), where 𝑌 = (𝑦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=',𝑦𝑇𝑦) is the ground truth summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For the extractive part, UTS-ext targets at generating a score vector ˆ𝐿 = {ˆ𝑙1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' , ˆ𝑙𝑇𝑦𝑠 } for each sentence, where each score denotes the sentence’s extracting probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We convert the human-written summaries to gold label vector 𝐿 = {𝑙1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=',𝑙𝑇𝑦𝑠 }, where 𝑙𝑖 ∈ {0, 1} denotes whether the 𝑖-th sentence is selected (1) or not (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' During the training process, the cross-entropy loss is calculated between 𝐿 and ˆ𝐿, which is minimized to optimize ˆ𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4 MODEL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1 Overview In this section, we introduce our Unified Timeline Summarizer (UTS) in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The overview of UTS is shown in Figure 1 and can be split into two parts, one aims to generate an abstractive summary, and one targets selecting important sentences as a summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Abstractive part includes: (1) Event Embedding Module (See § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2): To obtain the vector representations for each event, we employ a recurrent network with Selective Reading Units (SRU) to learn the local representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (2) Graph-based Encoder (See § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='3): The representations learned in the last module do not incorporate interaction between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 8 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Comparision between MTS and UTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' MTS UTS Event Embedding Module SRU SRU Graph-based Encoder Transformer Time-Event Memory Key-Value Memory Key-Value Memory Summary Generator Editing Gate Editing Gate Sentence Embedding Module SRU Sentence Extractor RNN Unifier Inconsistency loss Hence, we propose a graph-based encoder to learn the global representation of each event incorporating the information from other events and the relationship between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (3) Time-Event Memory (See § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4): we propose a time-event memory, which stores the local and global event representation, with time position keys together forming a timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (4) Summary Generator (See § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5): eventually, we use an RNN-based decoder to generate the summary under the guidance of event-level attention and word-level attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Extractive part includes: (5) Sentence Embedding Module (See § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='6): this module embeds the sentence to a vector representation in a similar way to the event embedding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (6) Sentence Extractor (See § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7): the sentence extractor selects the salient sentences as the summary following the sequential time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Additionally, we propose (7) Chronological-Attention Unifier (See § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='8), to let the two parts complement each other by unifying the attention distributions of abstractive parts and extractive parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, we propose a time-aware inconsistency loss to penalize the inconsistency between these two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Although some encoder and decoder modules in MTS are similar to UTS, there are three significant differences in our UTS model compared with MTS: (1) MTS encodes each event independently, without considering the information interaction between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' While in UTS, we propose a graph encoder, which learns global representations for input events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (2) We propose a unified timeline framework that can not only generate an abstractive summary, but also an extractive summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' That is, only UTS includes the extractive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (3) We propose to unify the abstractive and extractive parts together, where the two tasks can benefit each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Specifically, we show the comparison between MTS and UTS in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2 Event Embedding Module We first propose an event embedding module to obtain the word-level and event-level vector representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' To begin with, we use an embedding matrix 𝑒 to map a one-hot representation of each word in 𝑥𝑖 into a high-dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We denote 𝑒(𝑤𝑖 𝑡) as the embedding representation of word 𝑤𝑖 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We then employ a bi-directional recurrent neural network (Bi-RNN) to model the temporal interactions between words: ←− ℎ𝑖 𝑡 = LSTMenc([𝑒(𝑤𝑖 𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑝𝑖], ←−−− ℎ𝑖 𝑡−1), (1) −→ ℎ𝑖 𝑡 = LSTMenc([𝑒(𝑤𝑖 𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑝𝑖], −−−→ ℎ𝑖 𝑡−1), (2) ℎ𝑖 𝑡 = −→ ℎ𝑖 𝑡 + ←− ℎ𝑖 𝑡, (3) Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 9 Chronological Event-level Attention Chronological Sentence-level Attention .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Event1 (1) Event Embedding Module (1) Event Embedding Module (1) Event Embedding Module Key Local Value Global Value Time1 Time2 Time3 (4) Summary Generator (3) Time-Event Memory Abstractive Timeline Summary (2) Graph-based Encoder Sentences (5) Sentence Embedding Module (5) Sentence Embedding Module (5) Sentence Embedding Module (6) Summary Extractor Extractive Timeline Summary (7) Chronological-Attention Unifier .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Event2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Event3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Overview of UTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We divide our model into abstractive summarization part and extractive summarization part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Abstractive part includes: (1) Event Embedding Module, (2) Graph-based encoder, (3) Time-Event Memory, and (4) Summary Generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Extractive part includes: (5) Sentence Embedding Module and (6) Sentence Extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Additionally, there is a (7) Chronological Attention Unifier that unifies the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' where “;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' denotes the concatenation between vectors, and ℎ𝑖 𝑡 denotes the hidden state of 𝑡-th word in Bi-RNN for event 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' To capture the sequential information of events, we randomly initialize a time position encoding vector 𝑝𝑖 of 𝑖-th event to be included in the Bi-RNN input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Apart from obtaining word representation ℎ𝑖 𝑡, we also need to gain event representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Simply taking the final state of Bi-RNN ℎ𝑖 𝑇 𝑖𝑤 as the representation of the whole event cannot fully capture the feature of the whole event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Thus, we employ the selective reading module consisted of SRU proposed in [11] to gain new event representation 𝑎𝑖: 𝑠𝑖 𝑡 = SRU(𝑠𝑖 𝑡−1, [ℎ𝑖 𝑡,ℎ𝑖 𝑇 𝑖𝑤]), (4) 𝑎𝑖 = 𝑠𝑖 𝑇 𝑖𝑤, (5) where 𝑠𝑖 𝑡 is the hidden state of 𝑡-th SRU cell in 𝑖-th event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' At the high level, SRU is a modified version of GRU, which replaces the update gate in original GRU [15] with a new gate taking each input ℎ𝑖 𝑡 and coarse event representation ℎ𝑖 𝑇𝑤 into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We omit the details here due to limited space and readers can refer to [11] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' So far, we obtain the representation of 𝑖-th event 𝑎𝑖 and 𝑡-th word in 𝑎𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', ℎ𝑖 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='3 Graph-based Encoder The event representation 𝑎𝑖 in the previous section is calculated independently, without considering the information flow between different events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, the importance of each event and whether it should be included in the summary does not only depend on itself but also is related to other events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For example, in Table 1, Jackson releases his first solo album might be an important event, but its importance is weakened by his “Thriller” album that breaks the racial Manuscript submitted to ACM 10 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, we propose a graph-based encoder to learn the relationship between events and obtain a global event representation that incorporates such information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' As shown in Figure 1, to embed relationship information, we set up the relation edges in our document modeling graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The relation edge in our graph is firstly initialized by the event representation: 𝑟𝑖,𝑗 = MLP𝑎([𝑎𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑎𝑗]), (6) where MLP is a multi-layer perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Next, during the relation-aware encoding process, we incorporate the relation edge 𝑟𝑖,𝑗 into the final event represen- tation by self attention operation: 𝑏𝑖 = RE(𝑎𝑖,𝑎∗,𝑟𝑖,∗), (7) where ∗ denotes all indexes between 1 and𝑇𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This module is based on Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Thus, we first introduce Transformer: 𝑏𝑖′ = Transformer(𝑎𝑖,𝑎∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (8) Concretely, the first input is for query and the second input is for keys and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Each output element, 𝑏𝑖′, is computed as weighted sum of a linearly transformed input values: 𝑏𝑖′ = 𝑇𝑒 ∑︁ 𝑗=1 𝛼𝑖,𝑗 𝑔 � 𝑎𝑗𝑊 𝑉 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (9) Each weight coefficient, 𝛼𝑖,𝑗 𝑔 , is computed using a softmax function: 𝛼𝑖,𝑗 𝑔 = exp � 𝛽𝑖,𝑗 𝑔 � �𝑇𝑒 𝑘=1 exp � 𝛽𝑖,𝑘 𝑔 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (10) 𝛽𝑖,𝑗 𝑔 is computed using a compatibility function that compares two input elements: 𝛽𝑖,𝑗 𝑔 = � 𝑎𝑖𝑊 𝑄� � 𝑎𝑗𝑊 𝐾 �𝑇 √ 𝑑 , (11) where 𝑑 is the hidden dimension, and 𝑊 𝑄,𝑊 𝐾,𝑊 𝑉 ∈ R𝑑×𝑑 are parameter matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RE is similar to Transformer, with two changes in Equation 9 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Specifically, we modify Equation 9 to propagate edge information to the sub-layer output: 𝑏𝑖 = 𝑇𝑒 ∑︁ 𝑗=1 𝛼𝑖,𝑗 𝑔 � 𝑎𝑗𝑊 𝑉 𝑟 + 𝑟𝑖,𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (12) In this way, the representation of each event is more comprehensive, consisting of its relation dependency information with other events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the meantime, when deciding the weight of each edge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', 𝛽𝑖,𝑗 𝑔 , we also incorporate relation edge information, since close relationships can have a great impact on edge weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, Equation 11 is changed to: 𝛽𝑖,𝑗 𝑔 = � 𝑎𝑖𝑊 𝑄 𝑟 � � 𝑎𝑗𝑊 𝐾 𝑟 + 𝑟𝑖,𝑗 �𝑇 √ 𝑑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (13) Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 11 Word-attention MJ was born in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Event1 Event2 Event3 Chronological event-attention Initial State Key Time1 Time2 Time3 Local Value Global Value Update Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' An overview of the summary generator in the abstractive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The summary generator generates the next word based on word-level and event-level attention, as well as the key-value memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The intuition for Transformer architecture is that each input is not isolated, and its representation depends on other inputs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In our augmented Transformer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', graph-based encoder, the polished event representation 𝑏𝑖 follows the same idea and expands the dependency between input documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 𝑏𝑖 does not only depend on its corresponding content but also depends on other inputs, as well as the relationships with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4 Time-Event Memory As stated in the Introduction, in the timeline dataset, the generated summary should capture the time-series information to guide the chronological generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, we propose a key-value memory module where keys together form a timeline, and this time series information is used to guide the generation process as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The key in this memory is the time position encoding 𝑝𝑖 introduced in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We will use this key as time guidance to extract information from the value part in the memory, which will be introduced in detail in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The value part stores event information of local aspect in local value and global aspect in global value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Local value simply stores the event representation 𝑎𝑖, which means that only captures information from the current event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' On the other hand, the global value is responsible for learning the event feature from a global perspective, not only based on itself but also its relationship with other events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, it stores the graph-based encoder output, 𝑏𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 Summary Generator To generate a consistent and informative summary, so as to avoid mixing information from different time stamps due to unawareness of correct timeline, we propose an RNN-based decoder that incorporates outputs of time-event memory module and event representation as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 12 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan Following [35], we randomly initialize an LSTM cell taking the concatenation of all event representations as input, and use the output as decoder initial state: ℎ′ 0 = LSTMini � ℎ𝑐, [𝑎1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑎𝑇𝑒 ] � , (14) where ℎ𝑐 is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1 Word-level attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Next, following traditional attention mechanism in [4], we summarize the input document into context vector 𝑐′ 𝑡−1 dynamically, and the 𝑡-th decoding step is calculated as: ℎ′ 𝑡 = LSTMdec �ℎ′ 𝑡−1, [𝑐′ 𝑡−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑒(𝑦𝑡−1)]� , (15) where ℎ′ 𝑡 is the hidden state of 𝑡-th decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Context vector 𝑐′ 𝑡−1 is calculated as: 𝛼𝑡′ 𝑖,𝑗 = 𝑊 ⊺ 𝑎 tanh � 𝑊𝑏ℎ′ 𝑡−1 +𝑊ℎℎ𝑖 𝑗 � , (16) 𝛼𝑡 𝑖,𝑗 = exp � 𝛼𝑡′ 𝑖,𝑗 � / 𝑇𝑒 ∑︁ 𝑘=1 �� � 𝑇 𝑖 𝑤 ∑︁ 𝑗=1 exp � 𝛼𝑡′ 𝑘,𝑗 ��� � , (17) 𝑐′ 𝑡−1 = 𝑇𝑒 ∑︁ 𝑖=1 �� � 𝑇 𝑖 𝑤 ∑︁ 𝑗=1 𝛼𝑡 𝑖,𝑗ℎ𝑖 𝑗 �� � , (18) where we first use the decoder state ℎ′ 𝑡−1 to attend to each states ℎ𝑖 𝑗 which results in the attention distribution 𝛼𝑡 𝑖,𝑗, shown in Equation 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ℎ𝑖 𝑗 denotes the representation of 𝑗-th word in event 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Then we use the attention distribution 𝛼𝑡 𝑖,𝑗 to obtain the weighted sum of document states as the context vector 𝑐′ 𝑡−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Context vector 𝑐′ 𝑡−1 here only takes the word-level attention into consideration without considering event-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, in timeline summarization, it is important for the model to be aware of which event it is currently describing, or it may confuse information from different events and result in an unfaithful summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, we introduce an event-level attention 𝛽 similar to the calculation of word-level attention and use it to adjust word-level attention: 𝛽𝑡′ 𝑖 = 𝑊 ⊺ 𝑐 tanh � 𝑊𝑑ℎ′ 𝑡−1 +𝑊𝑒𝑎𝑖� , (19) 𝛽𝑡 𝑖 = exp � 𝛽𝑡′ 𝑖 � / 𝑇𝑒 ∑︁ 𝑗=1 exp � 𝛽𝑡′ 𝑗 � , (20) 𝛾𝑡 𝑖,𝑗 = 𝛼𝑡 𝑖,𝑗𝛽𝑡 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (21) The new context vector 𝑐𝑡 (replacing 𝑐′ 𝑡 in Equation 15) is now calculated as: 𝑐𝑡 = 𝑇𝑒 ∑︁ 𝑖=1 �� � 𝑇 𝑖 𝑤 ∑︁ 𝑗=1 𝛾𝑡 𝑖,𝑗ℎ𝑖 𝑗 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (22) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2 Event-level attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Apart from using event-level attention to directly guide word-level attention, we also use it to obtain the weighted sum of event representation to be concatenated in the projection layer in Equation 31: 𝑒𝑡 = 𝑇𝑒 ∑︁ 𝑖=1 𝛽𝑡 𝑖 𝑎𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (23) Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='3 Memory guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' So far, we have finished the calculation of the context vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Next, we introduce how to incorporate the guidance from memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We first use hidden state ℎ′ 𝑡 to attend to each key in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' As stated in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4, keys, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', time position embeddings, conform the timeline that represents the time series information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Thus, we let the model take advantage of this sequential information, and calculate the relevance between position encoding and current state as time-attention 𝜋(𝑝𝑖,ℎ′ 𝑡): 𝜋(𝑝𝑖,ℎ′ 𝑡) = exp(ℎ′ 𝑡𝑊𝑒𝑝𝑖)/ 𝑇𝑒 ∑︁ 𝑗=1 exp(ℎ′ 𝑡𝑊𝑒𝑝 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (24) Time-attention is then used to gain the weighted sum of local value 𝑣1 and global value 𝑣2 in the memory: 𝑚1′ 𝑡 = 𝑇𝑒 ∑︁ 𝑖=1 𝜋(𝑝𝑖,ℎ′ 𝑡)𝑣𝑖 1, (25) 𝑚2′ 𝑡 = 𝑇𝑒 ∑︁ 𝑖=1 𝜋(𝑝𝑖,ℎ′ 𝑡)𝑣𝑖 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (26) 𝑚1′ 𝑡 and 𝑚2′ 𝑡 stores information from different level, thus should play different roles in generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' By a fusion gate, local value 𝑚1′ 𝑡 is changed to 𝑚1 𝑡 and will be incorporated into the projection layer in Euqation 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 𝑔1 𝑡 = 𝑊𝑜 ([ℎ′ 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑐𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑚1′ 𝑡 ]), (27) 𝑚1 𝑡 = 𝑔1 𝑡 · 𝑚1′ 𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (28) We place the local value in the projection layer since 𝑚1 𝑡 stores the detailed information rather than the global feature in the input, thus should play an important part when generating each word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' As for the global value 𝑚2′ 𝑡 , it stores the global feature of the event in a different position, thus should influence the whole generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, information from 𝑚2′ 𝑡 is fusioned into the decoding state ℎ′ 𝑡 by a gate: 𝑔2 𝑡 = 𝑊𝑛([ℎ′ 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑐𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑚2′ 𝑡 ]), (29) ℎ′ 𝑡 = 𝑔2 𝑡 · ℎ′ 𝑡 + (1 − 𝑔2 𝑡 ) · 𝑚2′ 𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (30) Finally, an output projection layer is applied to get the final generating distribution 𝑃𝑣 over vocabulary: 𝑃𝑣 = softmax � 𝑊𝑣 [𝑚1 𝑡 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='ℎ′ 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑐𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='𝑒𝑡] + 𝑏𝑣 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (31) We concatenate the output of decoder LSTM ℎ′ 𝑡, the word context vector 𝑐𝑡, the event context vector 𝑒𝑡, and memory vector 𝑚1 𝑡 as the input of the output projection layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In order to handle the out-of-vocabulary (OOV) problem, we equip the pointer network [27, 56] with our decoder, which enables the decoder capable of copying words from the source text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The design of the pointer network is the same as the model used in [56], thus we omit this procedure due to limited space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Our objective function in the abstractive part is the negative log likelihood of the target word𝑦𝑡, shown in Equation 32: Labs = − 𝑇𝑦 ∑︁ 𝑡=1 log 𝑃𝑣(𝑦𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (32) The gradient descent method is employed to update the parameters in the abstractive part to minimize this loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 14 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan Word-level Encoding SRU Sentence-level Encoding 0 SRU SRU Sentence Embedding Module Summary Extractor xN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' An overview of the sentence extractor in the extractive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In each decoding step, a sentence is to be included in the summary in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='6 Sentence Embedding Module So far, we introduce the abstractive timeline summarization part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Next, we will introduce the extractive summarization part in UTS, and how to unify these two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Our sentence embedding module takes inspiration from [11], where the embedding module also takes the form of a hierarchical structure and consists of an iterative polishing process to better encoder the input document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, we employ a new Bi-RNN to process each sentence and obtain the representation ˆℎ𝑖 𝑡, denoting the 𝑡-th word in 𝑖-th sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We use the last hidden state to represent the overall sentence representation, denoted as ˆℎ𝑖 𝑇𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The document representation is initialized as the average of all sentence representations: 𝐷1 = tanh �� � 𝑊 1 𝑇𝑦𝑠 𝑇𝑦𝑠 ∑︁ 𝑖=1 � ˆℎ𝑖 𝑇𝑤 � + 𝑏�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (33) Next, to model the sequential relationship between sentences and obtain a more comprehensive sentence representation, we iteratively polish the sentence and document representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For brevity, we take the first iteration as an example to illustrate the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, there is an RNN based on SRU (introduced in Equation 4) in the iteration: ˆ𝑎𝑖 1 = SRU( ˆ𝑎𝑖−1 0 , [ ˆℎ𝑖−1 𝑇𝑤 , 𝐷1]), (34) 𝐷2 = GRUiter( ˆ𝑎𝑇𝑦𝑠 1 , 𝐷1), (35) where ˆ𝑎𝑖 1 is the hidden state of 𝑖-th SRU cell in the first iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In this way, we iteratively polish the sentence and document representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We use 𝐼 to denote the iteration number, thus the final representation for 𝑖-th sentence is ˆ𝑎𝑖 𝐼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7 Sentence Extractor Different from previous work that builds a classifier to assign importance score to each sentence, we use an RNN consisting of LSTM cells to select sentences, wherein each step a sentence is selected as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Following traditional attention mechanism in [4], we summarize the input document sentences into context vector𝑐ext dynamically, Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 15 Selected Sentence Number Sentence Number Tile Time-aware Inconsistency Loss Selected Sentence Number Selected Sentence Number Event Number Decode Step Number Convolution Pooling Event-level Attention Map in Summary Generator Sentence-level Attention Map in Sentence Extractor Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The illustration of the Chronological-Attention Unifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' After a convolution and a tile operation, the event-level attention in the summary generation is compared with the sentence-level attention in sentence extractor, where a novel inconsistency loss function is introduced to penalize the inconsistency between these two levels of attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' and the 𝑡-th decoding step is calculated as: ˜ℎ𝑡+1 = LSTMext( ˜ℎ𝑡, [𝑐ext 𝑡 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ˆ𝑎𝑜𝑡 𝐼 ]), (36) 𝑐ext 𝑡 = 𝑇𝑦𝑠 ∑︁ 𝑖=1 ˆ𝛽𝑖 𝑡 ˆ𝑎𝑖 𝐼, (37) 𝑜𝑡 = argmax(MLP( ˜ℎ𝑡)), (38) where 𝑜𝑡 is the index of the selected in 𝑡 step, and ˆ𝑎𝑜𝑡 𝐼 is the hidden state of the previously selected sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ˆ𝛽𝑖 𝑡 is the attention weight on 𝑖-th sentence in 𝑡-th step, and is computed in a similar way to §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Thus, the details are omitted here due to limited space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In this way, our extracted summary is generated in sequence, so as to better capture the sequential information in the input: Lext = − 𝑇𝑦𝑠 ∑︁ 𝑖=1 log 𝑃𝑠 (𝑙𝑖) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (39) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='8 Chronological-Attention Unifier In both abstractive and extractive timeline summarization tasks, the attention on the input document should both follow the time sequential order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, it is intuitive to encourage these two levels of attention to be mostly consistent with each other during training as an intrinsic learning target for free (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', without additional human annotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1, we propose the event-level attention 𝛽 in abstractive part, while in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7, the extractor pays sentence-level attention ˆ𝛽 on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The event-level attention evolves each time a new word is predicted in the summary generator, while the sentence-level attention evolves when a new sentence is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, we first use a convolutional neural network (CNN) to extract the evolving event attention feature from the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, as shown in Figure 4, a Manuscript submitted to ACM 16 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan convolution along the decode step number axis is conducted on the event-level attention map, and the new attention matrix with the sentence-numbered axis is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Then, for each sentence-select step, we duplicate the event-level attention 𝛽𝑖 𝑡 multiple times, where the duplicate number is the sentence number in the 𝑖-th event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Finally, we would like the event-level attention to be high when the sentence-level attention is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, we design the following time-aware inconsistency loss: Linc = − 1 𝑇𝑦 𝑇𝑦𝑠 ∑︁ 𝑡=1 log � 1 |K| ∑︁ 𝑡 ∈K ˆ𝛽𝑡 × 𝛽𝑡 � , (40) where K is the set of top K attended sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This implicitly encourages the distribution of the sentence-level attentions to be sharp and event-level attention to be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' To avoid the degenerated solution for the distribution of sentence attention to be one-hot and event attention to be high, we include the original loss functions for training the extractor (Lext in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7) and abstracter (Labs in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Note that this module is the only part that the extractor is interacting with the abstracter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Our time-aware inconsistency loss facilitates our end-to-end trained unified model to be mutually beneficial to both the extractor and abstracter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 5 EXPERIMENTAL SETUP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1 Research Questions We list seven research questions that guide the experiments: RQ1 (See § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1): What is the overall performance of UTS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Does it outperform other baselines on multilingual datasets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ2 (See § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2): What is the performance of our model on out-of-domain classic timeline summarization dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ3 (See § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='3): What is the effect of each module in UTS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Does our multi-task framework help better summarization performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ4 (See § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4): Is the time position embedding useful so that the summary generator can attend to correct information in the time-event memory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ5 (See § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5): Can event-level attention correctly guide word-level attention in decoding process in the abstractive part?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ6 (See § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='6): Are the chronological attentions successfuly unified in the abstractive and extractive summarization tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ7 (See § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7): What is the influence of the parameter settings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2 Dataset To our best knowledge, there are no large-scale summarization datasets for timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, in our previous work [10], we collect a large-scale timeline dataset from the world’s largest Chinese encyclopedia4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The character subsection of this website consists of celebrities at all times and in all countries or lands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' On each website page, there is a timeline summary for each character, and in the character experience section of this page, each event is set as a paragraph with explanation and details, which is selected as an input article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the previous timeline works [72], they did not pre-select important sentences from the event news articles as a way to test the summarization ability of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, in our work, we did not preprocess the event paragraph as well, since these event paragraphs are similar to news articles in content and in style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We filter out irrelevant content such as cited sources and figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We did not specifically extract time information from the input, because our model learns the time information in an implicit way, instead of particularly encoding it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In total, the training dataset amounts to 169,423 samples with 5,000 evaluation and 5,000 test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' On average, there are 353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='79 words and 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='19 words in the article and summary respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 4https://baike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='com/ Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 17 Datasets # docs (train/val/test) avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' document length avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' summary length vocabulary size words sentences words sentences document summary TL17 4,650 1,252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='33 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='76 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='73 102,099 6,725 Celebrity TS 169,423/5,000/5,000 353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='79 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='76 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='97 444,725 191,334 Event TS 83,188/3,000/3,000 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='19 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='73 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='05 1,083,249 368,619 Wiki TS 140,000/5,000/5,000 606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='65 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='79 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='19 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='89 1,029,617 438,011 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Comparison of summarization datasets with respect to overall corpus size, size of training, validation, and test set, average document (source) and summary (target) length (in terms of words and sentences), and vocabulary size on both on source and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' TS denotes Timeline Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Furthermore, in this work, we first augment the previous dataset with event timeline summarization cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' On the Chinese encyclopedia, there is also a social event subsection that includes the developments of related events over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, in the development history section of each page, there are event paragraphs that describe the development of the corresponding event, and there is also a corresponding timeline summary for these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' After the same cleaning operation, we have 83,188 training cases, 3,000 validation, and 3,000 test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' On average, there are 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='19 words and 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='62 words in the article and summary respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Note that the above two datasets are both in the Chinese language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' To test the performance of our model on multi- lingual datasets, we collect an English timeline summarization dataset from Wikipedia websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Since there are no character or event subsections in Wikipedia, we filter timeline pages by checking if there are multiple timestamps in the summary and document on each website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Other preprocesses are similar to the Chinese encyclopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' A human evaluation on 200 sampled cases from the collected corpus shows that 196 cases are timeline document-summary pairs, 145 of which are about characters, and 51 are about events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Since we have large-scale English summarization datasets, we can test the generalization ability of our model on classic timeline summarization datasets, which are small-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, we report the performance of UTS on out-of-domain dataset Timeline 17 (TL17) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' TL17 contains human-written timelines about topics such as civil wars or the British Petroleum oil disaster, collected from major news outlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Each topic also has a set of related news articles scraped from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The statistics of the four datasets are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We also give timeline statistic information in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' It can be seen that compared with TL17 dataset, our three datasets are significantly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This again demonstrates the necessity of our datasets, which are large enough to train a neural-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In terms of timeline-related attributes, the summaries in our datasets have more date stamps in each sentence on average, which requires the summarization model to be more time-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The average date number in the document input is smaller in our datasets, this is because that our input document is shorter than TL17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, the average sentences/dates ratio of our datasets is comparable to TL17, proving the time attribute of our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='3 Comparison Methods We first conduct an ablation study to prove the effectiveness of each module in UTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Then, to evaluate the performance of our proposed dataset and model, we compare it with the following baselines: Abstractive baselines: (1) Pointer-Gen [56] is an RNN based model with an attention mechanism and allows the system to copy words from the source text via pointing for abstractive summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 18 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan Datasets Document Summary length Compression Avg dates Avg sents/dates Avg dates Avg sents/dates Sent Date TL17 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='14 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='70 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='46 Celebrity TS 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='50 Event TS 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='36 Wiki TS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='74 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Timeline-specific statistic attributes of our datasets and TL17 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (2) FTSum leverages open information extraction and dependency parse technologies to extract actual fact descriptions from the source text [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Since there is no open information extraction tool in Chinese, we use POS tagging to extract entities and verbs to replace them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (3) Unified is a unified model combining the strength of extractive and abstractive summarization proposed in [28], where sentence-level attention is used to modulate the word-level attention such that words in less attended sentences are less likely to be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (4) GPG is a model proposed by Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [57] which generates summaries by “editing” pointed tokens instead of hard copying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The editing is performed by transforming the pointed word vector into a target space with a learned relation embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (5) SAGCopy is an augmented Transformer with a self-attention guided copy mechanism, which was proposed by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Specifically, they first identify the importance of each source word based on the degree centrality with a directed graph built by the self-attention layer in the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' They then use the centrality of each source word to guide the copy process explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (6) MTS is the first abstractive timeline summarization framework proposed in our previous work [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This method achieves state-of-the-art performance on the celebrity timeline summarization dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Extractive baselines: (1) Lead3 is an extractive baseline that concatenates the first-3 sentences of each source document as a summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (2) TextRank [44] is an unsupervised algorithm while sentence importance scores are computed based on eigenvector centrality within weighted-graphs for extractive sentence summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (3) ITS One of state-of-the-art extractive summarization models proposed in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ITS iteratively polishes the document representation on many passes through the document, so as to extract better summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For testing our models on out-of-domain dataset, we compare with a number of traditional timeline summarization baselines: (1) Chieu [14] is an unsupervised baseline based on direct summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (2) Martschat [41] greedily selects a combination of sentences from the entire collection, which maximizes submodular functions for content coverage, textual and temporal diversity and a high count of date references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (3) Tran[5] is an original date-wise timeline summarization approach, using regression for both date selection and summarization, and using all sentences of a date as candidate sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (4) Pubcount [26] is a simple date-wise baseline that uses the publication count to rank dates, and all sentences published on a date for candidate selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (5) Datawise [26] uses supervised date selection, PM-MEAN for candidate selection and CENTROID-OPT for summa- rization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 19 (6) Clust [26] uses DATEMENTIONCOUNT to rank clusters, and CENTROID-OPT for summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The performance of these baselines are consistent with the result from [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4 Evaluation Metrics For evaluation metrics, we adopt ROUGE F1 score in [37] which is widely applied for summarization evaluation [11, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The ROUGE metrics compare the generated summary with the reference summary by computing overlapping lexical units, including ROUGE-1 (unigram), ROUGE-2 (bi-gram), and ROUGE-L (longest common subsequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For the out-of-domain test dataset, we follow [26], and use the specific timeline evaluation metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=', Alignment- based ROUGE F1-score, and Date F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Alignment-based ROUGE F1-score compares the textual overlap between a system and a ground-truth timeline, while also considering the assignments of dates to texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Date F1-score compares only the dates of a system and a ground-truth timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [55] notes that only using the ROUGE metric to evaluate summarization quality can be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Therefore, we also evaluate our model by human evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Three highly educated participants are asked to score 100 randomly sampled summaries generated by GPG, SAGCopy, MTS, and UTS-abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Statistical significance of observed differences between the performance of two runs are tested using a two-tailed paired t-test and is denoted using ▲(or ▼) for strong significance for 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 Implementation Details We implement our experiments in TensorFlow [1] on NVIDIA GTX 1080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For all experiments, our model has 256-dimensional hidden states and 128-dimensional word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Following See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' [56], we do not pretrain the word embeddings, instead, they are learned from scratch during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We use a vocabulary of 50k words for both source and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For time-event memory, the dimension of the key, global value, and local value are 128, 512, and 256 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We initialize all of the parameters randomly using a uniform distribution in [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The batch size is set to 16, and the event number is set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For the abstractive part, during training and at test time we truncate the article to 400 tokens and limit the length of the summary to 70 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For the extractive part, we used a greedy algorithm similar to [47] to obtain an oracle summary for each document to train extractive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The algorithm generates an oracle consisting of multiple sentences which maximize the ROUGE-2 score against the gold summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We limit the input sentence number to 24, the length of each input sentence to 20, and the number of selected sentences to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For the chronological-attention unifier, we set 𝐾 to 3 for computing Linc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We use Adagrad optimizer [17] as our optimizing algorithm and the learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (This was found to work best of Stochastic Gradient Descent, Adadelta, Momentum, Adam, and RMSProp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We use gradient clipping with a maximum gradient norm of 2, but do not use any form of regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We use loss on the validation set to implement early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In decoding, we employ a beam search with beam size 4 to generate a more fluent summary sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' When testing our model on the out-of-domain Timeline 17 dataset, for each example with S source input documents, we take the first 400/S tokens from each source document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For the training efficiency, it takes about 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7 hours to train an epoch, and our model reaches the best performance after only 3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' While for baseline Pointer-Gen, it takes 7 hours to train an epoch, but it reaches the best performance after 7 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In particular, our model makes much quicker progress in the early phases of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This demonstrates the effectiveness of our unified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In terms of testing, it takes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='06 hours to generate summaries for all the cases in the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We selected the top-3 checkpoints based on the evaluation loss on the validation set, and report the averaged results on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 20 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan Models Celebrity Timeline Dataset Event Timeline Dataset Wiki Timeline Dataset RG-1 RG-2 RG-L RG-1 RG-2 RG-L RG-1 RG-2 RG-L Sentence extraction methods Lead3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='36 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='96 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='99 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='47 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='26 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='73 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='94 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='56 TextRank 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='27 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='34 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='86 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='89 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='43 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='66 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='47 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='40 ITS 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='03 18.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='37 Unified-ext 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='29 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='16 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='06 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='39 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='47 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='82 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='28 UTS-ext 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='81 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='26 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='03 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='12 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='01 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='06 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='64 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='81 Abstractive methods Pointer-Gen 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='61 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='35 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='51 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='56 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='84 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='00 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='07 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='65 FTSum 37.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='80 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='05 Unified-abs 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='24 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='95 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='42 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='58 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='93 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='95 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='84 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='37 GPG 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='43 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='59 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='38 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='38 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='77 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='81 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='80 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='85 SAGCopy 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='64 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='84 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='41 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='40 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='95 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='72 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='84 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='01 MTS 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='78 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='24 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='69 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='89 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='38 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='97 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='68 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='88 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='18 UTS-abs 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='56 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='95 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='18 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='30 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='63 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='28 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='92 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='62 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ1: ROUGE scores comparison between baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Models and baselines in the top half are extractive, while those in the bottom half are abstractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' All our ROUGE scores have a 95% confidence interval of at most ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='24 as reported by the official ROUGE script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 6 EXPERIMENTAL RESULTS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1 Overall Performance For research question RQ1, we examine the performance of our model and baselines in terms of ROUGE as shown in table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Firstly, on the celebrity timeline dataset, abstractive models outperform extractive models by a substantial margin on our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We attribute this result to the observation that the gold summary of this dataset tends to use new expressions to summarize the original input documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This demonstrates the necessity of abstractive timeline summarization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Secondly, we compare our previous model MTS with recently-proposed baselines including SAGCopy and GPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' These two baselines obtain lower ROUGE scores on our datasets than MTS, which demonstrates the effectiveness of our previous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Finally, based on MTS, our augmented model UTS-ext and UTS-abs achieves even better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Concretely, for the abstractive part, UTS-abs outperforms SAGCopy by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='56%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='92% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='61%, and outperforms MTS by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='47%, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='68% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='95% in terms of ROUGE-1, ROUGE-2 and ROUGE-L respectively on celebrity timeline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For the extractive part, our extractive method achieves about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='23% points improvement on ROUGE-2 compared with ITS on the celebrity timeline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We attribute the improvement to two aspects: Firstly, the abstractive objective can promote the recognition of important sentences for the extractive model with the chronological attention unifier network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Besides, while extractive gold label sequences are obtained by greedily optimizing ROUGE-2 on the gold- standard summary, gold labels may not be accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Joint learning of two objectives may correct some biases for the extractive model due to the inaccurate labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The above results prove the superiority of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Note that we mainly compare our model with ITS, because our extractive part is mostly based on ITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Our framework can be applied to other extractive models, and theoretically, will bring benefits for both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We leave it as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Our human evaluation study assessed the overall quality of the summaries on the celebrity timeline dataset by asking three highly educated participants to rank them taking into account the following criteria: Fluency (is the Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 21 Fluency Informativeness Fidelity GPG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='39 SAGCopy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='43 MTS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='61 UTS-abs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='77▲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='62▲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='65▲ Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ1: Human evaluation comparison with main baselines on celebrity timeline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' summary fluent and grammatical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ), Informativeness (does the summary convey important facts about the topic in question?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ), and Fidelity (is the summary faithful to the input?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We pick SAGCopy and GPG as baselines since their performance is relatively high compared to other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The rating score ranges from 1 to 3 and 3 is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The results are presented in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We can see that our model performs much better than all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the fluency indicator, our model achieves a high score of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='77, which is higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='59 of GPG and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='64 of SAGCopy, indicating that our model can reduce the grammatical errors and improve the readability of the summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the informativeness indicator, our model is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='05 better than SAGCopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' It indicates that our model can effectively capture salient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the fidelity indicator, UTS-abs outperforms all baselines by a large margin, which indicates the multi-granularity semantic information and joint learning with extractive summarization does help to avoid the unfaithful information of the generated summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' It is worth noticing that the infidelity problem is a serious problem existing in timeline summarization, and MTS and UTS-abs greatly alleviates such problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We also conduct the paired student t-test between our model and SAGCopy (the row with shaded background), and the result demonstrates the significance of the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The kappa statistics is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='46 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='49 respectively, which indicates moderate agreement between annotators5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' To prove the significance of these results, we also conduct the paired student t-test between our model and SAGCopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We obtain a p-value of 3 × 10−8, 8 × 10−12, and 9 × 10−11 for fluency, informativeness, and fidelity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We also show a case study in Table 9 with translated version in Table 8 selected from celebrity timelime dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The case is about James Cameron’s career as a director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We omit unimportant information in the input document due to limited space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The input document includes most of his works, and the detailed information of each event, while the summary reference only introduces the main event of his experience, omitting those details and unimportant events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' It can be seen that the summary generated by UTS-abs successfully captures the important events, and introduces them in the correct order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The output of our UTS-ext has a high overlap with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' As for baseline GPG, it fails to capture the most important events, but includes irrelevant information such as details in filming “Piranha II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' For baseline SAGCopy, it also generates unimportant descriptions including information “The dark angel of the last world”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Moreover, our extractive and abstractive summary show consistent behavior with the high overlap, which further indicates that the two methods can jointly promote the recognition of important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Compared with the extracted summary, the generated summary is more concise and coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2 Out of Domain Test Next, we address research question RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In Table 10, we present the performance of UTS on the classic timeline summarization TL17 dataset as an out-of-domain test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' It can be seen that both of our models outperform existing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Specifically, UTS-ext outperforms the best baseline Datawise by 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4% on AR1-F score, demonstrating the 5[32] characterize kappa values < 0 as no agreement, 0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='20 as slight, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='21-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='40 as fair, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='41-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='60 as moderate, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='61-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='80 as substantial, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='81-1 as almost perfect agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 22 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan In 1981, James Cameron directed his first film, “Piranha II”, which was shot entirely in Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Cameron didn’t get along well with an Italian speaking staff, and the producers didn’t let him participate in the final editing of the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1984, Cameron released his first self-made and self-directed film “Terminator”, which costs only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 million dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1986, James Cameron’s second self-made work, “Alien 2”, was published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1987, “Alien 2” won seven Academy Award nominations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' James Cameron won the best director award at the 14th Saturn Awards for this film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1989, Cameron wrote and directed his third film, “The Abyss”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1991, his film “Terminator 2” made 200 million dollars at the box office in the United States, and he also won the 18th Saturn Awards for best director and best screenwriter for this film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1997, Cameron directed the film “Titanic”, which wins 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='84 billion at the box office, and starred Leonardo DiCaprio and Kate Winslet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1998, the film won 14 nominations and 11 awards at the 70th Academy Awards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 2000, he directed and supervised the TV series “The dark angel of the last world” with the theme of gene therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' reference In 1981, he directed the first film “Piranha II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1984, he became famous for his science fiction film “Terminator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1986, he wrote and directed the film “Alien 2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1991, he won the best director award and best screenwriter award at the 18th Saturn awards for his film “Terminator 2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1997, his film “Titanic” won 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='84 billion dollars at the box office, breaking the global box office record;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' it won 11 awards including best picture at the 70th Academy Awards, and James Cameron won the best director award at the Oscars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' GPG In 1981, James Cameron directed the film “Piranha II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Cameron Cameron didn’t get along well with an Italian speaking staff, and the producers didn’t let him participate in the final editing of the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1984, he became famous for his science fiction film “Terminator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1986, he wrote and directed the film “Alien 2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1986, he wrote and directed the film “Alien 2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1991, he won the best director award and best screenwriter award at the 18th Saturn Awards for his film “Terminator 2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1997, his film Titanic won 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='84 billion US dollars at the box office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' SAGCopy In 1981, James Cameron directed his first work, “Piranha II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1984, Cameron released his first film, “Terminator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1986, James Cameron wrote and directed his second work, “Alien 2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1997, James Cameron directed the film “Titanic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 2000, he directed and supervised the TV series “The dark angel of the last world” with the theme of gene therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' UTS-ext In 1981, James Cameron directed his first film, “Piranha II”, which was shot entirely in Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1984, Cameron released his first self-made and self-directed film “Terminator”, which cost only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 million dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1986, James Cameron’s second self-made work, “Alien 2”, was published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' in 1997, Cameron directed the film “Titanic”, which wins 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='84 billion at the box office, and starred Leonardo DiCaprio and Kate Winslet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' UTS-abs In 1981, Cameron directed his first work, “Piranha II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1984, he released his first film “Terminator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1986, his second film, “Alien 2”, was published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1987, “Alien 2” won the 14th Saturn Award for best director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1991, his film “Terminator 2” made 200 million dollars in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1997, he directed the film “Titanic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In 1998, the film won 14 Academy Awards nominations and 11 of them at the 70th Academy Awards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ1: Examples of the generated answers by UTS-abs, UTS-ext and baselines (translated version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' effectiveness of the neural network in the traditional extractive style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' UTS-abs performs similar to UTS-ext, improving the AR1-F score of Pubcount by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This demonstrates that the abstractive methods can be adapted to out-of-domain small-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Specifically, since our original WikiTS dataset is in encyclopedia style, while Timeline 17 is a news dataset, this demonstrates that our model can be applied to datasets of different language styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='3 Ablation Study Next, we turn to research question RQ3, where we perform an ablation study on the test set to investigate the influence of different modules in our proposed UTS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Modules are tested in four ways: (1) we remove the sentence extractor Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 23 1981年，詹姆斯·卡梅隆执导了第一部作品《食人鱼2》，影片完全在意大利拍摄。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='卡梅隆和一口意大利 语的工作人员相处得并不愉快，而拍摄完毕后制片方不让他参与影片的最终剪辑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1984年，卡梅隆推出了 他第一部自编自导的影片《终结者》，这部影片的拍摄只花了650万美元.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1986年，詹姆斯·卡梅隆自编 自导的第二部作品《异形2》问世.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1987年，《异形2》获得了七项奥斯卡奖提名.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='詹姆斯·卡梅隆凭借此 片获得了第14届土星奖最佳导演奖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1989年，卡梅隆自编自导了第三部电影《深渊》.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1991年，他执导的 电影《终结者2》在美国上映后取得了2亿美元的票房，他也凭借该片获得了第18届土星奖最佳导演奖以 及最佳编剧奖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1997年，詹姆斯·卡梅隆执导了电影《泰坦尼克号》，该片获得18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4亿美元的票房，由莱 昂纳多·迪卡普里奥、凯特·温斯莱特等主演.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1998年，在第70届奥斯卡金像奖上这部影片获得了14个奥 斯卡奖的提名并获得了其中的11个奖项.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2000年，他执导并监制了以基因治疗为题材的电视剧《末世黑天 使》.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' reference 1981年，詹姆斯卡梅隆执导首部电影《食人鱼2》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1984年，因自编自导科幻电影《终结者》 成名。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1986年，自编自导电影《异形2》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1991年，凭借电影《终结者2》获得第18届土星奖最 佳导演奖以及最佳编剧奖。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1997年，他执导的电影《泰坦尼克号》取得了18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4亿美元的票房， 打破全球影史票房纪录;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='该片在第70届奥斯卡金像奖上获得了包括最佳影片在内的11个奖项，詹 姆斯·卡梅隆凭借该片获得了奥斯卡奖最佳导演奖。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' GPG 1981年，詹姆斯·卡梅隆执导了部作品《食人鱼2》，卡梅隆卡梅隆和一口意大利语的工作 人员相处得并不愉快，而拍摄完毕后制片方不让他参与影片的最终剪辑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 1984年，卡梅隆凭 借科幻电影《终结者》出名。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1986年，他自编自导了电影《异形2》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1986年，他自编自导 了电影《异形2》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1991年，他凭借《终结者2》获得了第18届土星奖最佳导演奖和最佳编剧 奖。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1997年，他的电影《泰坦尼克号》在美国获得了18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4亿票房。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' SAGCopy 1981年，詹姆斯·卡梅隆执导了第一部作品《食人鱼2》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1984年，卡梅隆推出了他第一 部自编自导的影片《终结者》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1986年，詹姆斯·卡梅隆自编自导的第二部作品《异 形2》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1997年，詹姆斯·卡梅隆执导了电影《泰坦尼克号》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2000年，他执导并监制了以 基因治疗为题材的电视剧《末世黑天使》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' UTS-ext 1981年，詹姆斯·卡梅隆执导了第一部作品《食人鱼2》，影片完全在意大利拍摄。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1984年， 卡 梅 隆 推 出 了 他 第 一 部 自 编 自 导 的 影 片 《 终 结 者 》.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='这 部 影 片 的 拍 摄 只 花 了650万 美 元。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1986年，詹姆斯·卡梅隆自编自导的第二部作品《异形2》问世。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1997年，詹姆斯·卡 梅隆执导了电影《泰坦尼克号》，该片获得18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4亿美元的票房，由莱昂纳多·迪卡普里奥、凯 特·温斯莱特等主演。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' UTS-abs 1981年，卡梅隆执导了第一部作品《食人鱼2》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1984年，卡梅隆推出了他第一部自编自导的 影片《终结者》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1986年，自编自导的第二部作品《异形2》问世。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1987年，《异形2》获得了 第14届土星奖最佳导演奖。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1991年，他执导的电影《终结者2》在美国上映后取得了2亿美元的 票房。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1997年，执导了电影《泰坦尼克号》。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1998年，在第70届奥斯卡金像奖上这部影片获得 了14个奥斯卡奖的提名并获得了其中的11个奖项。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ1: Examples of the generated answers by UTS-abs, UTS-ext and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ3: Visualizations of time-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The figure in the left part is the attention map in the first decoding step, and the figure in the right part is in the final decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' and only train the generator to verify the effectiveness of joint learning on the abstractive summarization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (2) we remove the summary generator part and only train the sentence extractor to verify the effectiveness of joint learning on the extractive summarization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (3) we remove the graph-based encoder and only stores the local representation in the Manuscript submitted to ACM 24 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan AR1-F AR2-F Date-F1 Chieu 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1 Martschat 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4 Tran 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7 Pubcount 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1 Datewise 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4 Clust 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7 UTS-ext 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='08 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='9 UTS-abs 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='51 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='6 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ2: ROUGE scores on out-of-domain TL17 summarization dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ROUGE-1 ROUGE-2 ROUGE-L UTS-abs 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='56 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='95 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='18 without multitask 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='58 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='54 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='55 without global 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='66 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='87 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='76 without local 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='14 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='15 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='00 UTS-ext 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='81 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='26 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='03 without multitask 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='78 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='09 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='03 without global 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='00 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='89 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='07 without local 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='28 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='98 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='69 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ3: ROUGE scores of different ablation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' memory to verify the effectiveness of global representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' (4) we remove the time-event memory entirely to verify the importance of global and local representation further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Table 11 presents the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We find that the ROUGE-2 score of extractive summarization drops by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='26% after the summary generator is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This indicates that the joint learning method helps extractive summarization to benefit from abstractive summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ROUGE-2 score of abstractive summarization drops by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='54% after the sentence extractor is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This indicates that extractive summarization does help abstractive summarization identify important sentences during the interactive decoding phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ROUGE-2 score of extractive summarization drops by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='72%, while the ROUGE-2 score of abstractive summarization drops by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='25% after the global representation is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' It indicates establishing the graph-based encoder to simulate the relationships between events is necessary to improve the performance of both extractive and abstractive summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ROUGE-2 score drops by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='72% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='45% compared with UTS-abs after removing the global representation and the local representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' It indicates the semantic information of the time-event memory is of great importance to encode multiple events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='4 Analysis of Time Position Embedding We then address RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The usefulness of time position embedding is reflected by time-attention in the memory, denoted as 𝜋 as introduced in Equation 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' If the time position embedding successfully encodes the time information, then the time-attention should obey the development of the input document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We visualize the attention map of two randomly sampled examples as shown in Figure 5 from the celebrity timeline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The figure on the left is the attention map in the first decoding step, and the figure on the right is in the final decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The darker the color is, the higher Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ5: Visualizations of two level attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The figure above is the event-level attention and the three figures below are the word-level attentions of first lead three events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' the attention is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Due to limited space, we omit the corresponding event descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' When decoding starts, UTS-abs learns to pay attention to the first two events, which always consist of parallel information such as the birthplace and birth date of the character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The attentions on the last several events are low since it does not need this information in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' When decoding ends, UTS-abs focuses more on the last several events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, it also pays attention to the first few events, since timeline summarization is a process of information accumulation, and later sentences should consider previous information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The above example demonstrates the effectiveness of time position embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 Analysis of Event-level Attention We now turn to RQ5, whether event-level attention can guide word-level attention in the abstractive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We first conduct a case study to visualize the two-level attention, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The figure above is the event-level attention, and the three figures below are word-level attention corresponding to the first three events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We only show the first 11 words in an event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The result shows that the third event is the most important event in this decoding step, and the weights of the words in this event are also greater than other words on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The above observation demonstrates that event-level attention gives the correct guidance for word-level attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Apart from the visualization, we also conduct a quantitative analysis to measure how greatly the word-level attention is influenced by event-level information, which is reflected by inconsistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We adjust the inconsistency loss proposed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='8 to evaluate the inconsistency between event attention and word attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The new consistency loss at 𝑡-th decoding step is the negative log-likelihood of the product of attention value of most attended words and their corresponding event-level attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The intuition is to verify whether the event-level attention is high too when word-level attention is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' When training starts, the inconsistency loss is around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='3, and when training ends, the loss drops to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This means that event-level information greatly influences the word-level attention and the model learns to unify these two attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We did not directly add inconsistency loss to training because we found that made UTS perform worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Instead, we let the model learn by itself to unify these two attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='6 Analysis of the Unified Chronological Attentions We then address RQ6, examining whether the chronological attentions in the abstractive and extractive parts are indeed unified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Remember that we come up with a time-aware inconsistency loss to unify the two attentions, thus, by looking at the loss curve, we can examine the effectiveness of this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM 26 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan 0k 2k 4k 6k 8k 10k Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='0 Time-aware Inconsistency Loss Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' RQ6: Time-aware inconsistency loss curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The loss curve of the inconsistency is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We can see that when the training begins, the inconsistency loss fluctuates from time to time, probably because the model aims to train the extractor and generator separately at the beginning of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, the average of the inconsistency loss presents a falling tendency, which means that the extractor and generator unify during the whole training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the end, the time-aware inconsistency loss drops from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='7 Robustness of Parameter Setting 64 128 256 448 512 Hidden Size 0 5 10 15 20 25 30 35 40 ROUGE score ROUGE-1 ROUGE-2 ROUGE-L Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Performance of UTS-abs with different parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Manuscript submitted to ACM Follow the Timeline!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Generating Abstractive and Extractive Timeline Summary in Chronological Order 27 Finally, we turn to address RQ7 to investigate the robustness of parameter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We train our model in different parameter settings as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The hidden size of the RNN is tuned from 64 to 512, and we use the ROUGE 𝐹1 score to evaluate each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' As the hidden size grows larger from 64 to 256, the performance rises along with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The increment of hidden size improves the ROUGE-1 and ROUGE-L scores by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='54 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='77 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' When the hidden size continuously goes larger from 256 to 512, the performance is declined slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The increment of hidden size leads to a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='15% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content='25% drop in terms of ROUGE-1 and ROUGE-L respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Nonetheless, we can find that each metric is maintained at a stable interval, which demonstrates that our UTS is robust in terms of different parameter sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 7 CONCLUSION AND FUTURE WORK In our previous work, we propose a framework named MTS which aims to generate summaries that concisely summarize the evolution trajectory along the timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' However, in this method, the time information is captured in an implicit and indirect way, where it is hard to verify and ensure the decoder indeed captures the time-sequential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Hence, in this work, we propose a novel Unified Timeline Summarizer (UTS) that can generate abstractive and extractive timeline summaries in time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Specifically, in the encoder part, we propose a graph-based event encoder that relates multiple events according to their content dependency and learns a representation of each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the decoder part, to ensure the chronological order of the abstractive summary, we propose to extract the feature of event-level attention in its generation process with sequential information remained and use it to simulate the evolutionary attention of the ground truth summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' The event-level attention can also be used to assist in extracting summary, where we devise a time-aware inconsistency loss function to penalize the inconsistency between abstractive attention and extractive attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Note that the extractive summary is generated one by one, thus the extracted summary also comes in time sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' We augment the character timeline summarization dataset proposed in our previous work with the event timeline summarization corpus and English corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' Experimental results on these datasets and on out-of-domain Timeline 17 dataset show that our UTS model can significantly outperform the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' In the near future, we aim to propose a multi-modal time-aware timeline summarization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank the anonymous reviewers for their constructive comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' This work was supported by National Key Research and Development Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 2020YFB1406702), National Natural Science Foundation of China (NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 62122089 & No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdAyT4oBgHgl3EQf8_qr/content/2301.00867v1.pdf'} +page_content=' 61876196) Manuscript submitted to ACM 28 Xiuying Chen, Mingzhe Li, Shen Gao, Zhangming Chan, Dongyan Zhao, Xin Gao, Xiangliang Zhang, and Rui Yan REFERENCES [1] Martín Abadi, Paul Barham, 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b/_dFRT4oBgHgl3EQfszdk/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8e0de96b588b6153ac8f73c6d498fb3986393911197130d83bb26f2b669db822 +size 1651298 diff --git a/a9E1T4oBgHgl3EQfdAS_/content/tmp_files/2301.03191v1.pdf.txt b/a9E1T4oBgHgl3EQfdAS_/content/tmp_files/2301.03191v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5dee778c8a0e84b7dba4f90bf44349296426f76c --- /dev/null +++ b/a9E1T4oBgHgl3EQfdAS_/content/tmp_files/2301.03191v1.pdf.txt @@ -0,0 +1,1503 @@ +Line-Torus Intersection for Ray Tracing: Alternative Formulations + +VACLAV SKALA +Department of Computer Science and Engineering +University of West Bohemia +Univerzitni 8, CZ 30614 Plzen +CZECH REPUBLIC +http://www.VaclavSkala.eu + + +Abstract: - Intersection algorithms are very important in computation of geometrical problems. Algorithms for +a line intersection with linear or quadratic surfaces are quite efficient. However, algorithms for a line +intersection with other surfaces are more complex and time consuming. In this case the object is usually closed +into a simple bounding volume to speed up the cases when the given line cannot intersect the given object. +In this paper new formulations of the line-torus intersection problem are given and new specification of the +bounding volume for a torus is given as well. The presented approach is based on an idea of a line intersection +with an envelope of rotating sphere that forms a torus. Due to this approach new bounding volume can be +formulated which is more effective as it enables to detect cases when the line passes the “hole” of a torus, too. + + +Key-Words: Line clipping; torus line intersection, CAD systems + +1 Introduction +Intersection algorithms play a significant role in all +geometric problems and CAD/CAM systems. +Intersection algorithms are well documented for +linear cases, e.g. line-plane or line-triangle etc., and +also for some specific non-linear surfaces like line- +sphere intersection etc. However, there are other +objects like bicubic parametric patches, torus etc. In +this case computation of intersection points is more +complex and usually complex formula or iterative +formula are to be used. + + +Figure 1: Torus +(Courtesy of Wikipedia) + +Intersection of a line and closed surface can be +considered as generalized well known clipping +problem. Intersection of a line or ray with a surface +is the key problem solved in all ray-tracing +techniques. Due to the computational complexity a +bounding volumes are used to detect cases when a +line cannot intersect the given object. +In this paper we present torus-line intersection +problem [1] [2], which leads to a quartic equation +[3] +in +principle, +and +show +other +possible +formulations of the line-torus intersection problem +which offer quite different representations of the +problem. These reformulations lead to a formulation +of a new problem – generalized line clipping by an +envelope (convex or non-convex) of parametric +closed surfaces. + + +2 Torus Line Intersection +Torus-line intersection is actually a solution of a line +in E3 usually given in a parametric form as + + + +(1) +and a torus, which is generally a surface of the +4th order and can be given as : + + + +(2) +An alternative formulation +Note that the axis is the rotational axis. The torus +equation can be reformulated as + + + +(4) +where + + + + + + + + + + + + +(5) + + + + +(3) +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +288 +Issue 7, Volume 12, July 2013 + + + +As there will be some geometric transformations +used latter on we can also scale the given torus and +a line so that , i.e. the torus is “normalized”. +Now the intersection of a line and the torus is +given as a solution of equations: + + +(6) +and + + +(7) + +Substituting Eq.5 to Eq.6 we get + + + + +(8) +and finally we get + + + + + + +(9) +This equations is quite complex, but by detailed +evaluation we get a quartic equation + + +(10) +where: + + + + + + + + + + + + + + +(11) +It can be seen that the computation can be simplified +for the case, when , i.e. the directional +vector of the line is normalized or the equation is +divided by . +It means that we are getting a quartic equation in +the from [4] + + +(12) +which can be simplified by substitution + + + +(13) +to + + +(14) +where + + + + + + + + + + +(15) +If solution for is found, then the solution of the +original equation is given by Eq.12. To get a +solution for the following a qubic equation has to +be solved + + + + + +(16) +Then the values can be computed from real +solution of the equation above as two quadratic +equations as follows: +If then + + + +(17) +If then + + + +(18) +It can be seen that the solution itself is not simple, +but the formula is closed. +On the opposite, an iterative method like +Bisection or Newton method can be used. However +there are up to 4 intersections of the line and the +torus, so it is necessary to find relevant intervals +for , with one intersection only. + + +2.1 Alternative Torus Representation +There are other formulations of a torus as follows, +but they are not convenient for our purposes. + + + + +(19) +or a parametric form as + + + + +(20) +It can be seen that a solution of a line-torus +intersection is not a simple task and it leads to a +non-trivial computational problem. +However, there are some other geometrically +equivalent formulations which could be used for +finding a solution. In the following we will consider +only circular torus. + + +2.2 Geometric Transformations +Geometric transformations with points are defined +in the projective space using homogeneous +coordinates, i.e. in the projective extension of the +Euclidean space. A point in the +Euclidean +coordinates +has +homogeneous +coordinates ; is the homogeneous +coordinate. The conversion between the projective +space and the Euclidean space is defined as + + + + +(21) +It means that the projective representation is +actually a one parametric set. A point in the +Euclidean space E2 is represented as a line with the +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +289 +Issue 7, Volume 12, July 2013 + + + +origin of the coordinate system excluded in the +projective space. Geometric transformations with +points like rotation, translation, mirroring etc. can be +than described by the matrix as + + +(22) +Note that might have some physical meaning +and units, e.g. [m], while has no unit, it is just a +“scaling factor”. That’s why we used “:” to separate +the values in the vector notation. +A line in E2 determined by two points given in +the homogeneous coordinates can be computed +using the cross product as [5], [6]. + + + + + + + + + + + + + + +(23) +Intersection of two lines and in E2 can be +computes as + + + + + + + + + + + + + +(24) +We can see that both computations are in the E2 case +“dual”, i.e. line and points are dual [7]. In the E3 +case a point is dual to a plane and vice versa. It can +be shown that a plane given by three points can be +determined by the extended cross product as + + + + + + + + + + + + + + + + + + + + + + +(25) +Again, an intersection of three planes can be +computed as, see [7], [8], [9] for details + + + + + + + + + + + + + + + + + + + + + +(26) +This approach is simple, easy to implement and +convenient for GPU implementation as well. +However, matrix transformations for points +cannot be used for geometric transformations with +lines in the E2 case nor with planes in the E3 case. It +can be shown [6] that if a line is given by two +points and and those points are geometrically +transformed using the matrix, i.e. + + +(27) +and + + + + +(28) +then + + +(29) +It can be shown that the matrix is defined as + + +(30) +Because are coefficients of an implicit equation +we can simply write + + +(31) +As the implicit form is used, coefficients of a line +can be multiplied by any non-zero constant and the +line will be same. Therefore + + +(32) +where means protectively equal. Similarly for a +plane + + +(33) +It means that we can correctly manipulate with lines +and planes, now. + +2.3 Bounding Volume +Let us assume that the torus lies in the plane, +i.e. the -axis is its rotational axis. Bounding +volume, defined in [1], is based on an idea that torus +is bounded by an intersection of a sphere and two +half-spaces, Fig.2. + +Figure 2: Bounding volume + +The radius of the enclosing sphere is given as + + +(34) +The bounding test computes intersection of a line +with a sphere. If such intersections and +exist then the line does not intersect the torus if the +following condition is valid [1] + + + + + +(35) +It can be seen that the test does not eliminate cases +when a line: + +is passing the “hole inside of the torus” without +touching or intersecting the torus – line + +nearly touches the torus – line – but there is +a small probability +r +x +tmin +pA +pC +pB +pD +tmax +R1 +R +y +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +290 +Issue 7, Volume 12, July 2013 + + + +It should be noted that the Fig.2 presents general +situation in the E3 case. + + +2.4 Torus Transformation +So far we have dealt with a general situation +expecting that the torus is in its basic position, i.e. it +lies in the plane and the axis is the +rotational axis. In the case of torus general position +the following transformations can be used: + + + + + + + + + + + + + + + + + + + +(36) +where: defines axis of the torus, defines +axis of the torus, is used to get an +orthonormal basis, and is the torus centre. +It can be seen that there are some interesting +properties of the line-torus intersection problem, +like + +torus rotational symmetry, + +if mirroring operation is used only one quadrant +can be considered to solve the intersection +problem. +We will explore if those properties can contribute to +simplification of computation in the following part. + +2.5 Intersections Classification +As a torus is rotationally invariant we can rotate the +given line about axis so that it lies in a plane + , i.e. in a plane parallel to the plane. +There is no significant computational expense as the +transformation matrix is accumulated with the +matrix. Now we can distinguish three fundamentally +different cases according to the value: +a) : generally intersection with two +independent parts have to be considered, i.e. for + and and due to convexity each +part could have up to 2 intersections only +(2 convex envelopes are generated), +b) : this case is more complex as +the envelope has only one part, but it is not +convex as it can have an inflexion point and 3 +intersection points can be generated, +c) : the simplest case as only one +convex envelope is generated. + + +Figure 3: Torus plane intersection for + +The +above +mentioned +three +cases +differ +significantly. Unfortunately the envelope is not +convex in all the cases. +2.6 Vieta’s Formula +Let us assume that is a polynomial of degree + + + +(37) +Then according to the Vieta’s formula the roots +satisfy equations + + + + +( + + + + + + + + +(38) +In the quadratic equation case + + +(39) +we obtain + + + + +(40) +These formulas are not well known and will be used +latter on. In the following we will show different +approaches to the line – torus intersection problem. + + +3 New Intersection Formulations +In the previous part we have presented the +“traditional” approach to the line–torus intersection +detection +and +computation. +Now, +different +equivalent formulations, which could lead to +simpler and faster solutions, will be formulated in +the following part. They can be briefly classified as +follows: + a sphere is rotating about axis (the envelope +forms a torus) and intersection with the line in E3 +is computed directly, + a sphere is fixed on the axis and intersection +with the line rotating about axis (i.e. it is +actually an intersection of a sphere and double +cone) in E3 is computed directly, + a sphere is rotating about axis and intersection +with the plane in E3, on which the given +line lies, results into circles in this plane, i.e. +circles in E2, forming an envelope, i.e. a curve is +given as an intersection of a torus with a plane. +An intersection of the envelope of all circles and +the line is computed in E2. +This is actually a generalized line-clipping +problem. +Let us explore the first possible formulation more +in detail, now. +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +291 +Issue 7, Volume 12, July 2013 + + + + + +3.1 Sphere Rotation - Intersection in E3 +Let us consider a situation in which a torus and line +are in the same relative position, but using the above +mentioned geometric transformation, the torus is in +its basic position, i.e. in the plane. +A torus can be represented as a union all spheres +with a radius rotating about axis in the +plane with a radius . It means that the torus can be +defined as a union, i.e. an envelope, of all rotating +spheres about axis as + + + + +(41) +where: , +Now the problem line-torus intersection is +transformed to a generalized line clipping problem, +when a line is clipped by an envelope of all rotating +spheres which forms the torus, i.e. + + + + + + +(42) +where and are given constants of the torus. +Due to the rotational symmetry about the axis, +the torus and the line can be rotated about axis so +that the line will lie in a plane parallel to the +plane. +Now, the given line is defined as + + +(43) +A point and a directional vector of the line are +defined as + + +(44) +where: as the line lies in a plane parallel to +the plane, i.e. . +The problem of a line-torus intersection problem +is transformed to generalized line clipping problem +in E2 actually, when a line is clipped by a parametric +envelope. +A line is given in the case of E3 as + + +(45) +and a sphere + + +(46) +substituting we get + + + +(47) +i.e. + + + +(48) +where + + + + +(49) + + + + + + + +(50) +where: + + + + +(51) +and + + + + + + +(52) +and + + + + + +(53) +Therefore + + + + + + + +(54) +The quadratic equation is now + + + + + + + + + +(55) +In the case of the normalized directional vector , +i.e. , resp. , we get a quadratic +equation parameterized by as follows + + + + +(56) +i.e. + + + +(57) + where + + + + + + + + + + + +(58) +and + + +(59) + +If the Vieta’s formula is used we get the following +equivalent equations + + + + + + +If a quadratic equation is considered as a quadratic +function of , then the extreme value +is given as + + + + +(60) +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +292 +Issue 7, Volume 12, July 2013 + + + + + + + + + + + + + + +The point is inside of the envelope; +see Fig.4, if and only if . +Substituting to the function + + + +(61) +we get + + + + +(62) +i.e. + + +(63) +Substituting + + + +(64) +This leads to: + + +(65) +Therefore + + +(66) +where is an identity matrix and is a tensor +product producing a matrix. + +F(t)=at +bt+c +2 +x1 +x2 +x +y + +Figure 4: Rotating sphere plane intersection and +envelope +As we recently set in the quadratic equation, +we can write + + +(67) +where and are the line parameter values +for line sphere intersection. +The second Vieta’s [2] equation can be used to +determine intervals for φ with one root only for +iterative solvers. + +In the classified case: + +ad a) we can use mirroring operations and solve +the intersection in one quadrant only twice for +non-mirrored and for mirrored cases as there +might be two tuples of intersections, + +ad b) situation is complex as the envelope has +an inflection point so there might be three +intersections in one quadrant + +ad c) this case is similar to the previous but only +two intersection points might occur +z=const +R-r +R+r +z +x1 +x2 +x0 +x +0 +R + +Figure 5: Rotating spheres + +However the intersection computation is still too +complex. + + +3.2 Line Rotation – Intersection in E3 +Another alternative approach is based on a fixed +sphere position on the axis and the given line +rotates about axis generally in E3. This approach is +actually “dual” in some sense to the previous one +and leads to an envelope given as an intersection of +a sphere and double cone. +There are two possible equivalent formulations: +the center of a sphere is on the axis and the +rotating line is in a general position in E3 or +geometric transformation is made so that the +rotating line rotates about axis and the vertex of a +double cone is in the origin of the coordinate +system; the center of a sphere is in the plane, +i.e. was moved up. + +A line in E3 is defined as + + + +(68) +and a sphere on the axis is defined as + + + +(69) +As the line is rotated about y axis the rotation matrix + is expressed as +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +293 +Issue 7, Volume 12, July 2013 + + + + + + + + + + + + + + + +(70) +Then the rotating line forming a double cone in E3 +can be expressed as + + + +(71) +Substituting we get + + + + +(72) +or + + + + +(73) +It means that a quadratic equation is obtained again, +i.e. + + + + +(74) +As +the +matrix + +is +orthonormal, +i.e. + and directional vector can be +normalized, i.e. then we get a significant +simplification + + + +(75) +Let us explore coefficients of this quadratic equation +more in a detail. + + + + + + +(76) +As we get + + + +(77) +Using cross product symmetry we get + + + + +(78) +Now there is another simplification possible as + and + + + + + + + + + + + + + + + + + + + +(79) +Now the last term of the equation + + + + + + + +(80) +As + + + + + + + +(81) +Using cross product symmetry we get + + + + +(82) +Again, there is another simplification possible as + and + + + + + + + + + + + + + + + + + + + + + + + + + +(83) +Putting all together we get + + + + + +(84) +i.e. + + + + + +(85) + +ρ +z=const +xS +x +zS +z +R +0 +r +f + +Figure 6: Intersection plane-rotating sphere + + + +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +294 +Issue 7, Volume 12, July 2013 + + + +3.3 Intersection with a Plane - Solution in E2 +It this part we will concentrate on the case, when +sphere rotates about axis and intersect a plane on +which the given line lies and is parallel to the +plane +As the given line lies in a plane parallel to the + plane the rotating sphere intersect the plane, +Fig.5, which results into circles in the plane, +Fig.6. + + +(86) + +Let us consider the line formulation. + + + + + +(87) +A sphere is rotating about axis is described by +i.e. + + + +(88) +A plane on which the given line lies is defined +as . Then + + + + + + +(89) +As we get + + + + + +(90) + +As the given line is defined as + + + +(91) +we get + + + + + + + +(92) +i.e. a quadratic equation has a form + + + + + + + + +(93) +In the case of the normalized directional vector , +i.e. , resp. , we get a quadratic +equation parameterized by as follows + + + + + + + + +(94) + +3.4 Hybrid method +Let torus is represented as an envelope of rotating +spheres about axis again. Spheres intersect the +plane on which the given line lies and form +circles in the plane , on the plane parallel to + plane. Those circles on the plane are +described by an equation +As all the circles are on the plane the +equation can be simplified to + + +(95) +where + + +(96) +Note that represents rotation of the sphere about + axis, resulting circle is on the plane. The + radius of a circle is given + + + +(97) +The envelope of a plane-torus intersection is given +as + + + + + +(98) +Let us consider the case, when , Fig.7. + +Figure 7: An envelope given as union +of plane-rotating sphere intersections + +Angles are determined as follows + + + + +(99) +The angle is an angle when the first circle that +contributes to an envelope; the angle is for the +last circle that contributes to the envelope and the +angle is for the largest circle inside the envelope. +The given line lies in the plane and is +defined as + + + +(100) + +The line can be re-parameterized so that +then circles are defined as: + + + +(101) + +Now the problem is effectively transferred to E2. + + +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +295 +Issue 7, Volume 12, July 2013 + + + +3.5 New Bounding Volume +The “standard” bounding volume [1] is based on a +sphere in E3 and an intersection of two half spaces, +Fig.2. As the line lies in the plane for +we can distinguish following fundamental cases: + +ad a) we can use mirroring operations and solve +the intersection in one quadrant only twice for +non-mirrored and for mirrored cases as there +might be two tuples of intersections, + +ad b) situation is complex as the envelope has +an inflection point so there might be three +intersections in one quadrant, + +ad c) this case is similar to the previous but +only two intersection points might occur. +However +if +many +lines-torus +intersections +computation are needed, like in the ray tracing +rendering technique, the more precise bounding +volume is needed to increase the efficiency of +computation. The “standard” bounding volume +works fine for the case ad b). On the other hand it +can be seen that + +in the case ad a), i.e. when a line passes +through the torus, i.e. through a “hole” and +does not intersect the torus, detailed +computation has to be made, that is +computationally expensive. + +in the case ad c), i.e. when a line intersects +the torus in its “outer part”, i.e. + the distance between two planes can +be smaller than . +Let us explore the first case as it leads to higher +efficiency. + + +A +B +xA +x’A +xB +x’B +x +y +k +k’ + +Figure 8: Torus-plane intersection and a ray + +Fig.8 presents an intersection plane-torus for + . It can be seen that a circle (as +we are in E2), with the center at with the radius +forms bounding surfaces together with the mirrored + circle by axis. The center of the circle is +defined as follows: + + + + +(102) + +where + + + +(103) + +or + + + +(104) + +It can be seen that in the case of a +special case is obtained as there is no “hole” at all, +Fig.9 +xA +x’A +x +y + +Figure 9: A boundary situation + + + + +Figure 10: Line-torus intersection for + , i.e. the case ad b) + +The test for the ad a) case can be formulated as: if +the line intersects the axis in the interval + +and does not intersect the circle nor the circle , +then the line does not intersect the given torus. Fig.6 +presents two lines, in the case A, the line is not +considered for intersection computation with torus, +while in the cases B, the detailed intersection +test/computation has to be made. + + +Figure 11: Line-torus intersection and bounding for + , i.e. the case ad c) + + + +A +xA +x’A +x +y +x +y +yA +y’A +e +A +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +296 +Issue 7, Volume 12, July 2013 + + + +The test for the ad b) test remains as the original, +Fig.10, as up to 3 intersections can occur in one +quadrant as there is a point of inflexion. +In the case ad c), i.e. , there are +only 2 intersection points possible, Fig.11. It can be +seen that the distance between two planes, given by + and values is now smaller than the original +distance . It can be seen that the new distance is +given as + + +(105) + + + +4 Conclusion +New +alternative +formulations +for +line-torus +intersection +problem +have +been +presented. +Unfortunately +all +the +presented +alternative +formulations do not lead to simpler computational +formulas. It seems to that an implicit form for the +line-torus intersection is the most efficient one. +There is still one possibility to use toroidal +coordinate system; however the computational +expense is too high. +As a result of new geometrically equivalent +formulations a new bounding object, actually circles +in E2, for the line-torus intersection has been +developed and described. +The new bounding object increases line-torus +intersection computation efficiency significantly as +it also detects the cases when a line or ray is passing +a “hole” of the torus. The efficiency of the new +torus bounding test grows with the ratio + . + + +Acknowledgment +The author would like to thank to colleagues and +students at the VSB-Technical University of Ostrava +and University of West Bohemia for their critical +comments, suggestions and hints. Thanks belong +also to anonymous reviewers for corrections and +comments. +This research was supported by the Ministry of +Education +of +the +Czech +Republic, +projects +No.LH12181, LG13047. + +References: +[1] Cychosz,J.M.: Intersecting a Ray with An +Elliptical Torus, Graphics Gems II (Ed. James +Arvo), p. 251-256, Academic Press, 1991 +[2] Hazewinkel,M.(Ed.): Viète +theorem, +Encyclopedia of Mathematics, Springer, 2001 +[3] Herbison-Evans,D.: Solving +Quartic +and +Cubics for Graphics, Graphics Gems, pp.1-15, +Academic Press, 1995 +[4] Lengyel,E.: Mathematics +for +3D +Game +Programming and Computer Graphics, Course +Technology, pp.147-148, 2012 +[5] Skala,V.: A new approach to line and line +segment clipping in homogeneous coordinates, +The Visual Computer, ISSN 0178-2789, +Vol.21, No.11, pp.905-914, Springer Verlag, +2005 +[6] Skala,V.: Length, +Area +and +Volume +Computation in Homogeneous Coordinates, +International Journal of Image and Graphics, +Vol.6., No.4, pp.625-639, ISSN 0219-4678, +2006 +[7] Skala,V.: Barycentric Coordinates Computation +in Homogeneous Coordinates, Computers & +Graphics, Elsevier, ISSN 0097-8493, Vol. 32, +No.1, pp.120-127, 2008 +[8] Skala,V: Duality and Intersection Computation +in Projective Space with GPU Support, WSEAS +Trans.on +Mathematics, +ISSN +1109-2769, +Vol.9,No.6, pp.407-416, 2010 +[9] Skala,V.: Geometry, +Duality +and +Robust +Computing in Engineering, WSEAS Trans.on +Computers, Vol.11, No.9, ISSN 1109-2742, +pp.275-291, 2012 +WSEAS TRANSACTIONS on COMPUTERS +Vaclav Skala +E-ISSN: 2224-2872 +297 +Issue 7, Volume 12, July 2013 + diff --git a/a9E1T4oBgHgl3EQfdAS_/content/tmp_files/load_file.txt b/a9E1T4oBgHgl3EQfdAS_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28087e493c25fd80a3f7b62412e39d8532484861 --- /dev/null +++ b/a9E1T4oBgHgl3EQfdAS_/content/tmp_files/load_file.txt @@ -0,0 +1,260 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf,len=259 +page_content='Line-Torus Intersection for Ray Tracing: Alternative Formulations VACLAV SKALA Department of Computer Science and Engineering University of West Bohemia Univerzitni 8, CZ 30614 Plzen CZECH REPUBLIC http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='VaclavSkala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='eu Abstract: - Intersection algorithms are very important in computation of geometrical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Algorithms for a line intersection with linear or quadratic surfaces are quite efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' However, algorithms for a line intersection with other surfaces are more complex and time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In this case the object is usually closed into a simple bounding volume to speed up the cases when the given line cannot intersect the given object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In this paper new formulations of the line-torus intersection problem are given and new specification of the bounding volume for a torus is given as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The presented approach is based on an idea of a line intersection with an envelope of rotating sphere that forms a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Due to this approach new bounding volume can be formulated which is more effective as it enables to detect cases when the line passes the “hole” of a torus, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Key-Words: Line clipping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' torus line intersection, CAD systems 1 Introduction Intersection algorithms play a significant role in all geometric problems and CAD/CAM systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Intersection algorithms are well documented for linear cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' line-plane or line-triangle etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=', and also for some specific non-linear surfaces like line- sphere intersection etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' However, there are other objects like bicubic parametric patches, torus etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In this case computation of intersection points is more complex and usually complex formula or iterative formula are to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Figure 1: Torus (Courtesy of Wikipedia) Intersection of a line and closed surface can be considered as generalized well known clipping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Intersection of a line or ray with a surface is the key problem solved in all ray-tracing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Due to the computational complexity a bounding volumes are used to detect cases when a line cannot intersect the given object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In this paper we present torus-line intersection problem [1] [2], which leads to a quartic equation [3] in principle, and show other possible formulations of the line-torus intersection problem which offer quite different representations of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' These reformulations lead to a formulation of a new problem – generalized line clipping by an envelope (convex or non-convex) of parametric closed surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 2 Torus Line Intersection Torus-line intersection is actually a solution of a line in E3 usually given in a parametric form as (1) and a torus, which is generally a surface of the 4th order and can be given as : (2) An alternative formulation Note that the axis is the rotational axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The torus equation can be reformulated as (4) where (5) (3) WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 288 Issue 7, Volume 12, July 2013 As there will be some geometric transformations used latter on we can also scale the given torus and a line so that , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' the torus is “normalized”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Now the intersection of a line and the torus is given as a solution of equations: (6) and (7) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='5 to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='6 we get (8) and finally we get (9) This equations is quite complex, but by detailed evaluation we get a quartic equation (10) where: (11) It can be seen that the computation can be simplified for the case, when , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' the directional vector of the line is normalized or the equation is divided by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It means that we are getting a quartic equation in the from [4] (12) which can be simplified by substitution (13) to (14) where (15) If solution for is found, then the solution of the original equation is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' To get a solution for the following a qubic equation has to be solved (16) Then the values can be computed from real solution of the equation above as two quadratic equations as follows: If then (17) If then (18) It can be seen that the solution itself is not simple, but the formula is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' On the opposite, an iterative method like Bisection or Newton method can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' However there are up to 4 intersections of the line and the torus, so it is necessary to find relevant intervals for , with one intersection only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='1 Alternative Torus Representation There are other formulations of a torus as follows, but they are not convenient for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (19) or a parametric form as (20) It can be seen that a solution of a line-torus intersection is not a simple task and it leads to a non-trivial computational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' However, there are some other geometrically equivalent formulations which could be used for finding a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In the following we will consider only circular torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='2 Geometric Transformations Geometric transformations with points are defined in the projective space using homogeneous coordinates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' in the projective extension of the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' A point in the Euclidean coordinates has homogeneous coordinates ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' is the homogeneous coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The conversion between the projective space and the Euclidean space is defined as (21) It means that the projective representation is actually a one parametric set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' A point in the Euclidean space E2 is represented as a line with the WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 289 Issue 7, Volume 12, July 2013 origin of the coordinate system excluded in the projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Geometric transformations with points like rotation, translation, mirroring etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' can be than described by the matrix as (22) Note that might have some physical meaning and units, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' [m], while has no unit, it is just a “scaling factor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' That’s why we used “:” to separate the values in the vector notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' A line in E2 determined by two points given in the homogeneous coordinates can be computed using the cross product as [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (23) Intersection of two lines and in E2 can be computes as (24) We can see that both computations are in the E2 case “dual”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' line and points are dual [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In the E3 case a point is dual to a plane and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It can be shown that a plane given by three points can be determined by the extended cross product as (25) Again, an intersection of three planes can be computed as, see [7], [8], [9] for details (26) This approach is simple, easy to implement and convenient for GPU implementation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' However, matrix transformations for points cannot be used for geometric transformations with lines in the E2 case nor with planes in the E3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It can be shown [6] that if a line is given by two points and and those points are geometrically transformed using the matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (27) and (28) then (29) It can be shown that the matrix is defined as (30) Because are coefficients of an implicit equation we can simply write (31) As the implicit form is used, coefficients of a line can be multiplied by any non-zero constant and the line will be same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Therefore (32) where means protectively equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Similarly for a plane (33) It means that we can correctly manipulate with lines and planes, now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='3 Bounding Volume Let us assume that the torus lies in the plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' the -axis is its rotational axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Bounding volume, defined in [1], is based on an idea that torus is bounded by an intersection of a sphere and two half-spaces, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Figure 2: Bounding volume The radius of the enclosing sphere is given as (34) The bounding test computes intersection of a line with a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' If such intersections and exist then the line does not intersect the torus if the following condition is valid [1] (35) It can be seen that the test does not eliminate cases when a line: \uf0b7 is passing the “hole inside of the torus” without touching or intersecting the torus – line \uf0b7 nearly touches the torus – line – but there is a small probability r x tmin pA pC pB pD tmax R1 R y WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 290 Issue 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Volume 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' July 2013 It should be noted that the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='2 presents general situation in the E3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='4 Torus Transformation So far we have dealt with a general situation expecting that the torus is in its basic position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' it lies in the plane and the axis is the rotational axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In the case of torus general position the following transformations can be used: (36) where: defines axis of the torus, defines axis of the torus, is used to get an orthonormal basis, and is the torus centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It can be seen that there are some interesting properties of the line-torus intersection problem, like \uf0b7 torus rotational symmetry, \uf0b7 if mirroring operation is used only one quadrant can be considered to solve the intersection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' We will explore if those properties can contribute to simplification of computation in the following part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='5 Intersections Classification As a torus is rotationally invariant we can rotate the given line about axis so that it lies in a plane , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' in a plane parallel to the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' There is no significant computational expense as the transformation matrix is accumulated with the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Now we can distinguish three fundamentally different cases according to the value: a) : generally intersection with two independent parts have to be considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' for and and due to convexity each part could have up to 2 intersections only (2 convex envelopes are generated), b) : this case is more complex as the envelope has only one part, but it is not convex as it can have an inflexion point and 3 intersection points can be generated, c) : the simplest case as only one convex envelope is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Figure 3: Torus plane intersection for The above mentioned three cases differ significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Unfortunately the envelope is not convex in all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='6 Vieta’s Formula Let us assume that is a polynomial of degree (37) Then according to the Vieta’s formula the roots satisfy equations ( (38) In the quadratic equation case (39) we obtain (40) These formulas are not well known and will be used latter on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In the following we will show different approaches to the line – torus intersection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 3 New Intersection Formulations In the previous part we have presented the “traditional” approach to the line–torus intersection detection and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Now, different equivalent formulations, which could lead to simpler and faster solutions, will be formulated in the following part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' They can be briefly classified as follows: \uf0b7 a sphere is rotating about axis (the envelope forms a torus) and intersection with the line in E3 is computed directly, \uf0b7 a sphere is fixed on the axis and intersection with the line rotating about axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' it is actually an intersection of a sphere and double cone) in E3 is computed directly, \uf0b7 a sphere is rotating about axis and intersection with the plane in E3, on which the given line lies, results into circles in this plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' circles in E2, forming an envelope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' a curve is given as an intersection of a torus with a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' An intersection of the envelope of all circles and the line is computed in E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' This is actually a generalized line-clipping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Let us explore the first possible formulation more in detail, now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 291 Issue 7, Volume 12, July 2013 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='1 Sphere Rotation - Intersection in E3 Let us consider a situation in which a torus and line are in the same relative position, but using the above mentioned geometric transformation, the torus is in its basic position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' A torus can be represented as a union all spheres with a radius rotating about axis in the plane with a radius .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It means that the torus can be defined as a union, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' an envelope, of all rotating spheres about axis as (41) where: , Now the problem line-torus intersection is transformed to a generalized line clipping problem, when a line is clipped by an envelope of all rotating spheres which forms the torus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (42) where and are given constants of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Due to the rotational symmetry about the axis, the torus and the line can be rotated about axis so that the line will lie in a plane parallel to the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Now, the given line is defined as (43) A point and a directional vector of the line are defined as (44) where: as the line lies in a plane parallel to the plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The problem of a line-torus intersection problem is transformed to generalized line clipping problem in E2 actually, when a line is clipped by a parametric envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' A line is given in the case of E3 as (45) and a sphere (46) substituting we get (47) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (48) where (49) (50) where: (51) and (52) and (53) Therefore (54) The quadratic equation is now (55) In the case of the normalized directional vector , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' , resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' , we get a quadratic equation parameterized by as follows (56) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (57) where (58) and (59) If the Vieta’s formula is used we get the following equivalent equations If a quadratic equation is considered as a quadratic function of , then the extreme value is given as (60) WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 292 Issue 7, Volume 12, July 2013 The point is inside of the envelope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='4, if and only if .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Substituting to the function (61) we get (62) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (63) Substituting (64) This leads to: (65) Therefore (66) where is an identity matrix and is a tensor product producing a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' F(t)=at +bt+c 2 x1 x2 x y Figure 4: Rotating sphere plane intersection and envelope As we recently set in the quadratic equation, we can write (67) where and are the line parameter values for line sphere intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The second Vieta’s [2] equation can be used to determine intervals for φ with one root only for iterative solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In the classified case: \uf0b7 ad a) we can use mirroring operations and solve the intersection in one quadrant only twice for non-mirrored and for mirrored cases as there might be two tuples of intersections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' \uf0b7 ad b) situation is complex as the envelope has an inflection point so there might be three intersections in one quadrant \uf0b7 ad c) this case is similar to the previous but only two intersection points might occur z=const R-r R+r z x1 x2 x0 x 0 R Figure 5: Rotating spheres However the intersection computation is still too complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='2 Line Rotation – Intersection in E3 Another alternative approach is based on a fixed sphere position on the axis and the given line rotates about axis generally in E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' This approach is actually “dual” in some sense to the previous one and leads to an envelope given as an intersection of a sphere and double cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' There are two possible equivalent formulations: the center of a sphere is on the axis and the rotating line is in a general position in E3 or geometric transformation is made so that the rotating line rotates about axis and the vertex of a double cone is in the origin of the coordinate system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' the center of a sphere is in the plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' was moved up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' A line in E3 is defined as (68) and a sphere on the axis is defined as (69) As the line is rotated about y axis the rotation matrix is expressed as WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 293 Issue 7, Volume 12, July 2013 (70) Then the rotating line forming a double cone in E3 can be expressed as (71) Substituting we get (72) or (73) It means that a quadratic equation is obtained again, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (74) As the matrix is orthonormal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' and directional vector can be normalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' then we get a significant simplification (75) Let us explore coefficients of this quadratic equation more in a detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (76) As we get (77) Using cross product symmetry we get (78) Now there is another simplification possible as and (79) Now the last term of the equation (80) As (81) Using cross product symmetry we get (82) Again, there is another simplification possible as and (83) Putting all together we get (84) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (85) ρ z=const xS x zS z R 0 r f Figure 6: Intersection plane rotating sphere WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 294 Issue 7, Volume 12, July 2013 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='3 Intersection with a Plane - Solution in E2 It this part we will concentrate on the case, when sphere rotates about axis and intersect a plane on which the given line lies and is parallel to the plane As the given line lies in a plane parallel to the plane the rotating sphere intersect the plane, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='5, which results into circles in the plane, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (86) Let us consider the line formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (87) A sphere is rotating about axis is described by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' (88) A plane on which the given line lies is defined as .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Then (89) As we get (90) As the given line is defined as (91) we get (92) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' a quadratic equation has a form + (93) In the case of the normalized directional vector , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' , resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' , we get a quadratic equation parameterized by as follows + (94) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='4 Hybrid method Let torus is represented as an envelope of rotating spheres about axis again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Spheres intersect the plane on which the given line lies and form circles in the plane , on the plane parallel to plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Those circles on the plane are described by an equation As all the circles are on the plane the equation can be simplified to (95) where (96) Note that represents rotation of the sphere about axis, resulting circle is on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The radius of a circle is given (97) The envelope of a plane-torus intersection is given as (98) Let us consider the case, when , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Figure 7: An envelope given as union of plane-rotating sphere intersections Angles are determined as follows (99) The angle is an angle when the first circle that contributes to an envelope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' the angle is for the last circle that contributes to the envelope and the angle is for the largest circle inside the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The given line lies in the plane and is defined as (100) The line can be re-parameterized so that then circles are defined as: (101) Now the problem is effectively transferred to E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 295 Issue 7, Volume 12, July 2013 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='5 New Bounding Volume The “standard” bounding volume [1] is based on a sphere in E3 and an intersection of two half spaces, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' As the line lies in the plane for we can distinguish following fundamental cases: \uf0b7 ad a) we can use mirroring operations and solve the intersection in one quadrant only twice for non-mirrored and for mirrored cases as there might be two tuples of intersections, \uf0b7 ad b) situation is complex as the envelope has an inflection point so there might be three intersections in one quadrant, \uf0b7 ad c) this case is similar to the previous but only two intersection points might occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' However if many lines-torus intersections computation are needed, like in the ray tracing rendering technique, the more precise bounding volume is needed to increase the efficiency of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The “standard” bounding volume works fine for the case ad b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' On the other hand it can be seen that \uf0b7 in the case ad a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' when a line passes through the torus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' through a “hole” and does not intersect the torus, detailed computation has to be made, that is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' \uf0b7 in the case ad c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' when a line intersects the torus in its “outer part”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' the distance between two planes can be smaller than .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Let us explore the first case as it leads to higher efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' A B xA x’A xB x’B x y k k’ Figure 8: Torus-plane intersection and a ray Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='8 presents an intersection plane-torus for .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It can be seen that a circle (as we are in E2), with the center at with the radius forms bounding surfaces together with the mirrored circle by axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The center of the circle is defined as follows: (102) where (103) or (104) It can be seen that in the case of a special case is obtained as there is no “hole” at all, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='9 xA x’A x y Figure 9: A boundary situation Figure 10: Line-torus intersection for , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' the case ad b) The test for the ad a) case can be formulated as: if the line intersects the axis in the interval and does not intersect the circle nor the circle , then the line does not intersect the given torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='6 presents two lines, in the case A, the line is not considered for intersection computation with torus, while in the cases B, the detailed intersection test/computation has to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Figure 11: Line-torus intersection and bounding for , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' the case ad c) A xA x’A x y x y yA y’A e A WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 296 Issue 7, Volume 12, July 2013 The test for the ad b) test remains as the original, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='10, as up to 3 intersections can occur in one quadrant as there is a point of inflexion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' In the case ad c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' , there are only 2 intersection points possible, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It can be seen that the distance between two planes, given by and values is now smaller than the original distance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It can be seen that the new distance is given as (105) 4 Conclusion New alternative formulations for line-torus intersection problem have been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Unfortunately all the presented alternative formulations do not lead to simpler computational formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' It seems to that an implicit form for the line-torus intersection is the most efficient one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' There is still one possibility to use toroidal coordinate system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' however the computational expense is too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' As a result of new geometrically equivalent formulations a new bounding object, actually circles in E2, for the line-torus intersection has been developed and described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The new bounding object increases line-torus intersection computation efficiency significantly as it also detects the cases when a line or ray is passing a “hole” of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' The efficiency of the new torus bounding test grows with the ratio .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Acknowledgment The author would like to thank to colleagues and students at the VSB-Technical University of Ostrava and University of West Bohemia for their critical comments, suggestions and hints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' Thanks belong also to anonymous reviewers for corrections and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' This research was supported by the Ministry of Education of the Czech Republic, projects No.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='407-416, 2010 [9] Skala,V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content=' : Geometry, Duality and Robust Computing in Engineering, WSEAS Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='on Computers, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='11, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='9, ISSN 1109-2742, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} +page_content='275-291, 2012 WSEAS TRANSACTIONS on COMPUTERS Vaclav Skala E-ISSN: 2224-2872 297 Issue 7, Volume 12, July 2013' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E1T4oBgHgl3EQfdAS_/content/2301.03191v1.pdf'} diff --git a/bNFAT4oBgHgl3EQfXR3e/content/tmp_files/2301.08533v1.pdf.txt b/bNFAT4oBgHgl3EQfXR3e/content/tmp_files/2301.08533v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6261de8cb5ab429f5f14283c0259cfa5939196ba --- /dev/null +++ b/bNFAT4oBgHgl3EQfXR3e/content/tmp_files/2301.08533v1.pdf.txt @@ -0,0 +1,575 @@ +LEARNING FREQUENCY-SPECIFIC QUANTIZATION SCALING IN VVC FOR +STANDARD-COMPLIANT TASK-DRIVEN IMAGE CODING +Kristian Fischer, Fabian Brand, Christian Herglotz, and Andr´e Kaup +Multimedia Communications and Signal Processing +Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg (FAU) +Cauerstr. 7, 91058 Erlangen, Germany +{Kristian.Fischer, Fabian.Brand, Christian.Herglotz, Andre.Kaup}@fau.de +©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or +reuse of any copyrighted component of this work in other works. DOI: ICIP46576.2022.9897987 +ABSTRACT +Today, visual data is often analyzed by a neural network +without any human being involved, which demands for spe- +cialized codecs. For standard-compliant codec adaptations +towards certain information sinks, HEVC or VVC provide +the possibility of frequency-specific quantization with scal- +ing lists. This is a well-known method for the human visual +system, where scaling lists are derived from psycho-visual +models. In this work, we employ scaling lists when perform- +ing VVC intra coding for neural networks as information +sink. To this end, we propose a novel data-driven method +to obtain optimal scaling lists for arbitrary neural networks. +Experiments with Mask R-CNN as information sink reveal +that coding the Cityscapes dataset with the proposed scaling +lists result in peak bitrate savings of 8.9 % over VVC with +constant quantization. +By that, our approach also outper- +forms scaling lists optimized for the human visual system. +The generated scaling lists can be found under https: +//github.com/FAU-LMS/VCM_scaling_lists. +Index Terms— Video Coding for Machines, Scaling +Lists, Adapted Quantization, VVC, Instance Segmentation +1. INTRODUCTION +Modern hybrid video codecs such as HEVC [1] and VVC [2] +allow for a frequency-specific quantization of the transform +coefficients by providing scaling lists to the coder. In total, a +scaling list in VVC consists of 28 scaling matrices for each +prediction mode, blocksize, and color component [3]. Scal- +ing lists have mostly been employed to optimize the coding +quality for the human visual system (HVS). In 1999, Chang +et al. [4] proposed JPEG quantization tables that are opti- +mized for human perception modeled by a non-linear point +transformation and a modulation transfer function. Later, the +same model was adopted to derive the default quantization +matrices for HEVC [5, 6]. Prangnell et al. [7] proposed quan- +tization matrices for HEVC that are optimized for the human +The authors gratefully acknowledge that this work has been funded by +the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) +under project number 426084215. +perception on high-resolution multimedia data. Those HVS- +optimized matrices commonly exploit that the HVS is less +sensitive to high spatial frequency in visual content. Thus, +high frequencies are quantized more coarsely to save bitrate +while keeping a similar visual quality. +Due to the rapid progress in the field of neural networks +solving tasks such as object detection, semantic segmenta- +tion, or tracking, the amount of coding scenarios where the +visual data is directly analyzed by a neural network rather +than being observed by a human is constantly increasing. +This requires new, optimized coding schemes. As a result +of this, MPEG introduced an ad-hoc group on video coding +for machines (VCM) in 2019 [8] that aims at standardiz- +ing efficient bitstreams for such machine-to-machine (M2M) +scenarios. Previous approaches to improve the coding effi- +ciency of HEVC or VVC for VCM scenarios mainly consist +of adding spatial saliency information to the encoding pro- +cess [9, 10, 11]. In addition, we proposed a feature-based +rate distortion optimization for VVC in [12]. +Optimizing +scaling lists for algorithms as information sink has been pro- +posed in [13] and [14] for JPEG. A broad analysis on the +frequency sensitivity of image classification networks has +been made in [15] by adding perturbations based on Fourier +basis functions. +Motivated by this development, this paper proposes a +data-driven method to train optimal scaling matrices when +coding images for an arbitrary neural network. Eventually, +this scaling list is added to the VVC encoding process in +order to reduce the amount of bits that is spent for frequency +coefficients that are less important for the applied instance +segmentation network Mask R-CNN [16]. +2. BACKGROUND +In hybrid video codecs, the error signal x of height H and +width W between the prediction and the original signal is first +transformed into X in order to condense the signal energy +on few coefficients. To that end, a 2D frequency transform +such as the discrete cosine transform (DCT) is applied. Af- +terwards, X is uniformly quantized resulting in quantization +arXiv:2301.08533v1 [eess.IV] 20 Jan 2023 + +x +DCT ++ +IDCT +˜x +Analysis +Network +Predictions +Ltask +· S +16 +c · U(−0.5, 0.5) +X +˜X +Fig. 1. Proposed method to train the scaling matrix S for the DCT blocksize B. X and ˜X are of size 3 × B2 × H/B × W/B. +S is of size B2 and added to each RGB color channel. IDCT denotes the inverse DCT. +indices ˆ +X that are eventually entropy coded and transmitted. +At the decoder, the whole process is reverted and the trans- +form coefficients ˆ +X are transformed back into the decoded +signal ˆx. The quantization step size ∆ is derived from the +user-defined quantization parameter (QP) by +∆ = 2(QP−4)/6 · 2β−8, +(1) +with β defining the bit depth of the signal x to ensure a sim- +ilar quality for all bit depths [3]. Normally, each transform +coefficient Xk at position k is quantized with the same step +size ∆. However, HEVC and VVC also provide the possi- +bility to define a scaling matrix S that allows for a specific +quantization of each frequency coefficient [3, 5]. By that, the +quantization of Xk changes to +ˆ +Xk = +�Xk +∆k ++ 0.5 +� +, +with +∆k = ∆ · Sk +16 . +(2) +The coefficients in the scaling matrix are defined as positive +integers, with 0 < Sk < 16 resulting in a smaller quantization +interval and Sk > 16 resulting in a coarser quantization step +size than the initial quantization step size ∆ derived from QP. +3. PROPOSED SCALING LIST GENERATION +3.1. Scaling Matrix Generation +To train the VCM-optimized scaling matrix S, we build up +the framework depicted in Fig. 1, which emulates the signal +flow of the coding chain. Contrary to the real hybrid coding +chain, we applied three adaptations in order to make an end- +to-end training possible. First, we utilized RGB images as +input data x instead of error signals. Second, we focused +on the DCT transformation, since it is the most important +representative of the VVC transforms. Third, quantization +is not applicable to gradient-descent optimization due to its +non-differentiable characteristic. Thus, we emulate the quan- +tization by adding uniform noise similar to the field of learned +end-to-end image compression [17]. The strength of the noise +is steered by the user-defined constant c. Hence, the distorted +transform coefficients ˜ +X are calculated by +˜ +X = X + U(−0.5, 0.5) · c · S +16. +(3) +With the trainable scaling matrix S, the noise can separately +be amplified for each frequency coefficient Xk. Per design, +we bound S between 16 and 128 by applying a Sigmoid non- +linearity: +S = 16 + 112 · Sigmoid(S′). +(4) +Here, S′ is the actual trainable parameter resulting in the +bounded scaling matrix S. The borders of 16 and 128 are +inspired by the minimum and maximum values given for the +default HEVC matrix [6] and the work in [7]. By that, we +only allow the network to increase ∆k. With the maximum +S of 128, ∆k is limited to eight times the initial step size ∆. +When training the scaling matrix, two contrary targets are +pursued. The main goal of the proposed method is to increase +S in order to enlarge the quantization step size, and thus to +ultimately reduce the bitrate during encoding in the inference +case. To achieve this goal, we introduce a loss +Lrate(S) = +16 +mean(S), +(5) +approximating the likely rate savings later in inference and +forcing the network to increase the coefficients of S. Due to +the mean, the loss is independent of the DCT blocksize B. +Naturally, only training on Lrate would result in the maxi- +mum possible value of S, neither considering the output qual- +ity resulting from S in inference nor weighting the different +frequency coefficients. Therefore, we also consider the task +loss Ltask of the analysis network depending on the present +noise in ˜x. With that, the network is pushed to increase the +noise for the coefficients that are less harmful for the analysis +network. The overall minimization problem is defined as +S = arg min +S +Ltask(˜x|S) + λ · Lrate(S), +(6) +where the parameter λ weights the training towards one of the +competing goals of a large scaling matrix, presumably result- +ing in a lower bitrate, or a high task performance. +3.2. Training Setup +As analysis network in training, we employed the state-of- +the-art instance segmentation network Mask R-CNN [16] +with its proposed loss as Ltask. +Its pre-trained weights +were taken from the Detectron2 library [18]. +We trained +our framework on the 2965 RGB Cityscapes [19] training +images, which were cropped to patches of 512 × 1024 pixels. +The scaling list was optimized by the Adam optimizer for + +person86% +rsonc = 4 +c = 16 +c = 64 +λ = 0.01 +λ = 0.1 +λ = 1 +λ = 10 +16 +32 +48 +64 +80 +96 +112 +128 +Sk +Fig. 2. Resulting 8×8 scaling matrices for different combina- +tions of initial noise strength c and loss weighting parameter +λ. The direct component is located in the top left corner. A +yellow value corresponds to a larger scaling matrix value. +20 epochs. After 10 epochs, the initial learning rate of 0.01 +was decreased to 0.001. The batch size was set to 16. With +each training run, we trained one scaling matrix S for one +DCT blocksize B, a given noise strength c, and one λ value. +The scaling matrix was applied to each RGB color channel. +Finally, the learned scaling matrix is rounded to the nearest +integer. +3.3. Resulting Scaling Matrices +In Fig. 2, the scaling matrices generated by our proposed +framework are depicted for a DCT blocksize of 8 × 8 and +several combinations of c and λ. The smaller c, and therewith +the smaller the initial noise, the higher the network increases +the values in the scaling matrix due to the lower impact on the +task accuracy. For a high noise strength, only very few coef- +ficients are increased, eventually resulting in only very few +quantization step sizes ∆k to be enlarged, when applying the +scaling matrix for coding. +Fig. 2 also shows the influence of the loss weighting. For a +low λ, the task loss has a high priority which results in the ma- +jority of scaling coefficients to be near the minimum value of +16 to avoid a noise amplification. With increasing λ, the rate +loss Lrate is more prioritized. Therefore, the training results +in larger scaling coefficients, since it is more important to in- +crease the scaling coefficients at the drawback of an increased +noise leading to a higher task loss. To put it into a nutshell, +the proposed method reveals that the evaluated Mask R-CNN +model is less sensitive to deterioration in the high frequency +components, which is in line with the findings in [15]. +4. PERFORMANCE EVALUATION IN VVC +In this section, we demonstrate that the scaling lists derived +from our measurement framework result in coding gains over +conventional coding without frequency-adaptive quantization +and over HVS-optimized scaling lists. +4.1. Evaluation Setup +For evaluation, we coded the 500 Cityscapes validation im- +ages with the standard-compliant VVC test model (VTM) [20] +version 10.0 following the work in [21] and the MPEG VCM +CTCs [22]. We selected QP values of 12, 17, 22, and 27 in +order to obtain a high task accuracy close to the performance +of uncompressed images, which would typically be required +for practical applications. To measure the inference task per- +formance, we utilized the weighted average precision (wAP) +metric as proposed in [21], which weights the state-of-the-art +object detection metric, average precision, according to the +class frequency to level class imbalances in the Cityscapes +dataset. To quantify the coding gains of our proposed method, +we calculate the Bjøntegaard delta rate (BDR) [23] with the +wAP as quality metric. It measures the bitrate change com- +pared to the anchor codec VTM-10.0 without scaling lists at +the same accuracy of Mask R-CNN applied to the decoded +images. +In order to obtain the scaling lists, we trained the scal- +ing matrices with our proposed framework as described in +Section 3 for squared blocks of size 2, 4, 8, 16, 32, and 64. +The resulting scaling matrices measured in RGB colorspace +were taken for the luma and the chroma channels respectively, +and combined to one scaling list as described for VTM by +JVET [24]. There, it is also described how to interpolate the +scaling lists for rectangular blocks. +4.2. Influence of Training Parametrization on Coding +Performance +As shown in Section 3, the trained scaling matrices are highly +influenced by the initial choice of the noise strength c and +the loss weighting λ. +In order to find the best configura- +tion in terms of coding gains, we measured the coding per- +formance of VTM-10.0 with each scaling list derived from +the twelve combinations shown in Fig. 2. The resulting rate- +wAP points are depicted in Fig. 3. In most cases, the result- +ing points from the scaling lists show a lower rate or a higher +wAP than VTM without frequency-specific quantization. The + +0.5 +1 +2 +0.34 +0.36 +0.38 +0.4 +QP=12 +QP=17 +QP=22 +QP=27 +Avg bitstream size per frame in MBit +Weighted AP +uncompressed +λ = 0.01 +c = 4 +without scaling list +λ = 0.1 +c = 16 +optimal scaling list +λ = 1 +c = 64 +λ = 10 +Fig. 3. Coding performance of VTM-10.0 depending on the +used scaling list for the Cityscapes validation set and Mask R- +CNN as analysis network. The black dotted line corresponds +to the wAP on uncompressed input data. The marker shape +and color denote the selected λ and c values, respectively, +when training the scaling lists with the proposed method. +BDR-values listed in Table 1 confirm this observation that +VTM coding with the proposed VCM-optimized scaling lists +results in bitrate savings over VTM coding without scaling +lists for nearly all investigated combinations. At best, the +coding gains achieve 7.7 % of bitrate savings for c = 16 and +λ = 10. +Another observation from Fig. 3 is that the optimal scaling +list depends on the selected QP. As a rule of thumb for choos- +ing c, it can be said that the noise strength c in training shall +roughly be similar to the quantization step size ∆ defined via +the QP as in (1). This effect is especially observable for a +QP of 27, where the points for c = 4 result in a worse coding +performance than for c = 64. Such a relationship cannot be +found for λ. There, our experiments suggest testing multiple +combinations for practical applications to find the best scaling +list for each QP. By doing so, we obtain the hand-optimized +scaling list resulting in the black curve in Fig. 3 and BDR +savings of 8.9 % over VTM-10.0 without scaling lists. +4.3. Comparison with HVS-optimized Scaling Lists +The default HVS-optimized scaling lists for JPEG and HEVC +also increase the quantization step size for large DCT- +frequency coefficients similar to our proposed scaling lists. +Thus, we also measure their VCM coding performance as +comparison. +Table 2 shows the BDR values for the qual- +ity metrics of PSNR, VMAF, and wAP for two HVS-based +Table 1. BDR values in % for the measurement points in +Fig. 3. VTM without scaling list is taken as anchor. Negative +values denote bitrate savings over the anchor. The markers +correspond to the markers used in Fig. 3. +c +4 +16 +64 +λ +0.01 +-1.4 +-1.3 +-4.7 +0.1 +-6.0 +-3.5 +-4.6 +1 +-0.6 +-3.4 +-4.6 +10 +15.1 +-7.7 +-7.0 +Table 2. +BDR values in % for Cityscapes validation im- +ages for VTM with the corresponding scaling list for the three +quality metrics PSNR, VMAF, and wAP. VTM without scal- +ing list is taken as anchor. +Codec +Scaling list +BDR +PSNR +BDR +VMAF +BDR +wAP +VTM +JPEG-like [25] +8.3 +2.6 +-4.4 +VTM +HEVC-default [25] +1.5 +0.0 +-4.2 +VTM +optimal scaling list +5.4 +5.9 +-8.9 +and our proposed optimal scaling list. +Both scaling lists +were taken from [25]. The BDR values show that the HVS- +optimized scaling lists result in bitrate savings around 4 % for +the investigated VCM-scenario because they also favor low +frequencies. However, due to its large optimization towards +the final analysis network, encoding the Cityscapes dataset +with our proposed scaling lists results in more than twice the +bitrate savings. The large coding gains for the VCM scenario +come with coding losses for the classic coding scenarios +measured by PSNR and VMAF for all three scaling lists. +5. CONCLUSION +In this paper, we proposed a novel data-driven method to ob- +tain trained scaling lists that are optimized for VVC intra +coding in M2M scenarios. Our measurements revealed that +the applied Mask R-CNN network is less sensitive to high +DCT frequencies similar to the HVS. Thus, a larger quantiza- +tion step size was assigned to those coefficients. By that, our +standard-compliant optimization results in up to 7.7 % of bi- +trate savings over conventional VVC coding with static quan- +tization intervals. When selecting the optimal scaling list for +each QP by hand, we were able to further increase the cod- +ing gains to 8.9 %. Future research has to show, whether the +proposed method also achieves similar coding gains for video +coding with low-delay P or randomaccess. Furthermore, we +plan to evaluate whether our proposed method also results in +coding gains for other analysis networks and tasks. Moreover, +we believe that our method could also be adapted for the HVS +by substituting Ltask with a suitable metric representing HVS. + +6. REFERENCES +[1] G. J. Sullivan, J.-R. Ohm, W.-J. Han, and T. Wiegand, +“Overview of the high efficiency video coding (HEVC) +standard,” IEEE Transactions on Circuits and Systems +for Video Technology, vol. 22, no. 12, pp. 1649–1668, +Sept. 2012. +[2] B. Bross, Y.-K. Wang, Y. Ye, S. Liu, J. Chen, G. J. Sulli- +van, and J.-R. Ohm, “Overview of the versatile video +coding (VVC) standard and its applications,” +IEEE +Transactions on Circuits and Systems for Video Tech- +nology, vol. 31, no. 10, pp. 3736–3764, Oct. 2021. +[3] H. Schwarz, M. Coban, M. Karczewicz, T.-D. Chuang, +F. Bossen, A. Alshin, J. Lainema, C. R. Helmrich, and +T. 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Rep., +Joint Video Exploration Team (JVET) of ITU-T SG 16 +WP 3 and ISO/IEC JTC 1/SC 29/WG 11, Oct. 2019. + diff --git a/bNFAT4oBgHgl3EQfXR3e/content/tmp_files/load_file.txt b/bNFAT4oBgHgl3EQfXR3e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0144f0ca74a6c778dccaa9398bbce0035493e52 --- /dev/null +++ b/bNFAT4oBgHgl3EQfXR3e/content/tmp_files/load_file.txt @@ -0,0 +1,433 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf,len=432 +page_content='LEARNING FREQUENCY-SPECIFIC QUANTIZATION SCALING IN VVC FOR STANDARD-COMPLIANT TASK-DRIVEN IMAGE CODING Kristian Fischer, Fabian Brand, Christian Herglotz, and Andr´e Kaup Multimedia Communications and Signal Processing Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg (FAU) Cauerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 7, 91058 Erlangen, Germany {Kristian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='Fischer, Fabian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='Brand, Christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='Herglotz, Andre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='Kaup}@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='de ©2022 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' DOI: ICIP46576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='9897987 ABSTRACT Today, visual data is often analyzed by a neural network without any human being involved, which demands for spe- cialized codecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' For standard-compliant codec adaptations towards certain information sinks, HEVC or VVC provide the possibility of frequency-specific quantization with scal- ing lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' This is a well-known method for the human visual system, where scaling lists are derived from psycho-visual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' In this work, we employ scaling lists when perform- ing VVC intra coding for neural networks as information sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' To this end, we propose a novel data-driven method to obtain optimal scaling lists for arbitrary neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Experiments with Mask R-CNN as information sink reveal that coding the Cityscapes dataset with the proposed scaling lists result in peak bitrate savings of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='9 % over VVC with constant quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' By that, our approach also outper- forms scaling lists optimized for the human visual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The generated scaling lists can be found under https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='com/FAU-LMS/VCM_scaling_lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Index Terms— Video Coding for Machines, Scaling Lists, Adapted Quantization, VVC, Instance Segmentation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' INTRODUCTION Modern hybrid video codecs such as HEVC [1] and VVC [2] allow for a frequency-specific quantization of the transform coefficients by providing scaling lists to the coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' In total, a scaling list in VVC consists of 28 scaling matrices for each prediction mode, blocksize, and color component [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Scal- ing lists have mostly been employed to optimize the coding quality for the human visual system (HVS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' In 1999, Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' [4] proposed JPEG quantization tables that are opti- mized for human perception modeled by a non-linear point transformation and a modulation transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Later, the same model was adopted to derive the default quantization matrices for HEVC [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Prangnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' [7] proposed quan- tization matrices for HEVC that are optimized for the human The authors gratefully acknowledge that this work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 426084215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' perception on high-resolution multimedia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Those HVS- optimized matrices commonly exploit that the HVS is less sensitive to high spatial frequency in visual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Thus, high frequencies are quantized more coarsely to save bitrate while keeping a similar visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Due to the rapid progress in the field of neural networks solving tasks such as object detection, semantic segmenta- tion, or tracking, the amount of coding scenarios where the visual data is directly analyzed by a neural network rather than being observed by a human is constantly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' This requires new, optimized coding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' As a result of this, MPEG introduced an ad-hoc group on video coding for machines (VCM) in 2019 [8] that aims at standardiz- ing efficient bitstreams for such machine-to-machine (M2M) scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Previous approaches to improve the coding effi- ciency of HEVC or VVC for VCM scenarios mainly consist of adding spatial saliency information to the encoding pro- cess [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' In addition, we proposed a feature-based rate distortion optimization for VVC in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Optimizing scaling lists for algorithms as information sink has been pro- posed in [13] and [14] for JPEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' A broad analysis on the frequency sensitivity of image classification networks has been made in [15] by adding perturbations based on Fourier basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Motivated by this development, this paper proposes a data-driven method to train optimal scaling matrices when coding images for an arbitrary neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Eventually, this scaling list is added to the VVC encoding process in order to reduce the amount of bits that is spent for frequency coefficients that are less important for the applied instance segmentation network Mask R-CNN [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' BACKGROUND In hybrid video codecs, the error signal x of height H and width W between the prediction and the original signal is first transformed into X in order to condense the signal energy on few coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' To that end, a 2D frequency transform such as the discrete cosine transform (DCT) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Af- terwards, X is uniformly quantized resulting in quantization arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='08533v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='IV] 20 Jan 2023 x DCT + IDCT ˜x Analysis Network Predictions Ltask S 16 c · U(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='5) X ˜X Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Proposed method to train the scaling matrix S for the DCT blocksize B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' X and ˜X are of size 3 × B2 × H/B × W/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' S is of size B2 and added to each RGB color channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' IDCT denotes the inverse DCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' indices ˆ X that are eventually entropy coded and transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' At the decoder, the whole process is reverted and the trans- form coefficients ˆ X are transformed back into the decoded signal ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The quantization step size ∆ is derived from the user-defined quantization parameter (QP) by ∆ = 2(QP−4)/6 · 2β−8, (1) with β defining the bit depth of the signal x to ensure a sim- ilar quality for all bit depths [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Normally, each transform coefficient Xk at position k is quantized with the same step size ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' However, HEVC and VVC also provide the possi- bility to define a scaling matrix S that allows for a specific quantization of each frequency coefficient [3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' By that, the quantization of Xk changes to ˆ Xk = �Xk ∆k + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='5 � , with ∆k = ∆ · Sk 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' (2) The coefficients in the scaling matrix are defined as positive integers, with 0 < Sk < 16 resulting in a smaller quantization interval and Sk > 16 resulting in a coarser quantization step size than the initial quantization step size ∆ derived from QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' PROPOSED SCALING LIST GENERATION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Scaling Matrix Generation To train the VCM-optimized scaling matrix S, we build up the framework depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 1, which emulates the signal flow of the coding chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Contrary to the real hybrid coding chain, we applied three adaptations in order to make an end- to-end training possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' First, we utilized RGB images as input data x instead of error signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Second, we focused on the DCT transformation, since it is the most important representative of the VVC transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Third, quantization is not applicable to gradient-descent optimization due to its non-differentiable characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Thus, we emulate the quan- tization by adding uniform noise similar to the field of learned end-to-end image compression [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The strength of the noise is steered by the user-defined constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Hence, the distorted transform coefficients ˜ X are calculated by ˜ X = X + U(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='5) · c · S 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' (3) With the trainable scaling matrix S, the noise can separately be amplified for each frequency coefficient Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Per design, we bound S between 16 and 128 by applying a Sigmoid non- linearity: S = 16 + 112 · Sigmoid(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' (4) Here, S′ is the actual trainable parameter resulting in the bounded scaling matrix S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The borders of 16 and 128 are inspired by the minimum and maximum values given for the default HEVC matrix [6] and the work in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' By that, we only allow the network to increase ∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' With the maximum S of 128, ∆k is limited to eight times the initial step size ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' When training the scaling matrix, two contrary targets are pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The main goal of the proposed method is to increase S in order to enlarge the quantization step size, and thus to ultimately reduce the bitrate during encoding in the inference case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' To achieve this goal, we introduce a loss Lrate(S) = 16 mean(S), (5) approximating the likely rate savings later in inference and forcing the network to increase the coefficients of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Due to the mean, the loss is independent of the DCT blocksize B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Naturally, only training on Lrate would result in the maxi- mum possible value of S, neither considering the output qual- ity resulting from S in inference nor weighting the different frequency coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Therefore, we also consider the task loss Ltask of the analysis network depending on the present noise in ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' With that, the network is pushed to increase the noise for the coefficients that are less harmful for the analysis network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The overall minimization problem is defined as S = arg min S Ltask(˜x|S) + λ · Lrate(S), (6) where the parameter λ weights the training towards one of the competing goals of a large scaling matrix, presumably result- ing in a lower bitrate, or a high task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Training Setup As analysis network in training, we employed the state-of- the-art instance segmentation network Mask R-CNN [16] with its proposed loss as Ltask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Its pre-trained weights were taken from the Detectron2 library [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' We trained our framework on the 2965 RGB Cityscapes [19] training images, which were cropped to patches of 512 × 1024 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The scaling list was optimized by the Adam optimizer for person86% rsonc = 4 c = 16 c = 64 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='01 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='1 λ = 1 λ = 10 16 32 48 64 80 96 112 128 Sk Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Resulting 8×8 scaling matrices for different combina- tions of initial noise strength c and loss weighting parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The direct component is located in the top left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' A yellow value corresponds to a larger scaling matrix value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' After 10 epochs, the initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='01 was decreased to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The batch size was set to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' With each training run, we trained one scaling matrix S for one DCT blocksize B, a given noise strength c, and one λ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The scaling matrix was applied to each RGB color channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Finally, the learned scaling matrix is rounded to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Resulting Scaling Matrices In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 2, the scaling matrices generated by our proposed framework are depicted for a DCT blocksize of 8 × 8 and several combinations of c and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The smaller c, and therewith the smaller the initial noise, the higher the network increases the values in the scaling matrix due to the lower impact on the task accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' For a high noise strength, only very few coef- ficients are increased, eventually resulting in only very few quantization step sizes ∆k to be enlarged, when applying the scaling matrix for coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 2 also shows the influence of the loss weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' For a low λ, the task loss has a high priority which results in the ma- jority of scaling coefficients to be near the minimum value of 16 to avoid a noise amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' With increasing λ, the rate loss Lrate is more prioritized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Therefore, the training results in larger scaling coefficients, since it is more important to in- crease the scaling coefficients at the drawback of an increased noise leading to a higher task loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' To put it into a nutshell, the proposed method reveals that the evaluated Mask R-CNN model is less sensitive to deterioration in the high frequency components, which is in line with the findings in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' PERFORMANCE EVALUATION IN VVC In this section, we demonstrate that the scaling lists derived from our measurement framework result in coding gains over conventional coding without frequency-adaptive quantization and over HVS-optimized scaling lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Evaluation Setup For evaluation, we coded the 500 Cityscapes validation im- ages with the standard-compliant VVC test model (VTM) [20] version 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='0 following the work in [21] and the MPEG VCM CTCs [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' We selected QP values of 12, 17, 22, and 27 in order to obtain a high task accuracy close to the performance of uncompressed images, which would typically be required for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' To measure the inference task per- formance, we utilized the weighted average precision (wAP) metric as proposed in [21], which weights the state-of-the-art object detection metric, average precision, according to the class frequency to level class imbalances in the Cityscapes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' To quantify the coding gains of our proposed method, we calculate the Bjøntegaard delta rate (BDR) [23] with the wAP as quality metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' It measures the bitrate change com- pared to the anchor codec VTM-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='0 without scaling lists at the same accuracy of Mask R-CNN applied to the decoded images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' In order to obtain the scaling lists, we trained the scal- ing matrices with our proposed framework as described in Section 3 for squared blocks of size 2, 4, 8, 16, 32, and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The resulting scaling matrices measured in RGB colorspace were taken for the luma and the chroma channels respectively, and combined to one scaling list as described for VTM by JVET [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' There, it is also described how to interpolate the scaling lists for rectangular blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Influence of Training Parametrization on Coding Performance As shown in Section 3, the trained scaling matrices are highly influenced by the initial choice of the noise strength c and the loss weighting λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' In order to find the best configura- tion in terms of coding gains, we measured the coding per- formance of VTM-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='0 with each scaling list derived from the twelve combinations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The resulting rate- wAP points are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' In most cases, the result- ing points from the scaling lists show a lower rate or a higher wAP than VTM without frequency-specific quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='5 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='4 QP=12 QP=17 QP=22 QP=27 Avg bitstream size per frame in MBit Weighted AP uncompressed λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='01 c = 4 without scaling list λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='1 c = 16 optimal scaling list λ = 1 c = 64 λ = 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Coding performance of VTM-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='0 depending on the used scaling list for the Cityscapes validation set and Mask R- CNN as analysis network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The black dotted line corresponds to the wAP on uncompressed input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The marker shape and color denote the selected λ and c values, respectively, when training the scaling lists with the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' BDR-values listed in Table 1 confirm this observation that VTM coding with the proposed VCM-optimized scaling lists results in bitrate savings over VTM coding without scaling lists for nearly all investigated combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' At best, the coding gains achieve 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='7 % of bitrate savings for c = 16 and λ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Another observation from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3 is that the optimal scaling list depends on the selected QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' As a rule of thumb for choos- ing c, it can be said that the noise strength c in training shall roughly be similar to the quantization step size ∆ defined via the QP as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' This effect is especially observable for a QP of 27, where the points for c = 4 result in a worse coding performance than for c = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Such a relationship cannot be found for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' There, our experiments suggest testing multiple combinations for practical applications to find the best scaling list for each QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' By doing so, we obtain the hand-optimized scaling list resulting in the black curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3 and BDR savings of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='9 % over VTM-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='0 without scaling lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Comparison with HVS-optimized Scaling Lists The default HVS-optimized scaling lists for JPEG and HEVC also increase the quantization step size for large DCT- frequency coefficients similar to our proposed scaling lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Thus, we also measure their VCM coding performance as comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Table 2 shows the BDR values for the qual- ity metrics of PSNR, VMAF, and wAP for two HVS-based Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' BDR values in % for the measurement points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' VTM without scaling list is taken as anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Negative values denote bitrate savings over the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The markers correspond to the markers used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' c 4 16 64 λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='6 10 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='0 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' BDR values in % for Cityscapes validation im- ages for VTM with the corresponding scaling list for the three quality metrics PSNR, VMAF, and wAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' VTM without scal- ing list is taken as anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Codec Scaling list BDR PSNR BDR VMAF BDR wAP VTM JPEG-like [25] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='4 VTM HEVC-default [25] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='2 VTM optimal scaling list 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='9 and our proposed optimal scaling list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Both scaling lists were taken from [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The BDR values show that the HVS- optimized scaling lists result in bitrate savings around 4 % for the investigated VCM-scenario because they also favor low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' However, due to its large optimization towards the final analysis network, encoding the Cityscapes dataset with our proposed scaling lists results in more than twice the bitrate savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' The large coding gains for the VCM scenario come with coding losses for the classic coding scenarios measured by PSNR and VMAF for all three scaling lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a novel data-driven method to ob- tain trained scaling lists that are optimized for VVC intra coding in M2M scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Our measurements revealed that the applied Mask R-CNN network is less sensitive to high DCT frequencies similar to the HVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Thus, a larger quantiza- tion step size was assigned to those coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' By that, our standard-compliant optimization results in up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='7 % of bi- trate savings over conventional VVC coding with static quan- tization intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' When selecting the optimal scaling list for each QP by hand, we were able to further increase the cod- ing gains to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='9 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Future research has to show, whether the proposed method also achieves similar coding gains for video coding with low-delay P or randomaccess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Furthermore, we plan to evaluate whether our proposed method also results in coding gains for other analysis networks and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Moreover, we believe that our method could also be adapted for the HVS by substituting Ltask with a suitable metric representing HVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' REFERENCES [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Sullivan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content=' Ohm, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} +page_content='-J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFAT4oBgHgl3EQfXR3e/content/2301.08533v1.pdf'} diff --git a/bNFPT4oBgHgl3EQfAjSZ/vector_store/index.pkl b/bNFPT4oBgHgl3EQfAjSZ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f8b969990d6ad9f3bc78f334a0d3d9bf0ee55bde --- /dev/null +++ b/bNFPT4oBgHgl3EQfAjSZ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:36d102f3c3405d3ac4e3113c96d597405a10db12a1c8124bea05f9c4b4c02211 +size 101344 diff --git a/btFAT4oBgHgl3EQfXx0t/content/tmp_files/2301.08535v1.pdf.txt b/btFAT4oBgHgl3EQfXx0t/content/tmp_files/2301.08535v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a23d34b0b63e771d11d0f443451db4a000ce872 --- /dev/null +++ b/btFAT4oBgHgl3EQfXx0t/content/tmp_files/2301.08535v1.pdf.txt @@ -0,0 +1,1196 @@ +Globular clusters as indicators of Galactic evolution +N. R. Arakelyan ∗1 and S. V. Pilipenko1 +1Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 117997 Russia +We have studied the system of globular clusters (GCs) that formed in other galaxies and +eventually accreted onto the Milky Way. Thus, the samples of GCs belonging to different +tidal streams, obtained on the basis of the latest data from the Gaia observatory, were +taken from the literature. We measured the anisotropy of the distribution of these GCs +using the gyration tensor and found that the distribution of GCs in the streams is isotropic. +Nevertheless, it can be seen that some of the accreted GCs included into existing samples +actually belong to the disk of the Galaxy. To clarify the origin of GCs, we investigated the +“age–metallicity” relation. This dependence demonstrates bimodality and its two different +branches clearly show the difference between the clusters formed in the streams and in the +disk of the Galaxy. Furthermore, we have studied the influence of the large–scale environment +of the Galaxy (i.e., the Local Supercluster) on the distribution of satellite galaxies and +Galactic GCs. The satellite galaxies of the Milky Way are known to form an anisotropic +planar structure, so we included them in our analysis too. An inspection has shown that +the plane of the satellite galaxies is perpendicular both to the disk of the Galaxy and the +supergalactic plane. For GCs more distant than 100 Kpc, a similar picture is observed. +Key words: (Galaxy:)globular clusters: general – Galaxy: structure – galaxies: dwarf +I. +INTRODUCTION +Globular clusters (GCs) belong to the old- +est objects inhabiting galaxies. +According to +the hierarchical theory, the galaxies are formed +by merger of low–mass and then by larger mass +objects [1]. +Whenthe galaxies of very differ- +ent masses merge, as a rule, the lower mass +galaxy disrupts gradually. As it continues its or- +bital motion, a tidal tail of gas, dust, stars, and +GCs forms behind the galaxy, thus enriching the +larger mass galaxy. This type of mergers have +∗E-mail: n.rubenovna@mail.ru +occurred and still occur with our Galaxy too. +According to Bland–Hawthorn and Gerhard [2], +about 100 satellite galaxies have accreted onto +the Galaxy during the life time of the Universe. +But out of these satellite galaxies, only massive +ones contribute GCs to our Galaxy, since galax- +ies with stellar mass 107 M⊙ have very few GCs +[3]. +Myeong et al. +[4] argue that the clusters +with the critical energy E ≥ −1.6 × 105 km2 +s−2 were accreted from dwarf galaxies. Cosmo- +logical hydrodynamic simulations show that 15– +40% of the stars in the halo were formed outside +the Galaxy (ex–situ), that is, in dwarf satellite +arXiv:2301.08535v1 [astro-ph.GA] 20 Jan 2023 + +2 +galaxies, and then were accreted [5, 6]. In the re- +sults of studies by different authors, the percent- +age of GCs formed ex–situ differs. For example, +Forbes [7] claims that 54% of GCs (87 out of 160 +clusters) were formed ex–situ and then accreted, +while Kruijssenn et al. [8] claim that the per- +centage of accreted clusters is 43. According to +Massari et al. [9], this value reaches 60%. +Thus, a significant part of the GCs of the +Galaxy was accreted from the outside. Informa- +tion about the origin of GCs can be preserved +both in the properties of the stellar population +of GCs and in the spatial distribution and dy- +namics of GCs themselves. In particular, it is +well known that both satellite galaxies and GCs +exhibit a disk–like structure perpendicular to the +disk of the Galaxy [see, e.g. 10–12]. This struc- +ture may be a result of the accretion of several +galaxies that arrived in our Galaxy mainly from +polar directions. According to Zeldovich theory +[13], formation of the large–scale structure of the +Universe occurs through independent contrac- +tion or expansion of the matter in the three mu- +tually perpendicular directions. A good example +of a structure contractingin one direction and +expanding in two other directions is the Local +Supercluster of Galaxies, which looks likea typi- +cal Zeldovich “pancake”. This structure sets the +preferred direction in the vicinity of the Galaxy, +and therefore can affect the preferrential direc- +tion of accretion and distribution of accreted ma- +terial in our Galaxy. +Tidal streams of the Galaxy are actively dis- +cussed in the literature [e.g. 14–35]. Recent mea- +surement of the GCs proper motions using GAIA +data made it possible to identify GCs belonging +to specific tidal streams. The problem regard- +ing the difference in the physical properties of +GCs formed in–situ and ex–situ was studied in +detail in [36]. It was shown there that accreted +GCs differ by the abundance of alpha–elements, +as well as by the range of masses. The purpose +of our study is to check the spatial orientation +of the GCs system belonging to the streams, +i.e., admittedly accreted onto the Galaxy from +outside. +For this, the orientation of the sys- +tems identified by different authors was com- +pared with the disk of the Galaxy, as well as +with the plane of the Local Supercluster. In ad- +dition, the “age–metallicity” relation (AMR) for +GCs belonging to the streams was analyzed and +the relation between GCs colors and their origin +was discussed. +The paper is organized as follows. +In Sec- +tion 2, the studied GC samples are described +and the anisotropy of their distribution is inves- +tigated. In Section 3 the AMR for GCs is dis- +cussed. In Section 4, the influence of the Local +Supercluster is measured. Conclusions are pre- +sented in Section 5. + +3 +II. +MILKY WAY GLOBULAR CLUSTERS +IN TIDAL STREAMS +Our Galaxy contains at least 157 GCs [46, +2010 edition]1 and [45]. +As time goes by, the +papers regarding new clusters in the Milky Way +appear (e.g., FSR 1716 [47], FSR 1758 [40, 48], +V V V −CL001 [49], V V V −CL002 [50], BH 140 +[51], Gran 1 [52], Pfleiderer 2 [53], ESO 93– +8 [54], Mercer 5 [55], Segue 3 [56], Ryu 059, +Ryu 879 [57], Kim 3 [58], Crater/Laevens 1 +[59, 60], Laevens 3 [61] and BLISS 1 [62]). +Although even before obtaining high–precision +GAIA data, there were attempts to identify GCs +belonging to the tidal streams [33, 37–39], but +after the appearance of GAIA data, these at- +tempts significantly advanced [4, 7, 9, 40–44]. In +this paper, we draw our attention to three stud- +ies: [9] (hereinafter, Massari), [40] (hereinafter, +Myeong), and [7] (hereinafter, Forbes), which +contain the most complete lists of GCs belonging +to different tidal streams. +Forbes et al. +list 76 clusters that belong +to five progenitors–satellite galaxies. To check +membership, the authors used integrals of mo- +tion (IOM), AMR, and alpha–elements depen- +dencies. Out of these 76 GCs, nine belong to the +well–known Sagittarius dwarf spheroidal galaxy +(Sgr dSph), 28 belong to the Gaia–Enceladus +dwarf galaxy, nine GCs – to the Sequoia dwarf +galaxy, 21 clusters – to the low–energy satellite +1 http://physwww.mcmaster.ca/~harris/Databases. +html +Koala, nine clusters – to a low–mass satellite, the +Helmi streams. Out of the remaining 36 clusters, +they are either “Low–energy” ones (25 objects) +or “High–energy” ones (11 clusters). These clus- +ters have high energies and a wide range of an- +gular momenta, which suggests that they origi- +nated from different progenitors. +Massari et al. examined 151 GCs for which +they collected complete kinematic information. +They concluded that 62 of these clusters were +most likely formed in the Galaxy (in–situ), while +the remaining clusters (89 clusters) were most +likely formed ex–situ and then accreted. Basi- +cally, accreted clusters are associated with four +known merger events: Gaia–Enceladus – 26 GCs +(+6 candidates), the Sagittarius dwarf galaxy – +eight GCs, the Helmi stream (H99) – 10 GCs, +and the Sequoia galaxy – seven GCs. The re- +maining 36 clusters are classified as “Low en- +ergy” ones (25 GCs) or “High energy” ones (11 +GCs). Association of the clusters with any group +is uncertain, due to the partial overlap of the de- +bris of different progenitor galaxies. +Myeong et al. considered 34 GCs, which ac- +creted onto the Galaxy. To verify this, the au- +thors used kinematic data of GAIA [63] in com- +bination with photometry from DECaPS (DE- +Cam Plane Survey [64]). In opinion of Mueong +et al., 6 GCs belong to the Sagittarius dwarf +spheroidal galaxy, 7 – to Sequoia galaxy, 21 – +to the Gaia Sausage. Summing up the results +of three above–mentioned papers, we obtain the +main list of tidal streams from which accreted a + +4 +considerable fraction of GCs: +(1) Sagittarius dwarf spheroidal galaxy (Sgr +dSph) with the nucleus NGC 6715 (M54). +(2) +Sequoia +galaxy +with +the +nucleus +NGC 5139 (Omega Centauri (ω Cen)). +(3) Helmi stream (H99). +(4) +Gaia–Enceladus +with +the +nucleus +NGC 1851. +Other possible variations of the +name of this stream – Gaia Sausage or Canis +Major (CMa). +(5) Low–energy progenitor Coala, to which +Kraken may be equivalent, and also a low– +energy group (E < −1.86 × 105 km2 s−2). +(6) High energy group (E > −1.5 × 105 km2 +s−2). +A. +Anisotropy of the distribution of +globular clusters +The number of GCs belonging to different +streams, according to the classification of the au- +thors considered in this paper, is as follows: ac- +cording to Forbes – 87 GCs, according to Massari +– 89 (these clusters are located at the distance +from 1.42 to 144.77 kpc from the center of the +Galaxy) and according to Myeong – 34 GCs (at +the distance from 2.42 to 71.36 kpc). +In order to understand whether there is any +difference in the distributions of GCs belonging +to the streams and the objects formed in the +Milky Way, we decided to check the anisotropy of +the distribution of these GCs using the gyration +tensor, as in [12]. The tensor is constructed as +follows: +Sij = 1 +N +N +� +k=1 +xk +i xk +j , +(1) +where S – gyration tensor, N – the number of +objects, xk +i – the distance of k th object to the +Galactic center along coordinate axis i. +Stan- +dard mathematical operations for determination +of the eigenvalues and eigenvectors of a tensor +allow us to characterize the anisotropy of the +distribution. +The eigenvalues a, b, and c, for +convenience, are sorted in ascending order, so +that a > b > c . The degree of anisotropy is +characterized by the ratios of the eigenvalues c/a +and b/a, which, in the case of an isotropic dis- +tribution, approach 1. The eigenvectors of the +inertia tensor determine the orientation of the +anisotropic distribution in space. +To check the statistical significance of the +found parameters of the GCs system, we gener- +ated 10 000 random samples with the same radial +distribution and number of objects as in the data +for observed objects, and measured the median +value and the root-mean-square value of the ra- +tio of the eigenvalues of the tensors. Anisotropy +is statistically significant if the ratio of the eigen- +values of the tensor for the real catalogs differs +from the median of the random samples by more +than 3σ. Random samples are constructed by +fixing the distances (R) from the real sample +and assigning random angular coordinates to the +GCs. +In Fig. 1, we show the results of measuring + +5 +the anisotropy for GCs using the gyration ten- +sor. The panels show the ratios c/a and b/a as +functions of R , calculated for all GCs with a dis- +tance < R . The distributions of real objects are +shown by dots, solid line represents the median +result for 10 000 random samples, and dashed +lines represent the median ±3σ. The “angle” in +these panels is measured between the normal to +the Galactic plane and the minor (green trian- +gles) or the major (blue dots) axis of distribu- +tion. +From the measurements of the anisotropy us- +ing the gyration tensor for all samples of GCs +in the streams from the three above-mentioned +papers, it follows that the distribution of GCs in +the streams is isotropic. Thus, none of the sam- +ples exhibit statistically significant anisotropy. +In [12] [p. 7, Fig. 7], for the entire sample of +GCs at the distance from 2 to 10 kpc, statis- +tically significant anisotropy is observed, which +the authors associated with GCs belonging to +the disk of the Galaxy, that is, formed in–situ. +In this paper, we studied spatial distributions +of GCs, which, according to a number of au- +thors, belong to the tidal streams, that is, were +formed ex–situ. As seen in Fig. 1, for all sam- +ples, the spatial distribution of GCs belonging to +the tidal streams is isotropic. This is consistent +with the conclusion of Arakelyan et al. [12] that +the statistically significant anisotropy for the en- +tire GCs sample is due to the clusters that were +most likely formed in the Galaxy or have been +interacting with the Galaxy disk for a very long +time. It is also important that the clusters that +belong to the tidal streams do not exhibit signif- +icant structure, which we might expect, first, be- +cause clustering occurs mainly along the distin- +guished directions associated with the walls and +filaments of a large–scale structure, and, second, +because anisotropic distribution is observed for +satellite galaxies. +Nevertheless, it is seen in Fig. 1 that for +all three samples (Forbes, Massari, Myeong) for +GCs that belong to the streams, the major axis +of the gyration tensor is in the disk, at the dis- +tances from about 3 to 10—20 kpc. It seems un- +likely that such a situation can arise for a ran- +dom isotropic GCs distribution. The distribu- +tion of the directions of the axes of the tensor +as in Fig. 1, one can expect if a part of the GCs +in each of the samples belongs to the disk. We +demonstrate this below using random catalogs. +To check the probability of entering of the +GCs from the disk into the GCs sample from +the tidal streams, we generate random catalogs +containing the same number of GCs as the real +samples. Moreover, we take the galactocentric +GC distances from the real samples. +Angular +coordinates are assigned randomly. To simulate +a situation in which some of the clusters belong +to the disk, for n clusters the height above the +disk is set to zero (Cartesian coordinate z). +Using such models, we calculated the con- +ditional probability of obtaining a distribution +similar to the right-hand column in Fig. 1, i.e., +when the major axes of the gyration tensor in the + +6 +FIG. 1: Galactic clusters anisotropy, quantified by the gyration tensor for cluster samples Forbes (top), +Massari (center), and Myeong (bottom). The left and middle columns show the ratios c/a and b/a as a +function of the galactocentric cluster distance, respectively. Each blue dot represents the ratio of the tensor +eigenvalues calculated for all clusters at the distance smaller than R from the Galactic center. Solid green +line represents the median eigenvalues ratios for 10 000 random samples. Dashed lines indicate the deviations +±3σ of such random distributions. The right column shows the angles, measured in degrees, between the +Milky Way’s Galactic pole and the major (blue dots) and minor (green triangles) axis of the gyration tensor. +Green triangles close to 90◦, indicate the polar plane. +distance range from 3.5 to 20 kpc are located at +an angle of more than 70◦ to the direction to +the Galactic pole, provided that n clusters be- +long to the disk. If the distribution is isotropic, +i.e. n = 0, this probability is equal to 4.5, 0.6, +and 1.1% for the samples of Forbes, Massari and +Myeong, respectively. +For this probability to exceed, for example, +10%, the disk must contain n = 6, 16, and 8 +GCs for the Forbes, Massari, and Myeong sam- +ples, respectively. From this, we can conclude +that a part of the GCs, formed to the opinion of +these authors outside our Galaxy, actually be- +longs to its disk. It should be noted that in [36], +based on the analysis of the abundance of alpha– +elements, it was shown that the group of Low + +1.0 +0.8 +0.6 +S(c/a) +0.4 +0.2 +0.0 +1 +10 +100 +R/kpc1.0 +0.8 +0.6 +S(b/a) +0.4 +0.2 +0.0 +1 +10 +100 +R/kpcAAA +80 +60 +Angle(deg) +40 +20 +10 +100 +R/kpc1.0 +0.8 +0.6 +S(c/a) +0.4 +0.2 +0.0 +- +1 +10 +100 +R/kpo1.0 +0.8 +0.6 +S(b/a) +0.4 +0.2 +0.0 +1 +10 +100 +R/kpo90 +80 +70 +60 +Angle(deg) +50 +40 +30 +20 +10 +0 +10 +100 +R/kpc1.0 +0.8 +0.6 +S(c/a) +0.4 +0.2 +0.0 +1 +10 +100 +R/kpc1.0 +0.8 +0.6 +S(b/a) +0.4 +0.2 +0.0 +1 +10 +100 +R/kpo90 +80 +70 +60 +Angle(deg) +50 +40 +30 +20 +10 +0 +1 +10 +100 +R/kpc7 +energy clusters from the Massari work was most +likely formed in–situ, which also indicates the in- +accuracy of the in–situ/ex–situ separation in the +Massari sample. In order to verify further the +origin of the GCs, we use the “age–metallicity” +diagram. +III. +TWO BRANCHES OF THE GSS IN +THE “AGE–METALLICITY” RELATION +The literature discusses the fact that the pop- +ulation of the GCs of the Milky Way exhibits +bimodality of colors: +there are blue and red +clusters [65–77]. +This is due to the bimodal- +ity of metallicity [73, 75, 78–87]. +Blue clus- +ters are found mostly in the halo of the Galaxy. +These clusters probably previously belonged to +the satellite galaxies. +At the same time, red +clusters are spatially concentrated towards the +Galactic center and rotate with it. Blue clusters +are old and metal-poor, while red clusters are +younger and metal–rich. The ratio [Fe/H] peaks +for blue and red clusters in the Milky Way are +approximately –1.5 and –0.5, respectively. Such +a bimodality assumes two mechanisms of GCs +formation. The authors of [76, 77, 80] argue that +red clusters are formed in–situ, while blue ones +were accreted either as a result of the merger of +satellite galaxies with the Galaxy or as a result +of tidal capture of the clusters themselves. +To understand the difference between in–situ +clusters and ex–situ clusters, we plotted respec- +tive samples from Massari, Forbes, and Myeong +in the “age–metallicity” diagram. +The results +are shown in Fig. 2. +The “age–metallicity” dependence clearly +shows that GCs have two branches. The low- +metals branch contains mainly clusters that be- +long to different tidal streams formed by the par- +tial destruction of satellite galaxies. The clusters +in this sequence show a wide variation in age and +metallicity, but there are no clusters less than +6 Gyr old. The clusters of a more metals-rich +branch, formed in–situ, also have a scatter in +metallicity, but all clusters are more than 11 Gyr +old. +It is worth noting that the in–situ clusters +were formed not in the Galaxy as we know it, +but in its progenitor. In the hierarchical model +of the formation of galaxies, the mass of a galaxy +is accumulated gradually due to mergers, and +galaxies as a whole do not have a clearly defined +moment of formation. Therefore, for the objects +formed long ago, it is difficult to distinguish be- +tween the concepts of in–situ and ex–situ. How- +ever, specifically for our Galaxy, it is believed +that it did not experience mergers with the ob- +jects of comparable mass since z = 2 or less than +10.5 Gyr ago [88]. By that time, it had gained +only 1/5 of its current total mass (including the +dark halo). Six Gyr ago (the age of the youngest +GCs), its mass was about 60% ofthe current one +[89]. + +8 +FIG. 2: Dependence of the age of GCs on metallicity for cluster samples Forbes, Massari, and Myeong, +the left, middle, and right panels, respectively. Blue dots represent ex–situ clusters and the red ones show +in–situ clusters. +IV. +THE ROLE OF THE LOCAL +SUPERCLUSTER +In the hierarchical model of galaxies forma- +tion, accretion of the matter is controlled by +large–scalef lows, which are also responsible for +the formation of a cellular structure–“pancakes” +and filaments. +In accordance with Zeldovich +theory [13], a “pancake” is formed from a uni- +formly filled volume if compression occurs in +one of the three mutually perpendicular direc- +tions, and expansion — in two other direc- +tions. +Thus, the large–scale structure is asso- +ciated with anisotropic motions of matter, and +this anisotropy can also affect the distribution +of the matter in the galaxies. Our Galaxy, to- +gether with the Local Group, is located within +the Local Supercluster (LSC) [90–95], a well– +visible pancake–like structure with dimensions +of tens of Mpc. +We have tested the influence of the Local Su- +percluster on the spatial distribution of GCs, +as well as dwarf satellite galaxies of the Milky +Way. +The satellite galaxies were a priori ac- +creted onto our Galaxy from the outside. At the +same time, they form a clear–cut flat structure +[10–12]. Therefore, we did not limit ourselves to +analyzing the GCs distribution, but also consid- +ered satellite galaxies. For this, the angle distri- +butions between the axes of the gyration tensor +(1) and the plane of the Local Supercluster were +obtained for dwarf satellite galaxies (27 satel- +lites [96]) and for: (i) for the entire GCs sample +(157 GCs [46, 2010 edition], [45]); (ii) for the +GCs from the Forbes list; (iii) for the GCs from +the Massari list, and (iv) for the GCs from the +Myeong list. In Fig. 3, the “Angle” is presented +as a function of galactocentric distance for the +GCs and satellites of the Galaxy. The “Angle” is +measured between the plane of the Local Super- +cluster and the minor (green triangles) or major +(blue dots) axes of the distribution of GCs. +Fig. 3 shows that for satellite galaxies (top +row, left) at the largest distances, the major +and minor axes are located in the supergalac- +tic plane. At the same time, the minor axis is +located in the disk of the Milky Way and the +major one is perpendicular to the disk. +This + +-0.5 +-1.0: +[Fe/H] +-1.5 +-2.0 - +6 +8 +10 +12 +14 +16 +Age(Gyr)-0.5 +-1.0: +[Fe/H] +-1.5 +-2.0 - +6 +8 +10 +12 +14 +16 +Age(Gyr)-0.5 +-1.0: +[Fe/H] +-1.5 +-2.0: +6 +8 +10 +12 +14 +16 +Age(Gyr)9 +FIG. 3: The “Angle” as a function of galactocentric distance for satellite galaxies for entire GCs sample +(top row, left to right) and for cluster samples Forbes, Massari, and Myeong (bottom row, left to right, +respectively). The “Angle” is measured between the LSC plane and the small (green triangles) or large +(blue dots) axes of the distribution of GCs. +means that the plane of satellite galaxies is per- +pendicular to both the disk of the Galaxy and +the supergalactic plane. +We can say the following regarding all 157 +GCs (top row, on the right). At small distances, +up to 4 kpc, we are not interested in the result, +since these are GCs in the central part of the +Galaxy. From 4 to 20 kpc, the minor axis of the +system is perpendicular to the disk of the Galaxy +and is in the supergalactic plane. The major axis +is in the disk of the Galaxy and is perpendicular +to the supergalactic plane at a distance of about +20 kpc. Thus, in the range from 4 to 20 kpc, +the orientation of the GC system corresponds +to the disk of the Galaxy, the influence of the +Supercluster does not manifest itself. The same +can be said for the clusters from streams, but +only with a caveat that they have a larger noise. +The minor axis shows a large scatter; this may +be due to the fact that there are fewer clusters +belonging to the disk of the Galaxy in the sample +of objects from the streams. +At the distance of about 100 kpc, the picture +for GCs resembles that of satellite galaxies for all +samples, i.e., the GC system is oriented perpen- +dicular both to the disk and the Supercluster. +It is worth noting that only six clusters are ob- +served at such distances, which is not enough for +reliable conclusions. +At a distance of about 30 kpc in all samples +except Myeong, the major axis is in the super- +galactic plane, while the minor one makes with + +80 +A +60 +Angle(deg) +40 +20 +0 +20 +30 +50 +100 +200 +R/kpc80 +60 +Angle(deg) +K +40 +20 +V +10 +100 +R/kpc80 +W +V +60 +Angle(deg) +40 +20 +XV +10 +100 +R/kpc80 +△公 +60 +Angle(deg) +40 +20 +XV +10 +100 +R/kpc80 +60 +Angle(deg) +40 +V +AA +20 +V +1 +10 +100 +R/kpc10 +the large axis an angle of about 60◦ for all GCs +and for Forbes sample; for the Massari sample, +the angle is within 45◦. +At a distance of 25– +40 +kpc, there are only 10 GCs, of which 10 +and 9 GCs belong to the streams of the Forbes +and Massari samples, respectively. Thus, in the +Forbes and Massari samples at these distances, +there may be signs of the influence of the Super- +cluster on the orientation of the system of ac- +creted GCs, but the reliability of this conclusion +is low. +V. +DISCUSSION AND CONCLUSIONS +In this paper, we studied the GC system that +was formed outside the Galactic disk. To do this, +we took from the literature the samples of GCs +that were formed in different tidal streams. We +chose the works of Forbes, Massari, and Myeong, +since their lists of GCs belonging to the differ- +ent streams are the most complete and are based +on the latest data from the GAIA observatory. +Having studied a number of works, including +those mentioned above, we obtained the main +list of tidal streams in which GCs belonged and +later were accreted: Sagittarius dwarf spheroidal +galaxy (Sgr dSph), Sequoia Galaxy (Sequoia), +Helmi Stream (H99), Gaia–Enceladus (possibly +Gaia Sausage or CMa), the low energy group +(possibly Koala or Kraken), and the High en- +ergy group. +It is believed that the accretion onto the +Galaxy was anisotropic, which is manifested, +for example, as a disk–like structure of satel- +lite galaxies. +We measured the anisotropy of +the distribution of GCs that belonged to the +streams using the gyration tensor. The measure- +ment result showed that no statistically signif- +icant anisotropy is observed for accreted GCs. +Having obtained this result, we can state that +the anisotropic structure that is observed for the +complete sample of GCs (see [12], p. 7, Fig. 7) is +due to the presence of many GCs in the Galactic +disk, and is associated with the clusters formed +in–situ. +However, in Fig. 1 for the three samples of +the accreted GCs, the major axis of the gyration +tensor at a distance from 3 to 20 kpc is in the +disk. This may be due to the fact that the sam- +ples contain a significant number of GCs that +have formed in the disk of the Galaxy. To esti- +mate their number, the distribution of GCs with +random angular coordinates was modeled and it +was shown that the probability of a random re- +alization of such a distribution, in which there +are no GCs belonging to the disk, is 4.5, 0.6, +and 1.1% for the Forbes, Massari, and Myeong +samples, respectively. This conclusion is consis- +tent with the conclusion of Marsakov et al. [36], +who had shown that some of the clusters from +the Massari catalog claimed to be ex–situ are in +fact genetically related to our Galaxy. +We also checked how the clusters formed in– +situ and ex–situ behave respective to the AMR +(Fig. 2). +Two branches can be easily distin- +guished; the low–metals branch contains mainly + +11 +clusters belonging to different streams, and they +have a large spread in the age and metallicity. At +the same time, the clusters in the more metallic +branch, which most likely formed in the Galaxy, +have a scatter in metallicity, but their age is over +11 Gyr. +To check the likely influence of the Local Su- +percluster on the distribution of satellite galax- +ies and GCs of the Milky Way, we presented the +Figures, which show the angle between the LSC +plane and the axes of distribution of GCs sys- +tems or satellite galaxies, as a function of the +galactocentric distance. +Fig. 3 (top row, left) +shows that the plane of the satellite galaxies is +both perpendicular to the disk of the Galaxy +and to the supergalactic plane. For GCs at the +distances of up to 20 kpc, only the influence of +the Galactic disk is traced; at the distances of +about 30 kpc, the orientation of the GCs sys- +tem may coincide with the supergalactic plane, +and at larger distances (more than 100 kpc), the +orientation resembles that for satellite galaxies. +[1] P. J. E. Peebles, Astrophys. J. 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J. 144, 4 (2012), +1204.1562. + diff --git a/btFAT4oBgHgl3EQfXx0t/content/tmp_files/load_file.txt b/btFAT4oBgHgl3EQfXx0t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b6291e5592093255d74e5894a164a4f1dafc2ae --- /dev/null +++ b/btFAT4oBgHgl3EQfXx0t/content/tmp_files/load_file.txt @@ -0,0 +1,1399 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf,len=1398 +page_content='Globular clusters as indicators of Galactic evolution N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Arakelyan ∗1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Pilipenko1 1Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 117997 Russia We have studied the system of globular clusters (GCs) that formed in other galaxies and eventually accreted onto the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Thus, the samples of GCs belonging to different tidal streams, obtained on the basis of the latest data from the Gaia observatory, were taken from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' We measured the anisotropy of the distribution of these GCs using the gyration tensor and found that the distribution of GCs in the streams is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Nevertheless, it can be seen that some of the accreted GCs included into existing samples actually belong to the disk of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To clarify the origin of GCs, we investigated the “age–metallicity” relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This dependence demonstrates bimodality and its two different branches clearly show the difference between the clusters formed in the streams and in the disk of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Furthermore, we have studied the influence of the large–scale environment of the Galaxy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=', the Local Supercluster) on the distribution of satellite galaxies and Galactic GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The satellite galaxies of the Milky Way are known to form an anisotropic planar structure, so we included them in our analysis too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' An inspection has shown that the plane of the satellite galaxies is perpendicular both to the disk of the Galaxy and the supergalactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' For GCs more distant than 100 Kpc, a similar picture is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Key words: (Galaxy:)globular clusters: general – Galaxy: structure – galaxies: dwarf I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' INTRODUCTION Globular clusters (GCs) belong to the old- est objects inhabiting galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' According to the hierarchical theory, the galaxies are formed by merger of low–mass and then by larger mass objects [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Whenthe galaxies of very differ- ent masses merge, as a rule, the lower mass galaxy disrupts gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' As it continues its or- bital motion, a tidal tail of gas, dust, stars, and GCs forms behind the galaxy, thus enriching the larger mass galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This type of mergers have ∗E-mail: n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='rubenovna@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='ru occurred and still occur with our Galaxy too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' According to Bland–Hawthorn and Gerhard [2], about 100 satellite galaxies have accreted onto the Galaxy during the life time of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' But out of these satellite galaxies, only massive ones contribute GCs to our Galaxy, since galax- ies with stellar mass 107 M⊙ have very few GCs [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Myeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [4] argue that the clusters with the critical energy E ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6 × 105 km2 s−2 were accreted from dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Cosmo- logical hydrodynamic simulations show that 15– 40% of the stars in the halo were formed outside the Galaxy (ex–situ), that is, in dwarf satellite arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='08535v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='GA] 20 Jan 2023 2 galaxies, and then were accreted [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In the re- sults of studies by different authors, the percent- age of GCs formed ex–situ differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' For example, Forbes [7] claims that 54% of GCs (87 out of 160 clusters) were formed ex–situ and then accreted, while Kruijssenn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [8] claim that the per- centage of accreted clusters is 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' According to Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [9], this value reaches 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Thus, a significant part of the GCs of the Galaxy was accreted from the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Informa- tion about the origin of GCs can be preserved both in the properties of the stellar population of GCs and in the spatial distribution and dy- namics of GCs themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In particular, it is well known that both satellite galaxies and GCs exhibit a disk–like structure perpendicular to the disk of the Galaxy [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This struc- ture may be a result of the accretion of several galaxies that arrived in our Galaxy mainly from polar directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' According to Zeldovich theory [13], formation of the large–scale structure of the Universe occurs through independent contrac- tion or expansion of the matter in the three mu- tually perpendicular directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' A good example of a structure contractingin one direction and expanding in two other directions is the Local Supercluster of Galaxies, which looks likea typi- cal Zeldovich “pancake”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This structure sets the preferred direction in the vicinity of the Galaxy, and therefore can affect the preferrential direc- tion of accretion and distribution of accreted ma- terial in our Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Tidal streams of the Galaxy are actively dis- cussed in the literature [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 14–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Recent mea- surement of the GCs proper motions using GAIA data made it possible to identify GCs belonging to specific tidal streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The problem regard- ing the difference in the physical properties of GCs formed in–situ and ex–situ was studied in detail in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' It was shown there that accreted GCs differ by the abundance of alpha–elements, as well as by the range of masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The purpose of our study is to check the spatial orientation of the GCs system belonging to the streams, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=', admittedly accreted onto the Galaxy from outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' For this, the orientation of the sys- tems identified by different authors was com- pared with the disk of the Galaxy, as well as with the plane of the Local Supercluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In ad- dition, the “age–metallicity” relation (AMR) for GCs belonging to the streams was analyzed and the relation between GCs colors and their origin was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In Sec- tion 2, the studied GC samples are described and the anisotropy of their distribution is inves- tigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In Section 3 the AMR for GCs is dis- cussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In Section 4, the influence of the Local Supercluster is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Conclusions are pre- sented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' MILKY WAY GLOBULAR CLUSTERS IN TIDAL STREAMS Our Galaxy contains at least 157 GCs [46, 2010 edition]1 and [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' As time goes by, the papers regarding new clusters in the Milky Way appear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=', FSR 1716 [47], FSR 1758 [40, 48], V V V −CL001 [49], V V V −CL002 [50], BH 140 [51], Gran 1 [52], Pfleiderer 2 [53], ESO 93– 8 [54], Mercer 5 [55], Segue 3 [56], Ryu 059, Ryu 879 [57], Kim 3 [58], Crater/Laevens 1 [59, 60], Laevens 3 [61] and BLISS 1 [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Although even before obtaining high–precision GAIA data, there were attempts to identify GCs belonging to the tidal streams [33, 37–39], but after the appearance of GAIA data, these at- tempts significantly advanced [4, 7, 9, 40–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In this paper, we draw our attention to three stud- ies: [9] (hereinafter, Massari), [40] (hereinafter, Myeong), and [7] (hereinafter, Forbes), which contain the most complete lists of GCs belonging to different tidal streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' list 76 clusters that belong to five progenitors–satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To check membership, the authors used integrals of mo- tion (IOM), AMR, and alpha–elements depen- dencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Out of these 76 GCs, nine belong to the well–known Sagittarius dwarf spheroidal galaxy (Sgr dSph), 28 belong to the Gaia–Enceladus dwarf galaxy, nine GCs – to the Sequoia dwarf galaxy, 21 clusters – to the low–energy satellite 1 http://physwww.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='ca/~harris/Databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' html Koala, nine clusters – to a low–mass satellite, the Helmi streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Out of the remaining 36 clusters, they are either “Low–energy” ones (25 objects) or “High–energy” ones (11 clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' These clus- ters have high energies and a wide range of an- gular momenta, which suggests that they origi- nated from different progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' examined 151 GCs for which they collected complete kinematic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' They concluded that 62 of these clusters were most likely formed in the Galaxy (in–situ), while the remaining clusters (89 clusters) were most likely formed ex–situ and then accreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Basi- cally, accreted clusters are associated with four known merger events: Gaia–Enceladus – 26 GCs (+6 candidates), the Sagittarius dwarf galaxy – eight GCs, the Helmi stream (H99) – 10 GCs, and the Sequoia galaxy – seven GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The re- maining 36 clusters are classified as “Low en- ergy” ones (25 GCs) or “High energy” ones (11 GCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Association of the clusters with any group is uncertain, due to the partial overlap of the de- bris of different progenitor galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Myeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' considered 34 GCs, which ac- creted onto the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To verify this, the au- thors used kinematic data of GAIA [63] in com- bination with photometry from DECaPS (DE- Cam Plane Survey [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In opinion of Mueong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=', 6 GCs belong to the Sagittarius dwarf spheroidal galaxy, 7 – to Sequoia galaxy, 21 – to the Gaia Sausage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Summing up the results of three above–mentioned papers, we obtain the main list of tidal streams from which accreted a 4 considerable fraction of GCs: (1) Sagittarius dwarf spheroidal galaxy (Sgr dSph) with the nucleus NGC 6715 (M54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' (2) Sequoia galaxy with the nucleus NGC 5139 (Omega Centauri (ω Cen)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' (3) Helmi stream (H99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' (4) Gaia–Enceladus with the nucleus NGC 1851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Other possible variations of the name of this stream – Gaia Sausage or Canis Major (CMa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' (5) Low–energy progenitor Coala, to which Kraken may be equivalent, and also a low– energy group (E < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='86 × 105 km2 s−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' (6) High energy group (E > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 × 105 km2 s−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Anisotropy of the distribution of globular clusters The number of GCs belonging to different streams, according to the classification of the au- thors considered in this paper, is as follows: ac- cording to Forbes – 87 GCs, according to Massari – 89 (these clusters are located at the distance from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='42 to 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='77 kpc from the center of the Galaxy) and according to Myeong – 34 GCs (at the distance from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='42 to 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='36 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In order to understand whether there is any difference in the distributions of GCs belonging to the streams and the objects formed in the Milky Way, we decided to check the anisotropy of the distribution of these GCs using the gyration tensor, as in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The tensor is constructed as follows: Sij = 1 N N � k=1 xk i xk j , (1) where S – gyration tensor, N – the number of objects, xk i – the distance of k th object to the Galactic center along coordinate axis i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Stan- dard mathematical operations for determination of the eigenvalues and eigenvectors of a tensor allow us to characterize the anisotropy of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The eigenvalues a, b, and c, for convenience, are sorted in ascending order, so that a > b > c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The degree of anisotropy is characterized by the ratios of the eigenvalues c/a and b/a, which, in the case of an isotropic dis- tribution, approach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The eigenvectors of the inertia tensor determine the orientation of the anisotropic distribution in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To check the statistical significance of the found parameters of the GCs system, we gener- ated 10 000 random samples with the same radial distribution and number of objects as in the data for observed objects, and measured the median value and the root-mean-square value of the ra- tio of the eigenvalues of the tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Anisotropy is statistically significant if the ratio of the eigen- values of the tensor for the real catalogs differs from the median of the random samples by more than 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Random samples are constructed by fixing the distances (R) from the real sample and assigning random angular coordinates to the GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 1, we show the results of measuring 5 the anisotropy for GCs using the gyration ten- sor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The panels show the ratios c/a and b/a as functions of R , calculated for all GCs with a dis- tance < R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The distributions of real objects are shown by dots, solid line represents the median result for 10 000 random samples, and dashed lines represent the median ±3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The “angle” in these panels is measured between the normal to the Galactic plane and the minor (green trian- gles) or the major (blue dots) axis of distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' From the measurements of the anisotropy us- ing the gyration tensor for all samples of GCs in the streams from the three above-mentioned papers, it follows that the distribution of GCs in the streams is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Thus, none of the sam- ples exhibit statistically significant anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In [12] [p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 7], for the entire sample of GCs at the distance from 2 to 10 kpc, statis- tically significant anisotropy is observed, which the authors associated with GCs belonging to the disk of the Galaxy, that is, formed in–situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In this paper, we studied spatial distributions of GCs, which, according to a number of au- thors, belong to the tidal streams, that is, were formed ex–situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 1, for all sam- ples, the spatial distribution of GCs belonging to the tidal streams is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This is consistent with the conclusion of Arakelyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [12] that the statistically significant anisotropy for the en- tire GCs sample is due to the clusters that were most likely formed in the Galaxy or have been interacting with the Galaxy disk for a very long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' It is also important that the clusters that belong to the tidal streams do not exhibit signif- icant structure, which we might expect, first, be- cause clustering occurs mainly along the distin- guished directions associated with the walls and filaments of a large–scale structure, and, second, because anisotropic distribution is observed for satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Nevertheless, it is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 1 that for all three samples (Forbes, Massari, Myeong) for GCs that belong to the streams, the major axis of the gyration tensor is in the disk, at the dis- tances from about 3 to 10—20 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' It seems un- likely that such a situation can arise for a ran- dom isotropic GCs distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The distribu- tion of the directions of the axes of the tensor as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 1, one can expect if a part of the GCs in each of the samples belongs to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' We demonstrate this below using random catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To check the probability of entering of the GCs from the disk into the GCs sample from the tidal streams, we generate random catalogs containing the same number of GCs as the real samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Moreover, we take the galactocentric GC distances from the real samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Angular coordinates are assigned randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To simulate a situation in which some of the clusters belong to the disk, for n clusters the height above the disk is set to zero (Cartesian coordinate z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Using such models, we calculated the con- ditional probability of obtaining a distribution similar to the right-hand column in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=', when the major axes of the gyration tensor in the 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 1: Galactic clusters anisotropy, quantified by the gyration tensor for cluster samples Forbes (top), Massari (center), and Myeong (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The left and middle columns show the ratios c/a and b/a as a function of the galactocentric cluster distance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Each blue dot represents the ratio of the tensor eigenvalues calculated for all clusters at the distance smaller than R from the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Solid green line represents the median eigenvalues ratios for 10 000 random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Dashed lines indicate the deviations ±3σ of such random distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The right column shows the angles, measured in degrees, between the Milky Way’s Galactic pole and the major (blue dots) and minor (green triangles) axis of the gyration tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Green triangles close to 90◦, indicate the polar plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' distance range from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 to 20 kpc are located at an angle of more than 70◦ to the direction to the Galactic pole, provided that n clusters be- long to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' If the distribution is isotropic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' n = 0, this probability is equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='1% for the samples of Forbes, Massari and Myeong, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' For this probability to exceed, for example, 10%, the disk must contain n = 6, 16, and 8 GCs for the Forbes, Massari, and Myeong sam- ples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' From this, we can conclude that a part of the GCs, formed to the opinion of these authors outside our Galaxy, actually be- longs to its disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' It should be noted that in [36], based on the analysis of the abundance of alpha– elements, it was shown that the group of Low 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6 S(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 1 10 100 R/kpc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6 S(b/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 1 10 100 R/kpcAAA 80 60 Angle(deg) 40 20 10 100 R/kpc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6 S(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 1 10 100 R/kpo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6 S(b/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 1 10 100 R/kpo90 80 70 60 Angle(deg) 50 40 30 20 10 0 10 100 R/kpc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6 S(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 1 10 100 R/kpc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6 S(b/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 1 10 100 R/kpo90 80 70 60 Angle(deg) 50 40 30 20 10 0 1 10 100 R/kpc7 energy clusters from the Massari work was most likely formed in–situ, which also indicates the in- accuracy of the in–situ/ex–situ separation in the Massari sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In order to verify further the origin of the GCs, we use the “age–metallicity” diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' TWO BRANCHES OF THE GSS IN THE “AGE–METALLICITY” RELATION The literature discusses the fact that the pop- ulation of the GCs of the Milky Way exhibits bimodality of colors: there are blue and red clusters [65–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This is due to the bimodal- ity of metallicity [73, 75, 78–87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Blue clus- ters are found mostly in the halo of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' These clusters probably previously belonged to the satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' At the same time, red clusters are spatially concentrated towards the Galactic center and rotate with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Blue clusters are old and metal-poor, while red clusters are younger and metal–rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The ratio [Fe/H] peaks for blue and red clusters in the Milky Way are approximately –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 and –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Such a bimodality assumes two mechanisms of GCs formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The authors of [76, 77, 80] argue that red clusters are formed in–situ, while blue ones were accreted either as a result of the merger of satellite galaxies with the Galaxy or as a result of tidal capture of the clusters themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To understand the difference between in–situ clusters and ex–situ clusters, we plotted respec- tive samples from Massari, Forbes, and Myeong in the “age–metallicity” diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The “age–metallicity” dependence clearly shows that GCs have two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The low- metals branch contains mainly clusters that be- long to different tidal streams formed by the par- tial destruction of satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The clusters in this sequence show a wide variation in age and metallicity, but there are no clusters less than 6 Gyr old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The clusters of a more metals-rich branch, formed in–situ, also have a scatter in metallicity, but all clusters are more than 11 Gyr old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' It is worth noting that the in–situ clusters were formed not in the Galaxy as we know it, but in its progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In the hierarchical model of the formation of galaxies, the mass of a galaxy is accumulated gradually due to mergers, and galaxies as a whole do not have a clearly defined moment of formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Therefore, for the objects formed long ago, it is difficult to distinguish be- tween the concepts of in–situ and ex–situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' How- ever, specifically for our Galaxy, it is believed that it did not experience mergers with the ob- jects of comparable mass since z = 2 or less than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 Gyr ago [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' By that time, it had gained only 1/5 of its current total mass (including the dark halo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Six Gyr ago (the age of the youngest GCs), its mass was about 60% ofthe current one [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 2: Dependence of the age of GCs on metallicity for cluster samples Forbes, Massari, and Myeong, the left, middle, and right panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Blue dots represent ex–situ clusters and the red ones show in–situ clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' THE ROLE OF THE LOCAL SUPERCLUSTER In the hierarchical model of galaxies forma- tion, accretion of the matter is controlled by large–scalef lows, which are also responsible for the formation of a cellular structure–“pancakes” and filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In accordance with Zeldovich theory [13], a “pancake” is formed from a uni- formly filled volume if compression occurs in one of the three mutually perpendicular direc- tions, and expansion — in two other direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Thus, the large–scale structure is asso- ciated with anisotropic motions of matter, and this anisotropy can also affect the distribution of the matter in the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Our Galaxy, to- gether with the Local Group, is located within the Local Supercluster (LSC) [90–95], a well– visible pancake–like structure with dimensions of tens of Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' We have tested the influence of the Local Su- percluster on the spatial distribution of GCs, as well as dwarf satellite galaxies of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The satellite galaxies were a priori ac- creted onto our Galaxy from the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' At the same time, they form a clear–cut flat structure [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Therefore, we did not limit ourselves to analyzing the GCs distribution, but also consid- ered satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' For this, the angle distri- butions between the axes of the gyration tensor (1) and the plane of the Local Supercluster were obtained for dwarf satellite galaxies (27 satel- lites [96]) and for: (i) for the entire GCs sample (157 GCs [46, 2010 edition], [45]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' (ii) for the GCs from the Forbes list;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' (iii) for the GCs from the Massari list, and (iv) for the GCs from the Myeong list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 3, the “Angle” is presented as a function of galactocentric distance for the GCs and satellites of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The “Angle” is measured between the plane of the Local Super- cluster and the minor (green triangles) or major (blue dots) axes of the distribution of GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 3 shows that for satellite galaxies (top row, left) at the largest distances, the major and minor axes are located in the supergalac- tic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' At the same time, the minor axis is located in the disk of the Milky Way and the major one is perpendicular to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0: [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 - 6 8 10 12 14 16 Age(Gyr)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0: [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0 - 6 8 10 12 14 16 Age(Gyr)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0: [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='0: 6 8 10 12 14 16 Age(Gyr)9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 3: The “Angle” as a function of galactocentric distance for satellite galaxies for entire GCs sample (top row, left to right) and for cluster samples Forbes, Massari, and Myeong (bottom row, left to right, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The “Angle” is measured between the LSC plane and the small (green triangles) or large (blue dots) axes of the distribution of GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' means that the plane of satellite galaxies is per- pendicular to both the disk of the Galaxy and the supergalactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' We can say the following regarding all 157 GCs (top row, on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' At small distances, up to 4 kpc, we are not interested in the result, since these are GCs in the central part of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' From 4 to 20 kpc, the minor axis of the system is perpendicular to the disk of the Galaxy and is in the supergalactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The major axis is in the disk of the Galaxy and is perpendicular to the supergalactic plane at a distance of about 20 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Thus, in the range from 4 to 20 kpc, the orientation of the GC system corresponds to the disk of the Galaxy, the influence of the Supercluster does not manifest itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The same can be said for the clusters from streams, but only with a caveat that they have a larger noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The minor axis shows a large scatter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' this may be due to the fact that there are fewer clusters belonging to the disk of the Galaxy in the sample of objects from the streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' At the distance of about 100 kpc, the picture for GCs resembles that of satellite galaxies for all samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=', the GC system is oriented perpen- dicular both to the disk and the Supercluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' It is worth noting that only six clusters are ob- served at such distances, which is not enough for reliable conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' At a distance of about 30 kpc in all samples except Myeong, the major axis is in the super- galactic plane, while the minor one makes with 80 A 60 Angle(deg) 40 20 0 20 30 50 100 200 R/kpc80 60 Angle(deg) K 40 20 V 10 100 R/kpc80 W V 60 Angle(deg) 40 20 XV 10 100 R/kpc80 △公 60 Angle(deg) 40 20 XV 10 100 R/kpc80 60 Angle(deg) 40 V AA 20 V 1 10 100 R/kpc10 the large axis an angle of about 60◦ for all GCs and for Forbes sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' for the Massari sample, the angle is within 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' At a distance of 25– 40 kpc, there are only 10 GCs, of which 10 and 9 GCs belong to the streams of the Forbes and Massari samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Thus, in the Forbes and Massari samples at these distances, there may be signs of the influence of the Super- cluster on the orientation of the system of ac- creted GCs, but the reliability of this conclusion is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS In this paper, we studied the GC system that was formed outside the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To do this, we took from the literature the samples of GCs that were formed in different tidal streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' We chose the works of Forbes, Massari, and Myeong, since their lists of GCs belonging to the differ- ent streams are the most complete and are based on the latest data from the GAIA observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Having studied a number of works, including those mentioned above, we obtained the main list of tidal streams in which GCs belonged and later were accreted: Sagittarius dwarf spheroidal galaxy (Sgr dSph), Sequoia Galaxy (Sequoia), Helmi Stream (H99), Gaia–Enceladus (possibly Gaia Sausage or CMa), the low energy group (possibly Koala or Kraken), and the High en- ergy group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' It is believed that the accretion onto the Galaxy was anisotropic, which is manifested, for example, as a disk–like structure of satel- lite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' We measured the anisotropy of the distribution of GCs that belonged to the streams using the gyration tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' The measure- ment result showed that no statistically signif- icant anisotropy is observed for accreted GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Having obtained this result, we can state that the anisotropic structure that is observed for the complete sample of GCs (see [12], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 7) is due to the presence of many GCs in the Galactic disk, and is associated with the clusters formed in–situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' However, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 1 for the three samples of the accreted GCs, the major axis of the gyration tensor at a distance from 3 to 20 kpc is in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This may be due to the fact that the sam- ples contain a significant number of GCs that have formed in the disk of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To esti- mate their number, the distribution of GCs with random angular coordinates was modeled and it was shown that the probability of a random re- alization of such a distribution, in which there are no GCs belonging to the disk, is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='6, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='1% for the Forbes, Massari, and Myeong samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' This conclusion is consis- tent with the conclusion of Marsakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [36], who had shown that some of the clusters from the Massari catalog claimed to be ex–situ are in fact genetically related to our Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' We also checked how the clusters formed in– situ and ex–situ behave respective to the AMR (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Two branches can be easily distin- guished;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' the low–metals branch contains mainly 11 clusters belonging to different streams, and they have a large spread in the age and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' At the same time, the clusters in the more metallic branch, which most likely formed in the Galaxy, have a scatter in metallicity, but their age is over 11 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' To check the likely influence of the Local Su- percluster on the distribution of satellite galax- ies and GCs of the Milky Way, we presented the Figures, which show the angle between the LSC plane and the axes of distribution of GCs sys- tems or satellite galaxies, as a function of the galactocentric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 3 (top row, left) shows that the plane of the satellite galaxies is both perpendicular to the disk of the Galaxy and to the supergalactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' For GCs at the distances of up to 20 kpc, only the influence of the Galactic disk is traced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' at the distances of about 30 kpc, the orientation of the GCs sys- tem may coincide with the supergalactic plane, and at larger distances (more than 100 kpc), the orientation resembles that for satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Peebles, Astrophys.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Bastian, Monthly Notices Royal Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 486, 3180 (2019), 1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='05680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Massari, H.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Libe- skind, Monthly Notices Royal Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 481, 918 (2018), 1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='04770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' B.' metadata={'source': 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A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Rosenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Bedin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=', Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 667, 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+page_content=' and Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 527, A81 (2011), 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='2450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [50] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Moni Bidin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Mauro, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Geisler, D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Alonso-Garc´ıa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Borissova, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=', Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' and Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 535, A33 (2011), 1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='1854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [51] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Cantat-Gaudin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Jordi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Vallenari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Bragaglia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Balaguer-N´u˜nez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 618, A93 (2018), 1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='08726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [52] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Gran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Zoccali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Contreras Ramos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Valenti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Rojas-Arriagada, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Carballo- Bello, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Alonso-Garcia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Minniti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Re- jkuba, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Surot, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' and Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 628, A45 (2019), 1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='10872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [53] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Ortolani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Bonatto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Bica, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Barbuy, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 138, 889 (2009), 0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content='1225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' [54] E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Babler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Indebetouw, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Whitney, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Wat- son, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' 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Gilmore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Irwin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Just, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Koposov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Mateo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} +page_content=' Olszewski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQfXx0t/content/2301.08535v1.pdf'} 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Pek¨oz, and Rhonda Righter +University of California, Berkeley Department of Statistics, Boston University +Questrom School of Business, and University of California, Berkeley Department +of Industrial Engineering and Operations Research +Abstract +In Gambler’s Ruin when both players start with the same amount of money, we +show the playing time stochastically increases when the games are made more fair. +We give two different arguments for this fact that extend results from Pek¨oz & Ross +(2021). We then use this to show that the exit time from a symmetric interval for +Brownian motion with drift stochastically increases as the drift moves closer to zero; +this result is not easily obtainable from available explicit formulas for the density. +1 +INTRODUCTION +For a simple random walk Sn = �n +i=1 Xi, where each Xi is +1 with probability p and +−1 with probability 1 − p, we are interested in T |k|, the time until the random walk hits +either k or −k, where k is a specified positive integer. It should be noted that T |k| is the +duration of the Gambler’s Ruin when both players start with k. It was recently shown in +Pek¨oz & Ross (2021) that T |k| for any k is stochastically maximized when p = 1/2, and +previously in Zhang & Ross (2021) where the weaker result in expectation was shown. In +this paper we prove the stronger result that T |k| stochastically increases as p gets closer to +1/2 using two very different arguments. We then use this to show that the exit time from a +symmetric interval for Brownian motion with drift stochastically increases when the drift +moves closer to zero, which appears also to be a new result and not easily obtainable from +available formulas for the exit time probability density function. We also show that, for +all p and k, T |k| is independent of which player wins the game. It should be noted that +the technical challenges involved have left a scarcity of results for Gambler’s Ruin with +more than two players, see Diaconis & Ethier (2021) and the references therein. +The Gambler’s Ruin problem is one of the oldest problems in probability. As told by +Song & Song (2013), computing the chances each player wins was solved by Pascal and +Fermat and appeared with four other problems in the earliest book published on probabil- +ity theory (Huygens (1657)). Study of the distribution of the duration of the game started +with de Moivre (1711) and was taken up in modern times with Feller (1968)[Chapter 14.5], +who derived the formula +P(T |k| = n) =k−12n+1[p(n+k)/2(1 − p)(n−k)/2 + p(n−k)/2(1 − p)(n+k)/2] +× +k−1 +� +j=1 +cosn−1 +�πj +2k +� +sin +�πj +2k +� +sin +�πj +2 +� +1 + +for point probabilities when n − k is even, and Karni (1977), who derived +P(T |k| = n) += +�� n − 1 +1 +2(n − k) +� +− +� +n − 1 +1 +2(n − 3k) +� +− +� n − 1 +1 +2(n + k) +� ++ +� +n − 1 +1 +2(n − 5k) +� ++ +� +n − 1 +1 +2(n + 3k) +�� +× [p(n+k)/2(1 − p)(n−k)/2 + p(n−k)/2(1 − p)(n+k)/2] +in the case where n ⩾ 5k and gave adjustments for k ⩽ n < 5k. +Neither of the expressions above, however, seems amenable to showing that the tail +probabilities +P(T |k| > n) increase (i.e., that T |k| stochastically increases) as p moves +towards 1/2. +The expressions can be easily seen to be increasing as p moves towards +1/2 for sufficiently large n (the derivative with respect to p of either expression has the +same sign as n(1 − 2p)(pk + (1 − p)k) + k(pk − (1 − p)k)), and hence the corresponding +tail probabilities increase too. +The expressions for the point probabilities above can, +however, decrease as p moves towards 1/2 when n is small, and thus not much can be +easily said about tail probabilities in general. This leaves stochastic inequalities difficult +to obtain from these expressions. Our main result, which we state next, bypasses the +use of these expressions and shows that moving p towards 1/2 stochastically increases the +game duration. +Theorem 1.1. +P(T |k| > n) is increasing in p for 0 ⩽ p ⩽ 1/2 and decreasing in p for +1/2 ⩽ p ⩽ 1. +We prove this using two different approaches in the next sections. +2 +A DECOMPOSITION ARGUMENT +Let T0 be the first time (after time 0) the walk revisits state 0. As was done in Pek¨oz & +Ross (2021), we can use the decomposition +T |k| = Z + +N−1 +� +i=1 +Yi +where N ∼ Geometric(P(T0 > T |k|)), N − 1 is distributed as the number of times the +walk returns to 0 before it first hits ±k, Z ∼ (T |k||T0 > T |k|) is the time to go from 0 +to ±k given the walk does not return to 0, and Yi ∼ (T0|T0 < T |k|), i = 1, 2, ... are i.i.d. +random variables representing the times to return to 0 given the walk returns to 0 before +hitting ±k. All of these component random variables are independent. We will show the +following lemma, from which Theorem 1.1 follows immediately. +Lemma 2.1. Yi, Z, and N are all stochastically increasing (decreasing) in p for 0 ⩽ p ⩽ +1/2 (1/2 ⩽ p ⩽ 1). +To prove this lemma, let ui(p) be the chance the walk goes up next when it is at level i +given it returns to 0 before reaching either +k or −k, for i = −k+1, −k+2, ..., k−1. Note +that, by symmetry, ui(p) = 1 − u−i(1 − p) is the chance the walk goes down next from +level −i given it returns to 0 before reaching ±k, where now the probability of the walk +going up is 1 − p. We show in Lemma 2.2 (2) that ui(p) = ui(1 − p), which then implies +that the probability of moving away from 0 given the walk returns to 0 before reaching +±k depends only on the player imbalance, |p − (1 − p)| = 2|p − 1/2|, not on the direction +of imbalance. Moreover, from Lemma 2.2 (3), this probability is decreasing in the player +imbalance. +2 + +Lemma 2.2. The following hold for 0 ⩽ p ⩽ 1: +1. +P(T0 < T |k|) is strictly increasing (decreasing) in p when p < 1/2 (when p > 1/2). +2. ui(p) = ui(1 − p) for −k + 1 ⩽ i ⩽ k − 1. +3. When 1 ⩽ i ⩽ k − 2, ui(p) is strictly increasing (decreasing) in p when p < 1/2 +(when p > 1/2). +Proof of Lemma 2.2. For Part 1, note that any path of length 2n that starts and ends at +0 without first visiting k or −k must contain n upward steps and n downward steps, and +thus its probability equals (p(1 − p))n, which is strictly increasing in p for p < 1/2 and +strictly decreasing in p for p > 1/2. The result for +P(T0 < T |k|) follows, since it is the +sum of the probabilities of many such paths. This gives Part 1. +For Parts 2 and 3, we use the notation +Pi to denote the measure conditioned on starting +the walk at level i. From our observation about the symmetry of ui(p), pick i > 0 without +loss of generality, and let T j(p) be the time to hit level j ⩾ 0. We have +ui(p) += +p Pi+1(T0 < T k) +Pi(T0 < T k) += p Pi+1(T i < T k)Pi(T0 < T k) +Pi(T0 < T k) += +p Pi+1(T i < T k), +which we combine with +Pi+1(T i < T k) = 1 − p + p Pi+2(T i+1 < T k)Pi+1(T i < T k), +to get +ui(p) = p(1 − p) + ui+1(p)ui(p). +(2.1) +For Part 2 we use backward induction. First note the boundary case uk−1(p) = uk−1(1 − +p) = 0. We use the induction hypothesis ui+1(p) = ui+1(1 − p) in the second equality +below and (2.1) in the first equality below to get +ui(p) = +p(1 − p) +1 − ui+1(p) = +p(1 − p) +1 − ui+1(1 − p) = ui(1 − p) +and thus Part 2 holds for all i > 0 and then for all i looking at the walk reflected across +the origin. +For Part 3, we again use backward induction. First note that uk−1(p) = 0 and (2.1) +with i = k −2 gives uk−2(p) = p(1−p), which is increasing in p for p < 1/2 and decreasing +for p > 1/2. Taking the derivative, from equation (2.1) we get +u′ +i(p) = 1 − 2p + u′ +i(p)ui+1(p) + ui(p)u′ +i+1(p) = 1 − 2p + ui(p)u′ +i+1(p) +1 − ui+1(p) +which is positive when p < 1/2 and negative when p > 1/2 using the induction hypothesis +that u′ +i+1(p) > 0 for p < 1/2 and u′ +i+1(p) < 0 for p > 1/2. Thus Part 3 holds. +Proof of Lemma 2.1. Let Yi(p) be the time for the walk to go from level i to 0 given it +hits 0 before it hits ±k, and for fixed p, 0 < i < k. Let Zi(p) be the time for the walk to +3 + +go from level i to ±k given it hits ±k before it hits 0, and for fixed p, 0 < i < k. Then, +from the structure of the random walk, and with I(p) ∼ Bernoulli(p), +Y (j) +=st +I(p)Y1(p) + (1 − I(p))Y−1(p) +Z(j) +=st +I(p)Z1(p) + (1 − I(p))Z−1(p) +Yi(p) +=st +Y−i(1 − p) +Zi(p) +=st +Y−(k−i)(1 − p). +The results for Y (j) and Z will then follow once we show that Yi(p) ⩽st Yi(p′) for 0 ⩽ p ⩽ +p′ ⩽ 1/2 and 0 < i < k, and that Y1(p) =st Y−1(p) and Z1(p) =st Z−1(p) for all p. That +Y1(p) =st Y−1(p) and Z1(p) =st Z−1(p) follows from Part 2 of Lemma 2.2 and the last +two equations above. We will show that Yi(p) ⩽st Yi(p′) by coupling the walks starting +in state i so that Yi(p) ⩽ Yi(p′) with probability 1. Whenever the walks are at different +levels, we let them step independently; since the walks both started at the same level, +they will remain an even number of steps apart and thus after this step can meet but not +cross. When the walks are at the same level, by Part 3 of Lemma 2.2 the chance of a step +up is at least as large for the second walk as it is for the first walk, so we can couple the +steps so the second walk is always no closer to the origin than the first walk. This means +we will have Yi ⩽ Yi(p′) and thus Yi ⩽st Yi(p′). The result for N follows from Part 1 of +Lemma 2.2. +Proof of Theorem 1.1. We can similarly couple T |k|(p′) together with T |k|(p) on the same +probability space (making the dependency on p explicit in the notation), using our decom- +position and Lemma 2.2, so that T |k|(p′) ⩾ T |k| almost surely and the Theorem follows. +3 +AN INDUCTIVE ARGUMENT +We will use induction on k in our second proof of Theorem 1. Consider the following, +alternative, decomposition, in which the time to play a game when the boundary is ±k +(call this a size k game) is a random sum of times to play smaller games. In particular, +we first play a size 1 game, at which time the random walk is at ±1 with respective +probabilities p and 1−p. Then we play a size k −1 game, at the end of which the position +of the random walk will be in the set {−k, −(k − 2), k − 2, k}, depending on whether it +was at +1 or −1 at the end of the first game, and depending on whether it went up or +down by k − 1 steps from there. If the position at the end of the second game is ±(k − 2), +then we play a third game of size 2, and so on, until we finally reach ±k and stop. Let +N be the total number of such games, and let r(n) = +P{N = n|N ⩾ n} be its hazard +rate. +Let y(i) be such that at the end of the i’th game, the possible positions of the +random walk are ±k and ±y(i) ̸= ±k and let d(i) be the size of the i’th game. That +is, let y(1) = d(1) = 1, and, for i = 2, 3..., if y(i − 1) > 0 let d(i) = k − y(i − 1) and +y(i) = |y(i − 1) − d(i)|, and if y(i − 1) = 0 let d(i) = y(i) = 1. Note that for fixed k, y(i) +and d(i) are deterministic sequences. We therefore have, by construction and from the +Markov property of our random walk, the following. +Lemma 3.1. For k > 1, +T |k| = +N +� +i=1 +T |d(i)| +where the T |d(i)|’s are independent. +We note that y(i), d(i), r(n), (and N) all depend on k, but we suppress this dependency +in the notation. +4 + +Let π+ +k (p) = +P(T k < T −k) be the probability that the walk S = {Sn}∞ +n=0, with S0 = 0, +ends at +k before hitting −k, so π+ +1 (p) = p and π+ +k (p) = pk/(pk+(1−p)k), from the classic +Gambler’s Ruin formula (Feller (1968)). Let τ(j) = �j +i=1 T |d(i)||N > j be the time to play +j subgames, conditioned on S having not hit ±k in the first j games (so Sτ(j) = ±y(j)). +Lemma 3.2. For any j, τ(j) is independent of the event {Sτ(j) = +y(j)}, and +P(Sτ(j) = +y(j)) = +py(j) +py(j) + (1 − p)y(j) = π+ +y(j)(p). +Proof. Fix j, Condition on N > j and τ(j) = t, and for ease of notation, let y = y(j). Let +ρ+ be an arbitrary path of S that ends up at +y at time t without hitting ±k before time +t. Then, on path ρ+, S has a total of (t − y)/2 + y upward steps and (t − y)/2 downward +steps, so the probability of path ρ+ is p(t−y)/2(1 − p)(t−y)/2py. Let ρ− be the same path +reflected through the origin (interchanging up and down steps), so the path ρ− ends up +at −y at time t, also without hitting ±k before time t. The probability of path ρ− is +p(t−y)/2(1 − p)(t−y)/2(1 − p)y. Let A(ρ±) be the event that S takes path ρ+ or ρ− before +hitting ±k. Then +P(Sτ(j) += ++y, A(ρ±)|τ(j) = t) += +P(Sτ(j) = +y|A(ρ±), τ(j) = t)P(A(ρ±)|τ(j) = t) += +p(t−y)/2(1 − p)(t−y)/2py +p(t−y)/2(1 − p)(t−y)/2py + p(t−y)/2(1 − p)(t−y)/2(1 − p)y +P(A(ρ±)|τ(j) = t) += +py +py + (1 − p)y +P(A(ρ±)|τ(j) = t). +Summing over all such pairs of reflected paths, ρ+ and ρ−, of length t, we have for each t, +P(Sτ(j) = +y|τ(j) = t) = +py +py + (1 − p)y = π+ +y (p). +Thus, the event that Sτ(j) = +y is independent of τ(j), and +P(Sτ(j) = +y) = π+ +y (p). +A similar argument gives the following corollary. +Corollary 3.3. The random variable T |k|(p) is independent of whether the walk first hits ++k or −k for all p and for any k. +The following are also corollaries of Lemma 3.2. +Corollary 3.4. For 0 ⩽ p ⩽ 1/2, π+ +k (p) is increasing in p, from π+ +k (0) = 0 to π+ +k (1/2) = +1/2. +Corollary 3.5. In the decomposition of Lemma 3.1, for fixed k > 1, N is independent of +the T |d(i)|’s, and, for n = 1, 2, ..., +r(n) = +� π+ +y(n−1)(p)π+ +d(n)(p) + (1 − π+ +y(n−1)(p))(1 − π+ +d(n)(p)) +if y(n − 1) > 0 +0 +if y(n − 1) = 0 +� +. +Proof. That N is independent of the T |d(i)|’s follows from Corollary 3.3, because, for each +subgame, the probability that it is the last game is independent of the time to play the +game. That r(n) = 0 if y(n − 1) = 0 is immediate because then y(n) = d(n) = 1. From +Lemma 3.2, for y(n − 1) > 0 and given N ⩾ n, the probability that the walk is at y(n − 1) +5 + +(respectively, −y(n−1)) at the end of the n−1’st game, i.e., at time τ(n−1), is π+ +y(n−1)(p) +(respectively, 1−π+ +y(n−1)(p)). The probability that the walk hits ±k at the end of the n’th +game given it has not yet hit ±k, is the probability of going up by d(n) from y(n − 1) or +going down by d(n) from −y(n − 1), i.e., +r(n) = +P{N = n|N ⩾ n} = π+ +y(n−1)(p)π+ +d(n)(p) + (1 − π+ +y(n−1)(p))(1 − π+ +d(n)(p)). +Proof of Theorem 1.1. We use induction on k, where the result follows trivially for T |1| ≡ +1. Fix k and assume the result holds for all j = 1, ..., k−1, and therefore for each subgame i +because d(i) < k. That is, assume T |d(i)| is stochastically increasing in p (for 0 ⩽ p ⩽ 1/2). +Then we will have the result for k, i.e., T |k| is stochastically increasing in p, from Lemma +3.1 and the induction hypothesis, as long as N is stochastically increasing in p. We will +show the stronger result, that N gets larger in the hazard rate sense as p increases, i.e., +its hazard rate r(n) is decreasing (nonstrictly) in p for all n. If y(n − 1) = 0 then r(n) = 0 +is trivially decreasing in p. Define, for fixed n such that y(n − 1) ̸= 0, +gn(p) = 1 − r(n) = π+ +y(n−1)(p)(1 − π+ +d(n)(p)) + (1 − π+ +y(n−1)(p))π+ +d(n)(p). +Then, taking the derivative with respect to p and using Corollary 3.4 for 0 ⩽ p ⩽ 1/2, +g′ +n(p) += +π+′ +y(n−1)(p) + π+′ +d(n)(p) − 2π+′ +y(n−1)(p)π+ +d(n)(p) − 2π+ +y(n−1)(p)π+′ +d(n)(p) +⩾ +π+′ +y(n−1)(p) + π+′ +d(n)(p) − π+′ +y(n−1)(p) − π+′ +d(n)(p) = 0. +Remark 3.6. We note that the decomposition can be simplified when k is even by choosing +y(1) = k/2 instead of y(1) = 1: +T |k|(p) = +N +� +i=1 +� +T |k/2| +i ++ ˜T |k/2| +i +� +where T |k/2| +i +=st ˜T |k/2| +i +=st T |k/2|, and N, T |k/2| +i +, and ˜T |k/2| +i +for all i are independent, and +N ∼ Geometric((π+ +k/2(p))2 + (1 − π+ +k/2(p))2). +4 +A COROLLARY FOR BROWNIAN MOTION WITH DRIFT +Letting B(t) denote standard Brownian motion at time t, for k > 0 consider the time +Tµ = inf{t ⩾ 0 : µt + B(t) /∈ [−k, k]} +that Brownian motion with drift µ first exits the interval [−k, k]. Borodin, A. & Salminen, +P. (2002)[p. 309 equation (3.0.2), and p. 641] gives the probability density function for +Tµ as +P(Tµ ∈ dt) = (e−µk + eµk)e−µ2t/2 +∞ +� +i=−∞ +k + 4ik +√ +2πt3/2 e−(k+4ik)2/(2t)dt. +(4.1) +Using weak convergence of the paths of asymmetric simple random walk to Brownian +motion with drift, we obtain the following immediate corollary from Theorem 1.1. +Corollary 4.1. With the definitions above, if 0 ⩽ µ ⩽ µ′ then Tµ ⩾st Tµ′. +This result shows that the exit time from a symmetric interval for Brownian motion +with drift is stochastically increased as the drift is moved towards zero. In particular, it +is stochastically maximized when there is no drift. As far as we can tell, this result is new +and does not seem easily obtainable from the explicit form (4.1) of the density. +6 + +5 +ACKNOWLEDGEMENTS +We greatly appreciate the feedback from the referees, which improved the presentation. +REFERENCES +Andrei N. Borodin & Paavo Salminen. Handbook of Brownian Motion - Facts and Formu- +lae, Second Edition, Springer 2002. +P. +Diaconis +& +S. +N. +Ethier. +Gambler’s +Ruin +and +the +ICM, +preprint +on +https://arxiv.org/abs/2011.07610 +Feller, W. (1968). An introduction to probability theory and its applications. Vol. I. New +York: John Wiley & Sons Inc. +Huygens, C. (1657). De Ratiociniis in Ludo Aleae, printed in Exercitationum Mathemati- +carum by F. van Schooten, Elsevirii, Leiden. Reprinted in Oeuvres, 14, (1920). +Karni, E. (1977). The Probability Distribution of the Duration of the Game in the Classical +Ruin Problem. Journal of Applied Probability, 14(2), 416-420. +Moivre, A. de (1711). De Mensura Sortis, seu, de Probabilitate Eventuum in Ludis a Casu +Fortuito Pendentibus, Philosophical Transactions, 27, 213–264. +Pek¨oz, E. & Ross, S., Fair Gambler’s Ruin stochastically maximizes playing time. Accepted +to appear, Advances in Applied Probability (2022). +Ross, Sheldon M. (2019). Introduction to Probability Models, 12th Edition. Amsterdam: +Academic Press. Print. +Song, Seongjoo, & Song, Jongwoo. (2013). A Note on the History of the Gambler’s Ruin +Problem. Communications for Statistical Applications and Methods, 20(2), 157–168. +Zhang, Z., and Ross, S. (2021). Finding the Best Dueler, Preprint. +7 + diff --git a/ctE_T4oBgHgl3EQf0RyG/content/tmp_files/load_file.txt b/ctE_T4oBgHgl3EQf0RyG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c36ad937989423beb3a16e96ba4cbdf7134893b --- /dev/null +++ b/ctE_T4oBgHgl3EQf0RyG/content/tmp_files/load_file.txt @@ -0,0 +1,243 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf,len=242 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='08328v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='PR] 19 Jan 2023 INCREASING GAMBLER’S RUIN DURATION AND BROWNIAN MOTION EXIT TIMES Steven Evans, Erol A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Pek¨oz, and Rhonda Righter University of California, Berkeley Department of Statistics, Boston University Questrom School of Business, and University of California, Berkeley Department of Industrial Engineering and Operations Research Abstract In Gambler’s Ruin when both players start with the same amount of money, we show the playing time stochastically increases when the games are made more fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We give two different arguments for this fact that extend results from Pek¨oz & Ross (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We then use this to show that the exit time from a symmetric interval for Brownian motion with drift stochastically increases as the drift moves closer to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' this result is not easily obtainable from available explicit formulas for the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 1 INTRODUCTION For a simple random walk Sn = �n i=1 Xi, where each Xi is +1 with probability p and −1 with probability 1 − p, we are interested in T |k|, the time until the random walk hits either k or −k, where k is a specified positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' It should be noted that T |k| is the duration of the Gambler’s Ruin when both players start with k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' It was recently shown in Pek¨oz & Ross (2021) that T |k| for any k is stochastically maximized when p = 1/2, and previously in Zhang & Ross (2021) where the weaker result in expectation was shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' In this paper we prove the stronger result that T |k| stochastically increases as p gets closer to 1/2 using two very different arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We then use this to show that the exit time from a symmetric interval for Brownian motion with drift stochastically increases when the drift moves closer to zero, which appears also to be a new result and not easily obtainable from available formulas for the exit time probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We also show that, for all p and k, T |k| is independent of which player wins the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' It should be noted that the technical challenges involved have left a scarcity of results for Gambler’s Ruin with more than two players, see Diaconis & Ethier (2021) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The Gambler’s Ruin problem is one of the oldest problems in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' As told by Song & Song (2013), computing the chances each player wins was solved by Pascal and Fermat and appeared with four other problems in the earliest book published on probabil- ity theory (Huygens (1657)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Study of the distribution of the duration of the game started with de Moivre (1711) and was taken up in modern times with Feller (1968)[Chapter 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' who derived the formula P(T |k| = n) =k−12n+1[p(n+k)/2(1 − p)(n−k)/2 + p(n−k)/2(1 − p)(n+k)/2] × k−1 � j=1 cosn−1 �πj 2k � sin �πj 2k � sin �πj 2 � 1 for point probabilities when n − k is even,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' and Karni (1977),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' who derived P(T |k| = n) = �� n − 1 1 2(n − k) � − � n − 1 1 2(n − 3k) � − � n − 1 1 2(n + k) � + � n − 1 1 2(n − 5k) � + � n − 1 1 2(n + 3k) �� × [p(n+k)/2(1 − p)(n−k)/2 + p(n−k)/2(1 − p)(n+k)/2] in the case where n ⩾ 5k and gave adjustments for k ⩽ n < 5k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Neither of the expressions above, however, seems amenable to showing that the tail probabilities P(T |k| > n) increase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', that T |k| stochastically increases) as p moves towards 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The expressions can be easily seen to be increasing as p moves towards 1/2 for sufficiently large n (the derivative with respect to p of either expression has the same sign as n(1 − 2p)(pk + (1 − p)k) + k(pk − (1 − p)k)), and hence the corresponding tail probabilities increase too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The expressions for the point probabilities above can, however, decrease as p moves towards 1/2 when n is small, and thus not much can be easily said about tail probabilities in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' This leaves stochastic inequalities difficult to obtain from these expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Our main result, which we state next, bypasses the use of these expressions and shows that moving p towards 1/2 stochastically increases the game duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' P(T |k| > n) is increasing in p for 0 ⩽ p ⩽ 1/2 and decreasing in p for 1/2 ⩽ p ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We prove this using two different approaches in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 2 A DECOMPOSITION ARGUMENT Let T0 be the first time (after time 0) the walk revisits state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' As was done in Pek¨oz & Ross (2021), we can use the decomposition T |k| = Z + N−1 � i=1 Yi where N ∼ Geometric(P(T0 > T |k|)), N − 1 is distributed as the number of times the walk returns to 0 before it first hits ±k, Z ∼ (T |k||T0 > T |k|) is the time to go from 0 to ±k given the walk does not return to 0, and Yi ∼ (T0|T0 < T |k|), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' random variables representing the times to return to 0 given the walk returns to 0 before hitting ±k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' All of these component random variables are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We will show the following lemma, from which Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1 follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Yi, Z, and N are all stochastically increasing (decreasing) in p for 0 ⩽ p ⩽ 1/2 (1/2 ⩽ p ⩽ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' To prove this lemma, let ui(p) be the chance the walk goes up next when it is at level i given it returns to 0 before reaching either +k or −k, for i = −k+1, −k+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Note that, by symmetry, ui(p) = 1 − u−i(1 − p) is the chance the walk goes down next from level −i given it returns to 0 before reaching ±k, where now the probability of the walk going up is 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We show in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2 (2) that ui(p) = ui(1 − p), which then implies that the probability of moving away from 0 given the walk returns to 0 before reaching ±k depends only on the player imbalance, |p − (1 − p)| = 2|p − 1/2|, not on the direction of imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Moreover, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2 (3), this probability is decreasing in the player imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 2 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The following hold for 0 ⩽ p ⩽ 1: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' P(T0 < T |k|) is strictly increasing (decreasing) in p when p < 1/2 (when p > 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' ui(p) = ui(1 − p) for −k + 1 ⩽ i ⩽ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' When 1 ⩽ i ⩽ k − 2, ui(p) is strictly increasing (decreasing) in p when p < 1/2 (when p > 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' For Part 1, note that any path of length 2n that starts and ends at 0 without first visiting k or −k must contain n upward steps and n downward steps, and thus its probability equals (p(1 − p))n, which is strictly increasing in p for p < 1/2 and strictly decreasing in p for p > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The result for P(T0 < T |k|) follows, since it is the sum of the probabilities of many such paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' This gives Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' For Parts 2 and 3, we use the notation Pi to denote the measure conditioned on starting the walk at level i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' From our observation about the symmetry of ui(p), pick i > 0 without loss of generality, and let T j(p) be the time to hit level j ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We have ui(p) = p Pi+1(T0 < T k) Pi(T0 < T k) = p Pi+1(T i < T k)Pi(T0 < T k) Pi(T0 < T k) = p Pi+1(T i < T k), which we combine with Pi+1(T i < T k) = 1 − p + p Pi+2(T i+1 < T k)Pi+1(T i < T k), to get ui(p) = p(1 − p) + ui+1(p)ui(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1) For Part 2 we use backward induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' First note the boundary case uk−1(p) = uk−1(1 − p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We use the induction hypothesis ui+1(p) = ui+1(1 − p) in the second equality below and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1) in the first equality below to get ui(p) = p(1 − p) 1 − ui+1(p) = p(1 − p) 1 − ui+1(1 − p) = ui(1 − p) and thus Part 2 holds for all i > 0 and then for all i looking at the walk reflected across the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' For Part 3, we again use backward induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' First note that uk−1(p) = 0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1) with i = k −2 gives uk−2(p) = p(1−p), which is increasing in p for p < 1/2 and decreasing for p > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Taking the derivative, from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1) we get u′ i(p) = 1 − 2p + u′ i(p)ui+1(p) + ui(p)u′ i+1(p) = 1 − 2p + ui(p)u′ i+1(p) 1 − ui+1(p) which is positive when p < 1/2 and negative when p > 1/2 using the induction hypothesis that u′ i+1(p) > 0 for p < 1/2 and u′ i+1(p) < 0 for p > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Thus Part 3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Let Yi(p) be the time for the walk to go from level i to 0 given it hits 0 before it hits ±k, and for fixed p, 0 < i < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Let Zi(p) be the time for the walk to 3 go from level i to ±k given it hits ±k before it hits 0, and for fixed p, 0 < i < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Then, from the structure of the random walk, and with I(p) ∼ Bernoulli(p), Y (j) =st I(p)Y1(p) + (1 − I(p))Y−1(p) Z(j) =st I(p)Z1(p) + (1 − I(p))Z−1(p) Yi(p) =st Y−i(1 − p) Zi(p) =st Y−(k−i)(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The results for Y (j) and Z will then follow once we show that Yi(p) ⩽st Yi(p′) for 0 ⩽ p ⩽ p′ ⩽ 1/2 and 0 < i < k, and that Y1(p) =st Y−1(p) and Z1(p) =st Z−1(p) for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' That Y1(p) =st Y−1(p) and Z1(p) =st Z−1(p) follows from Part 2 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2 and the last two equations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We will show that Yi(p) ⩽st Yi(p′) by coupling the walks starting in state i so that Yi(p) ⩽ Yi(p′) with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Whenever the walks are at different levels, we let them step independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' since the walks both started at the same level, they will remain an even number of steps apart and thus after this step can meet but not cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' When the walks are at the same level, by Part 3 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2 the chance of a step up is at least as large for the second walk as it is for the first walk, so we can couple the steps so the second walk is always no closer to the origin than the first walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' This means we will have Yi ⩽ Yi(p′) and thus Yi ⩽st Yi(p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The result for N follows from Part 1 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We can similarly couple T |k|(p′) together with T |k|(p) on the same probability space (making the dependency on p explicit in the notation), using our decom- position and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2, so that T |k|(p′) ⩾ T |k| almost surely and the Theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 3 AN INDUCTIVE ARGUMENT We will use induction on k in our second proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Consider the following, alternative, decomposition, in which the time to play a game when the boundary is ±k (call this a size k game) is a random sum of times to play smaller games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' In particular, we first play a size 1 game, at which time the random walk is at ±1 with respective probabilities p and 1−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Then we play a size k −1 game, at the end of which the position of the random walk will be in the set {−k, −(k − 2), k − 2, k}, depending on whether it was at +1 or −1 at the end of the first game, and depending on whether it went up or down by k − 1 steps from there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' If the position at the end of the second game is ±(k − 2), then we play a third game of size 2, and so on, until we finally reach ±k and stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Let N be the total number of such games, and let r(n) = P{N = n|N ⩾ n} be its hazard rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Let y(i) be such that at the end of the i’th game, the possible positions of the random walk are ±k and ±y(i) ̸= ±k and let d(i) be the size of the i’th game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' That is, let y(1) = d(1) = 1, and, for i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', if y(i − 1) > 0 let d(i) = k − y(i − 1) and y(i) = |y(i − 1) − d(i)|, and if y(i − 1) = 0 let d(i) = y(i) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Note that for fixed k, y(i) and d(i) are deterministic sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We therefore have, by construction and from the Markov property of our random walk, the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' For k > 1, T |k| = N � i=1 T |d(i)| where the T |d(i)|’s are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We note that y(i), d(i), r(n), (and N) all depend on k, but we suppress this dependency in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 4 Let π+ k (p) = P(T k < T −k) be the probability that the walk S = {Sn}∞ n=0, with S0 = 0, ends at +k before hitting −k, so π+ 1 (p) = p and π+ k (p) = pk/(pk+(1−p)k), from the classic Gambler’s Ruin formula (Feller (1968)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Let τ(j) = �j i=1 T |d(i)||N > j be the time to play j subgames, conditioned on S having not hit ±k in the first j games (so Sτ(j) = ±y(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' For any j, τ(j) is independent of the event {Sτ(j) = +y(j)}, and P(Sτ(j) = +y(j)) = py(j) py(j) + (1 − p)y(j) = π+ y(j)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Fix j, Condition on N > j and τ(j) = t, and for ease of notation, let y = y(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Let ρ+ be an arbitrary path of S that ends up at +y at time t without hitting ±k before time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Then, on path ρ+, S has a total of (t − y)/2 + y upward steps and (t − y)/2 downward steps, so the probability of path ρ+ is p(t−y)/2(1 − p)(t−y)/2py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Let ρ− be the same path reflected through the origin (interchanging up and down steps), so the path ρ− ends up at −y at time t, also without hitting ±k before time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The probability of path ρ− is p(t−y)/2(1 − p)(t−y)/2(1 − p)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Let A(ρ±) be the event that S takes path ρ+ or ρ− before hitting ±k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Then P(Sτ(j) = +y, A(ρ±)|τ(j) = t) = P(Sτ(j) = +y|A(ρ±), τ(j) = t)P(A(ρ±)|τ(j) = t) = p(t−y)/2(1 − p)(t−y)/2py p(t−y)/2(1 − p)(t−y)/2py + p(t−y)/2(1 − p)(t−y)/2(1 − p)y P(A(ρ±)|τ(j) = t) = py py + (1 − p)y P(A(ρ±)|τ(j) = t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Summing over all such pairs of reflected paths, ρ+ and ρ−, of length t, we have for each t, P(Sτ(j) = +y|τ(j) = t) = py py + (1 − p)y = π+ y (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Thus, the event that Sτ(j) = +y is independent of τ(j), and P(Sτ(j) = +y) = π+ y (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' A similar argument gives the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The random variable T |k|(p) is independent of whether the walk first hits +k or −k for all p and for any k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The following are also corollaries of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' For 0 ⩽ p ⩽ 1/2, π+ k (p) is increasing in p, from π+ k (0) = 0 to π+ k (1/2) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' In the decomposition of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1, for fixed k > 1, N is independent of the T |d(i)|’s, and, for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', r(n) = � π+ y(n−1)(p)π+ d(n)(p) + (1 − π+ y(n−1)(p))(1 − π+ d(n)(p)) if y(n − 1) > 0 0 if y(n − 1) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' That N is independent of the T |d(i)|’s follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='3, because, for each subgame, the probability that it is the last game is independent of the time to play the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' That r(n) = 0 if y(n − 1) = 0 is immediate because then y(n) = d(n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2, for y(n − 1) > 0 and given N ⩾ n, the probability that the walk is at y(n − 1) 5 (respectively, −y(n−1)) at the end of the n−1’st game, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', at time τ(n−1), is π+ y(n−1)(p) (respectively, 1−π+ y(n−1)(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' The probability that the walk hits ±k at the end of the n’th game given it has not yet hit ±k, is the probability of going up by d(n) from y(n − 1) or going down by d(n) from −y(n − 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', r(n) = P{N = n|N ⩾ n} = π+ y(n−1)(p)π+ d(n)(p) + (1 − π+ y(n−1)(p))(1 − π+ d(n)(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We use induction on k, where the result follows trivially for T |1| ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Fix k and assume the result holds for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', k−1, and therefore for each subgame i because d(i) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' That is, assume T |d(i)| is stochastically increasing in p (for 0 ⩽ p ⩽ 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Then we will have the result for k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', T |k| is stochastically increasing in p, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1 and the induction hypothesis, as long as N is stochastically increasing in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We will show the stronger result, that N gets larger in the hazard rate sense as p increases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', its hazard rate r(n) is decreasing (nonstrictly) in p for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' If y(n − 1) = 0 then r(n) = 0 is trivially decreasing in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Define, for fixed n such that y(n − 1) ̸= 0, gn(p) = 1 − r(n) = π+ y(n−1)(p)(1 − π+ d(n)(p)) + (1 − π+ y(n−1)(p))π+ d(n)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Then, taking the derivative with respect to p and using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='4 for 0 ⩽ p ⩽ 1/2, g′ n(p) = π+′ y(n−1)(p) + π+′ d(n)(p) − 2π+′ y(n−1)(p)π+ d(n)(p) − 2π+ y(n−1)(p)π+′ d(n)(p) ⩾ π+′ y(n−1)(p) + π+′ d(n)(p) − π+′ y(n−1)(p) − π+′ d(n)(p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' We note that the decomposition can be simplified when k is even by choosing y(1) = k/2 instead of y(1) = 1: T |k|(p) = N � i=1 � T |k/2| i + ˜T |k/2| i � where T |k/2| i =st ˜T |k/2| i =st T |k/2|, and N, T |k/2| i , and ˜T |k/2| i for all i are independent, and N ∼ Geometric((π+ k/2(p))2 + (1 − π+ k/2(p))2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 4 A COROLLARY FOR BROWNIAN MOTION WITH DRIFT Letting B(t) denote standard Brownian motion at time t, for k > 0 consider the time Tµ = inf{t ⩾ 0 : µt + B(t) /∈ [−k, k]} that Brownian motion with drift µ first exits the interval [−k, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Borodin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' & Salminen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' (2002)[p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 309 equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='2), and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 641] gives the probability density function for Tµ as P(Tµ ∈ dt) = (e−µk + eµk)e−µ2t/2 ∞ � i=−∞ k + 4ik √ 2πt3/2 e−(k+4ik)2/(2t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1) Using weak convergence of the paths of asymmetric simple random walk to Brownian motion with drift, we obtain the following immediate corollary from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' With the definitions above, if 0 ⩽ µ ⩽ µ′ then Tµ ⩾st Tµ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' This result shows that the exit time from a symmetric interval for Brownian motion with drift is stochastically increased as the drift is moved towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' In particular, it is stochastically maximized when there is no drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' As far as we can tell, this result is new and does not seem easily obtainable from the explicit form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content='1) of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 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157–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=', and Ross, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' Finding the Best Dueler, Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE_T4oBgHgl3EQf0RyG/content/2301.08328v1.pdf'} diff --git a/dNE1T4oBgHgl3EQfxwWp/content/tmp_files/2301.03426v1.pdf.txt b/dNE1T4oBgHgl3EQfxwWp/content/tmp_files/2301.03426v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..275a593c158de60b70a714512673476dad906d09 --- /dev/null +++ b/dNE1T4oBgHgl3EQfxwWp/content/tmp_files/2301.03426v1.pdf.txt @@ -0,0 +1,1056 @@ +End-to-end Unsupervised Learning of Long-Term 3D Stable objects +Ibrahim Hroob∗, Sergi Molina, Riccardo Polvara, Grzegorz Cielniak and Marc Hanheide +Abstract— 3D point cloud semantic classification is an im- +portant task in robotics as it enables a better understanding +of the mapped environment. This work proposes to learn the +long-term stability of the 3D objects using a neural network +based on PointNet++, where the long-term stable object refers +to a static object that cannot move on its own (e.g. tree, pole, +building). The training data is generated in an unsupervised +manner by assigning a continuous label to individual points by +exploiting multiple time slices of the same environment. Instead +of using discrete labels, i.e. static/dynamic, we propose to use +a continuous label value indicating point temporal stability +to train a regression PointNet++ network. We evaluated our +approach on point cloud data of two parking lots from the +NCLT dataset. The experiments’ performance reveals that static +vs dynamic object classification is best performed by training +a regression model, followed by thresholding, compared to +directly training a classification model. +I. INTRODUCTION +3D point cloud maps are one of the standard map formats +used for vehicle localization [1], [2]; such maps represent +a snapshot of the static environment around the mobile +robot at the time of acquiring the scans. We are interested +in a learning-based approach for detecting potential long- +term stable objects that are invariant across time in the +3D point cloud maps. The importance of such objects is +that they enable robust localization over an extended time, +also known as long-term localization. However, the raw +maps are unsuitable for long-term operations and may cause +performance degradation in pose estimation if not a complete +failure in the localization system in the long-term [3]. That +is mainly due to the following two reasons: (i) capturing +dynamic objects as static, e.g. parked cars, and (ii) the raw +maps suffer from the ”flying ghost” artefacts [4], which is +caused by a moving object while recording the data, e.g. +pedestrians or cars in motion. +Many solutions are proposed to detect moving objects in +point cloud maps based on geometrical methods, as in [1], +[2], [4], [5], or based on deep learning methods. The latter +can achieve dense full class segmentation [6], [7], [8], which +could be used to infer and detect dynamic objects. Thus, +one could directly exploit the object semantics in the map to +segment dynamic objects and identify the potential long-term +stable objects. +The issue with achieving full class semantics is that they +rely heavily on the supervised annotated training data, which +is not always available and expensive to generate. However, +*Corresponding author: ihroob@lincoln.ac.uk +All the authors are within the Lincoln Centre for Autonomous Systems +(LCAS), University of Lincoln, Lincoln, UK, LN6 7TS. This work has +been supported by the European Commission as part of H2020 under grant +number 871704 (BACCHUS). +Fig. 1: Overview of our proposed point-to-point regression +model for estimating point-wise long-term stability in a 3D +point cloud map. The input is a set of points X ∈ R3, with +the estimated 3D surface normal’s Nx. The output is a point- +wise stability estimation score l ∈ R1 bounded between +[0, 1]. The data-loader block is responsible for dividing the +input map into smaller sub-maps to feed them into the +network. The green area is the sampling submap, which +moves with a fixed grid in the x and y directions. +for some tasks, as in static/dynamic segmentation, an entire +class annotation may not be required. Therefore, several +solutions are presented to address this binary segmentation +problem in an unsupervised data-driven manner [9], [10], +[11], [12]. The main problem with the unsupervised and +geometrical approaches is that they only segment dynamic +objects in the current scene; thus, they can not infer the long- +term stable objects since they do not explore the history of +the environment. For instance, a parked car will be classified +as static, which is not valid in the long-term perspective. +To tackle these problems, we propose an end-to-end unsu- +pervised learning approach that can learn the local geometry +of the long-term stable objects on the raw 3D point cloud +maps. The output of our approach is a point-wise temporal +stability score, where higher values indicate that the point +belongs to a dynamic object (e.g. car, bike, pedestrian), +and lower values represent the most stable points from +a long-term perspective (e.g. tree trunks, pole, building). +Since we are interested in an unsupervised learning method, +we propose an automatic labelling algorithm to generate +training data with a point-wise stability score by exploit- +ing different time instances of the environment. Regarding +the learning step, we propose a regression neural network +taking advantage of the pioneering work of the hierarchical +PointNet++[13]. +arXiv:2301.03426v1 [cs.CV] 9 Jan 2023 + +Input: N,(d + C) +Output: N,l +"flying ghost"' artefacts +Dynamic +Point-wise stability +score represented as +a heat-map +Parked cars +Static + Submap size +x gria +Optional +Overlapping area +layer (mean) +PointNet++ +Encoder +PointNet++ +decoder +Sigmoid +Skip +activation +connections +layer +Per-point binary +Data loader +PointNet++ regression +classification +NetworkIn summary, the main contributions of this paper are +three-fold: (i) an unsupervised automatic labelling algorithm +exploiting long-term observations for a given environment, +(ii) a regression network based on PointNet++ for point-wise +long-term stability score estimation (see Fig. 1) , and (iii) +a comprehensive evaluation of the proposed auto-labelling +algorithm and the regression neural network using real-world +data, where the results demonstrate the effectiveness and +the convenience of the proposed approach since it does not +require manual annotation. +II. RELATED WORK +Static/dynamic object segmentation is an active research +area with methods broadly classified into geometry-based +and deep learning approaches. +Geometrical approaches are based on: motion cues [2], +[14], [15], ray tracing [4], [16] or voxel traversal [5]. Motion +cues (visibility-based) approaches identify dynamic points +by comparing the current laser scan with the previous scans. +For instance, Pomerleau et al. [14] infer the dynamic part +of a scene by comparing the incoming scan with a global +map based on visibility assumptions; that is, if a point +is observed behind a previously seen point, the old point +might belong to a dynamic object. Ray tracing methods +rely on shooting rays and checking for occlusions. Usually, +those methods are computationally expensive and run offline. +An example of a method using this strategy is OctoMap +[16], which is a probabilistic 3D mapping framework based +on the octree data structure. The nodes in OctoMap store +an occupancy probability p that indicates if the node is +free, occupied, or unknown. The likelihood of the nodes is +repeatedly updated for each scan using ray-tracing; thus, this +method can naturally filter out dynamic voxels by updating +their probability score. The peopleremover [5] filters the +dynamic points using a voxel grid instead of octree to store +the identifier of all laser rays that hit the voxel. It uses +voxel traversal [17] to update voxel occupancy instead of +ray tracing. The fully built occupancy grid could be used as +a Static/Dynamic binary classifier to filter the points from +the actual point cloud map. +Deep learning approaches could be either supervised or +unsupervised. Supervised methods [7], [8], [18], [19], [20], +[21], can achieve full classes semantic segmentation, which +one could use directly to detect all long-term stable instances. +However, such methods currently rely heavily on hand- +annotated data and are prone to human error or unknown +classes [22]. While on the other hand, unsupervised methods +are a more interesting choice for learning object semantics, +usually in the form of dynamic or static binary objects; those +methods are data-driven methods that require minimal or +no supervision. For example, [23] segment the indoor scene +into foreground and background classes, the segmentation is +performed w.r.t the floor plan, so any point that does not +match with the floor plan is labelled as dynamic, else static. +Those labels are used in a neural network model to improve +agent localization. Recently, scene flow [11], [24], [25], [26] +approaches are being applied to point cloud directly in an +unsupervised way to label the points into moving or rigid +objects between lidar frames. Those methods are paired with +a deep neural network as in [27], [28], [29] for an end-to-end +object semantic estimation. +However, most of the methods mentioned above require +motion information to infer dynamic objects; therefore, they +cannot detect objects that can potentially move but are static +in the current observation, for instance, a parked car. In +contrast to other works, our approach is more focused on +identifying the long-term stable objects in a given environ- +ment, as those objects are a key landmark to guarantee long- +term localization without degradation in performance [30]. +Our method is unsupervised and does not require human +input as it implicitly learns the long-term stable objects in an +environment by exploiting previous temporal observations. +III. PROPOSED METHOD +Our proposed method aims at classifying static and dy- +namic objects in a given environment from a 3D point cloud +map by considering the long-term perspective. The input to +the unsupervised labelling algorithm is a set of observations +O0:k, where Oi is a set of points in the 3D Euclidean metric +space {x0, x1, ..., xn} ∈ R3, with their estimated normal’s +{n0, n1, ..., nn}. The output is a set of point-wise labelled +maps ML +0:k, where the label value is continuous indicating +point long-term stability bounded between [0, 1], where a +lower score close to 0 means the point belongs to a long- +term stable object like a tree trunk or light post. While higher +values close to 1 indicate that the point may belong to a +dynamic object such as pedestrians or cars. The values in +between represent the slow dynamics in the environment as +in seasonal changes. The following subsections explain how +the point cloud maps are labelled. Then we detail the network +architecture used for point-wise stability estimation. +A. Unsupervised data labelling +In this work, we are interested in labelling the point cloud +in an unsupervised fashion. Instead of using discrete labels, +we assign a continuous value to the points ranging between +[0, 1], indicating points’ temporal stability by exploring the +spatio-temporal dependency for a point across different ob- +servations of the environment. The motivation behind using +continuous values is to remove any bias when classifying +the points into static/dynamic on the unsupervised labels, +i.e. having the threshold value for classification, which means +fewer parameters to tune. Furthermore, the continuous labels +are more suitable to indicate slowly moving objects because +they are difficult to label as static or dynamic (i.e. they +can better utilize the 3D spatial information in the point +cloud data); thus, this leads to better learning by the neural +network. We detail the proposed automatic labelling method +in the following subsections (the entire labelling pipeline is +illustrated in Fig. 2). +Data pre-processing: +The input to this step is the raw +observations of the environment in which we denote them +as O0:k. The pre-processing output is a set of filtered and +aligned maps w.r.t the first map denoted as M0:k. Our + +Fig. 2: The unsupervised data labelling pipeline has two main parts: the data pre-processing block that filters the raw +observations O0:k and aligns them w.r.t the first map. Then the annotation block generates a point-wise stability score for +each map by exploiting other maps. +pre-processing data pipeline has three main steps. First, +for each observation Oi, the ground plane is segmented +and removed using the Cloth Simulation Filtering (CSF) +algorithm [4]. The motivation behind removing the ground +plane is: (i) the ground plane is stable, (ii) to increase the +points label disparity when extracting the distance to the +nearest neighbour point in other observations/maps at later +stages, and (iii) to reduce the overall map size (during our +experiments we found that the number of points that belongs +to the ground plane is roughly between 30 − 50%, which +helps to reduce the computational time at later stages). The +second step is to remove the outliers using the Statistical +Outlier Removal (SOR) Filter [31]. In the final step, the maps +are registered w.r.t the first map using Iterative Closest Point +[32]. This step is essential to ensure robust data association +between spatio-temporal points across all observations. By +the end of this pre-processing data pipeline, we end up with +a set of clean and aligned maps denoted M0:k. +Automatic labelling: The labelling algorithm (shown in +Algorithm 1) assigns a spatio-temporal stability label for +each point in a given reference map Mref by exploiting +the distances to the nearest neighbour points (KNN) using a +multidimensional binary search tree (kd-tree) w.r.t all other +time slices of the environment M0:k. The motivation behind +using KNN is that the long-term stable object, for instance, +will remain in the same place in all snap-shots of the +environment; thus, the distance to the nearest neighbour point +is small in all-time instances, while on the other hand, a +dynamic object may not appear in the same place in all +maps. Therefore the associated distance for each point in +the dynamic object will be higher than the stable object. +The algorithm works as follows: it iterates across all maps, +for each selected map mRef, it go through all the points, +where for each point p it finds the euclidean distance +d = +��3 +i=1(qi − pi)2 to the closest point q on other +map mOther. d is the distance vector w.r.t all other maps. +Finally, the feature used to represent the point stability is +the maximum distance dmax in d, however dmax is not +bounded i.e. dmax ∈ [0, ∞); to bound the label value +between [0, 1] we used the Cumulative Distribution Function +of an exponential function: +f(dmax, λ) = 1 − e−λdmax, +(1) +Where λ is a hyper-parameter that controls the sensitivity +of the function, after mapping the labels using Eq. 1, the +dynamic objects are squeezed to 1. +Algorithm 1 Unsupervised point-wise labelling algorithm +Input: A set of filtered and aligned maps M0:k +for each mRef ∈ M0:k do ▷ Set the current map mRef +as the reference map +for each p ∈ mRef do +Initialize closest distance vector: d = {} +for each mOther ∈ M0:k do +if mOther is not mRef then +q ← KNN(mOther, p) +d.insert( +��3 +i=1(qi − pi)2) +end if +end for +Point label: p.l = 1 − e−λ. max(d) +end for +end for +Output: A set of labeled maps ML +0:k +B. Network +Our work aims to learn the local geometry of the stable +objects from the unsupervised labelled point cloud generated +using Algorithm 1. We exploit the hierarchical PointNet +(PointNet++) developed by Qi et al. [13]. PointNet++ is +a deep hierarchical neural network that can learn feature +geometry at different levels of abstraction. +In this work, instead of training the network as a +static/dynamic binary classifier, we train it as a regression +network on the continuous stability label value to better learn +the spatial-temporal information in the point cloud. Then + +00 +Mo +Mb +0 +0 +Ground +01 +Removal +M1 +Mi +(CSF) +Mo +&& +registration +Outliers +: +filter +. +. +(SOR) +ML +M,toMofeatures +Ok +Mk +Static +Mk +M. +association +Dynamic +SetM,asareference +Raw +Filter +ICP maps alignment +Processed +Labelled +Autolabellingalgorithm +observations +w.r.t the first map +maps +mapsat the inference stage, we use an optimal threshold value +ϵ to get the binary classification of the environment, where +this insight was not previously known in the literature for +the problem of static vs dynamic object classification. The +network is illustrated in Fig. 1. Similar to the original Point- +Net++ segmentation network, we used a set of abstraction +layers to extract the local and global features from the point +cloud and the same number of feature propagation layers. In +our implementation, we used 5 levels of abstraction layers. +The input number of points to each layer is as follows: +N1 : 1024, N2 : 512, N3 : 256, N4 : 128 and N5 : 32 +with the following sampling radiuses at each layer r1 : 0.1 +m, r2 : 0.2 m, r3 : 0.4 m, r4 : 0.8 m and r5 : 1.4 m. We +used Sigmoid as our activation function in the output layer +to bound the estimates between [0, 1]. +Label imbalance: To address the imbalance in continuous +labels for the regression network, we adopted a sample +weighting approach solution proposed by Steininger et al. +[33]. The weight for each sample is based on the rarity of +the label value, so the weight is inversely proportional to +the probability of the label value occurrence. This will help +the model better estimate the rare cases [34]. The weighting +function is defined as +fw(α, y) = +max(1 − αp +′(y), ϵ) +1 +N +�N +i=1(max(1 − αp +′(yi), ϵ)) +, +(2) +where p is the target variable density function, N is the +number of data points, y = y1, ...yN is the target values, +p +′ = ( p(y) − min(p(y)) )/( max(p(y)) − min(p(y)) ) is +the normalized density function ∈ [0, 1] , hyperparameter +α ∈ [0, ∞) which emphasize the weighting scheme, and ϵ is +a small positive real number to avoid negative or 0 weights. +For more details and experiments of the effectiveness of +this weighting scheme, we refer to the original density-based +weighting article [33]. +Loss function: We used a weighted Root Mean Square +Error (RMSE) as a cost function for the regression model +L = +� +� +� +� 1 +N +N +� +i=1 +fw(α, yi)( ˆyi − yi)2. +(3) +Combining the sample gradients with the weight fw(α, yi), +will lead to larger gradients for the rare cases; in this way, the +model is forced to have better estimates for the rare values +as discussed in [33]. +Data loader: +One could feed the whole map into the +network as a single shot. However, the issue with this +approach is that at the first layer of the network, it sub- +samples a fixed number of points (N1), which is 1024 in +this implementation; therefore, depending on the map size, +N1 points may not be representative enough to enable the +network to learn the local geometry of the environment [13]. +One solution is to divide the map into smaller subsets +of maps. Choosing the right submap size and the suitable +number of points is a challenging yet interesting problem +due to the uneven distribution of the data inside the point +cloud and the scale of the features. For instance, choosing +a small submap size will not enable the network to capture +high-level features. On the other hand, big submap size with +a sparse point will not help the network learn the low-level +features. In our implementation, we set the subsampled map +size to 10 × 10 m in the x, and y axis with no constraints +in the z axis, and the number of points is set to 4096. +Furthermore, two common ways of generating the submap +are either by randomly picking the submap centroid or by +generating submaps in a convolutional way moving with a +fixed grid in x and y axis (as illustrated in Fig. 1 data-loader +block). We found the latter approach to guarantee full map +coverage, thus capturing all the features compared to random +submap selection. The submap kernel moves to the next grid +only if it has subsampled all the points in that area. The grid +size for our experiments is 50% of the submap size, which +gives 50% overlap. +Voting layer: +At the inference stage of the network, +some points may be assigned multiple predictions due to +overlapping submaps. In PointNet [13], they use a voting +pool, where the class that gets more votes will be the +point label. However, in regression, the prediction value is +continuous; therefore, we take the mean of predictions to get +the point estimated label. +IV. EXPERIMENTS +Fig. 3: The two areas in NCLT dataset that we used to train +and evaluate our approach +A. Dataset and metric +For evaluating our approach, we used the North Campus +Long-Term (NCLT) dataset [35]. This dataset was collected +over 15 months containing 27 recordings, making it a good +candidate for studying and testing long-term operations. The +data was acquired using a two-wheeled robot on one of the +University of Michigan, USA campuses. The sensors used for +collecting the data are a Velodyne HDL-32E LiDAR, wheel +encoders, GPS, IMU and a gyroscope. To build the 3D point +cloud maps for our experiments, we used a Simultaneous +localization and mapping (SLAM) system FAST-LIO [36], +which requires only the 3D LiDAR data from the Velodyne +and the IMU data. Out of the 27 data recording sessions, +we selected 5 recordings due to the maximum overlap + +115 +Area 2 +202 +Area1 +429 +50 +526 +0 +m +-50 +A +-100 +-150 +-200 +-100 +0 +100 +200 +x(m)between them, which are the following: 2012-01-15, 2012- +02-02, 2012-04-29, 2012-05-26, and 2012-08-04. In those +observations, we selected two areas shown in Fig. 3, which +are two parking lots, where the size of Area 1 is 70 × 90 m, +and Area 2 is 40 × 30 m. We refer to the raw observations +as {O0, . . . , O4} and the processed observations as {M0, +... , M4} respectively tell the rest of this manuscript. +Ground truth binary maps: The ground truth maps were +manually labelled using CloudCompare software1 into binary +classes that are long-term stable and dynamic objects. Trees, +light posts and poles are labelled stable objects, whereas +everything else is dynamic. +Metric: +The metric we used to evaluate the baseline +segmentation model and the thresholded values of the re- +gression model is the mean intersection over union mIoU = +1 +N +�N +c=0 IoUc, where N is the number of classes, IoUc = +(|Pc ∩Gc|)/(Pc ∪Gc), c is the point class (stable/unstable), +Pc is the predicted set, Gc is the ground truth set. +B. Baseline and experimental scenarios +Baseline: +The baseline we used in our evaluation is a +PyTorch implementation of PointNet++ [37], which we used +as a static/dynamic binary classifier. The baseline uses the +weighted negative log-likelihood loss, where the weights are +based on the distribution of the classes. Furthermore, we use +the same submap size to generate data for the baseline, which +is 10 × 10 m in x and y with no limits in the z axis. The +baseline is trained and evaluated on the ground truth data. +In contrast, the regression model was trained on the stability +labels from the unsupervised auto-labelled data. +Scenarios: +We conducted two experiments: Test 1 and +Test 2. For Test 1, both models were trained on M0 of +Area 1 and evaluated in all other maps of both Areas. The +second test is similar to the first one, but both models were +trained on M0 of Area 2. The motivation behind those tests is +to evaluate the spatio-temporal generalization of the model, +i.e. to evaluate if the model can infer the stable objects at +different time slices of the environment. Or if it can infer +them in new unseen areas, where the spatial generalization +is useful when we want to infer object stability for an +environment that has no previous history (observations). +For accurate comparison between the baseline and the +regression model, we converted the regression output into +binary classes using a threshold value ϵ. An optimal threshold +for the regression model can be found with the Receiver +Operating Characteristic (ROC) curve, which could be found +by minimizing or maximizing a certain metric [38]. In our +case, the metric that we are trying to maximize is the +geometric mean [39], which is a metric for imbalanced +classification that, if optimized, gives a balance between +the sensitivity (true positives rate) of the model and the +specificity (inverse of false positive). +The optimal threshold ϵ for Test 1 is found using the ROC +curve of the inferred labels of (M0) of Area 1, which is +1CloudCompare (version 2.12) [GPL software]. (2022). Retrieved from +http://www.cloudcompare.org/ +equal to 0.269 (0.626 meters). Then for consistency and to +not over-fit the results, we used this value to convert the +regression output to binary for all evaluated maps in Test +1. Those binary labels are evaluated w.r.t the ground truth +labels. For Test 2, the optimal threshold was found for (M0) +of Area 2 that is equal to 0.3593 (0.89 meters), then we did +the same as for Test 1. +Implementation: We train and evaluate both the binary +classification (baseline) and regression models on a work- +station with Intel Core i7-6850K CPU, 64GB RAM and an +NVidia GTX 1080ti GPU with 12 GB RAM. The model is +implemented using PyTorch framework. We used a learning +rate of 0.001, momentum 0.9, and trained for 60 epochs for +both the baseline and our model (regression). The training +time for both models was 4 hours for training on M0 of +Area 1 and around 2 hours for training on M0 of Area 2. +C. Results on NCLT parking lot areas +1) Evaluating the unsupervised labelling: To evaluate the +accuracy of the auto-labelled data, we use the area under +the ROC curve, known as ROC AUC, which summarizes +the performance of the auto-labelling by a single number +with values between 1.0 (perfect labelling) and 0.5 (random +labelling). The ROC curves are computed by comparing the +stability scores with the ground truth binary data at different +thresholds. Tab. I summarize ROC AUC for Areas 1 and +2, which indicates a good performance of the auto-labelling +algorithm. +TABLE I: Auto labeling data noise evaluation using the area +under ROC curve +Map +M0 +M1 +M2 +M3 +M4 +ROC AUC Area 1 +0.997 +0.999 +0.995 +0.997 +0.988 +ROC AUC Area 2 +0.999 +0.999 +0.999 +0.999 +0.999 +2) Evaluating maps inference: As shown in Tab. II, when +both models were trained and evaluated in the same area, +they showed a comparable performance despite the fact that +the regression model (noted as ’Ours’) was trained on the +unsupervised labels only. The evaluation for both models is +w.r.t the ground truth labels. However, the interesting results +are when evaluating the models in the opposite area used +for training. For instance, in Test 1, the regression model +outperforms the binary segmentation model by a large margin +in most maps; on average, the mIoU score improved by +34.2% over the baseline. For the second Test on Area 1, +the regression model shows improvement over the baseline +with an average increase by 14.6% in the mIoU score. The +low score, when trained on Area 2 and evaluated on Area 1, +is because Area 2 is smaller and has fewer features than the +other Area. Overall, the results confirm that the continuous +labels can better utilize the 3D spatial information in the +point cloud data. A visualization of the best and worst results +of Test 1 are shown in Fig. 4. +D. Ablation studies +We conducted two ablation studies with results in Tab. III +to determine the regression model sensitivity to the two + +TABLE II: Performance comparison between the baseline and our approach, the metric used to compare both models is +mIoU expressed in %. The baseline is trained and evaluated on the ground truth data. Ours is trained on the unsupervised +labels and is evaluated w.r.t the ground truth binary labels. In addition, the regression RMSE loss is presented. The ’—’ +indicates training data. +Area 1 +Area 2 +Test +Model +Metric +M0 +M1 +M2 +M3 +M4 +M0 +M1 +M2 +M3 +M4 +Test 1 +Baseline +mIoU +— +0.85 +0.85 +0.89 +0.92 +0.36 +0.33 +0.34 +0.48 +0.47 +Ours +mIoU +— +0.98 +0.73 +0.88 +0.86 +0.78 +0.82 +0.78 +0.66 +0.65 +RMSE +— +0.15 +0.10 +0.12 +0.13 +0.26 +0.25 +0.25 +0.20 +0.22 +Test 2 +Baseline +mIoU +0.44 +0.28 +0.50 +0.49 +0.49 +— +0.95 +0.96 +0.84 +0.86 +Ours +mIoU +0.66 +0.54 +0.55 +0.57 +0.61 +— +0.95 +0.97 +0.86 +0.86 +RMSE +0.21 +0.37 +0.15 +0.15 +0.15 +— +0.16 +0.17 +0.11 +0.12 +(a) Visualization for M1 and M2 of Area 1 +(b) Visualization for M1 and M4 of Area 2 +Fig. 4: Visual evaluation of the inferred maps of both areas +in Test 1 +main hyperparameters, which are α that control the weights +of the labels Eq. 2, and λ, which controls the sensitivity +of the cumulative distributing function of an exponential +distribution Eq. 1. The metric we used for evaluation is +ROC AUC. When trained on Area 1, the model was less +sensitive to the change of those hyperparameters. However, +when trained on Area 2 and tested on 1, the hyperparameters’ +values have a noticeable impact on the model performance. +It is worth noting that changing the hyperparameters would +only require tuning ϵ to find the optimal threshold. +E. Limitations +Our unsupervised labelling algorithm relies on the accu- +racy of ICP map alignment; therefore, if this step fails, some +manual intervention is required. Furthermore, we assumed +(i) the scans are complete, e.g. there are no missing points +due to occlusion, and (ii) there are significant changes +between the observations, i.e. a dynamic object changed its +location at least once in the observations. Violating those +assumptions leads to significant label noise. We are confident +that the occlusion issue could be overcome by considering +the probability that the area of the environment is occupied, +TABLE III: Ablation studies the model performance by +varying α, which controls the label weights, and λ, which +adjusts the label transform function’s sensitivity. Results are +ROC AUC. ’—’ indicates training data. +Hyperparameter +alpha (α) +lambda (λ) +Test number +Test 1 +Test 2 +Test 1 +Test 2 +Area +Map +0.0 +0.5 +0.0 +0.5 +0.5 +1 +0.5 +1 +Area 1 +M0 +– +– +0.83 +0.93 +– +– +0.93 +0.85 +M1 +0.99 +0.99 +0.88 +0.91 +0.99 +0.99 +0.90 +0.85 +M2 +0.99 +0.99 +0.86 +0.96 +0.99 +0.99 +0.96 +0.82 +M3 +0.99 +0.99 +0.75 +0.95 +0.99 +0.99 +0.94 +0.79 +M4 +0.99 +0.99 +0.74 +0.93 +0.99 +0.99 +0.93 +0.77 +Area 2 +M0 +0.93 +0.92 +– +– +0.92 +0.92 +– +– +M1 +0.94 +0.92 +0.99 +0.99 +0.91 +0.91 +0.99 +0.99 +M2 +0.94 +0.94 +0.99 +0.99 +0.93 +0.93 +0.99 +0.99 +M3 +0.96 +0.96 +0.99 +0.99 +0.96 +0.96 +0.99 +0.99 +M4 +0.98 +0.99 +0.99 +0.98 +0.97 +0.98 +0.99 +0.99 +free or unknown [16]; thus, the unknown regions of the map +can be excluded when calculating the features. +V. CONCLUSION +We have presented a novel end-to-end unsupervised deep +learning method for estimating objects long-term stability in +a 3D point cloud map. Our approach has two parts; first, +an unsupervised labelling algorithm that generates a point- +wise stability score by utilizing the temporal observations +of a given environment. Second, the point-wise regression +network based on PointNet++ is trained on the stability +labels, which could be used to infer objects’ stability in +similar environments with no previous observations. +The experiments’ performance showed that the proposed +method could efficiently identify which points in a map could +belong to the long-term stable object, and this opens up the +possibility of refining robotics maps by only keeping the +stable points, which leads to an improvement in long-term +localization on environments subject to continuous changes. +In addition, it showed that long-term stable object classifi- +cation is best performed by training a regression model on +the stability scores followed by thresholding compared to +directly training a binary classifier; to our best knowledge, +this insight was not previously known in the literature for the +problem of object stability classification. 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O. Carlsson, and J. Cabrera, “Optimal +thresholds by maximizing or minimizing various metrics via roc-type +analysis,” Academic radiology, vol. 20, no. 7, pp. 807–815, 2013. +[39] J. D. Lawson and Y. Lim, “The geometric mean, matrices, metrics, +and more,” The American Mathematical Monthly, vol. 108, no. 9, pp. +797–812, 2001. + diff --git a/dNE1T4oBgHgl3EQfxwWp/content/tmp_files/load_file.txt b/dNE1T4oBgHgl3EQfxwWp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ece283f60d54e3b50b4b48db38c37ece5d456d77 --- /dev/null +++ b/dNE1T4oBgHgl3EQfxwWp/content/tmp_files/load_file.txt @@ -0,0 +1,740 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf,len=739 +page_content='End-to-end Unsupervised Learning of Long-Term 3D Stable objects Ibrahim Hroob∗, Sergi Molina, Riccardo Polvara, Grzegorz Cielniak and Marc Hanheide Abstract— 3D point cloud semantic classification is an im- portant task in robotics as it enables a better understanding of the mapped environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' This work proposes to learn the long-term stability of the 3D objects using a neural network based on PointNet++, where the long-term stable object refers to a static object that cannot move on its own (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' tree, pole, building).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The training data is generated in an unsupervised manner by assigning a continuous label to individual points by exploiting multiple time slices of the same environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Instead of using discrete labels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' static/dynamic, we propose to use a continuous label value indicating point temporal stability to train a regression PointNet++ network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We evaluated our approach on point cloud data of two parking lots from the NCLT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The experiments’ performance reveals that static vs dynamic object classification is best performed by training a regression model, followed by thresholding, compared to directly training a classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' INTRODUCTION 3D point cloud maps are one of the standard map formats used for vehicle localization [1], [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' such maps represent a snapshot of the static environment around the mobile robot at the time of acquiring the scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We are interested in a learning-based approach for detecting potential long- term stable objects that are invariant across time in the 3D point cloud maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The importance of such objects is that they enable robust localization over an extended time, also known as long-term localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' However, the raw maps are unsuitable for long-term operations and may cause performance degradation in pose estimation if not a complete failure in the localization system in the long-term [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' That is mainly due to the following two reasons: (i) capturing dynamic objects as static, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' parked cars, and (ii) the raw maps suffer from the ”flying ghost” artefacts [4], which is caused by a moving object while recording the data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' pedestrians or cars in motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Many solutions are proposed to detect moving objects in point cloud maps based on geometrical methods, as in [1], [2], [4], [5], or based on deep learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The latter can achieve dense full class segmentation [6], [7], [8], which could be used to infer and detect dynamic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Thus, one could directly exploit the object semantics in the map to segment dynamic objects and identify the potential long-term stable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The issue with achieving full class semantics is that they rely heavily on the supervised annotated training data, which is not always available and expensive to generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' However, Corresponding author: ihroob@lincoln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='uk All the authors are within the Lincoln Centre for Autonomous Systems (LCAS), University of Lincoln, Lincoln, UK, LN6 7TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' This work has been supported by the European Commission as part of H2020 under grant number 871704 (BACCHUS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 1: Overview of our proposed point-to-point regression model for estimating point-wise long-term stability in a 3D point cloud map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The input is a set of points X ∈ R3, with the estimated 3D surface normal’s Nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The output is a point- wise stability estimation score l ∈ R1 bounded between [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The data-loader block is responsible for dividing the input map into smaller sub-maps to feed them into the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The green area is the sampling submap, which moves with a fixed grid in the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' for some tasks, as in static/dynamic segmentation, an entire class annotation may not be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Therefore, several solutions are presented to address this binary segmentation problem in an unsupervised data-driven manner [9], [10], [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The main problem with the unsupervised and geometrical approaches is that they only segment dynamic objects in the current scene;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' thus, they can not infer the long- term stable objects since they do not explore the history of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For instance, a parked car will be classified as static, which is not valid in the long-term perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' To tackle these problems, we propose an end-to-end unsu- pervised learning approach that can learn the local geometry of the long-term stable objects on the raw 3D point cloud maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The output of our approach is a point-wise temporal stability score, where higher values indicate that the point belongs to a dynamic object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' car, bike, pedestrian), and lower values represent the most stable points from a long-term perspective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' tree trunks, pole, building).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Since we are interested in an unsupervised learning method, we propose an automatic labelling algorithm to generate training data with a point-wise stability score by exploit- ing different time instances of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Regarding the learning step, we propose a regression neural network taking advantage of the pioneering work of the hierarchical PointNet++[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='03426v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='CV] 9 Jan 2023 Input: N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='(d + C) Output: N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='l "flying ghost"\' artefacts Dynamic Point-wise stability score represented as a heat-map Parked cars Static Submap size x gria Optional Overlapping area layer (mean) PointNet++ Encoder PointNet++ decoder Sigmoid Skip activation connections layer Per-point binary Data loader PointNet++ regression classification NetworkIn summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' the main contributions of this paper are three-fold: (i) an unsupervised automatic labelling algorithm exploiting long-term observations for a given environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' (ii) a regression network based on PointNet++ for point-wise long-term stability score estimation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 1) , and (iii) a comprehensive evaluation of the proposed auto-labelling algorithm and the regression neural network using real-world data, where the results demonstrate the effectiveness and the convenience of the proposed approach since it does not require manual annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' RELATED WORK Static/dynamic object segmentation is an active research area with methods broadly classified into geometry-based and deep learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Geometrical approaches are based on: motion cues [2], [14], [15], ray tracing [4], [16] or voxel traversal [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Motion cues (visibility-based) approaches identify dynamic points by comparing the current laser scan with the previous scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For instance, Pomerleau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' [14] infer the dynamic part of a scene by comparing the incoming scan with a global map based on visibility assumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' that is, if a point is observed behind a previously seen point, the old point might belong to a dynamic object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Ray tracing methods rely on shooting rays and checking for occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Usually, those methods are computationally expensive and run offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' An example of a method using this strategy is OctoMap [16], which is a probabilistic 3D mapping framework based on the octree data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The nodes in OctoMap store an occupancy probability p that indicates if the node is free, occupied, or unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The likelihood of the nodes is repeatedly updated for each scan using ray-tracing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' thus, this method can naturally filter out dynamic voxels by updating their probability score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The peopleremover [5] filters the dynamic points using a voxel grid instead of octree to store the identifier of all laser rays that hit the voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' It uses voxel traversal [17] to update voxel occupancy instead of ray tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The fully built occupancy grid could be used as a Static/Dynamic binary classifier to filter the points from the actual point cloud map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Deep learning approaches could be either supervised or unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Supervised methods [7], [8], [18], [19], [20], [21], can achieve full classes semantic segmentation, which one could use directly to detect all long-term stable instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' However, such methods currently rely heavily on hand- annotated data and are prone to human error or unknown classes [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' While on the other hand, unsupervised methods are a more interesting choice for learning object semantics, usually in the form of dynamic or static binary objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' those methods are data-driven methods that require minimal or no supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For example, [23] segment the indoor scene into foreground and background classes, the segmentation is performed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t the floor plan, so any point that does not match with the floor plan is labelled as dynamic, else static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Those labels are used in a neural network model to improve agent localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Recently, scene flow [11], [24], [25], [26] approaches are being applied to point cloud directly in an unsupervised way to label the points into moving or rigid objects between lidar frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Those methods are paired with a deep neural network as in [27], [28], [29] for an end-to-end object semantic estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' However, most of the methods mentioned above require motion information to infer dynamic objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' therefore, they cannot detect objects that can potentially move but are static in the current observation, for instance, a parked car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In contrast to other works, our approach is more focused on identifying the long-term stable objects in a given environ- ment, as those objects are a key landmark to guarantee long- term localization without degradation in performance [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Our method is unsupervised and does not require human input as it implicitly learns the long-term stable objects in an environment by exploiting previous temporal observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' PROPOSED METHOD Our proposed method aims at classifying static and dy- namic objects in a given environment from a 3D point cloud map by considering the long-term perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The input to the unsupervised labelling algorithm is a set of observations O0:k, where Oi is a set of points in the 3D Euclidean metric space {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=', xn} ∈ R3, with their estimated normal’s {n0, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=', nn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The output is a set of point-wise labelled maps ML 0:k, where the label value is continuous indicating point long-term stability bounded between [0, 1], where a lower score close to 0 means the point belongs to a long- term stable object like a tree trunk or light post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' While higher values close to 1 indicate that the point may belong to a dynamic object such as pedestrians or cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The values in between represent the slow dynamics in the environment as in seasonal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The following subsections explain how the point cloud maps are labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Then we detail the network architecture used for point-wise stability estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Unsupervised data labelling In this work, we are interested in labelling the point cloud in an unsupervised fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Instead of using discrete labels, we assign a continuous value to the points ranging between [0, 1], indicating points’ temporal stability by exploring the spatio-temporal dependency for a point across different ob- servations of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The motivation behind using continuous values is to remove any bias when classifying the points into static/dynamic on the unsupervised labels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' having the threshold value for classification, which means fewer parameters to tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Furthermore, the continuous labels are more suitable to indicate slowly moving objects because they are difficult to label as static or dynamic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' they can better utilize the 3D spatial information in the point cloud data);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' thus, this leads to better learning by the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We detail the proposed automatic labelling method in the following subsections (the entire labelling pipeline is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Data pre-processing: The input to this step is the raw observations of the environment in which we denote them as O0:k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The pre-processing output is a set of filtered and aligned maps w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t the first map denoted as M0:k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 2: The unsupervised data labelling pipeline has two main parts: the data pre-processing block that filters the raw observations O0:k and aligns them w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t the first map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Then the annotation block generates a point-wise stability score for each map by exploiting other maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' pre-processing data pipeline has three main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' First, for each observation Oi, the ground plane is segmented and removed using the Cloth Simulation Filtering (CSF) algorithm [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The motivation behind removing the ground plane is: (i) the ground plane is stable, (ii) to increase the points label disparity when extracting the distance to the nearest neighbour point in other observations/maps at later stages, and (iii) to reduce the overall map size (during our experiments we found that the number of points that belongs to the ground plane is roughly between 30 − 50%, which helps to reduce the computational time at later stages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The second step is to remove the outliers using the Statistical Outlier Removal (SOR) Filter [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In the final step, the maps are registered w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t the first map using Iterative Closest Point [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' This step is essential to ensure robust data association between spatio-temporal points across all observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' By the end of this pre-processing data pipeline, we end up with a set of clean and aligned maps denoted M0:k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Automatic labelling: The labelling algorithm (shown in Algorithm 1) assigns a spatio-temporal stability label for each point in a given reference map Mref by exploiting the distances to the nearest neighbour points (KNN) using a multidimensional binary search tree (kd-tree) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t all other time slices of the environment M0:k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The motivation behind using KNN is that the long-term stable object, for instance, will remain in the same place in all snap-shots of the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' thus, the distance to the nearest neighbour point is small in all-time instances, while on the other hand, a dynamic object may not appear in the same place in all maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Therefore the associated distance for each point in the dynamic object will be higher than the stable object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The algorithm works as follows: it iterates across all maps, for each selected map mRef, it go through all the points, where for each point p it finds the euclidean distance d = ��3 i=1(qi − pi)2 to the closest point q on other map mOther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' d is the distance vector w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t all other maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Finally, the feature used to represent the point stability is the maximum distance dmax in d, however dmax is not bounded i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' dmax ∈ [0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' to bound the label value between [0, 1] we used the Cumulative Distribution Function of an exponential function: f(dmax, λ) = 1 − e−λdmax, (1) Where λ is a hyper-parameter that controls the sensitivity of the function, after mapping the labels using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 1, the dynamic objects are squeezed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Algorithm 1 Unsupervised point-wise labelling algorithm Input: A set of filtered and aligned maps M0:k for each mRef ∈ M0:k do ▷ Set the current map mRef as the reference map for each p ∈ mRef do Initialize closest distance vector: d = {} for each mOther ∈ M0:k do if mOther is not mRef then q ← KNN(mOther, p) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='insert( ��3 i=1(qi − pi)2) end if end for Point label: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='l = 1 − e−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' max(d) end for end for Output: A set of labeled maps ML 0:k B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Network Our work aims to learn the local geometry of the stable objects from the unsupervised labelled point cloud generated using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We exploit the hierarchical PointNet (PointNet++) developed by Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' PointNet++ is a deep hierarchical neural network that can learn feature geometry at different levels of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In this work, instead of training the network as a static/dynamic binary classifier, we train it as a regression network on the continuous stability label value to better learn the spatial-temporal information in the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Then 00 Mo Mb 0 0 Ground 01 Removal M1 Mi (CSF) Mo && registration Outliers : filter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' (SOR) ML M,toMofeatures Ok Mk Static Mk M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' association Dynamic SetM,asareference Raw Filter ICP maps alignment Processed Labelled Autolabellingalgorithm observations w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t the first map maps mapsat the inference stage, we use an optimal threshold value ϵ to get the binary classification of the environment, where this insight was not previously known in the literature for the problem of static vs dynamic object classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The network is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Similar to the original Point- Net++ segmentation network, we used a set of abstraction layers to extract the local and global features from the point cloud and the same number of feature propagation layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In our implementation, we used 5 levels of abstraction layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The input number of points to each layer is as follows: N1 : 1024, N2 : 512, N3 : 256, N4 : 128 and N5 : 32 with the following sampling radiuses at each layer r1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='1 m, r2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='2 m, r3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='4 m, r4 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='8 m and r5 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='4 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We used Sigmoid as our activation function in the output layer to bound the estimates between [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Label imbalance: To address the imbalance in continuous labels for the regression network, we adopted a sample weighting approach solution proposed by Steininger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The weight for each sample is based on the rarity of the label value, so the weight is inversely proportional to the probability of the label value occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' This will help the model better estimate the rare cases [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The weighting function is defined as fw(α, y) = max(1 − αp ′(y), ϵ) 1 N �N i=1(max(1 − αp ′(yi), ϵ)) , (2) where p is the target variable density function, N is the number of data points, y = y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='yN is the target values, p ′ = ( p(y) − min(p(y)) )/( max(p(y)) − min(p(y)) ) is the normalized density function ∈ [0, 1] , hyperparameter α ∈ [0, ∞) which emphasize the weighting scheme, and ϵ is a small positive real number to avoid negative or 0 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For more details and experiments of the effectiveness of this weighting scheme, we refer to the original density-based weighting article [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Loss function: We used a weighted Root Mean Square Error (RMSE) as a cost function for the regression model L = � � � � 1 N N � i=1 fw(α, yi)( ˆyi − yi)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' (3) Combining the sample gradients with the weight fw(α, yi), will lead to larger gradients for the rare cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' in this way, the model is forced to have better estimates for the rare values as discussed in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Data loader: One could feed the whole map into the network as a single shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' However, the issue with this approach is that at the first layer of the network, it sub- samples a fixed number of points (N1), which is 1024 in this implementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' therefore, depending on the map size, N1 points may not be representative enough to enable the network to learn the local geometry of the environment [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' One solution is to divide the map into smaller subsets of maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Choosing the right submap size and the suitable number of points is a challenging yet interesting problem due to the uneven distribution of the data inside the point cloud and the scale of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For instance, choosing a small submap size will not enable the network to capture high-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' On the other hand, big submap size with a sparse point will not help the network learn the low-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In our implementation, we set the subsampled map size to 10 × 10 m in the x, and y axis with no constraints in the z axis, and the number of points is set to 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Furthermore, two common ways of generating the submap are either by randomly picking the submap centroid or by generating submaps in a convolutional way moving with a fixed grid in x and y axis (as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 1 data-loader block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We found the latter approach to guarantee full map coverage, thus capturing all the features compared to random submap selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The submap kernel moves to the next grid only if it has subsampled all the points in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The grid size for our experiments is 50% of the submap size, which gives 50% overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Voting layer: At the inference stage of the network, some points may be assigned multiple predictions due to overlapping submaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In PointNet [13], they use a voting pool, where the class that gets more votes will be the point label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' However, in regression, the prediction value is continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' therefore, we take the mean of predictions to get the point estimated label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' EXPERIMENTS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 3: The two areas in NCLT dataset that we used to train and evaluate our approach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Dataset and metric For evaluating our approach, we used the North Campus Long-Term (NCLT) dataset [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' This dataset was collected over 15 months containing 27 recordings, making it a good candidate for studying and testing long-term operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The data was acquired using a two-wheeled robot on one of the University of Michigan, USA campuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The sensors used for collecting the data are a Velodyne HDL-32E LiDAR, wheel encoders, GPS, IMU and a gyroscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' To build the 3D point cloud maps for our experiments, we used a Simultaneous localization and mapping (SLAM) system FAST-LIO [36], which requires only the 3D LiDAR data from the Velodyne and the IMU data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Out of the 27 data recording sessions, we selected 5 recordings due to the maximum overlap 115 Area 2 202 Area1 429 50 526 0 m 50 A 100 150 200 100 0 100 200 x(m)between them, which are the following: 2012-01-15, 2012- 02-02, 2012-04-29, 2012-05-26, and 2012-08-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In those observations, we selected two areas shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 3, which are two parking lots, where the size of Area 1 is 70 × 90 m, and Area 2 is 40 × 30 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We refer to the raw observations as {O0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' , O4} and the processed observations as {M0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' , M4} respectively tell the rest of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Ground truth binary maps: The ground truth maps were manually labelled using CloudCompare software1 into binary classes that are long-term stable and dynamic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Trees, light posts and poles are labelled stable objects, whereas everything else is dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Metric: The metric we used to evaluate the baseline segmentation model and the thresholded values of the re- gression model is the mean intersection over union mIoU = 1 N �N c=0 IoUc, where N is the number of classes, IoUc = (|Pc ∩Gc|)/(Pc ∪Gc), c is the point class (stable/unstable), Pc is the predicted set, Gc is the ground truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Baseline and experimental scenarios Baseline: The baseline we used in our evaluation is a PyTorch implementation of PointNet++ [37], which we used as a static/dynamic binary classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The baseline uses the weighted negative log-likelihood loss, where the weights are based on the distribution of the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Furthermore, we use the same submap size to generate data for the baseline, which is 10 × 10 m in x and y with no limits in the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The baseline is trained and evaluated on the ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In contrast, the regression model was trained on the stability labels from the unsupervised auto-labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Scenarios: We conducted two experiments: Test 1 and Test 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For Test 1, both models were trained on M0 of Area 1 and evaluated in all other maps of both Areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The second test is similar to the first one, but both models were trained on M0 of Area 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The motivation behind those tests is to evaluate the spatio-temporal generalization of the model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' to evaluate if the model can infer the stable objects at different time slices of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Or if it can infer them in new unseen areas, where the spatial generalization is useful when we want to infer object stability for an environment that has no previous history (observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For accurate comparison between the baseline and the regression model, we converted the regression output into binary classes using a threshold value ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' An optimal threshold for the regression model can be found with the Receiver Operating Characteristic (ROC) curve, which could be found by minimizing or maximizing a certain metric [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In our case, the metric that we are trying to maximize is the geometric mean [39], which is a metric for imbalanced classification that, if optimized, gives a balance between the sensitivity (true positives rate) of the model and the specificity (inverse of false positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The optimal threshold ϵ for Test 1 is found using the ROC curve of the inferred labels of (M0) of Area 1, which is 1CloudCompare (version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='12) [GPL software].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Retrieved from http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='cloudcompare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='org/ equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='269 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='626 meters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Then for consistency and to not over-fit the results, we used this value to convert the regression output to binary for all evaluated maps in Test 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Those binary labels are evaluated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t the ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For Test 2, the optimal threshold was found for (M0) of Area 2 that is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='3593 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='89 meters), then we did the same as for Test 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Implementation: We train and evaluate both the binary classification (baseline) and regression models on a work- station with Intel Core i7-6850K CPU, 64GB RAM and an NVidia GTX 1080ti GPU with 12 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The model is implemented using PyTorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We used a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='001, momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='9, and trained for 60 epochs for both the baseline and our model (regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The training time for both models was 4 hours for training on M0 of Area 1 and around 2 hours for training on M0 of Area 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Results on NCLT parking lot areas 1) Evaluating the unsupervised labelling: To evaluate the accuracy of the auto-labelled data, we use the area under the ROC curve, known as ROC AUC, which summarizes the performance of the auto-labelling by a single number with values between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='0 (perfect labelling) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='5 (random labelling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The ROC curves are computed by comparing the stability scores with the ground truth binary data at different thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' I summarize ROC AUC for Areas 1 and 2, which indicates a good performance of the auto-labelling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' TABLE I: Auto labeling data noise evaluation using the area under ROC curve Map M0 M1 M2 M3 M4 ROC AUC Area 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='988 ROC AUC Area 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='999 2) Evaluating maps inference: As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' II, when both models were trained and evaluated in the same area, they showed a comparable performance despite the fact that the regression model (noted as ’Ours’) was trained on the unsupervised labels only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The evaluation for both models is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t the ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' However, the interesting results are when evaluating the models in the opposite area used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For instance, in Test 1, the regression model outperforms the binary segmentation model by a large margin in most maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' on average, the mIoU score improved by 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='2% over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' For the second Test on Area 1, the regression model shows improvement over the baseline with an average increase by 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='6% in the mIoU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The low score, when trained on Area 2 and evaluated on Area 1, is because Area 2 is smaller and has fewer features than the other Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Overall, the results confirm that the continuous labels can better utilize the 3D spatial information in the point cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' A visualization of the best and worst results of Test 1 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Ablation studies We conducted two ablation studies with results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' III to determine the regression model sensitivity to the two TABLE II: Performance comparison between the baseline and our approach, the metric used to compare both models is mIoU expressed in %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The baseline is trained and evaluated on the ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Ours is trained on the unsupervised labels and is evaluated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='t the ground truth binary labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In addition, the regression RMSE loss is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The ’—’ indicates training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Area 1 Area 2 Test Model Metric M0 M1 M2 M3 M4 M0 M1 M2 M3 M4 Test 1 Baseline mIoU — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='47 Ours mIoU — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='65 RMSE — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='22 Test 2 Baseline mIoU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='49 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='86 Ours mIoU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='61 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='86 RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='15 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='12 (a) Visualization for M1 and M2 of Area 1 (b) Visualization for M1 and M4 of Area 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 4: Visual evaluation of the inferred maps of both areas in Test 1 main hyperparameters, which are α that control the weights of the labels Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 2, and λ, which controls the sensitivity of the cumulative distributing function of an exponential distribution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The metric we used for evaluation is ROC AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' When trained on Area 1, the model was less sensitive to the change of those hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' However, when trained on Area 2 and tested on 1, the hyperparameters’ values have a noticeable impact on the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' It is worth noting that changing the hyperparameters would only require tuning ϵ to find the optimal threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Limitations Our unsupervised labelling algorithm relies on the accu- racy of ICP map alignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' therefore, if this step fails, some manual intervention is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Furthermore, we assumed (i) the scans are complete, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' there are no missing points due to occlusion, and (ii) there are significant changes between the observations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' a dynamic object changed its location at least once in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Violating those assumptions leads to significant label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' We are confident that the occlusion issue could be overcome by considering the probability that the area of the environment is occupied, TABLE III: Ablation studies the model performance by varying α, which controls the label weights, and λ, which adjusts the label transform function’s sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Results are ROC AUC.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content='99 free or unknown [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' thus, the unknown regions of the map can be excluded when calculating the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' CONCLUSION We have presented a novel end-to-end unsupervised deep learning method for estimating objects long-term stability in a 3D point cloud map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Our approach has two parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' first, an unsupervised labelling algorithm that generates a point- wise stability score by utilizing the temporal observations of a given environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Second, the point-wise regression network based on PointNet++ is trained on the stability labels, which could be used to infer objects’ stability in similar environments with no previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' The experiments’ performance showed that the proposed method could efficiently identify which points in a map could belong to the long-term stable object, and this opens up the possibility of refining robotics maps by only keeping the stable points, which leads to an improvement in long-term localization on environments subject to continuous changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' In addition, it showed that long-term stable object classifi- cation is best performed by training a regression model on the stability scores followed by thresholding compared to directly training a binary classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' to our best knowledge, this insight was not previously known in the literature for the problem of object stability classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' As for future work, we will address the limitations of the labelling algorithm and explore extracting the long-term stable objects directly from the 3D LiDAR scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE1T4oBgHgl3EQfxwWp/content/2301.03426v1.pdf'} +page_content=' Binary Ground Baseline Auto labelled Ours inference Ours inference Truth labels inference point cloud Continuous as binary M1 M2 Long-term Dvnamic stableBinary Ground Baseline Auto labelled Ours inference Ours inference Truthlabels inference point cloud Continuous as binary M M Long-term Dynamic stableREFERENCES [1] H.' metadata={'source': 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Asymmetric Mutual Learning +Qiong Wu, Jiahan Li, Pingyang Dai, Qixiang Ye, Senior Member, IEEE, Liujuan Cao, Member, IEEE, +Yongjian Wu, Rongrong Ji, Senior Member, IEEE +Abstract—Unsupervised +domain +adaptation +person +re- +identification (Re-ID) aims to identify pedestrian images within +an unlabeled target domain with an auxiliary labeled source- +domain dataset. Many existing works attempt to recover reliable +identity +information +by +considering +multiple +homogeneous +networks. And take these generated labels to train the model +in the target domain. However, these homogeneous networks +identify people in approximate subspaces and equally exchange +their knowledge with others or their mean net to improve their +ability, inevitably limiting the scope of available knowledge and +putting them into the same mistake. This paper proposes a Dual- +level Asymmetric Mutual Learning method (DAML) to learn +discriminative representations from a broader knowledge scope +with diverse embedding spaces. Specifically, two heterogeneous +networks mutually learn knowledge from asymmetric subspaces +through the pseudo label generation in a hard distillation +manner. The knowledge transfer between two networks is +based on an asymmetric mutual learning manner. The teacher +network learns to identify both the target and source domain +while adapting to the target domain distribution based on the +knowledge of the student. Meanwhile, the student network is +trained on the target dataset and employs the ground-truth label +through the knowledge of the teacher. Extensive experiments +in Market-1501, CUHK-SYSU, and MSMT17 public datasets +verified the superiority of DAML over state-of-the-arts. +Index Terms—Transfer learning, unsupervised domain adap- +tation, person re-identification, retrieval. +Qiong Wu is with Institute of Artificial Intelligence, and the Media +Analytics and Computing Laboratory, Department of Artificial Intelligence, +School of Informatics, Xiamen University, Xiamen 361005, China (e-mail: +qiong@stu.xmu.edu.cn). +Jiahan Li is with School of Information and Control Engineering, China +University of Mining and Technology, Xuzhou 221000, China (e-mail: jia- +han.li@cumt.edu.cn). +Pingyang Dai is with the Media Analytics and Computing Laboratory, De- +partment of Artificial Intelligence, School of Informatics, Xiamen University, +Xiamen 361005, China (e-mail: pydai@xmu.edu.cn). +Qixiang Ye is with the Peng Cheng Laboratory, Shenzhen 518066, China, +and also with the School of Electronics, Electrical and Communication +Engineering, University of Chinese Academy of Sciences, Beijing 100049, +China (e-mail: qxye@ucas.ac.cn). +Liujuan Cao is with the Media Analytics and Computing Lab, Department +of Computer Science, School of Informatics, Xiamen University, Xiamen +361005, China (e-mail: caoliujuan@xmu.edu.cn). +Yongjian Wu is with the Youtu Laboratory, Tencent, Shanghai 200233, +China. (e-mail: littlekenwu@tencent.com). +Rongrong Ji is with the Media Analytics and Computing Laboratory, De- +partment of Artificial Intelligence, School of Informatics, Xiamen University, +Xiamen 361005, China, also with the Fujian Engineering Research Center of +Trusted Artificial Intelligence Analysis and Application, Institute of Artificial +Intelligence, Xiamen University, Xiamen 361005, China, and also with the +Peng Cheng Laboratory, Shenzhen 518066, China. (e-mail: rrji@xmu.edu.cn). +Fig. 1. The statistics of common neighbors between different mod- +els. CNN-CNN and ViT-ViT curves denote the average number of +common neighbors in the k nearest neighbors of each instance. The +features are extracted by two homogeneous networks trained with +different initialization. Similarly, CNN-ViT represents the common +neighbors between CNN and ViT. Furthermore, Upbound refers to the +maximum number of neighbors to consider. These models are trained +on the CUHK-SYSU dataset in a supervised manner and cluster on +the Market1501 dataset. Compared to the CNN-CNN and ViT-ViT, +the CNN-ViT contains fewer common neighbors, and the ways they +distinguish two individuals are more different than homogeneous +networks. It demonstrates that heterogeneous networks address the +task in different patterns. +I. INTRODUCTION +P +ERSON re-identification (Re-ID) [1] aims at matching +individual pedestrian images from images captured by +different cameras according to identity. This task is chal- +lenging because the variations of viewpoints, body poses, +illuminations, and backgrounds will influence a person’s ap- +pearance. Recently, supervised person Re-ID methods [2]–[11] +made impressive progress. However, as the number of images +increases and the ensuing scene changes, regular supervised +learning approaches are losing their ability to adapt to complex +scenarios. The performance of person re-ID models trained on +existing datasets will evidently suffer for person images from a +new video surveillance system due to the domain gap. To avoid +time-consuming annotations on the new dataset, unsupervised +domain adaptation (UDA) is proposed to adapt the model +arXiv:2301.12439v1 [cs.CV] 29 Jan 2023 + +Upbound +17.5 +CNN-ViT +CNN-CNN +ViT-ViT +15.0 +Common Neighbor +12.5 +10.0 +7.5 +5.0 +2.5 +0.0 +0 +5 +10 +15 +20 +25 +Range of NeighborhoodJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +2 +trained on the labeled source-domain dataset to the unlabeled +target-domain dataset. +Generating trusted identity information on the target domain +is seen as the core of the UDA task. Some UDA Re-ID +methods [10], [12]–[15] directly apply GANs [16] to transfer +the style of pedestrian images from the source domain to +the target while keeping the identities to train the model. +However, the complexity of the human form and the limited +number of instance in a Re-ID dataset limit the quality of +generated images. After abandoning the image generation, +some methods [7], [17] introduce the attribute to bridge the +domain gap. These methods introduce additional annotation +information which defeats the purpose of the UDA Re-ID +task. Limited by the missing label on the target domain, +others [18]–[21] align the distributions of target and source +domains while only learning classifying on the source. To +better adapt the distribution of the target domain and train with +the target-domain identity knowledge, various methods [22]– +[24] apply a clustering algorithm in the target domain to +generate the pseudo labels for training in a supervised manner. +One of the keys to improving performance is alleviating the +influence of noisy labels. In this context, many methods [25]– +[27] based on clustering algorithms are proposed to rule out +the harmony from the noisy labels by introducing more than +one framework to predict pseudo labels. They aim to generate +knowledge with specific differences in samples and exchange +the knowledge among the networks to enhance their ability. +Despite encouraging progress, the benefits from the knowledge +mined by homogeneous networks are limited. As shown in +Fig. 1 CNN-CNN and ViT-ViT, these homogeneous networks +with similar structures identify pedestrians in a comparable +manner, and the relation among the instances are similar. +It suggests they use similar patterns to extract pedestrian +features, and networks may converge to equal each other. +Furthermore, this mode of operation makes it possible for the +networks to make the same mistakes and not be able to correct +them. Such a design limits the knowledge models can learn +from the training set and makes it possible for the networks +to repeat mistakes without being able to remedy them. As +a result, mining the information from different subspaces is +required to broaden the scope of knowledge and generate +reliable pseudo labels. +To tackle this problem, heterogeneous networks, as shown +in Fig. 1 CNN-ViT, can discover the information from multiple +subspaces and have more extensive latent knowledge. We +propose Dual-level Asymmetric Mutual Learning (DAML), a +novel unsupervised domain adaption method for person Re- +ID that broadens the scope of knowledge for the network +by exploiting information from two different subspaces and +selectively transferring information between heterogeneous +networks. The proposed DAML consists of a CNN that focuses +on identity learning as a teacher network and a ViT that +concentrates on adapting knowledge from the target domain +as a student network for embedding samples into different +subspaces and setting the constraints among the classifiers for +asymmetric mutual learning. +In particular, the CNN that works as a teacher will train +on both source and target datasets under the supervision of +ground-truth source-domain labels and pseudo-target-domain +labels. The former can provide reliable identity information +for extracting discriminative feature representation, while the +latter will assist the network in adapting the distribution of +the target domain. However, learning from the source domain +will harm the distribution that the network adapted limiting the +performance. To avoid this disadvantage, the ViT that works +as a student only trains with the guidance of pseudo-target- +domain labels and learns the knowledge from the teacher. In +the pseudo label generation stage, the relationship between +two samples is weighted according to their teacher and student +features similarity. Moreover, this process wholly exchanges +the knowledge learned from two different subspaces. After +predicting the identities of input images, the asymmetric +constraints between two heterogeneous networks selectively +exchange the knowledge. The student learns the identity +knowledge from the teacher network under the constraints +from the target-domain samples. Furthermore, for the student +can benefit more from the teacher and better utilize the ground- +truth labels, the source-domain identity knowledge learned +by the teacher is transferred to the target domain with the +constraints based on source-domain samples. In summary, the +DAML employs diverse subspaces to generate reliable pseudo +label in the target domain and help student adopt ground-truth +knowledge in the source domain. +Our main contributions are summarized below: +• We address the diverse subspaces learning and target- +domain identity learning for unsupervised domain adapta- +tion person Re-ID with proposed Dual-level Asymmetric +Mutual Learning (DAML). The former has rarely been +studied in the existing research, while the latter is crucial +for retrieving person in the target domain. +• We propose a novel Dual-level Asymmetric Mutual +Learning (DAML) method for unsupervised domain +adaptation person Re-ID. The asymmetric knowledge +learning between the teacher and the student helps them +play their roles better. +• To learn from diverse subspaces, the proposed DAML +introduces two heterogeneous networks to mine valuable +information from different subspaces and selectively ex- +change the information between them. +• To better utilize the knowledge mined by heterogeneous +networks and ensure the networks orient to the task, the +proposed DAML smoothly update the classifiers in a +hard distillation manner and exchange knowledge during +training in a soft distillation manner. +II. RELATED WORK +A. Unsupervised Domain Adaptation Person Re-ID +Unsupervised Domain Adaptation Person Re-ID has at- +tracted increasing attention in recent years due to its effec- +tiveness in reducing manual annotation costs. There are two +main categories of methods are proposed to address this issue. +Firstly, GAN-based methods aim to transfer samples from +the source domain to the target domain without altering their +identities. SPGAN [12] and PDA-Net [13] transfer images +directly from the source domain to the target domain while + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +3 +maintaining the original identity knowledge. The generated +images have a similar style to the target-domain images and +are used to train the model under the supervision of their +original labels in the source domain. To produce generated +images that are more realistic and have more detail, DG- +Net [28] and DG-Net++ [15] introduce disentanglement for +the generation stage. But the generation is expensive and +the style of generated images may not well fit the target +domain. Rather than transfer images from the source domain to +the target domain, HHL [10] transfers target-domain images +among the cameras to generate images that have the same +identity but contain the difference at the same time. Secondly, +the clustering-based methods clustering based methods do not +require expensive GAN networks for generation and have +achieved state-of-the-art performance to date. To reduce the +impact of noisy label, MMT [25] proposed a mutual learning +method providing soft labels. For more reliable pseudo labels, +SSG [29] clusters samples in three scales and validate each +other. MEB-Net [27] respectively introduces multiple groups +of prototypes or homogeneous networks to generate the pseudo +labels. UNRN [30] and GLT [26] design a memory bank +to save anchors for aligning the distribution and learning +identities in a contrastive learning manner. Limited by the +constraints in the feature level these methods rely on, the +models that collaborate to generate pseudo labels are ho- +mogeneous. These characteristics determine that the model +can only learn similar knowledge from others. Nevertheless, +these approaches alleviate the domain gap only considering +the single embedding space inevitably makes some mistakes. +B. Knowledge Distillation +Knowledge distillation makes a student network learns from +a strong teacher network to improve the student’s ability. The +common approaches can be summarized as hard distillation +and soft distillation. Soft distillation [31], [32] minimizes the +distribution difference between the prediction generated by +teacher and student. The soft label generated by the teacher +model can alleviate overfitting just like labels smoothing [33]. +Unlike the soft, hard-label distillation regards the prediction +result of the teacher as a valid label. And positive pairs pre- +dicted by the teacher are used to transfer identity knowledge +from the teacher to a student network in semi-supervised and +unsupervised learning tasks. Temporal ensembling [34] put +the former networks as the teacher and use memory saving +average predictions for each sample as supervision for the +unlabeled samples. To avoid storing predictions for saving +memory, Mean Teacher [35] averaged student model weights +as the parameter of the teacher. During the training, the +predictions made by the teacher are seen as supervision for +unlabeled samples. The models consider similar information +in these methods because the teacher and the student have the +same structure and similar initialization. It makes the networks +focus within a certain range and limit the knowledge student +can learn. The proposed DAML exchange knowledge utilizes +both soft and hard distillation in the different training stages. +Thanks to the heterogeneous networks, the proposed DAML +gives the student model a broader perspective and can generate +pseudo labels from different views. +C. CNN and ViT +Since AlexNet [36] achieve great success on ImageNet [37], +a variety of convolutional neural networks (CNN) [38]–[41] +is proposed to solve different tasks. As Transformers [42] +were proposed for machine translation and were seen with +significant results in many NLP tasks, the application of self- +attention to images is widely concerned. A new model without +any convolution, Vision Transformers (ViT) [43], has been +proposed for computer vision tasks and shows its potential. +During the calculation process, the CNN keeps the spatial +information and can only focus on the surrounding area in one +layer due to the nature of convolution. In contrast, ViT em- +phasizes the correlation between two patches, and its receptive +field involves the whole feature map. These differences make +the CNN and ViT learn different knowledge from the training +set for the same task. And in our paper, we take advantage +of this difference to achieve asymmetric distillation, making +ViT a better performer with our DAML. The ViT works as +a student because the receptive field of a patch in the ViT +covers the area that one convolution kernel can consider, not +vice-versa. +III. METHODOLOGY +A. Overview +The ultimate goal of the unsupervised domain adaptation +(UDA) person Re-ID is to gain a model work on a target- +domain dataset based on a labeled source-domain dataset and +an unlabeled target-domain dataset. Let S = {(xi +s, yi +s)}Ns +i=1 +and T += {xi +t}Nt +i=1 respectively denote the source-domain +images with ground-truth labels and the unlabeled target- +domain images, where Ns and Nt are the numbers of samples +from these two domains. +As shown in Fig. 2, the Dual-level Asymmetric Mutual +Learning (DAML) method trains the student to extract discrim- +inative representations from two different subspaces to perform +the UDA person Re-ID task. Firstly, DAML adopts two +heterogeneous networks: teacher CNN ET (·) and student ViT +ES(·) which are pre-trained on the source-domain dataset in +a supervised manner to extract features in different subspaces. +At each epoch, we first group target-domain samples into +K classes by the clustering algorithm. The distance between +two target-domain samples will be calculated according to the +features ET (xi +t) = tT +i ∈ RcT and ES(xi +t) = tS +i ∈ RcS ex- +tracted by the teacher and student models with corresponding +weights. The {ˆyi}Nt +i=1 are the pseudo labels for the target- +domain samples. Then, for each class center cy, we generate +its prediction with the classifiers C(·|WS +t ) and C(·|WT +t ) for +updating the parameter WS +t and WT +t in a smooth method. +After that, we train the teacher and the student models with +the pseudo labels in a supervised manner. For the teacher +model, classifier C(·|[WT +s , WT +t ]) will learn both source- +domain and target-domain knowledge. While the classifier +C(·|WS +t ) for the student model only directly learns the target- +domain knowledge. The constraints between two networks +transfer the identity knowledge learned by the teacher to the +target and help the student learn from diverse subspaces. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +4 +Source data +Both data +Target data +CNN +Teacher +ViT +Student +Classifier +[W��, W� +�] +source domain +target domain +𝐿�� +𝐿�� +𝐿�� +Classifier +[W� +�] +Classifier +[W� +�] +𝐿��� +Classifier +[W� +�, W� +�] +Update Classifier +[W� +�, W� +�] +Beginning of +Each Epoch +Cluster Base on Weighted Features +··· +··· +Features in +Different Spaces +Asymmetric Dual Networks +Predictions of +Cluster Centers +𝐿��� +𝐿�� +Fig. 2. +Overview of our Dual-level Asymmetric Mutual Learning method (DAML). The teacher network is trained under the supervision of +pseudo labels and ground-truth labels for target-domain and source-domain samples. And the student only directly learns knowledge from +target-domain samples with pseudo labels. At the beginning of epochs, we first generate the pseudo labels for target dataset, and update +the classifiers based on the predictions of cluster centers. To distill the different subspace knowledge from the teacher to the student, Lid +makes the student predictions of target-domain samples close to the teacher. Meanwhile, for student can better adopt the identity knowledge +learned by the teacher, we minimize the distribution differences of the same source-domain samples with Ldom. +Finally, we only adopt the features tS +i = ES(xi +t) extracted +by the student model for testing. +B. Smooth Classifier Update (SCU) +At the beginning of epochs, we extract the target-domain +features tT +i += ET (xi +t) and tS +i = ES(xi +t) with two hetero- +geneous networks. To better utilize the knowledge from the +two models, we first define the neighborhood of an instance +according to its relations in two different subspaces: +Ni = +� +xj +�����1 − +⟨tM +i , tM +j ⟩ +∥tM +i ∥2∥tM +j ∥2 +< α, M ∈ {T, S} +� +, +(1) +the α here is a hyper-parameter. With the limitation of neigh- +bor selection considering both teacher features and student +features simultaneously, the neighbors of an instance should +be close to it in both subspaces. The above constraint ensures +that instances with apparent differences will not be clustered +as the same identity because the patterns of the two models +applied to recognize an instance are different. +To exploit the information from two different subspaces and +make the pseudo labels more reliable, we combine features +from heterogeneous networks and define the distance between +two samples as: +di,j = +� +� +� +� +� +1 − +⟨[tT +i , tS +i ], [tT +j , tS +j ]⟩ +∥[tT +i , tS +i ]∥2∥[tT +j , tS +j ]∥2 +, +xi ∈ Nj and xj ∈ Ni +Inf, +Others +(2) +where [·, ·] represents the concatenation of two features, and +⟨·, ·⟩ denotes the dot product between two features. In short, +we define the similarity between two samples as the cos +similarity between the features constructed by concatenating +their teacher feature and student feature. Then the pseudo +labels can be generated based on the relationship among +instances with the clustering algorithm. +With the pseudo labels, some methods [25], [30] directly +update the classifier by replacing the classifier parameters with +the new class centers to adapt the count of classes change. +These methods will make the knowledge lost because the class +centers may not represent the corresponding class well. To +protect the knowledge involved in the classifiers, we update +the classifiers more smoothly as follows: +Wi +t = +ˆ +K +� +k=1 +ˆ +Wk +t · +epk +i +� ˆ +K +j=1 epj +i +, +(3) +where Wi +t is the parameters for the ith target-domain identity +in the next epoch, pi = C(ci| ˆ +Wt) is the prediction of class +center ci with the parameters ˆ +Wt from the last epoch which +includes +ˆK classes. Note that the momentum for SGD is +updated following the parameters in the process. +C. Identity Learning +The core of person re-identification is identifying the per- +sons. For two heterogeneous networks learning to extract +discriminative representation, there are two level objective + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +5 +functions are applied. Firstly, at the feature level, the triplet +loss: +Ltri(f) = 1 +n +n +� +i=1 +max{ρ + dp − dn, 0}, +(4) +is applied to guarantee the features can well represent their +corresponding samples. Where f represents a batch of the +features, n = |f| is the size of the batch, ρ is the tiniest +margin between the distance to the furthest positive instance +dp and the distance to the nearest negative instance dn. The +relationship between two instances from the source domain +depends on the ground-truth labels and the pseudo labels for +target-domain samples. Due to the different dimensions of the +features extracted by heterogeneous networks, the triplet loss +can only be applied in a certain subspace. +Then, in the logits level, we apply the cross-entropy loss +with classifiers: +LT tid = − 1 +n +n +� +i=1 +log P(ˆyi|C(tT +i |[WT +s , WT +t ])), +(5) +LStid = − 1 +n +n +� +i=1 +log P(ˆyi|C(tS +i |WS +t )), +(6) +where ˆyi is the pseudo label for target-domain example xi +t. +The trainable parameters WT +s , WT +t +and WS +t +respectively +denote the classifier parameters for the teacher classifying +source-domain samples, the teacher classifying target-domain +samples, and the student classifying target-domain samples. +Meanwhile, to take advantage of the ground-truth label, the +teacher also learns the source-domain knowledge by: +LT sid = − 1 +n +n +� +i=1 +log P(yi|C(sT +i |[WT +s , WT +t ])), +(7) +here, yi is the ground-truth label for source-domain sam- +ple xi +s. Note that, with Eq.(5) and Eq.(7), the classifier +C(tT +i |[WT +s , WT +t ]) in the teacher has learned both two domain +knowledge while classifier C(tT +i |WS +t ) for the student learns +the target-domain knowledge only. +D. Asymmetric Mutual Learning (AML) +Compare the structure of the Convolutional Neural Network +(CNN) and Vision Transformer (ViT), the most evident differ- +ence is that the ViT can capture long-range information with its +cascaded self-attention modules. However, CNN only focuses +on the local limited by the size of the convolution kernel. In +addition, the CNN inductive bias, which includes assumptions +of the data, can involve information that ViT may not con- +sider and the convolution kernel with a deterministic shape +guarantees spatial information. More intuitively, the features +extracted by the two networks have different dimensions. It +ensures the subspaces learned by the heterogeneous networks +are different but makes feature-level constraints unusable. +The asymmetric distillation benefit from the difference in the +patterns that two heterogeneous networks predict the identity. +And focus on twofold: to allow students access to knowledge +from the different subspaces and transfer the knowledge from +the source to the target. +To make the student learn from different subspaces and take +advantage of the reliable source-domain labels, the proposed +DAML transfers the identity knowledge from the teacher +by reducing the Kullback-Leibler divergence between the +predictions of the target-domain features as: +Lid = 1 +n +n +� +i=1 +C(tT +i |WT +t ) log C(tS +i |WS +t ) +C(tT +i |WT +t ), +(8) +with the above objective function, the student can learn the +knowledge from the teacher which adopts knowledge from +both source and target domain with Eq.(5) and Eq.(7). How- +ever, domain knowledge is also transferred to students and +may harm the performance in the target domain. The ideal +way to alleviate the distribution effect is to make the teacher +predict identities under the target domain. +Limited by the domain gap, the knowledge learned from the +source domain can not be directly applied to the target domain. +And the goal of the proposed asymmetric mutual learning is +to gain a student network that adapts to the target domain +while benefiting from the source-domain identity knowledge. +Making source-domain predictions from the teacher similar +to the student will transfer the classifying knowledge learned +from the source domain to the target domain: +Ldom = 1 +n +n +� +i=1 +C(sS +i |WS +t ) log C(sT +i |WT +t ) +C(sS +i |WS +t ) , +(9) +here, WT +t learns source-domain knowledge with Eq.(7) while +WS +t only learns the knowledge from target domain. Eq.(9) +focuses on making the teacher predict source-domain samples +in the same way as the student. In this way, the student +can better adopt identity knowledge from the source domain +without a domain gap as much as possible. Compared with +Eq.(8), the above equation distills the knowledge in a different +direction and together make the student model can distinguish +pedestrian in the target domain. +E. Optimization +The total loss L of DAML can be summarized as: +L = +� +LT tid + Ltri(tT ) +� ++ +� +LStid + Ltri(tS) +� +λ1 +� +LT sid + Ltri(sT ) +� ++ λ2Lid + λ3Ldom +(10) +where λ1, λ2, and λ3 are hype-parameters to balance the +contributions of individual loss terms. +IV. EXPERIMENTS +A. Datasets +Datasets We evaluated our method on three public datasets +Market-1501 [44], CUHK-SYSU [45] and MSMT17 [14]. +• Market-1501 contains 32, 668 labeled images captured +from 1, 501 identities by 6 cameras. The training set has +12, 936 images of 751 identities. In addition, 3, 368 query +images and 19, 732 gallery images from the other 750 +identities are used as the testing set. +• CUHK-SYSU includes 33, 901 labeled images of 8, 432 +identities taken in diverse scenes. The training set is +constructed with 5, 532 identities having 15, 088 images, + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +6 +TABLE I +COMPARISON OF CMC (%) AND mAP (%) PERFORMANCES WITH THE SOTA METHODS ON MARKET-1501, CUHK-SYSU AND MSMT17. +Method +Market-1501 → CUHK-SYSU +CUHK-SYSU → Market-1501 +mAP +R1 +R5 +R10 +mAP +R1 +R5 +R10 +Directly Transfer (IBN-ResNet-50) +74.1 +77.2 +85.7 +88.7 +38.8 +63.7 +79.4 +85.2 +Directly Transfer (ViT-Base) +86.0 +87.2 +94.1 +95.0 +36.2 +60.3 +76.5 +82.9 +UNRN [30](AAAI’21) +62.3 +64.1 +76.9 +82.0 +70.9 +86.7 +92.8 +94.5 +MMT(IBN-ResNet-50) [25](ICLR’20) +78.4 +81.0 +89.7 +92.2 +76.0 +88.8 +95.2 +97.0 +MEB-Net [27](ECCV’20) +81.1 +83.2 +90.9 +93.1 +69.3 +84.0 +92.9 +95.2 +DAML (Ours) +84.3 +86.2 +92.6 +94.6 +84.1 +93.1 +97.7 +98.2 +Supervised (IBN-ResNet-50) +90.8 +95.2 +96.6 +89.0 +83.0 +94.1 +97.4 +98.4 +Supervised (ViT-Base) +93.1 +97.2 +97.8 +92.1 +82.3 +93.2 +97.9 +98.8 +Method +Market-1501 → MSMT17 +CUHK-SYSU → MSMT17 +mAP +R1 +R5 +R10 +mAP +R1 +R5 +R10 +Directly Transfer (IBN-ResNet-50) +8.4 +23.8 +34.5 +39.6 +10.3 +26.3 +38.3 +44.3 +Directly Transfer (ViT-Base) +13.0 +33.3 +45.3 +51.1 +12.5 +28.2 +41.1 +47.5 +MEB-Net [27](ECCV’20) +20.6 +44.1 +58.3 +64.3 +21.3 +45.6 +59.5 +65.6 +UNRN [30](AAAI’21) +25.3 +52.4 +64.7 +69.7 +12.6 +31.1 +43.8 +49.7 +MMT(IBN-ResNet-50) [25](ICLR’20) +26.6 +54.4 +67.6 +72.9 +24.0 +49.0 +63.0 +68.6 +DAML (Ours) +41.4 +65.4 +76.0 +80.2 +44.0 +67.0 +78.0 +81.9 +Supervised (ViT-Base) +54.1 +76.6 +87.5 +90.8 +54.1 +76.6 +87.5 +90.8 +Supervised (IBN-ResNet-50) +49.9 +79.2 +88.2 +91.3 +49.9 +79.2 +88.2 +91.3 +TABLE II +ABLATION STUDY IN TERMS OF mAP (%) AND CMC (%) ON +CUHK-SYSU (CS) → MARKET-1501 (M). +Method +CS → M +mAP +R1 +IBN-ResNet-50(Directly) +38.8 +63.7 +ViT-Base(Directly) +36.2 +60.3 +DAML w/o LT sid + Ltri(sT ) +83.6 +92.8 +DAML w/o Lid +83.3 +92.3 +DAML w/o Ldom +83.6 +92.5 +DAML w/o SCU +81.0 +91.0 +DAML +84.1 +93.1 +IBN-ResNet-50(Supervised) +83.0 +94.1 +ViT-Base(Supervised) +82.3 +93.2 +and the rest is used for testing. There are 2, 900 images +for the query and 6, 978 images for the gallery in the +testing set. +• MSMT17 is a large-scale dataset consisting of 126, 441 +bounding boxes of 4, 101 identities caught on 12 outdoor +and 3 indoor cameras. Among them, 32, 621 images of +1, 041 identities are used for training and 93, 820 of 3, 060 +identities are used for testing. +B. Experiment Setting +Performance Metric: As a UDA task, we select one dataset +as the source-domain dataset and another as the target-domain +dataset. The model is trained with the labeled source-domain +training set and adapts the target domain through the unlabeled +target-domain training set. Then the performance is evaluated +according to the student network which work on the target- +domain testing set. In our experiments, following the standard +metrics, we employ the cumulative matching characteristic +(CMC) curve and the mean average precision (mAP) score. +Our experiments report rank-1, rank-5, and rank-10 accuracy +and mAP scores. +Implementation Details: In the most common setting, we +select IBN-ResNet-50 [41] as the teacher network and ViT- +Base [43] as the student network. The batch size is set to +64 for both source-domain and target-domain datasets. In one +batch, the sampler will select 16 identities and 4 images for +each identity according to the ground-truth label or pseudo +label for two domains. The input image has a fixed size of +256 × 128. +In the pre-training stage, we first train models 120 epochs +on the source-domain dataset. The teacher CNN model is +optimized by SGD with an initial learning rate of 1 × 10−2 +and weight decay of 5 × 10−4 with a learning rate decays at +40th and 70th epoch. The SGD optimizer is employed with +a momentum of 0.9 and the weight decay of 1 × 10−4 for +student ViT. The learning rate is set to 8 × 10−3, and the +cosine schedule is applied. The input images are augmented +with random flip and randomly erase with 50% probability. +In the fine-tuning stage, we adopted half the learning rate +of the previous stage. Specifically, the learning rate is set to +5 × 10−3 for teacher CNN and 4 × 10−3 for student ViT. +And the total number of training epochs is set to 60. The +input images for two heterogeneous networks are randomly +flipped and erased with 50% probability. When calculating the +neighbors of an instance, the maximum acceptable distance α +is 0.6. We generate the pseudo labels by DBSCAN [46]. For +DBSCAN, we select 0.6 as the maximum distance between + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +7 +TABLE III +INFLUENCE OF DIFFERENT BACKBONES IN TERMS OF mAP (%) AND RANK-1 (%) ON CUHK-SYSU (CS) → MARKET-1501 (M). +Method Backbone +Training +Parameter +Testing +Parameter +CS → M +mAP +R1 +R5 +R10 +MMT +IBN-ResNet-50 + IBN-ResNet-50 +99.8M +24.9M +76.0 +88.8 +95.2 +97.0 +MMT +ViT-Base + ViT-Base +345.2M +86.3M +75.2 +86.7 +94.2 +96.4 +UNRN +ResNet-50-NL +77.1M +38.5M +70.9 +86.7 +92.8 +94.5 +UNRN +ViT-Base +185.4M +86.3M +73.2 +86.8 +93.2 +95.2 +DAML +IBN-ResNet-50 + ViT-Base +111.2M +86.3M +84.1 +93.1 +97.7 +98.2 +neighbors and set the minimal number of neighbors to 2 for +CUHK-SYSU and 4 for others. The hype-parameters α, λ1, +λ2 and λ3 are set to 0.5, 0.1, 0.7 and 1.2, respectively. At the +feature level, the margin ρ for triplet loss is set to 1.2. +C. Comparison with State-of-the-art Methods +Since Duke University terminated the DukeMTMC [47] +dataset, which has been widely used for evaluation of unsu- +pervised domain adaptation person Re-ID task, the comparison +becomes difficult. To meet the moral and ethical requirements +and provide a new baseline for comparison, we evaluate the +performance of some representative works which have official +open-source codes based on the CUHK-SYSU dataset. And +the results in Market-1501 → MSMT17 setting is from the +authors’ reports. We compare our DAML with state-of-the- +art (SOTA) unsupervised domain adaptation person Re-ID +approaches. MMT [25] applies two networks that have the +same structure for learning from each other with both feature- +level and logit-level constraints. MEB-Net [27] introduces +three homogeneous networks, and the output of each network +is considered comprehensively in the pseudo label generation. +Moreover, UNRN [30] designs a memory bank storing class +centers from both source and target domains to mitigate the +influence of noise labels. As shown in Tab. I, we evaluate +the performance in four different manners, i.e., Market-1501 +→ CUHK-SYSU, CUHK-SYSU → Market-1501, Market- +1501 → MSMT17, and CUHK-SYSU → MSMT17. +Comparisons on large-scale datasets: The comparison +results on Market-1501 → MSMT17 and CUHK-SYSU → +MSMT17 are shown in the bottom of Table I. The proposed +DAML outperforms existing SOTAs by large margins. Specifi- +cally, DAML achieves the Rank-1 accuracy of 65.4% and mAP +of 41.4% in the Market-1501 → MSMT17 setting, signifi- +cantly improving the Rank-1 accuracy by 11.0% and mAP by +14.8% over the SOTA MMT. When compared to the SOTAs in +CUHK-SYSU → MSMT17 setting, the performance margin +between our DAML and MMT is also significantly, e.g., the +Rank-1 boost is 18.0%, and the mAP boost is 20.0%. +Comparisons on small-scale dataset: We also evalu- +ate DAML on two small-scale target-domain datasets set- +tings, Market-1501 → CUHK-SYSU and CUHK-SYSU → +Market-1501, as shown in the top of Table I. Similar to the re- +sults on large-scale datasets, DAML consistently outperforms +current SOTAs. Specifically, we achieve Rank-1 accuracy of +86.2% and mAP of 84.3% in Market-1501 → CUHK-SYSU +setting. Compared with the SOTA MEB-Net, the Rank-1 and +mAP respectively improved by 3.0% and 3.2%. Meanwhile +Rank-1 accuracy of 93.1% and mAP of 84.1% are gained +in CUHK-SYSU → Market-1501 setting. It improves the +Rank-1 accuracy and mAP by 4.3% and 8.1% compared with +the SOTA MMT. Note that the performance on Market-1501 +→ CUHK-SYSU setting is even worse than direct transfer. +Because there are only two samples per class in CUHK-SYSU +on average and it harms the pseudo label generation. We will +discuss this problem in section Samples Augment. +The above results demonstrate the outstanding performance +of DAML thanks to its ability to learn knowledge from differ- +ent subspaces and selectively transfer knowledge between two +heterogeneous networks for unsupervised domain adaptation +person Re-ID. +D. Ablation Study +In this section, we conduct ablation experiments on CUHK- +SYSU → Market-1501 setting to assess the contribution of +each component by separately removing them from DAML +for training and evaluation. +As shown in Table II, when removing LT sid+Ltri(sT ), the +Rank-1 accuracy drops by 0.3% and mAP drops by 0.5%, since +the reliable identity information is underutilized. It illustrates +that the ability to use the information in the source domain +effectively is an essential factor in determining the model’s +performance. It illustrates the essential to effectively use the +information in the source domain When removing Lid, which +helps student network to learn the identity knowledge from the +teacher, the performance drops of Rank-1 and mAP are 0.8% +and 0.8%, respectively, compared with the full DAML. The +performance drops due to the ignorance of the knowledge from +the different subspaces in the logit level. And the knowledge +can still transfer to each other through the pseudo label +generation. Similarly, to validate the effectiveness of Ldom, we +remove it from DAML. The result also shows the margin of the +Rank-1 accuracy by 0.6% and mAP by 0.5% to the complete +DAML, which demonstrates that Ldom effectively helps to +make the teacher network predict samples in The smooth +classifier update (SCU) saves the knowledge learned in the +last epoch. When it is removed, the Rank-1 accuracy drops by +2.1%, and mAP drops by 3.1%. The results prove that learning +the knowledge from different subspaces and taking advantage +of correct identity information from the source domain are the +two keys to solving UDA person Re-ID. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +8 +Pre-trained CNN +Input Image +Pre-trained ViT +CNN-ViT +CNN-CNN +ViT-CNN +ViT-ViT +Pre-trained CNN +Pre-trained ViT +Fig. 3. Visualization results of the models on Market-1501. For each line, we show an input image, the area considered by the pre-trained +ViT, the pre-trained CNN, and the different combinations of teacher and student in turn. +TABLE IV +ASYMMETRIC DISTILLATION ANALYSIS IN TERMS OF mAP (%) +AND RANK-1 (%) ON CUHK-SYSU (CS) → MARKET-1501 +(M). +Method +CS → M +Teacher +Student +mAP +R1 +IBN-ResNet-50 ViT-Base +84.1 +93.1 +ViT-Base +ViT-Base +82.0 +91.7 +ViT-Base +IBN-ResNet-50 +80.1 +91.3 +IBN-ResNet-50 IBN-ResNet-50 +79.7 +91.0 +E. Discussions +1) Influence of Backbone: To meet the requirement of +heterogeneous networks in the proposed DAML, we introduce +the ViT-Base, which contains more trainable parameters as the +backbone. To clarify the source of performance growth, we +repeat the experiments of MMT [25] and UNRN [30] while +replacing the backbone with ViT-Base. As shown in Tab. III, +”Backbone” represents the construction to extract features in +the testing stage. When the ViT-base replaces the backbone, +the optimization process follows the setting in TransReID [48]. +As shown in table. III, after replacing the backbone with ViT- +Base, there is no significant change in results. Limited by +the symmetrical design in mutual learning manner and the +high similarity in the classifying ways of ViTs, the Rank-1 +and mAP of MMT dropped 2.1% and 0.8%, respectively. For +the UNRN method, which focuses on making pseudo labels +reliable through the memory mechanism, the ViT brings 0.1% +and 2.3% in Rank-1 and mAP with much more parameters. +Based on the above experiments, we can conclude that the +heterogeneous networks and the asymmetric learning strategy +play a major role in the growth of performance. +2) Heterogeneous Networks Analysis: One of the keys to +improving UDA person Re-ID is learning knowledge from the +different subspaces. To illustrate the effect of heterogeneous +networks, we train the proposed DAML with different combi- +nations of teacher and student. From Table. IV, we can figure +out that the student with a heterogeneous teacher will achieve +better performance. Specifically, the performance of student +ViT improved by 0.7% and 1.1% in Rank-1 and mAP with the +asymmetric teacher. The results in the student CNN is similar, +Rank-1 and mAP are enhanced by 2.5% and 5.3%. These +experimental results strongly prove the necessity of using two +heterogeneous networks to work as the teacher and student. +And the knowledge from different subspaces has the capacity +to help the student to learn broader knowledge. +When ViT is seen as the student, the benefit from the +heterogeneous teacher is more evident than the improvement +that CNN works as the student. The heterogeneous teacher +for ViT brings in 1.4% and 2.1% on Rank-1 and mAP. While +it only improves Rank-1 and mAP in 0.3% and 0.4% for +the student CNN. This phenomenon can be ascribed to the +difference in the range of receptive field of these two networks +that the former can consider the relationship between any two +areas, but the size of convolution kernels limits the latter. It +gives ViT has the ability to learn the pattern that CNN applied +to classify identities but not vice versa. +3) Visualization: The proposed DAML can make the stu- +dent learn the knowledge from different subspaces. To further +illustrate the effectiveness of DAML, which can selectively +transfer the knowledge between two networks, we apply +Score-CAM [49] to visualize the pixel-wise attention areas +on CUHK-SYSU → Market-1501 setting. Fig. 3 visualizes +individual attention patterns for the three people from the +target domain, where each column represents the attention +area of the pre-trained CNN, ViT, and the different combi- + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +9 +TABLE V +INFLUENCE OF SAMPLE AUGMENT FOR CLUSTERING ALGORITHM +IN TERMS OF mAP (%) AND CMC (%) ON MARKET-1501 (M) → +CUHK-SYSU (CS). +Method +Repeat +Times +M → CS +mAP +R1 +R5 +R10 +Directly Transfer +- +86.0 +87.2 94.1 95.0 +DAML +0 +84.3 +86.2 92.6 94.6 ++ Random Crop +1 +89.3 +90.6 95.5 96.9 +2 +89.1 +90.4 95.4 96.6 ++ Random Erase +1 +88.3 +90.0 95.0 96.3 +2 +89.2 +90.6 95.5 96.6 ++ Random Crop ++ Random Erase +1 +88.5 +89.8 95.4 96.8 +2 +87.6 +89.0 95.2 96.6 +Supervised +- +90.8 +95.2 96.6 89.0 +nations of teacher-student. From the first two columns, we +can observe that the classifying patterns of CNN and ViT are +different, which states the difference between their embedding +spaces. With these discrepancies, the heterogeneous networks +can be improved by learning knowledge from heterogeneous +networks. In the last four columns, we can find that the +networks mutual learning with heterogeneous networks can +better consider the individual by the whole pedestrian while +also taking into account many details that identify the persons +more efficiently. On the contrary, the recognition patterns of +the networks that mutual learning with the same network +have no significant change. The visualization demonstrates the +function of DAML in learning the knowledge from different +subspaces improving the performance of the student. +4) Samples Augment: Due to the small number of samples +for each class in CUHK-SYSU, the performance of clustering +algorithm is severely limited The simplest and most direct way +to address this problem is by augmenting the samples with +random erase [50] and random crop, which can generate new +samples while keeping the original identity when extracting +features for the clustering algorithm. As shown in Tab. V, the +performance with augmented data is better than the original. +Specifically, the Rank-1 and mAP are enhanced by 4.4% +and 5.0% when applying the random crop method. Similarly, +the random erase improves Rank-1 and mAP by 4.4% and +4.9%. The above experiment results state the necessity of +enough samples for each class in the clustering algorithm. +Nevertheless, applying both random crop and random erase +is not as effective as applying only one. The Rank-1 and mAP +are only increased by 3.6% and 4.2%. It suggests that the +excessive augment method may harm identity knowledge and +reduce the benefits from the augmented samples. +Compared to datasets collected for research, the number of +identities and the number of samples in each identity are un- +known in a real-world system. And this information cannot be +counted on the raw data unless annotated on them. However, +one of the advantages of unsupervised domain adaptation Re- +ID is avoiding the annotation on the target domain, and it +means the class-dependent super parameters are not available. +Because clustering algorithms elapse a long time to run on +large datasets and take up most of the total training time, the +super parameter selection experiments may be unacceptable +for real-world systems. Thus, a method without any clustering +algorithm that still can mine identity knowledge from the +target domain may make more sense for applying to the real +world. On the other hand, an efficient data augment method +can restore the UDA methods based on clustering algorithm. +V. CONCLUSION +In this paper, we proposed the Dual-level Asymmetric +Mutual Learning, termed DAML, to learn knowledge from +a broader scope via asymmetric mutual learning with hetero- +geneous networks for unsupervised domain adaptation person +Re-ID. Our method aims to learn the knowledge from various +subspaces and transfer the identity knowledge from the source +to the target domain. The former can improve feature expres- +siveness while also rectifying potential faults during training. +The latter takes full advantage of the identity knowledge +from the source domain to improve the performance in the +target domain. Specifically, DAML first generates the pseudo +labels according to the features extracted by heterogeneous +networks, which are more reliable due to the consideration +of various subspaces. And the knowledge from two subspaces +can be exchanged in a hard distillation manner in this process. +With the smooth classifier update, the classifiers can maintain +the knowledge from the last epoch. Then, the teacher will +train on both source-domain and target-domain datasets to +utilize the ground-truth label and transfer the knowledge +to the target domain with the domain knowledge from the +student. To better adapt to the target domain, the student only +trained on the target-domain dataset and benefited from the +guidance from the teacher, which had learned the source- +domain knowledge. Experiments on four different experiment +settings prove essential to learn the knowledge from various +subspaces and demonstrate the effectiveness of the proposed +DAML for unsupervised domain adaptation person Re-ID. +REFERENCES +[1] S. Gong, M. Cristani, C. C. Loy, and T. M. Hospedales, “The re- +identification challenge,” in Person Re-Identification, 2014, pp. 1–20. +[2] W. Li, X. Zhu, and S. Gong, “Scalable person re-identification by +harmonious attention,” Int. J. Comput. Vis., pp. 1635–1653, 2020. +[3] Y. Sun, L. Zheng, Y. Li, Y. Yang, Q. Tian, and S. Wang, “Learning part- +based convolutional features for person re-identification,” IEEE Trans. +Pattern Anal. Mach. Intell., pp. 902–917, 2021. +[4] J. Yin, A. Wu, and W. Zheng, “Fine-grained person re-identification,” +Int. J. Comput. 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Yang, “Random erasing +data augmentation,” in AAAI, 2020, pp. 13 001–13 008. + diff --git a/e9FMT4oBgHgl3EQf1DG_/content/tmp_files/load_file.txt b/e9FMT4oBgHgl3EQf1DG_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ac1883a747577c46ba2b70dd3200293e5de3690 --- /dev/null +++ b/e9FMT4oBgHgl3EQf1DG_/content/tmp_files/load_file.txt @@ -0,0 +1,1035 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf,len=1034 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 1 Unsupervised Domain Adaptation on Person Re-Identification via Dual-level Asymmetric Mutual Learning Qiong Wu, Jiahan Li, Pingyang Dai, Qixiang Ye, Senior Member, IEEE, Liujuan Cao, Member, IEEE, Yongjian Wu, Rongrong Ji, Senior Member, IEEE Abstract—Unsupervised domain adaptation person re- identification (Re-ID) aims to identify pedestrian images within an unlabeled target domain with an auxiliary labeled source- domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Many existing works attempt to recover reliable identity information by considering multiple homogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And take these generated labels to train the model in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' However, these homogeneous networks identify people in approximate subspaces and equally exchange their knowledge with others or their mean net to improve their ability, inevitably limiting the scope of available knowledge and putting them into the same mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' This paper proposes a Dual- level Asymmetric Mutual Learning method (DAML) to learn discriminative representations from a broader knowledge scope with diverse embedding spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Specifically, two heterogeneous networks mutually learn knowledge from asymmetric subspaces through the pseudo label generation in a hard distillation manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The knowledge transfer between two networks is based on an asymmetric mutual learning manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The teacher network learns to identify both the target and source domain while adapting to the target domain distribution based on the knowledge of the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Meanwhile, the student network is trained on the target dataset and employs the ground-truth label through the knowledge of the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Extensive experiments in Market-1501, CUHK-SYSU, and MSMT17 public datasets verified the superiority of DAML over state-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Index Terms—Transfer learning, unsupervised domain adap- tation, person re-identification, retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Qiong Wu is with Institute of Artificial Intelligence, and the Media Analytics and Computing Laboratory, Department of Artificial Intelligence, School of Informatics, Xiamen University, Xiamen 361005, China (e-mail: qiong@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Jiahan Li is with School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221000, China (e-mail: jia- han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='li@cumt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Pingyang Dai is with the Media Analytics and Computing Laboratory, De- partment of Artificial Intelligence, School of Informatics, Xiamen University, Xiamen 361005, China (e-mail: pydai@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Qixiang Ye is with the Peng Cheng Laboratory, Shenzhen 518066, China, and also with the School of Electronics, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China (e-mail: qxye@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Liujuan Cao is with the Media Analytics and Computing Lab, Department of Computer Science, School of Informatics, Xiamen University, Xiamen 361005, China (e-mail: caoliujuan@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Yongjian Wu is with the Youtu Laboratory, Tencent, Shanghai 200233, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (e-mail: littlekenwu@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Rongrong Ji is with the Media Analytics and Computing Laboratory, De- partment of Artificial Intelligence, School of Informatics, Xiamen University, Xiamen 361005, China, also with the Fujian Engineering Research Center of Trusted Artificial Intelligence Analysis and Application, Institute of Artificial Intelligence, Xiamen University, Xiamen 361005, China, and also with the Peng Cheng Laboratory, Shenzhen 518066, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (e-mail: rrji@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The statistics of common neighbors between different mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' CNN-CNN and ViT-ViT curves denote the average number of common neighbors in the k nearest neighbors of each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The features are extracted by two homogeneous networks trained with different initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Similarly, CNN-ViT represents the common neighbors between CNN and ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Furthermore, Upbound refers to the maximum number of neighbors to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' These models are trained on the CUHK-SYSU dataset in a supervised manner and cluster on the Market1501 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Compared to the CNN-CNN and ViT-ViT, the CNN-ViT contains fewer common neighbors, and the ways they distinguish two individuals are more different than homogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It demonstrates that heterogeneous networks address the task in different patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' INTRODUCTION P ERSON re-identification (Re-ID) [1] aims at matching individual pedestrian images from images captured by different cameras according to identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' This task is chal- lenging because the variations of viewpoints, body poses, illuminations, and backgrounds will influence a person’s ap- pearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Recently, supervised person Re-ID methods [2]–[11] made impressive progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' However, as the number of images increases and the ensuing scene changes, regular supervised learning approaches are losing their ability to adapt to complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The performance of person re-ID models trained on existing datasets will evidently suffer for person images from a new video surveillance system due to the domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To avoid time-consuming annotations on the new dataset, unsupervised domain adaptation (UDA) is proposed to adapt the model arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='12439v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='CV] 29 Jan 2023 Upbound 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 CNN-ViT CNN-CNN ViT-ViT 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 Common Neighbor 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 0 5 10 15 20 25 Range of NeighborhoodJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 2 trained on the labeled source-domain dataset to the unlabeled target-domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Generating trusted identity information on the target domain is seen as the core of the UDA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Some UDA Re-ID methods [10], [12]–[15] directly apply GANs [16] to transfer the style of pedestrian images from the source domain to the target while keeping the identities to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' However, the complexity of the human form and the limited number of instance in a Re-ID dataset limit the quality of generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' After abandoning the image generation, some methods [7], [17] introduce the attribute to bridge the domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' These methods introduce additional annotation information which defeats the purpose of the UDA Re-ID task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Limited by the missing label on the target domain, others [18]–[21] align the distributions of target and source domains while only learning classifying on the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To better adapt the distribution of the target domain and train with the target-domain identity knowledge, various methods [22]– [24] apply a clustering algorithm in the target domain to generate the pseudo labels for training in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' One of the keys to improving performance is alleviating the influence of noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In this context, many methods [25]– [27] based on clustering algorithms are proposed to rule out the harmony from the noisy labels by introducing more than one framework to predict pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' They aim to generate knowledge with specific differences in samples and exchange the knowledge among the networks to enhance their ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Despite encouraging progress, the benefits from the knowledge mined by homogeneous networks are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 1 CNN-CNN and ViT-ViT, these homogeneous networks with similar structures identify pedestrians in a comparable manner, and the relation among the instances are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It suggests they use similar patterns to extract pedestrian features, and networks may converge to equal each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Furthermore, this mode of operation makes it possible for the networks to make the same mistakes and not be able to correct them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Such a design limits the knowledge models can learn from the training set and makes it possible for the networks to repeat mistakes without being able to remedy them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As a result, mining the information from different subspaces is required to broaden the scope of knowledge and generate reliable pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To tackle this problem, heterogeneous networks, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 1 CNN-ViT, can discover the information from multiple subspaces and have more extensive latent knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' We propose Dual-level Asymmetric Mutual Learning (DAML), a novel unsupervised domain adaption method for person Re- ID that broadens the scope of knowledge for the network by exploiting information from two different subspaces and selectively transferring information between heterogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The proposed DAML consists of a CNN that focuses on identity learning as a teacher network and a ViT that concentrates on adapting knowledge from the target domain as a student network for embedding samples into different subspaces and setting the constraints among the classifiers for asymmetric mutual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In particular, the CNN that works as a teacher will train on both source and target datasets under the supervision of ground-truth source-domain labels and pseudo-target-domain labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The former can provide reliable identity information for extracting discriminative feature representation, while the latter will assist the network in adapting the distribution of the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' However, learning from the source domain will harm the distribution that the network adapted limiting the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To avoid this disadvantage, the ViT that works as a student only trains with the guidance of pseudo-target- domain labels and learns the knowledge from the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In the pseudo label generation stage, the relationship between two samples is weighted according to their teacher and student features similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Moreover, this process wholly exchanges the knowledge learned from two different subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' After predicting the identities of input images, the asymmetric constraints between two heterogeneous networks selectively exchange the knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The student learns the identity knowledge from the teacher network under the constraints from the target-domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Furthermore, for the student can benefit more from the teacher and better utilize the ground- truth labels, the source-domain identity knowledge learned by the teacher is transferred to the target domain with the constraints based on source-domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In summary, the DAML employs diverse subspaces to generate reliable pseudo label in the target domain and help student adopt ground-truth knowledge in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Our main contributions are summarized below: We address the diverse subspaces learning and target- domain identity learning for unsupervised domain adapta- tion person Re-ID with proposed Dual-level Asymmetric Mutual Learning (DAML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The former has rarely been studied in the existing research, while the latter is crucial for retrieving person in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' We propose a novel Dual-level Asymmetric Mutual Learning (DAML) method for unsupervised domain adaptation person Re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The asymmetric knowledge learning between the teacher and the student helps them play their roles better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To learn from diverse subspaces, the proposed DAML introduces two heterogeneous networks to mine valuable information from different subspaces and selectively ex- change the information between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To better utilize the knowledge mined by heterogeneous networks and ensure the networks orient to the task, the proposed DAML smoothly update the classifiers in a hard distillation manner and exchange knowledge during training in a soft distillation manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Unsupervised Domain Adaptation Person Re-ID Unsupervised Domain Adaptation Person Re-ID has at- tracted increasing attention in recent years due to its effec- tiveness in reducing manual annotation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' There are two main categories of methods are proposed to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Firstly, GAN-based methods aim to transfer samples from the source domain to the target domain without altering their identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' SPGAN [12] and PDA-Net [13] transfer images directly from the source domain to the target domain while JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 3 maintaining the original identity knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The generated images have a similar style to the target-domain images and are used to train the model under the supervision of their original labels in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To produce generated images that are more realistic and have more detail, DG- Net [28] and DG-Net++ [15] introduce disentanglement for the generation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' But the generation is expensive and the style of generated images may not well fit the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Rather than transfer images from the source domain to the target domain, HHL [10] transfers target-domain images among the cameras to generate images that have the same identity but contain the difference at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Secondly, the clustering-based methods clustering based methods do not require expensive GAN networks for generation and have achieved state-of-the-art performance to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To reduce the impact of noisy label, MMT [25] proposed a mutual learning method providing soft labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' For more reliable pseudo labels, SSG [29] clusters samples in three scales and validate each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' MEB-Net [27] respectively introduces multiple groups of prototypes or homogeneous networks to generate the pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' UNRN [30] and GLT [26] design a memory bank to save anchors for aligning the distribution and learning identities in a contrastive learning manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Limited by the constraints in the feature level these methods rely on, the models that collaborate to generate pseudo labels are ho- mogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' These characteristics determine that the model can only learn similar knowledge from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Nevertheless, these approaches alleviate the domain gap only considering the single embedding space inevitably makes some mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Knowledge Distillation Knowledge distillation makes a student network learns from a strong teacher network to improve the student’s ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The common approaches can be summarized as hard distillation and soft distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Soft distillation [31], [32] minimizes the distribution difference between the prediction generated by teacher and student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The soft label generated by the teacher model can alleviate overfitting just like labels smoothing [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Unlike the soft, hard-label distillation regards the prediction result of the teacher as a valid label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And positive pairs pre- dicted by the teacher are used to transfer identity knowledge from the teacher to a student network in semi-supervised and unsupervised learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Temporal ensembling [34] put the former networks as the teacher and use memory saving average predictions for each sample as supervision for the unlabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To avoid storing predictions for saving memory, Mean Teacher [35] averaged student model weights as the parameter of the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' During the training, the predictions made by the teacher are seen as supervision for unlabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The models consider similar information in these methods because the teacher and the student have the same structure and similar initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It makes the networks focus within a certain range and limit the knowledge student can learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The proposed DAML exchange knowledge utilizes both soft and hard distillation in the different training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Thanks to the heterogeneous networks, the proposed DAML gives the student model a broader perspective and can generate pseudo labels from different views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' CNN and ViT Since AlexNet [36] achieve great success on ImageNet [37], a variety of convolutional neural networks (CNN) [38]–[41] is proposed to solve different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As Transformers [42] were proposed for machine translation and were seen with significant results in many NLP tasks, the application of self- attention to images is widely concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' A new model without any convolution, Vision Transformers (ViT) [43], has been proposed for computer vision tasks and shows its potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' During the calculation process, the CNN keeps the spatial information and can only focus on the surrounding area in one layer due to the nature of convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In contrast, ViT em- phasizes the correlation between two patches, and its receptive field involves the whole feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' These differences make the CNN and ViT learn different knowledge from the training set for the same task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And in our paper, we take advantage of this difference to achieve asymmetric distillation, making ViT a better performer with our DAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The ViT works as a student because the receptive field of a patch in the ViT covers the area that one convolution kernel can consider, not vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Overview The ultimate goal of the unsupervised domain adaptation (UDA) person Re-ID is to gain a model work on a target- domain dataset based on a labeled source-domain dataset and an unlabeled target-domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Let S = {(xi s, yi s)}Ns i=1 and T = {xi t}Nt i=1 respectively denote the source-domain images with ground-truth labels and the unlabeled target- domain images, where Ns and Nt are the numbers of samples from these two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 2, the Dual-level Asymmetric Mutual Learning (DAML) method trains the student to extract discrim- inative representations from two different subspaces to perform the UDA person Re-ID task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Firstly, DAML adopts two heterogeneous networks: teacher CNN ET (·) and student ViT ES(·) which are pre-trained on the source-domain dataset in a supervised manner to extract features in different subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' At each epoch, we first group target-domain samples into K classes by the clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The distance between two target-domain samples will be calculated according to the features ET (xi t) = tT i ∈ RcT and ES(xi t) = tS i ∈ RcS ex- tracted by the teacher and student models with corresponding weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The {ˆyi}Nt i=1 are the pseudo labels for the target- domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Then, for each class center cy, we generate its prediction with the classifiers C(·|WS t ) and C(·|WT t ) for updating the parameter WS t and WT t in a smooth method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' After that, we train the teacher and the student models with the pseudo labels in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' For the teacher model, classifier C(·|[WT s , WT t ]) will learn both source- domain and target-domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' While the classifier C(·|WS t ) for the student model only directly learns the target- domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The constraints between two networks transfer the identity knowledge learned by the teacher to the target and help the student learn from diverse subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 4 Source data Both data Target data CNN Teacher ViT Student Classifier [W��, W� �] source domain target domain 𝐿�� 𝐿�� 𝐿�� Classifier [W� �] Classifier [W� �] 𝐿��� Classifier [W� �, W� �] Update Classifier [W� �, W� �] Beginning of Each Epoch Cluster Base on Weighted Features ··· ··· Features in Different Spaces Asymmetric Dual Networks Predictions of Cluster Centers 𝐿��� 𝐿�� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Overview of our Dual-level Asymmetric Mutual Learning method (DAML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The teacher network is trained under the supervision of pseudo labels and ground-truth labels for target-domain and source-domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And the student only directly learns knowledge from target-domain samples with pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' At the beginning of epochs, we first generate the pseudo labels for target dataset, and update the classifiers based on the predictions of cluster centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To distill the different subspace knowledge from the teacher to the student, Lid makes the student predictions of target-domain samples close to the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Meanwhile, for student can better adopt the identity knowledge learned by the teacher, we minimize the distribution differences of the same source-domain samples with Ldom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Finally, we only adopt the features tS i = ES(xi t) extracted by the student model for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Smooth Classifier Update (SCU) At the beginning of epochs, we extract the target-domain features tT i = ET (xi t) and tS i = ES(xi t) with two hetero- geneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To better utilize the knowledge from the two models, we first define the neighborhood of an instance according to its relations in two different subspaces: Ni = � xj �����1 − ⟨tM i , tM j ⟩ ∥tM i ∥2∥tM j ∥2 < α, M ∈ {T, S} � , (1) the α here is a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' With the limitation of neigh- bor selection considering both teacher features and student features simultaneously, the neighbors of an instance should be close to it in both subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The above constraint ensures that instances with apparent differences will not be clustered as the same identity because the patterns of the two models applied to recognize an instance are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To exploit the information from two different subspaces and make the pseudo labels more reliable, we combine features from heterogeneous networks and define the distance between two samples as: di,j = � � � � � 1 − ⟨[tT i , tS i ], [tT j , tS j ]⟩ ∥[tT i , tS i ]∥2∥[tT j , tS j ]∥2 , xi ∈ Nj and xj ∈ Ni Inf, Others (2) where [·, ·] represents the concatenation of two features, and ⟨·, ·⟩ denotes the dot product between two features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In short, we define the similarity between two samples as the cos similarity between the features constructed by concatenating their teacher feature and student feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Then the pseudo labels can be generated based on the relationship among instances with the clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' With the pseudo labels, some methods [25], [30] directly update the classifier by replacing the classifier parameters with the new class centers to adapt the count of classes change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' These methods will make the knowledge lost because the class centers may not represent the corresponding class well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To protect the knowledge involved in the classifiers, we update the classifiers more smoothly as follows: Wi t = ˆ K � k=1 ˆ Wk t · epk i � ˆ K j=1 epj i , (3) where Wi t is the parameters for the ith target-domain identity in the next epoch, pi = C(ci| ˆ Wt) is the prediction of class center ci with the parameters ˆ Wt from the last epoch which includes ˆK classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Note that the momentum for SGD is updated following the parameters in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Identity Learning The core of person re-identification is identifying the per- sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' For two heterogeneous networks learning to extract discriminative representation, there are two level objective JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 5 functions are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Firstly, at the feature level, the triplet loss: Ltri(f) = 1 n n � i=1 max{ρ + dp − dn, 0}, (4) is applied to guarantee the features can well represent their corresponding samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Where f represents a batch of the features, n = |f| is the size of the batch, ρ is the tiniest margin between the distance to the furthest positive instance dp and the distance to the nearest negative instance dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The relationship between two instances from the source domain depends on the ground-truth labels and the pseudo labels for target-domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Due to the different dimensions of the features extracted by heterogeneous networks, the triplet loss can only be applied in a certain subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Then, in the logits level, we apply the cross-entropy loss with classifiers: LT tid = − 1 n n � i=1 log P(ˆyi|C(tT i |[WT s , WT t ])), (5) LStid = − 1 n n � i=1 log P(ˆyi|C(tS i |WS t )), (6) where ˆyi is the pseudo label for target-domain example xi t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The trainable parameters WT s , WT t and WS t respectively denote the classifier parameters for the teacher classifying source-domain samples, the teacher classifying target-domain samples, and the student classifying target-domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Meanwhile, to take advantage of the ground-truth label, the teacher also learns the source-domain knowledge by: LT sid = − 1 n n � i=1 log P(yi|C(sT i |[WT s , WT t ])), (7) here, yi is the ground-truth label for source-domain sam- ple xi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Note that, with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (7), the classifier C(tT i |[WT s , WT t ]) in the teacher has learned both two domain knowledge while classifier C(tT i |WS t ) for the student learns the target-domain knowledge only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Asymmetric Mutual Learning (AML) Compare the structure of the Convolutional Neural Network (CNN) and Vision Transformer (ViT), the most evident differ- ence is that the ViT can capture long-range information with its cascaded self-attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' However, CNN only focuses on the local limited by the size of the convolution kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In addition, the CNN inductive bias, which includes assumptions of the data, can involve information that ViT may not con- sider and the convolution kernel with a deterministic shape guarantees spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' More intuitively, the features extracted by the two networks have different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It ensures the subspaces learned by the heterogeneous networks are different but makes feature-level constraints unusable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The asymmetric distillation benefit from the difference in the patterns that two heterogeneous networks predict the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And focus on twofold: to allow students access to knowledge from the different subspaces and transfer the knowledge from the source to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To make the student learn from different subspaces and take advantage of the reliable source-domain labels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' the proposed DAML transfers the identity knowledge from the teacher by reducing the Kullback-Leibler divergence between the predictions of the target-domain features as: Lid = 1 n n � i=1 C(tT i |WT t ) log C(tS i |WS t ) C(tT i |WT t ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (8) with the above objective function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' the student can learn the knowledge from the teacher which adopts knowledge from both source and target domain with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' How- ever, domain knowledge is also transferred to students and may harm the performance in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The ideal way to alleviate the distribution effect is to make the teacher predict identities under the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Limited by the domain gap, the knowledge learned from the source domain can not be directly applied to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And the goal of the proposed asymmetric mutual learning is to gain a student network that adapts to the target domain while benefiting from the source-domain identity knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Making source-domain predictions from the teacher similar to the student will transfer the classifying knowledge learned from the source domain to the target domain: Ldom = 1 n n � i=1 C(sS i |WS t ) log C(sT i |WT t ) C(sS i |WS t ) , (9) here, WT t learns source-domain knowledge with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (7) while WS t only learns the knowledge from target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (9) focuses on making the teacher predict source-domain samples in the same way as the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In this way, the student can better adopt identity knowledge from the source domain without a domain gap as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Compared with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' (8), the above equation distills the knowledge in a different direction and together make the student model can distinguish pedestrian in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Optimization The total loss L of DAML can be summarized as: L = � LT tid + Ltri(tT ) � + � LStid + Ltri(tS) � λ1 � LT sid + Ltri(sT ) � + λ2Lid + λ3Ldom (10) where λ1, λ2, and λ3 are hype-parameters to balance the contributions of individual loss terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Datasets Datasets We evaluated our method on three public datasets Market-1501 [44], CUHK-SYSU [45] and MSMT17 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Market-1501 contains 32, 668 labeled images captured from 1, 501 identities by 6 cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The training set has 12, 936 images of 751 identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In addition, 3, 368 query images and 19, 732 gallery images from the other 750 identities are used as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' CUHK-SYSU includes 33, 901 labeled images of 8, 432 identities taken in diverse scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The training set is constructed with 5, 532 identities having 15, 088 images, JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 6 TABLE I COMPARISON OF CMC (%) AND mAP (%) PERFORMANCES WITH THE SOTA METHODS ON MARKET-1501, CUHK-SYSU AND MSMT17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Method Market-1501 → CUHK-SYSU CUHK-SYSU → Market-1501 mAP R1 R5 R10 mAP R1 R5 R10 Directly Transfer (IBN-ResNet-50) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 Directly Transfer (ViT-Base) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 UNRN [30](AAAI’21) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 MMT(IBN-ResNet-50) [25](ICLR’20) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 MEB-Net [27](ECCV’20) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 DAML (Ours) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 Supervised (IBN-ResNet-50) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 Supervised (ViT-Base) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 Method Market-1501 → MSMT17 CUHK-SYSU → MSMT17 mAP R1 R5 R10 mAP R1 R5 R10 Directly Transfer (IBN-ResNet-50) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 Directly Transfer (ViT-Base) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 MEB-Net [27](ECCV’20) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 UNRN [30](AAAI’21) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 MMT(IBN-ResNet-50) [25](ICLR’20) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 DAML (Ours) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 Supervised (ViT-Base) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 Supervised (IBN-ResNet-50) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 TABLE II ABLATION STUDY IN TERMS OF mAP (%) AND CMC (%) ON CUHK-SYSU (CS) → MARKET-1501 (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Method CS → M mAP R1 IBN-ResNet-50(Directly) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 ViT-Base(Directly) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 DAML w/o LT sid + Ltri(sT ) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 DAML w/o Lid 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 DAML w/o Ldom 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 DAML w/o SCU 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 DAML 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 IBN-ResNet-50(Supervised) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 ViT-Base(Supervised) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 and the rest is used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' There are 2, 900 images for the query and 6, 978 images for the gallery in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' MSMT17 is a large-scale dataset consisting of 126, 441 bounding boxes of 4, 101 identities caught on 12 outdoor and 3 indoor cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Among them, 32, 621 images of 1, 041 identities are used for training and 93, 820 of 3, 060 identities are used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Experiment Setting Performance Metric: As a UDA task, we select one dataset as the source-domain dataset and another as the target-domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The model is trained with the labeled source-domain training set and adapts the target domain through the unlabeled target-domain training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Then the performance is evaluated according to the student network which work on the target- domain testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In our experiments, following the standard metrics, we employ the cumulative matching characteristic (CMC) curve and the mean average precision (mAP) score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Our experiments report rank-1, rank-5, and rank-10 accuracy and mAP scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Implementation Details: In the most common setting, we select IBN-ResNet-50 [41] as the teacher network and ViT- Base [43] as the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The batch size is set to 64 for both source-domain and target-domain datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In one batch, the sampler will select 16 identities and 4 images for each identity according to the ground-truth label or pseudo label for two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The input image has a fixed size of 256 × 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In the pre-training stage, we first train models 120 epochs on the source-domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The teacher CNN model is optimized by SGD with an initial learning rate of 1 × 10−2 and weight decay of 5 × 10−4 with a learning rate decays at 40th and 70th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The SGD optimizer is employed with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 and the weight decay of 1 × 10−4 for student ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The learning rate is set to 8 × 10−3, and the cosine schedule is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The input images are augmented with random flip and randomly erase with 50% probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In the fine-tuning stage, we adopted half the learning rate of the previous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Specifically, the learning rate is set to 5 × 10−3 for teacher CNN and 4 × 10−3 for student ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And the total number of training epochs is set to 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The input images for two heterogeneous networks are randomly flipped and erased with 50% probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' When calculating the neighbors of an instance, the maximum acceptable distance α is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' We generate the pseudo labels by DBSCAN [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' For DBSCAN, we select 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 as the maximum distance between JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 7 TABLE III INFLUENCE OF DIFFERENT BACKBONES IN TERMS OF mAP (%) AND RANK-1 (%) ON CUHK-SYSU (CS) → MARKET-1501 (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Method Backbone Training Parameter Testing Parameter CS → M mAP R1 R5 R10 MMT IBN-ResNet-50 + IBN-ResNet-50 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8M 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 MMT ViT-Base + ViT-Base 345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2M 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 UNRN ResNet-50-NL 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1M 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5M 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 UNRN ViT-Base 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4M 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3M 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 DAML IBN-ResNet-50 + ViT-Base 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2M 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3M 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 neighbors and set the minimal number of neighbors to 2 for CUHK-SYSU and 4 for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The hype-parameters α, λ1, λ2 and λ3 are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' At the feature level, the margin ρ for triplet loss is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Comparison with State-of-the-art Methods Since Duke University terminated the DukeMTMC [47] dataset, which has been widely used for evaluation of unsu- pervised domain adaptation person Re-ID task, the comparison becomes difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To meet the moral and ethical requirements and provide a new baseline for comparison, we evaluate the performance of some representative works which have official open-source codes based on the CUHK-SYSU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And the results in Market-1501 → MSMT17 setting is from the authors’ reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' We compare our DAML with state-of-the- art (SOTA) unsupervised domain adaptation person Re-ID approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' MMT [25] applies two networks that have the same structure for learning from each other with both feature- level and logit-level constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' MEB-Net [27] introduces three homogeneous networks, and the output of each network is considered comprehensively in the pseudo label generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Moreover, UNRN [30] designs a memory bank storing class centers from both source and target domains to mitigate the influence of noise labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' I, we evaluate the performance in four different manners, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=', Market-1501 → CUHK-SYSU, CUHK-SYSU → Market-1501, Market- 1501 → MSMT17, and CUHK-SYSU → MSMT17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Comparisons on large-scale datasets: The comparison results on Market-1501 → MSMT17 and CUHK-SYSU → MSMT17 are shown in the bottom of Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The proposed DAML outperforms existing SOTAs by large margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Specifi- cally, DAML achieves the Rank-1 accuracy of 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4% and mAP of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4% in the Market-1501 → MSMT17 setting, signifi- cantly improving the Rank-1 accuracy by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0% and mAP by 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8% over the SOTA MMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' When compared to the SOTAs in CUHK-SYSU → MSMT17 setting, the performance margin between our DAML and MMT is also significantly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=', the Rank-1 boost is 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0%, and the mAP boost is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Comparisons on small-scale dataset: We also evalu- ate DAML on two small-scale target-domain datasets set- tings, Market-1501 → CUHK-SYSU and CUHK-SYSU → Market-1501, as shown in the top of Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Similar to the re- sults on large-scale datasets, DAML consistently outperforms current SOTAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Specifically, we achieve Rank-1 accuracy of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2% and mAP of 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3% in Market-1501 → CUHK-SYSU setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Compared with the SOTA MEB-Net, the Rank-1 and mAP respectively improved by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Meanwhile Rank-1 accuracy of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1% and mAP of 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1% are gained in CUHK-SYSU → Market-1501 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It improves the Rank-1 accuracy and mAP by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1% compared with the SOTA MMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Note that the performance on Market-1501 → CUHK-SYSU setting is even worse than direct transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Because there are only two samples per class in CUHK-SYSU on average and it harms the pseudo label generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' We will discuss this problem in section Samples Augment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The above results demonstrate the outstanding performance of DAML thanks to its ability to learn knowledge from differ- ent subspaces and selectively transfer knowledge between two heterogeneous networks for unsupervised domain adaptation person Re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Ablation Study In this section, we conduct ablation experiments on CUHK- SYSU → Market-1501 setting to assess the contribution of each component by separately removing them from DAML for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As shown in Table II, when removing LT sid+Ltri(sT ), the Rank-1 accuracy drops by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3% and mAP drops by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5%, since the reliable identity information is underutilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It illustrates that the ability to use the information in the source domain effectively is an essential factor in determining the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It illustrates the essential to effectively use the information in the source domain When removing Lid, which helps student network to learn the identity knowledge from the teacher, the performance drops of Rank-1 and mAP are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8%, respectively, compared with the full DAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The performance drops due to the ignorance of the knowledge from the different subspaces in the logit level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And the knowledge can still transfer to each other through the pseudo label generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Similarly, to validate the effectiveness of Ldom, we remove it from DAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The result also shows the margin of the Rank-1 accuracy by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6% and mAP by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5% to the complete DAML, which demonstrates that Ldom effectively helps to make the teacher network predict samples in The smooth classifier update (SCU) saves the knowledge learned in the last epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' When it is removed, the Rank-1 accuracy drops by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1%, and mAP drops by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The results prove that learning the knowledge from different subspaces and taking advantage of correct identity information from the source domain are the two keys to solving UDA person Re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 8 Pre-trained CNN Input Image Pre-trained ViT CNN-ViT CNN-CNN ViT-CNN ViT-ViT Pre-trained CNN Pre-trained ViT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Visualization results of the models on Market-1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' For each line, we show an input image, the area considered by the pre-trained ViT, the pre-trained CNN, and the different combinations of teacher and student in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' TABLE IV ASYMMETRIC DISTILLATION ANALYSIS IN TERMS OF mAP (%) AND RANK-1 (%) ON CUHK-SYSU (CS) → MARKET-1501 (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Method CS → M Teacher Student mAP R1 IBN-ResNet-50 ViT-Base 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 ViT-Base ViT-Base 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 ViT-Base IBN-ResNet-50 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 IBN-ResNet-50 IBN-ResNet-50 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Discussions 1) Influence of Backbone: To meet the requirement of heterogeneous networks in the proposed DAML, we introduce the ViT-Base, which contains more trainable parameters as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To clarify the source of performance growth, we repeat the experiments of MMT [25] and UNRN [30] while replacing the backbone with ViT-Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' III, ”Backbone” represents the construction to extract features in the testing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' When the ViT-base replaces the backbone, the optimization process follows the setting in TransReID [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As shown in table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' III, after replacing the backbone with ViT- Base, there is no significant change in results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Limited by the symmetrical design in mutual learning manner and the high similarity in the classifying ways of ViTs, the Rank-1 and mAP of MMT dropped 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' For the UNRN method, which focuses on making pseudo labels reliable through the memory mechanism, the ViT brings 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3% in Rank-1 and mAP with much more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Based on the above experiments, we can conclude that the heterogeneous networks and the asymmetric learning strategy play a major role in the growth of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 2) Heterogeneous Networks Analysis: One of the keys to improving UDA person Re-ID is learning knowledge from the different subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To illustrate the effect of heterogeneous networks, we train the proposed DAML with different combi- nations of teacher and student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' From Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' IV, we can figure out that the student with a heterogeneous teacher will achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Specifically, the performance of student ViT improved by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='7% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1% in Rank-1 and mAP with the asymmetric teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The results in the student CNN is similar, Rank-1 and mAP are enhanced by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' These experimental results strongly prove the necessity of using two heterogeneous networks to work as the teacher and student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And the knowledge from different subspaces has the capacity to help the student to learn broader knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' When ViT is seen as the student, the benefit from the heterogeneous teacher is more evident than the improvement that CNN works as the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The heterogeneous teacher for ViT brings in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1% on Rank-1 and mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' While it only improves Rank-1 and mAP in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4% for the student CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' This phenomenon can be ascribed to the difference in the range of receptive field of these two networks that the former can consider the relationship between any two areas, but the size of convolution kernels limits the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It gives ViT has the ability to learn the pattern that CNN applied to classify identities but not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 3) Visualization: The proposed DAML can make the stu- dent learn the knowledge from different subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To further illustrate the effectiveness of DAML, which can selectively transfer the knowledge between two networks, we apply Score-CAM [49] to visualize the pixel-wise attention areas on CUHK-SYSU → Market-1501 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 3 visualizes individual attention patterns for the three people from the target domain, where each column represents the attention area of the pre-trained CNN, ViT, and the different combi- JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 8, AUGUST 2021 9 TABLE V INFLUENCE OF SAMPLE AUGMENT FOR CLUSTERING ALGORITHM IN TERMS OF mAP (%) AND CMC (%) ON MARKET-1501 (M) → CUHK-SYSU (CS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Method Repeat Times M → CS mAP R1 R5 R10 Directly Transfer 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 DAML 0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 + Random Crop 1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9 2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 + Random Erase 1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='3 2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 + Random Crop + Random Erase 1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 Supervised 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0 nations of teacher-student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' From the first two columns, we can observe that the classifying patterns of CNN and ViT are different, which states the difference between their embedding spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' With these discrepancies, the heterogeneous networks can be improved by learning knowledge from heterogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' In the last four columns, we can find that the networks mutual learning with heterogeneous networks can better consider the individual by the whole pedestrian while also taking into account many details that identify the persons more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' On the contrary, the recognition patterns of the networks that mutual learning with the same network have no significant change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The visualization demonstrates the function of DAML in learning the knowledge from different subspaces improving the performance of the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' 4) Samples Augment: Due to the small number of samples for each class in CUHK-SYSU, the performance of clustering algorithm is severely limited The simplest and most direct way to address this problem is by augmenting the samples with random erase [50] and random crop, which can generate new samples while keeping the original identity when extracting features for the clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' V, the performance with augmented data is better than the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Specifically, the Rank-1 and mAP are enhanced by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='0% when applying the random crop method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Similarly, the random erase improves Rank-1 and mAP by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='4% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The above experiment results state the necessity of enough samples for each class in the clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Nevertheless, applying both random crop and random erase is not as effective as applying only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The Rank-1 and mAP are only increased by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='6% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' It suggests that the excessive augment method may harm identity knowledge and reduce the benefits from the augmented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Compared to datasets collected for research, the number of identities and the number of samples in each identity are un- known in a real-world system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And this information cannot be counted on the raw data unless annotated on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' However, one of the advantages of unsupervised domain adaptation Re- ID is avoiding the annotation on the target domain, and it means the class-dependent super parameters are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Because clustering algorithms elapse a long time to run on large datasets and take up most of the total training time, the super parameter selection experiments may be unacceptable for real-world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Thus, a method without any clustering algorithm that still can mine identity knowledge from the target domain may make more sense for applying to the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' On the other hand, an efficient data augment method can restore the UDA methods based on clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed the Dual-level Asymmetric Mutual Learning, termed DAML, to learn knowledge from a broader scope via asymmetric mutual learning with hetero- geneous networks for unsupervised domain adaptation person Re-ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Our method aims to learn the knowledge from various subspaces and transfer the identity knowledge from the source to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The former can improve feature expres- siveness while also rectifying potential faults during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' The latter takes full advantage of the identity knowledge from the source domain to improve the performance in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Specifically, DAML first generates the pseudo labels according to the features extracted by heterogeneous networks, which are more reliable due to the consideration of various subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' And the knowledge from two subspaces can be exchanged in a hard distillation manner in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' With the smooth classifier update, the classifiers can maintain the knowledge from the last epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Then, the teacher will train on both source-domain and target-domain datasets to utilize the ground-truth label and transfer the knowledge to the target domain with the domain knowledge from the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' To better adapt to the target domain, the student only trained on the target-domain dataset and benefited from the guidance from the teacher, which had learned the source- domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FMT4oBgHgl3EQf1DG_/content/2301.12439v1.pdf'} +page_content=' Experiments on four different experiment settings prove essential to learn the knowledge from various subspaces and demonstrate the 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nbellotto@dei.unipd.it +Abstract +Reconstructing accurate causal models of dynamic systems +from time-series of sensor data is a key problem in many real- +world scenarios. In this paper, we present an overview based +on our experience about practical challenges that the causal +analysis encounters when applied to autonomous robots and +how Continual Learning (CL) could help to overcome them. +We propose a possible way to leverage the CL paradigm +to make causal discovery feasible for robotics applications +where the computational resources are limited, while at the +same time exploiting the robot as an active agent that helps to +increase the quality of the reconstructed causal models. +1 +Introduction +Causal discovery approaches generally build the causal +model of the observed scenario from static or time-series +data collected and processed in advance. However, in many +real-world robotics applications, this approach could result +inefficient or even unfeasible. The link between Continual +Learning (CL) (Lesort et al. 2020) and Causality might rep- +resent a stepping stone towards the exploitation of causal +discovery algorithms (Glymour, Zhang, and Spirtes 2019) +that currently suffer many limitations in autonomous robots. +Causal inference is an active research area in different +fields, including robotics and autonomous systems (Hell- +str¨om 2021; Brawer, Qin, and Scassellati 2020; Cao et al. +2021; Katz et al. 2018; Angelov, Hristov, and Ramamoor- +thy 2019). However, most of these works overlooked some +key features that are important for real-world application, +i.e. the computational cost and the memory needed by causal +analysis when long time-series are processed to reconstruct +a causal model of the observed scenario. To this end, the +CL’s ability to enable the acquisition of more knowledge by +trained models without forgetting previous information, and +without using previous data recordings, might help to ad- +dress these problems and to achieve better result in terms +of quality of the causal analysis. For instance, a robot in +an automated warehouse with humans and various objects +(e.g. see Fig. 1) could observe and intervene in the interac- +tions among them (e.g. worker and shelf) in order to build +*This work has received funding from the EU H2020 research +& innovation programme – grant agreement 101017274 (DARKO). +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +a causal model and therefore a deep understanding of the +situation. Since the limited hardware resources though, the +robot’s causal analysis might be slow and based on a lim- +ited amount of data, leading to a low quality causal model. +The solutions suggested in this paper would allow the robot +to overcome its hardware limitations and, moreover, to im- +prove the quality of the causal models by continually feed- +ing new data for causal analysis, discarding the old col- +lected one. This would enable a more efficient use of the +robot’s memory and computing’s resources compared to ex- +isting causal discovery’s approaches. To summarise, this pa- +per proposes a Causal Robot Discovery (CRD) approach +to overcome current limitations in causal analysis for real- +world robotics applications, addressing in particular: +• the computing and memory hardware resources of the +robot, which may hinder its capability to perform mean- +ingful causal analysis; +• the update of previous causal models with new observa- +tional and interventional data from the robot to generate +more accurate ones. +The paper is structured as follows: related work about +continual learning and causal discovery are presented in Sec- +tion 2; Section 3 introduces our CRD approach and explains +how the integration of continual learning could help to over- +come the challenges of causal discovery in robotics; finally, +we conclude the paper in Section 4 discussing our current +and future work in this area. +2 +Related Work +Causal discovery: +Several methods have been developed +over the last few decades to derive causal relationships +from observational data, which can be categorized into two +main classes (Glymour, Zhang, and Spirtes 2019). The first +one includes constraint-based methods, such as Peter and +Clark (PC) and Fast Causal Inference (FCI), which rely on +conditional independence tests as constraint-satisfaction to +recover the causal graph. The second one includes score- +based methods, such as Greedy Equivalence Search (GES), +which assign a score to each Directed Acyclic Graph (DAG) +and perform a search in this score space. More recently, re- +inforcement learning-based methods have also been used to +discover causal structure (Zhu, Ng, and Chen 2020). How- +ever, many of these algorithms work only with static data +arXiv:2301.03886v1 [cs.RO] 10 Jan 2023 + +(i.e. no temporal information) and are not applicable to +time-series of sensor data in many robotics applications, +for which time-dependent causal discovery methods are in- +stead necessary. To this end, a variation of the PC algo- +rithm, called PCMCI, was adapted and applied to time-series +data (Runge 2018; Runge et al. 2019; Saetia, Yoshimura, and +Koike 2021). +Causal robotics: +Causal inference has been recently con- +sidered in robotics, for example to build and learn a Struc- +tural Causal Model (SCM) from a mix of observation and +self-supervised trials for tool affordance with a humanoid +robot (Brawer, Qin, and Scassellati 2020). Other applica- +tions include the use of PCMCI to derive the causal model +of an underwater robot trying to reach a target position (Cao +et al. 2021) or to predict human spatial interactions in a so- +cial robotics context (Castri et al. 2022). Further causality- +based approaches can be found in the robot imitation learn- +ing and manipulation area (Katz et al. 2018; Angelov, Hris- +tov, and Ramamoorthy 2019; Lee et al. 2021). However, all +these solutions rely on a fixed set of time-series for causal +analysis and do not consider the computational cost and +complexity for online update of the robot’s causal models. +Continual learning: +The concept of learning continually +from experience has been present in artificial intelligence +since early days (Weng et al. 2001). Recently this has been +explored more systematically in machine learning (Hadsell +et al. 2020; Parisi et al. 2019) and robotics (Lungarella et al. +2003; Lesort et al. 2020; Churamani, Kalkan, and Gunes +2020). To our knowledge though, few applications of the +continual learning paradigm can be found in the causality +field. Javed, White, and Bengio (2020) incorporate causality +and continual learning with an online algorithm that contin- +ually detects and removes spurious features from a causal +model. In (Kummerfeld and Danks 2012, 2013; Kocacoban +and Cussens 2019, 2020), instead, algorithms for online +causal structure learning are presented to deal with non- +stationary data. This is a key feature of data from real-world +environments, which is still under-investigated in robotics +and therefore motivates our approach proposed next. +3 +Causal Robot Discovery +A review of the literature revealed that the possible limita- +tions of autonomous robots doing causal discovery with their +own on-board sensors have not been taken into account. In- +deed, the computational and memory requirements for long +time-series of sensor data are often very demanding, making +the use of previous algorithms for causal inference unfeasi- +ble on such platforms. +Our approach is partially inspired by the works of Koca- +coban and Cussens (2019) for handling non-stationary data, +but differs from it in two ways. First of all, we adopt the +current state-of-the-art PCMCI method for causal discovery +from time-series data; second, we propose to re-learn the +causal model not only when the observed scenario changes, +but also at each new robot’s set of observations/interventions +(periodically, e.g. every few minutes). In particular, the in- +troduction of the CL paradigm could help the robot to over- +come the challenge of limited hardware resources and to +new +CM +observations +interventions +CRD +d +v +θ +old CM +p-values based +interventions +Figure 1: CRD approach: the robot provides observational +and interventional data about human-object interactions to +the CRD block. The latter generates a causal model, which +is stored and used to compare the next one built on subse- +quent robot’s observations and interventions. Based on the +p-values of the previous causal graph’s links, the CRD could +suggest the robot which links need to be better tested by fu- +ture interventions. +improve the quality of the causal analysis even with non- +stationary data. In addition, a CRD approach could benefit +from the fact that robots are physically embodied in the en- +vironment and can actively influence its dynamic processes +(i.e. by performing interventions). That is, CRD could im- +prove the accuracy of the causal model by enriching “pas- +sive” observational data from the sensors with “active” in- +terventional data from robot’s actions aimed at collecting +specific time-series for causal discovery. +Therefore, our goal is to decrease the need of hardware +resources – often scarce in autonomous systems – and to +increase the quality of the causal analysis by using the +robot as an active agent in the learning process. The use of +CRD would allow the creation of high quality causal mod- +els by continually updating them with new sensor data from +robot’s observations and interventions, without the need to +re-process the whole time-series but only new information in +an incremental fashion. The CRD system envisaged in this +paper is thought to limit the demand for hardware resources +and allow the robot to perform high quality causal discovery +in a reasonable time by using its own on-board sensor data. +The proposed approach is depicted in Fig. 1: (i) starting +from a prefixed set of variables, the robot collects meaning- +ful data by observing and intervening in the target scenario; +(ii) based on this data, a causal model is estimated using +PCMCI (Runge 2018), which computes test statistics and p- +values as causal strengths of the DAG’s links. At this stage, +differently from (Kocacoban and Cussens 2019), to increase +the accuracy of the causal discovery, the robot keeps on col- +lecting data by observing and intervening in the scenario to +create new causal models. Periodically then, the robot com- +pares the new causal models with the old ones, inheriting +from the latter only the links that minimise the p-values of +the DAG’s causal relations. By repeating this process until +the observed scenario changes, a stable version of the causal +model with minimum uncertainty levels would be reached. +This is useful not only for modeling the current scenario, +but also when it changes. Indeed, by re-using part of the +stored causal model for an initial scenario, the new causal + +discovery when the scenario changes can be significantly +sped up (Kocacoban and Cussens 2019). +Note that by iteratively discarding the old time-series +and storing only the built causal model helps to avoid the +combinatorial explosion otherwise affecting PCMCI, there- +fore allowing the robot to operate and compute new mod- +els within reasonable time. Furthermore, the catastrophic- +forgetting problem is mitigated by the fact that, during con- +tinuous operations, the robot observes similar processes with +only small incremental changes, which leads to sequences +of similar causal models reconstructed from relatively small +variations of previous ones. +The operations performed by our CRD approach are rep- +resented by the flowchart in Fig. 2 and described next. +1. The process starts with a first set of observational data +(i.e. sensors’ time-series) collected by the robot. An in- +ference matrix is estimated by performing conditional in- +dependence tests (e.g. correlation, transfer entropy) on +the time-series, producing an initial causal model. +2. Afterwards, since at the first attempt there are no stored +causal models, the “Interventions Strategy” (red block in +Fig. 2) is initially neglected and the “Causal Model Opti- +misation” (blue block) is used only to estimate and save +the causal model (CM), together with its test statistics +and p-values matrices. +3. Once the causal model of the observed scenario is ob- +tained, the robot can improve its quality by providing new +data to the CRD. In practice, the robot collects new time- +series of sensor data from the scenario so that a new infer- +ence matrix can be estimated. At this stage, two parallel +processes are executed. +- The first one, “Interventions Strategy”, compares the +CM obtained at the previous iteration with the in- +ference matrix just estimated. If the stored CM still +fits the estimated inference matrix, it means we are +in a stationary-data case and the robot is observing +the same scenario of the previous iteration. Therefore, +based on the p-values matrix of the stored CM, the +CRD might suggest the most “unrelaible” links that +need to be re-checked. These are used by the robot to +plan the next interventions. +- The second process, “Causal Model Optimisation”, +performs a fresh causal discovery on the new data +and compare the obtained CM with the one previously +stored. From this comparison, a new CM is derived that +inherits only the links minimising the p-values of the +causal graph. The result is then stored to be used at the +next iteration. +In case of stationary data, by repeating this procedure, +a stable version of the causal model with minimum uncer- +tainty can be reached. In case of non-stationary data, new +time-series will be provided to the CRD, which will detect +any significant variations by comparing the newly estimated +inference matrix with the one of the stored CM. In this case +there is no Interventions Strategy; instead, the Causal Model +Optimisation reconstructs the new model exploiting the sim- +ilarities with the stored CM to speed up the analysis. In par- +Observation +& +Intervention +Stored CM fit? +p-values based +interventions +Inference Matrix +Estimation +Causal Discovery +CM +Stored CM +Yes +Interventions +Strategy +Causal Model +Optimisation +Figure 2: CRD flowchart. +ticular, the last step is performed by comparing the new in- +ference matrix with the old CM’s one in order to identify +previous causal links that are still valid for the new model. +4 +Conclusion +In this paper we considered the hardware resource limita- +tions of autonomous robots, which are crucial to perform +causal inference, and proposed a new approach for causal +robot discovery to overcome some of the main challenges. +This includes improving the quality of the causal models by +using the robot as an active agent in the learning process. +To summarise, in both stationary and non-stationary data +cases, the CRD discards the time-series data after each new +CM reconstruction, allowing the robot to perform causal +discovery within reasonable time. Moreover, as already ex- +plained, in case of non-stationary data the old CM can be +partially exploited to speed up the reconstruction of the new +one. This favors not only the execution time of the causal +analysis but also the handling of catastrophic-forgetting phe- +nomena. Indeed, in case of non-stationary data, assuming +small and incremental variations of the observed scenario, +the new causal model is reconstructed by partially exploit- +ing the old one, thus reducing the possibility of completely +forgetting what was previously learnt. +Future work will be devoted to the implementation and +application of this approach to real-wold robotics prob- +lems, with a special interest in industrial scenarios involving +human-robot interaction and collaboration. + +References +Angelov, D.; Hristov, Y.; and Ramamoorthy, S. 2019. Using +causal analysis to learn specifications from task demonstra- +tions. In Proc. of the Int. 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Conf. on Learning Rep- +resentations, ICLR. + diff --git a/fNE2T4oBgHgl3EQfbgef/content/tmp_files/load_file.txt b/fNE2T4oBgHgl3EQfbgef/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e12eb7f5704b78474b399874234e15e381d31ab6 --- /dev/null +++ b/fNE2T4oBgHgl3EQfbgef/content/tmp_files/load_file.txt @@ -0,0 +1,377 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf,len=376 +page_content='From Continual Learning to Causal Discovery in Robotics * Luca Castri, 1 Sariah Mghames, 1 Nicola Bellotto 1,2 1University of Lincoln, UK 2University of Padua, Italy {lcastri, smghames}@lincoln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='uk, nbellotto@dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='it Abstract Reconstructing accurate causal models of dynamic systems from time-series of sensor data is a key problem in many real- world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In this paper, we present an overview based on our experience about practical challenges that the causal analysis encounters when applied to autonomous robots and how Continual Learning (CL) could help to overcome them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' We propose a possible way to leverage the CL paradigm to make causal discovery feasible for robotics applications where the computational resources are limited, while at the same time exploiting the robot as an active agent that helps to increase the quality of the reconstructed causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 1 Introduction Causal discovery approaches generally build the causal model of the observed scenario from static or time-series data collected and processed in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' However, in many real-world robotics applications, this approach could result inefficient or even unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The link between Continual Learning (CL) (Lesort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2020) and Causality might rep- resent a stepping stone towards the exploitation of causal discovery algorithms (Glymour, Zhang, and Spirtes 2019) that currently suffer many limitations in autonomous robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Causal inference is an active research area in different fields, including robotics and autonomous systems (Hell- str¨om 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Brawer, Qin, and Scassellati 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Angelov, Hristov, and Ramamoor- thy 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' However, most of these works overlooked some key features that are important for real-world application, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' the computational cost and the memory needed by causal analysis when long time-series are processed to reconstruct a causal model of the observed scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' To this end, the CL’s ability to enable the acquisition of more knowledge by trained models without forgetting previous information, and without using previous data recordings, might help to ad- dress these problems and to achieve better result in terms of quality of the causal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' For instance, a robot in an automated warehouse with humans and various objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 1) could observe and intervene in the interac- tions among them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' worker and shelf) in order to build This work has received funding from the EU H2020 research & innovation programme – grant agreement 101017274 (DARKO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' a causal model and therefore a deep understanding of the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Since the limited hardware resources though, the robot’s causal analysis might be slow and based on a lim- ited amount of data, leading to a low quality causal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The solutions suggested in this paper would allow the robot to overcome its hardware limitations and, moreover, to im- prove the quality of the causal models by continually feed- ing new data for causal analysis, discarding the old col- lected one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' This would enable a more efficient use of the robot’s memory and computing’s resources compared to ex- isting causal discovery’s approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' To summarise, this pa- per proposes a Causal Robot Discovery (CRD) approach to overcome current limitations in causal analysis for real- world robotics applications, addressing in particular: the computing and memory hardware resources of the robot, which may hinder its capability to perform mean- ingful causal analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' the update of previous causal models with new observa- tional and interventional data from the robot to generate more accurate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The paper is structured as follows: related work about continual learning and causal discovery are presented in Sec- tion 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Section 3 introduces our CRD approach and explains how the integration of continual learning could help to over- come the challenges of causal discovery in robotics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' finally, we conclude the paper in Section 4 discussing our current and future work in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2 Related Work Causal discovery: Several methods have been developed over the last few decades to derive causal relationships from observational data, which can be categorized into two main classes (Glymour, Zhang, and Spirtes 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The first one includes constraint-based methods, such as Peter and Clark (PC) and Fast Causal Inference (FCI), which rely on conditional independence tests as constraint-satisfaction to recover the causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The second one includes score- based methods, such as Greedy Equivalence Search (GES), which assign a score to each Directed Acyclic Graph (DAG) and perform a search in this score space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' More recently, re- inforcement learning-based methods have also been used to discover causal structure (Zhu, Ng, and Chen 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' How- ever, many of these algorithms work only with static data arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='03886v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='RO] 10 Jan 2023 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' no temporal information) and are not applicable to time-series of sensor data in many robotics applications, for which time-dependent causal discovery methods are in- stead necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' To this end, a variation of the PC algo- rithm, called PCMCI, was adapted and applied to time-series data (Runge 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Runge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Saetia, Yoshimura, and Koike 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Causal robotics: Causal inference has been recently con- sidered in robotics, for example to build and learn a Struc- tural Causal Model (SCM) from a mix of observation and self-supervised trials for tool affordance with a humanoid robot (Brawer, Qin, and Scassellati 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Other applica- tions include the use of PCMCI to derive the causal model of an underwater robot trying to reach a target position (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2021) or to predict human spatial interactions in a so- cial robotics context (Castri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Further causality- based approaches can be found in the robot imitation learn- ing and manipulation area (Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Angelov, Hris- tov, and Ramamoorthy 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' However, all these solutions rely on a fixed set of time-series for causal analysis and do not consider the computational cost and complexity for online update of the robot’s causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Continual learning: The concept of learning continually from experience has been present in artificial intelligence since early days (Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Recently this has been explored more systematically in machine learning (Hadsell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2019) and robotics (Lungarella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Lesort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Churamani, Kalkan, and Gunes 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' To our knowledge though, few applications of the continual learning paradigm can be found in the causality field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Javed, White, and Bengio (2020) incorporate causality and continual learning with an online algorithm that contin- ually detects and removes spurious features from a causal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In (Kummerfeld and Danks 2012, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Kocacoban and Cussens 2019, 2020), instead, algorithms for online causal structure learning are presented to deal with non- stationary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' This is a key feature of data from real-world environments, which is still under-investigated in robotics and therefore motivates our approach proposed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 3 Causal Robot Discovery A review of the literature revealed that the possible limita- tions of autonomous robots doing causal discovery with their own on-board sensors have not been taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In- deed, the computational and memory requirements for long time-series of sensor data are often very demanding, making the use of previous algorithms for causal inference unfeasi- ble on such platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Our approach is partially inspired by the works of Koca- coban and Cussens (2019) for handling non-stationary data, but differs from it in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' First of all, we adopt the current state-of-the-art PCMCI method for causal discovery from time-series data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' second, we propose to re-learn the causal model not only when the observed scenario changes, but also at each new robot’s set of observations/interventions (periodically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' every few minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In particular, the in- troduction of the CL paradigm could help the robot to over- come the challenge of limited hardware resources and to new CM observations interventions CRD d v θ old CM p-values based interventions Figure 1: CRD approach: the robot provides observational and interventional data about human-object interactions to the CRD block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The latter generates a causal model, which is stored and used to compare the next one built on subse- quent robot’s observations and interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Based on the p-values of the previous causal graph’s links, the CRD could suggest the robot which links need to be better tested by fu- ture interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' improve the quality of the causal analysis even with non- stationary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In addition, a CRD approach could benefit from the fact that robots are physically embodied in the en- vironment and can actively influence its dynamic processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' by performing interventions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' That is, CRD could im- prove the accuracy of the causal model by enriching “pas- sive” observational data from the sensors with “active” in- terventional data from robot’s actions aimed at collecting specific time-series for causal discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Therefore, our goal is to decrease the need of hardware resources – often scarce in autonomous systems – and to increase the quality of the causal analysis by using the robot as an active agent in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The use of CRD would allow the creation of high quality causal mod- els by continually updating them with new sensor data from robot’s observations and interventions, without the need to re-process the whole time-series but only new information in an incremental fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The CRD system envisaged in this paper is thought to limit the demand for hardware resources and allow the robot to perform high quality causal discovery in a reasonable time by using its own on-board sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The proposed approach is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 1: (i) starting from a prefixed set of variables, the robot collects meaning- ful data by observing and intervening in the target scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' (ii) based on this data, a causal model is estimated using PCMCI (Runge 2018), which computes test statistics and p- values as causal strengths of the DAG’s links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' At this stage, differently from (Kocacoban and Cussens 2019), to increase the accuracy of the causal discovery, the robot keeps on col- lecting data by observing and intervening in the scenario to create new causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Periodically then, the robot com- pares the new causal models with the old ones, inheriting from the latter only the links that minimise the p-values of the DAG’s causal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' By repeating this process until the observed scenario changes, a stable version of the causal model with minimum uncertainty levels would be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' This is useful not only for modeling the current scenario, but also when it changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Indeed, by re-using part of the stored causal model for an initial scenario, the new causal discovery when the scenario changes can be significantly sped up (Kocacoban and Cussens 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Note that by iteratively discarding the old time-series and storing only the built causal model helps to avoid the combinatorial explosion otherwise affecting PCMCI, there- fore allowing the robot to operate and compute new mod- els within reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Furthermore, the catastrophic- forgetting problem is mitigated by the fact that, during con- tinuous operations, the robot observes similar processes with only small incremental changes, which leads to sequences of similar causal models reconstructed from relatively small variations of previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The operations performed by our CRD approach are rep- resented by the flowchart in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2 and described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The process starts with a first set of observational data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' sensors’ time-series) collected by the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' An in- ference matrix is estimated by performing conditional in- dependence tests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' correlation, transfer entropy) on the time-series, producing an initial causal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Afterwards, since at the first attempt there are no stored causal models, the “Interventions Strategy” (red block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 2) is initially neglected and the “Causal Model Opti- misation” (blue block) is used only to estimate and save the causal model (CM), together with its test statistics and p-values matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Once the causal model of the observed scenario is ob- tained, the robot can improve its quality by providing new data to the CRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In practice, the robot collects new time- series of sensor data from the scenario so that a new infer- ence matrix can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' At this stage, two parallel processes are executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The first one, “Interventions Strategy”, compares the CM obtained at the previous iteration with the in- ference matrix just estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' If the stored CM still fits the estimated inference matrix, it means we are in a stationary-data case and the robot is observing the same scenario of the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Therefore, based on the p-values matrix of the stored CM, the CRD might suggest the most “unrelaible” links that need to be re-checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' These are used by the robot to plan the next interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The second process, “Causal Model Optimisation”, performs a fresh causal discovery on the new data and compare the obtained CM with the one previously stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' From this comparison, a new CM is derived that inherits only the links minimising the p-values of the causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' The result is then stored to be used at the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In case of stationary data, by repeating this procedure, a stable version of the causal model with minimum uncer- tainty can be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In case of non-stationary data, new time-series will be provided to the CRD, which will detect any significant variations by comparing the newly estimated inference matrix with the one of the stored CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In this case there is no Interventions Strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' instead, the Causal Model Optimisation reconstructs the new model exploiting the sim- ilarities with the stored CM to speed up the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' In par- Observation & Intervention Stored CM fit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' p-values based interventions Inference Matrix Estimation Causal Discovery CM Stored CM Yes Interventions Strategy Causal Model Optimisation Figure 2: CRD flowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' ticular, the last step is performed by comparing the new in- ference matrix with the old CM’s one in order to identify previous causal links that are still valid for the new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' 4 Conclusion In this paper we considered the hardware resource limita- tions of autonomous robots, which are crucial to perform causal inference, and proposed a new approach for causal robot discovery to overcome some of the main challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' This includes improving the quality of the causal models by using the robot as an active agent in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' To summarise, in both stationary and non-stationary data cases, the CRD discards the time-series data after each new CM reconstruction, allowing the robot to perform causal discovery within reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Moreover, as already ex- plained, in case of non-stationary data the old CM can be partially exploited to speed up the reconstruction of the new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' This favors not only the execution time of the causal analysis but also the handling of catastrophic-forgetting phe- nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Indeed, in case of non-stationary data, assuming small and incremental variations of the observed scenario, the new causal model is reconstructed by partially exploit- ing the old one, thus reducing the possibility of completely forgetting what was previously learnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Future work will be devoted to the implementation and application of this approach to real-wold 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Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} +page_content=' on Learning Rep- resentations, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE2T4oBgHgl3EQfbgef/content/2301.03886v1.pdf'} diff --git a/fdFJT4oBgHgl3EQfTyzf/content/2301.11506v1.pdf b/fdFJT4oBgHgl3EQfTyzf/content/2301.11506v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1bf867079aea570136cf8692660efd4aa7be11c8 --- /dev/null +++ b/fdFJT4oBgHgl3EQfTyzf/content/2301.11506v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1d3ef250a83384c032ebb8f0393a9c2d7d66063b0fa428240dca3709793472a7 +size 257118 diff --git a/fdFJT4oBgHgl3EQfTyzf/vector_store/index.faiss b/fdFJT4oBgHgl3EQfTyzf/vector_store/index.faiss new file mode 100644 index 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TIZÓN +DEPARTAMENTO DE MECÁNICA DE FLUIDOS Y PROPULSIÓN AEROESPACIAL, ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA +AERONÁUTICA Y DEL ESPACIO (ETSIAE), UNIVERSIDAD POLITÉCNICA DE MADRID (UPM), PZA. DEL CARDENAL CISNEROS 3, +28040 MADRID, SPAIN + + + + +Burnback Analysis of Solid Propellant Rocket Motors........................................................................................................................................................................................ 1 +1. Abstract ............................................................................................................................................................................................................................................. 1 +2. Introduction ....................................................................................................................................................................................................................................... 2 +3. Combustion front kinematics ............................................................................................................................................................................................................. 4 +3.1. Uniform recession rate ............................................................................................................................................................................................................ 10 +3.2. Cylindrical geometries ............................................................................................................................................................................................................ 12 +3.3. Non-regular geometries ........................................................................................................................................................................................................... 14 +3.3.1 Corners and cusp ............................................................................................................................................................................................................ 14 +3.3.2 Collisions ........................................................................................................................................................................................................................ 15 +4. Burnback analysis methods.............................................................................................................................................................................................................. 16 +4.1. Analytical methods ................................................................................................................................................................................................................. 17 +4.1.1 Simple/unique geometry................................................................................................................................................................................................. 17 +4.1.2 Combination of simple geometries .................................................................................................................................................................................. 17 +4.1.3 CAD based methods ....................................................................................................................................................................................................... 18 +4.2. Numerical methods ................................................................................................................................................................................................................. 18 +4.2.1 Direct surface tracking ................................................................................................................................................................................................... 19 +4.2.2 Minimum distance function (MDF)................................................................................................................................................................................ 19 +4.2.3 Theory of curve and surface evolution (PDE’s based).................................................................................................................................................... 20 +5. Burnback analytical solutions .......................................................................................................................................................................................................... 22 +5.1. Classic star ............................................................................................................................................................................................................................. 23 +5.1.5 Bipropellant star ............................................................................................................................................................................................................ 24 +5.2. Bipropellant burnback analysis ............................................................................................................................................................................................... 27 +6. Burnback numerical solution ........................................................................................................................................................................................................... 31 +6.1. Time marching method .......................................................................................................................................................................................................... 32 +6.2. Results and discussion ............................................................................................................................................................................................................ 34 +6.2.7 Error analysis ................................................................................................................................................................................................................. 35 +7. Conclusions ...................................................................................................................................................................................................................................... 37 +8. Acknowledgements ........................................................................................................................................................................................................................... 37 +9. References ........................................................................................................................................................................................................................................ 38 + +1. Abstract +Burnback analysis is a geometric exercise, whose correct solution leads to obtaining the thrust curve +of solid propellant rockets. Traditionally, Piobert's statement, which introduces a certain amount of +intuition, is used as an argument to construct analytical and numerical algorithms, although it is also +common to use numerical integration of differential equations, whose solution is free of ambiguities. +This paper presents a detailed study of the process experienced by the combustion surface that allows +enunciating the properties of the kinematics of the surface without the need to appeal to heuristic +considerations. To the author’s knowledge, although simple and usual in other disciplines, this kind of +analysis has not been presented previously in the field of the combustion process of a solid propellant. +A formal development of the theory allows us to identify the Eikonal equation as representative of the +physical process and the one that is necessary to solve to obtain a true problem description. Next, the +methods used throughout the technological development of solid propellant rockets are reviewed, from + +2 + +their beginnings, in which only analytical procedures and, at most, their automation were possible by +means of the first calculators, to modern methods, which obtain solutions to complex problems, based +on the numerical solution of PDE. Other methods are also reviewed, which are developed around some +of the properties presented by the solution, that is, methods of heuristic or phenomenological +foundation. As a result of the review, it becomes clear that the solution of the Eikonal equation for +burnback analysis is undertaken in the early 2000’s, clarifying the problem. However, all subsequent +developments, systematically, employ techniques based on the Level Set Method developed in the late +1990s. But LSM is applied to much more general and complex problems, and its use adds nothing new +to the problem solution. Finally, several examples of the capabilities of the most relevant methods are +provided, from the point of view of both efficiency and precision, presenting results in situations of +interest, in the field of propulsion by solid-propellant rockets. +2. Introduction +Solid-propellant rocket motors are the simplest high-performance propulsion system ever devised. It +consists of a structural vessel filled with a mixture of energetic solid components, which react +chemically at a high rate. This reaction produces gases at high temperature and pressure, which are +expelled at high speed through a nozzle, producing the consequent reaction force, that is, thrust. +When the solid propellant ignites and a combustion front is formed on its surface, it is gradually +consumed layer by layer. The combustion geometry determines the propulsive response of the system, +as it directly controls the mass released. By properly sizing the initially exposed area and anticipating +what its variation will be, the thrust variation capacity is anticipated in the geometric design of the +propellant (throttling by design). +From the economic point of view, solid propellant rocket engines are very effective propulsion systems +due to the simplicity of their configuration and the ease and safety in the tasks of handling, transport, +and use. From the propulsive point of view, the specific impulse they provide is modest, but in many +of the space and terrestrial applications this weakness is compensated by simplicity in design and +manufacturing economy. In addition, the solid propellant rocket motor has a very interesting impulse- +density value that makes them the ideal system in applications where the volume is limited. To ensure +the effective use of these systems and the fulfillment of the demanding requirements of the missions in +which they are used, design and simulation tools with high degree of fidelity are necessary. In this +sense, prediction of the thrust curve of the engine is essential. And, for this task, one must have +versatile, fast, reliable, and accurate tools for analyzing the evolution of the combustion surface. +The calculation of the burning surface area as a function of time is an essential step in the analysis +and design activities of solid propellant rocket engines. It is relatively easy to establish a heuristic +procedure, based on a set of simple rules, that determine the evolution of the combustion surface with +time for simple geometries, but only by a rigorous procedure can realistic and complex problems be +addressed: for any initial geometry, or when the combustion rate is not constant. +Towards the third decade of the nineteenth century the French general of artillery Guillaume Piobert +(1793-1871), military engineer and scientist, enunciated a hypothesis about what was the process that +followed the combustion of the substances used in the impulsion of projectiles: The combustion of the +inner parts of the gunpowder grains takes place only when the layers that cover them are consumed; +the speed with which the fire spreads from one cut to another, in the compound, has great influence +on the effects of the explosion (in his own words: "Rapidité de combustion. - La combustion des +parties intérieures des grains de poudre n'a lieu que lorsque les couches qui les recouvrent sont +consumées; la rapidité avec laquelle le feu se propage de tranche en tranche, dans la composition, a la + +3 + +plus grande influence sur les effets de l'explosion", this quote is from the publication Mémoires sur les +pouvoirs de guerre des différents procédés de fabrication: avec résumés des épreuves comparatives +faites sur ces poudres à Esquerdes en 1831 et 1832 et à Metz en 1836 et 1837 , printer-bookseller +Bachelier, 1844, Paris). That is, the propellant undergoes a local process, over the surface, and can be +described by a combustion front that consumes it by layers, sequentially. If the rate of combustion is +uniform, the layers have uniform thickness and the description of the evolution of the surface is +reduced to a geometric calculation, in which the time variable is proportional to the depth advanced +by the front. +In Figure 1, the photo corresponding to the geometry of a propellant in intermediate combustion +times is presented. To obtain the images it is necessary to quench the motor (a procedure can be the +sudden opening of the chamber, which causes a marked decrease in pressure that has as a consequence +that the chemical reaction freezes, stopping the process of consumption of the solid). The initial +geometry is an eight-pointed star. In the central photo the tips have been consumed, and the advance +of the combustion front has also continued in the valleys. Finally, in the last photograph, taken close +to the final moment, the combustion front is about to reach the engine casing, even though this will +happen earlier at some points than at others. All of these features are a direct consequence of the +initial geometry. Many of the aspects discussed in the preceding description have a marked influence +on the performance of the system. The geometry with edges, the complete consumption of geometric +entities (the tips) or the uneven consumption of the propellant that does not reach the casing +simultaneously are indicators that determine the efficiency of the process. + +Figure 1: Situation of the combustion surface in three instants, the initial +one, an intermediate state, and shortly before finishing the combustion +process. +In practice, with uniform surface recession rate (idealized situation in which the pressure of the +chamber must be uniform and the erosive combustion effects non-existent) the calculation of the +evolution of the combustion surface involves its displacement perpendicular to itself. That is, each +point on the surface is projected to a point on the new surface along the line perpendicular to the +original surface. The normal distance traveled by the combustion front at each point will be called +forward coordinate (symbol ������������). In the situation of constant burning rate, the value of the forward +coordinate is proportional to the burning time. In this text, the term pseudotime is used (symbol ������������) +when calculations are made with recession velocity equal to unity in the system of units in which the +geometry of the propellant has been stored. +This has led to burnback analysis being approached on many occasions through analytical procedures +with heuristic foundations, as in the well-known SPP© program [1], in which the initial surface is +formed by extracting simple geometric elements from the volume of the chamber, such as +parallelepipeds, spheres or cones whose combination and evolution reproduces the movement of a +complex surface. However, the complexity of some combustion surfaces and the possibility of the +process not taking place with constant recession velocity, make it advisable to establish a well-founded +general analysis that allows the problem to be addressed in any situation. + +4 + +Discrete methods should be used to assess the evolution of combustion surfaces in a general and +automatic manner. Although analytical methods can be very quick and immediate, their application +to complex geometries becomes complicated and laborious, or even unapproachable. Discrete methods +offer the possibility of representing arbitrary combustion surfaces and delivering results automatically +and repetitively. Usually, in the relevant literature, emphasis is placed on whether the methods solve +the problem quickly or not, that is, whether they are computationally efficient. This interest is +motivated because some methods employ search algorithms, which can slow them down if special +precautions are not taken, and others involve the numerical integration of differential equations. +Today, this aspect is of less relevance, because the power of computers has suffered a spectacular +increase in recent years and because the impact of the method used in burnback analysis is small, on a +global calculation of the design tasks. For application in the current context, the algorithms used to +calculate the combustion surface at different times must be flexible, reliable, robust, and accurate. +Flexible in the sense of allowing discretization of any surface and treatment of variable recession +velocity. Effective and robust when calculating solutions in which interference effects may appear, +such as caustics and rarefactions. And finally, accurate, which in principle could be regarded as a +consequence of the previous but is also achieved by using adequate algorithms and well-founded mesh +studies. In problems closely coupled with the resolution of the internal aerodynamics of the engine, the +calculation time of the combustion area can be a non-negligible fraction of the total time, but an inert +scalar in a domain of similar size should not exceed the fraction corresponding to the advection +calculation. In addition, the calculation of the combustion surface must not contain many mesh +points, when compared with those required in the detailed solution of a fluid field. +3. Combustion front kinematics +Mathematically, the problem is to determine the function ������������(������������, ������������, ������������) − ������������ = 0 that describes the +combustion surface at each point in time, in the domain initially occupied by the propellant, ������������, ������������, ������������ ∈ +������������, where ������������ ≥ 0 is the time elapsed since ignition. It can also be said that the expression allows us to +calculate the time (������������) it takes for the combustion front to reach the point (������������, ������������, ������������) at which, naturally, +������������(������������, ������������, ������������) = 0 defines the initial surface. For the correct approach to the problem, it is necessary to +provide sufficient information about the value of the burning rate at each point, and that means +knowing the recession velocity at all points of the volume initially occupied by the propellant, +although its calculation is a consequence of the geometry at each instant. + +Piobert's statement establishes that the combustion surface moves in the normal direction and +suggests that each point on the surface moves perpendicular to the surface itself, but what happens is +that the points disappear. The intuition of the scientist was correct, but it is worth developing a +procedure that can be followed with confidence in any situation. To do this, imagine that we can refer +to each point of the combustion surface ������������(������������, ������������, ������������) = ������������ by means of a position vector, ������������⃗������������(������������, ������������, ������������) where ������������ +and ������������ are two parameters, without specific physical dimensions, whose variation defines the surface. +Now, it is assumed that both parameters define the surface in the region of interest with values of +order unity, ������������~������������~1, although sometimes it may be convenient to parameterize the surface using the +arc lengths, which will be expressly indicated. All points on the surface are subjected to the +combustion process simultaneously and the geometry obtained is a consequence of this on the region +occupied by the propellant (for example, ������������ ≥ ������������ ∩ ������������). To correctly analyze the problem, we will use the +Huygens–Fresnel principle, which states that each point of a wavefront acts as a source point of a +spherical wavefront, and that the interaction of all of them forms the propagation of the original front. +Consider that the combustion process will affect only one point, ������������, at which the combustion process + +5 + +begins, as shown in Figure 2, and that the burning rate is constant and of value ������������̇������������. After a time ������������������������ +the material consumed will be the one inside the intersection between the propellant and the sphere of +center ������������ and radius ������������̇������������������������������������. If it is now considered that all points on the surface of the propellant +participate in the combustion process, each of them will be the center of a sphere that will have +consumed the propellant inside. Over time the propellant contained inside all spheres will have been +consumed and the combustion surface will be the envelope of the family of spheres internal to the +propellant. This is a generalized algorithm for the determination of the new position of the +combustion surface that can be applied whatever the shape of the combustion surface and that helps +to solve any complicated configuration. + +Figure 2: Diagram of the application of the Huygens–Fresnel principle to +the determination of the motion of the combustion surface. +The family of spheres that have their center at a point on the surface ������������ and radio ������������̇������������������������������������ is: +(������������⃗ − ������������⃗������������) ∙ (������������⃗ − ������������⃗������������) = �������������̇������������������������������������� +2 +(1) +The envelope of the family is obtained by canceling out the derivative of the equation of the surface +with respect to the parameters ������������ and ������������, and solving the generated system together with the equation +of the family itself (1). If, in general, it is assumed that the burning rate depends on the position, +differentiating yields +������������������������⃗������������ +������������������������ ∙ (������������⃗ − ������������⃗������������) = −�������������̇������������������������������������� +2 1 +������������̇������������ +������������������������̇������������ +������������������������ +(2) +������������������������⃗������������ +������������������������ ∙ (������������⃗ − ������������⃗������������) = −�������������̇������������������������������������� +2 1 +������������̇������������ +������������������������̇������������ +������������������������ +(3) +Replacing the parameters ������������ and ������������, equations (1), (2) and (3) provide the expression of the new +surface. Note that the evolution of the combustion surface must be smooth, at least, in this +development. As the combustion surface in time ������������ + ������������������������ is arbitrarily close to the original, using ������������������������⃗ = +������������⃗ − ������������⃗������������, it is obtained that |������������������������⃗|~������������̇������������������������������������, according to equation (1), which is small compared to the +characteristic size of the combustion surface ������������ ≫ |������������������������⃗|. Moreover, the left-hand side of equations (2) +and (3) is of the order of ������������|������������������������⃗|, while the right-hand side is of the order of |������������������������⃗|2, and since |������������������������⃗|2 ≪ +������������|������������������������⃗|, equations (2) y (3) must be replaced by +������������������������⃗������������ ������������������������ +⁄ +∙ (������������⃗ − ������������⃗������������) = 0 +(4) +������������������������⃗������������ ������������������������ +⁄ +∙ (������������⃗ − ������������⃗������������) = 0 +(5) +Consequently, the equations to be solved are (1), (4) and (5). Vectors ������������������������⃗������������ ������������������������ +⁄ + and ������������������������⃗������������ ������������������������ +⁄ + are tangent +to the surface ������������ and it is concluded that, both ������������������������⃗������������ ������������������������ +⁄ +∙ (������������⃗ − ������������⃗������������) = 0 and ������������������������⃗������������ ������������������������ +⁄ +∙ (������������⃗ − ������������⃗������������) = 0, are the +equations of planes perpendicular to the tangent vectors at the point ������������. +The above result cannot be applied on a combustion surface where the normal direction is not defined, +but the algorithm of the sphere family does not require the surfaces to be smooth and is very useful +when analyzing the evolution of the combustion surface in non-regular situations, with geometric +elements such as cusps or corners. Also, it is possible to easily analyze complex situations, for + +6 + +example, conductive cables embedded in the propellant or bipropellant situations, with different +burning rates. +The direction of advance of the surface is perpendicular to the surface ������������ and therefore parallel to the +gradient vector, ∇������������. The modulus of the vector is related to the speed of advance of the front since, by +the expression chosen at the beginning of this section, ������������(������������, ������������, ������������) − ������������ = 0, and in this way ������������������������ = ������������������������ or, +what is the same, +|������������������������| = 1 ������������̇������������ +⁄ + +(6) +Which is known as the Eikonal equation (word that, in Greek, means "image"). This equation is basic +in Geometric Optics because it allows the calculation of the trajectories of light rays, perpendicular to +the surfaces of the same optical path and, therefore, the calculation of the trajectories that reverse a +minimum time (Fermat's principle). In this case, the inverse of the burning rate plays the role of the +refractive index (ratio between the light speed in vacuum and that of the medium). This equation is +used not only in geometric optic applications, but also in other wave propagation problems, such as +electromagnetism or seismology. The solutions of the equation can exhibit geometric singularities +called caustic ("causticus" in Latin means “burnt”) or the well-known mirage phenomenon. +The vector ������������������������⃗ has the direction of ∇������������ and the modulus is the variation of the forward normal +coordinate, ������������������������ = ������������̇������������������������������������, with which equation (6) can be written as +������������������������ = 1 +������������̇������������ +������������������������⃗ +������������������������ +(7) +Differentiating with respect to ������������, +������������ +������������������������ [������������������������] = ������������ +������������������������ �1 +������������̇������������ +������������������������⃗ +������������������������� +(8) +The left-hand side can be transformed by the chain rule as follows, +������������ +������������������������ [������������������������] = ������������������������⃗ +������������������������ ∙ ������������[������������������������] = ������������̇������������������������������������ ∙ ������������[������������������������] = 1 +2 ������������̇������������������������[������������������������ ∙ ������������������������] = ������������ �1 +������������̇������������ +� +(9) +And equation (8) becomes +������������ +������������������������ �1 +������������̇������������ +������������������������⃗ +������������������������� = ������������ �1 +������������̇������������ +� +(10) +The equation with which the trajectory of the surface points can be calculated. Developing the +derivatives yields +1 +������������̇������������ +������������������������̇������������ +������������������������ +������������������������⃗ +������������������������ − ������������2������������⃗ +������������������������2 = ������������������������̇������������ +������������̇������������ + +(11) +From which certain interesting properties can be derived. The first one is that, if the burning rate is +constant, the surface points move along straight lines since the solution of ������������2������������⃗ ������������������������2 +⁄ += 0, is +������������⃗ = ������������⃗0 + ������������(������������������������ |������������������������| +⁄ +) +(12) +Where ������������⃗0 is the starting position and it has been used that ������������̇������������|∇������������| = 1. On the other hand, by +construction, ������������������������⃗ ������������������������ +⁄ + is a vector in the direction of the normal to the surface, whereas ������������2������������⃗ ������������������������2 +⁄ + is +perpendicular to it, so that the recession rate gradient can be broken down into a normal component +∇⊥������������̇������������ = ������������������������̇������������ ������������������������ +⁄ + and a parallel component ∇∥������������̇������������. Equation (11) can therefore be projected in the +directions perpendicular and parallel to the surface. In the direction perpendicular to the surface the +result is trivial (equation (12)) while in the parallel direction + +7 + +������������2������������⃗ +������������������������2 = − ������������∥������������̇������������ +������������̇������������ + +(13) +Which expresses that the trajectories only turn if there is a non-zero parallel gradient of the recession +rate. When the recession rate is constant the combustion surface can be reconstructed by simple +translations. For this reason, numerous heuristic algorithms have been developed over time to solve +this problem. +Some general results, related to geometric optics, of interest for the performances of rocket engines +have been reviewed. But the relevant thing is to calculate the combustion area at each moment, +because it allows us to determine the thrust curve. To have a means of assessing the area of +combustion, the surface must be parameterized with ������������⃗������������(������������, ������������, ������������) (note that the subscript will be ignored +hereafter), assuming that the values of ������������ y ������������ identify a point on the surface and, as long as the value +of the parameters is maintained, the point follows the trajectory described by (11). In other words, +parameterization complies with +������������������������⃗ +������������������������ = ������������̇�������������������������⃗ +(14) +Where the normal to the surface �������������⃗ is calculated as usual, +�������������⃗ = ������������⃗������������ × ������������⃗������������ +|������������⃗������������ × ������������⃗������������| +(15) +And ������������⃗������������ = ������������������������⃗ ������������������������ +⁄ + and ������������⃗������������ = ������������������������⃗ ������������������������ +⁄ + are used to simplify the notation. Equation (14) is equivalent to +equation (7), precursor of equation (10) that describes the trajectory, but, in this case, to express the +normal it is necessary to reconstruct the surface with the values of ������������⃗ near the considered ray. On the +other hand, the direction of the normal has been chosen in the direction of advance of the front, that +is, the same as ∇������������. +The combustion area, ������������������������(������������), at any given moment, is calculated by +������������������������ = � +|������������⃗������������ × ������������⃗������������| +������������(������������,������������) +������������������������ ������������������������ +(16) +traversing the set of parameters ������������(������������, ������������) that defines the combustion surface at each instant. +The temporal variation of the area is therefore +������������ +������������������������ (������������������������) = � +������������|������������⃗������������ × ������������⃗������������| +������������������������ +������������(������������,������������) +������������������������ ������������������������ +(17) +Differentiating the cross product yields +������������ +������������������������ (������������⃗������������ × ������������⃗������������) = ������������������������⃗������������ +������������������������ × ������������⃗������������ + ������������⃗������������ × ������������������������⃗������������ +������������������������ +(18) +The time derivatives of the position vector with respect to the parameters are obtained from equation +(14): +������������������������⃗������������ +������������������������ = ������������������������̇������������ +������������������������ �������������⃗ + ������������̇�������������������������⃗������������ +(19) +������������������������⃗������������ +������������������������ = ������������������������̇������������ +������������������������ �������������⃗ + ������������̇�������������������������⃗������������ +(20) +Where the nomenclature is �������������⃗������������ = �������������������������⃗ ������������������������ +⁄ + and �������������⃗������������ = �������������������������⃗ ������������������������ +⁄ + for the derivatives of the normal vector. +Substituting expressions (19) and (20) into (18), + +8 + +������������ +������������������������ (������������⃗������������ × ������������⃗������������) = ������������̇������������(�������������⃗������������ × ������������⃗������������ − �������������⃗������������ × ������������⃗������������) − �������������������������̇������������ +������������������������ ������������⃗������������ − ������������������������̇������������ +������������������������ ������������⃗������������� × �������������⃗ +(21) +Note that the first term in the right-hand side of equation (21) is a vector perpendicular to the +tangent plane (i.e. parallel to the normal direction) since both �������������⃗������������ and �������������⃗������������ are vectors contained in the +tangent plane defined by ������������⃗������������ and ������������⃗������������. However, the second term is a vector perpendicular to the +previous one, contained in the tangent plane. +Considering that ������������⃗������������ × ������������⃗������������ = |������������⃗������������ × ������������⃗������������|�������������⃗, it can also be written, +������������ +������������������������ (������������⃗������������ × ������������⃗������������) = ������������|������������⃗������������ × ������������⃗������������| +������������������������ +�������������⃗ + |������������⃗������������ × ������������⃗������������| �������������������������⃗ +������������������������ +(22) +and the comparison of equations (21) and (22) yields: +������������|������������⃗������������ × ������������⃗������������| +������������������������ += ������������̇������������(�������������⃗������������ × ������������⃗������������ − �������������⃗������������ × ������������⃗������������) ∙ �������������⃗ +(23) +�������������������������⃗ +������������������������ = − +1 +|������������⃗������������ × ������������⃗������������| �������������������������̇������������ +������������������������ ������������⃗������������ − ������������������������̇������������ +������������������������ ������������⃗������������� × �������������⃗ +(24) +Expression (23) evaluates the temporal evolution of the combustion area element, while expression +(24) determines whether the direction of propagation changes or not, which is a result that had +already been advanced, making use of the typical developments of geometric optics. These two +expressions summarize the behavior of the combustion surface. If the recession rate is uniform, the +direction of the normal at each point remains unchanged and the surface points move in a fixed +direction. Conversely, if the recession rate changes from one point to another on the surface, the +direction of the normal vector changes and the surface is distorted. +To further analyze expression (23), it is convenient to use some concepts of differential geometry of +surfaces. The vectors �������������⃗������������ and �������������⃗������������ are contained in the tangent plane and can be expressed as a linear +combination of the vectors ������������⃗������������ and ������������⃗������������, in the form +�������������⃗������������ = ������������11������������⃗������������ + ������������21������������⃗������������ +(25) +�������������⃗������������ = ������������12������������⃗������������ + ������������22������������⃗������������ +(26) +The matrix of coefficients is calculated by: +�������������11 +������������21 +������������12 +������������22� = − ������������� +������������ +������������ +������������� ������������� +������������ +������������ +������������� +−1 + +(27) +where the coefficients of the First Fundamental Form (which corresponds to the inner product ������������������������⃗ ∙ ������������������������⃗) +are ������������ = ������������⃗������������ ∙ ������������⃗������������, ������������ = ������������⃗������������ ∙ ������������⃗������������, ������������ = ������������⃗������������ ∙ ������������⃗������������, and are related to the area element by |������������⃗������������ × ������������⃗������������| = √������������������������ − ������������2. The +coefficients of the Second Fundamental Form (which corresponds to the inner product ������������������������⃗ ∙ �������������������������⃗) are ������������ = +−�������������⃗������������ ∙ ������������⃗������������ = �������������⃗ ∙ ������������⃗������������������������, ������������ = −�������������⃗������������ ∙ ������������⃗������������ = �������������⃗ ∙ ������������⃗������������������������ = �������������⃗ ∙ ������������⃗������������������������ = −�������������⃗������������ ∙ ������������⃗������������, ������������ = −�������������⃗������������ ∙ ������������⃗������������ = �������������⃗ ∙ ������������⃗������������������������, and are related to +the curvature of the surface. +The normal curvature of the surface is the ratio of both fundamental forms, ������������������������ = (������������������������⃗ ∙ �������������������������⃗) (������������������������⃗ ∙ ������������������������⃗) +⁄ +, +which is the component of the curvature vector ������������⃗ = ������������������������⃗ ������������������������ +⁄ + in the direction of the normal, where ������������⃗ = +������������������������⃗ ������������������������ +⁄ + is the tangent vector (in this case the parameter ������������ describes any curve contained in ������������ that +passes through the point in question). The normal curvature is independent of the curve on which it is +defined and depends only on the orientation of the tangent vector. Principal curvatures are the +maximum and minimum values of the normal curvatures of a given point. In particular, the main +curvatures, ������������1 and ������������2, of the surface are the eigenvalues of the matrix��������������������������������������, the average curvature, +������������ = +1 +2 (������������1 + ������������2), is half of the trace of the matrix with the sign changed, ������������ = − +1 +2 (������������11 + ������������22), and + +9 + +Gaussian curvature, ������������ = ������������1������������2, coincides with the determinant, ������������ = det��������������������������������������, which corresponds to the +intrinsic curvature of the surface. Naturally, all these values do not depend on the chosen parameters. +Substituting expressions (25) and (26) into (23), +������������|������������⃗������������ × ������������⃗������������| +������������������������ += ������������̇������������(������������11������������⃗������������ × ������������⃗������������ − ������������22������������⃗������������ × ������������⃗������������) ∙ �������������⃗ +(28) +That is, +������������|������������⃗������������ × ������������⃗������������| +������������������������ += −������������̇������������(������������1 + ������������2)|������������⃗������������ × ������������⃗������������| +(29) +Expression (29), which can be rewritten as ������������(������������������������) ������������������������ +⁄ += 2������������(������������������������), is a classical result in differential +geometry when one intends to obtain the variation of the area, ������������������������, of a family of surfaces, and is +directly related to very interesting topics, such as the plotting of surfaces of constant average +curvature, or obtaining surfaces of minimum area. In the current context, it provides a direct +geometric interpretation of how the combustion area evolves over time, due to the local value of the +recession rate and as a function of surface curvatures. At a symmetrical saddle point, ������������1 = −������������2, the +net variation of the combustion area is zero, whereas, if the surface concavity prevails at the point, +������������1 + ������������2 > 0, the area decreases, but if the surface is globally convex, ������������1 + ������������2 < 0, the combustion area +increases. +Similarly, equation (24) can be rewritten as +�������������������������⃗ +������������������������ = − +1 +|������������⃗������������ × ������������⃗������������|2 �������������������������̇������������ +������������������������ ������������⃗������������ × (������������⃗������������ × ������������⃗������������) − ������������������������̇������������ +������������������������ ������������⃗������������ × (������������⃗������������ × ������������⃗������������)� +(30) +The expression is apparently complicated, but if a new parameterization of the surface is used, being +(������������′, ������������′) the arc lengths, it is then verified that |������������⃗������������′| = |������������⃗������������′| = 1 and if, in addition, orthogonality is +required, i.e. ������������⃗������������′ ∙ ������������⃗������������′ = 0, this yields +�������������������������⃗ +������������������������ = − �������������������������̇������������ +������������������������′ ������������⃗������������′ + ������������������������̇������������ +������������������������′ ������������⃗������������′� ≡ −������������∥������������̇������������ +(31) +Where the vector identity ������������⃗ × ��������������⃗ × ������������⃗� = �������������⃗(������������⃗ ∙ ������������⃗) − ������������⃗�������������⃗ ∙ �������������⃗� has been used. To interpret the expression +more easily one can use ������������������������ = ������������̇������������������������������������ and write +�������������������������⃗ +������������������������ = − ������������∥������������̇������������ +������������̇������������ + +(32) +which is identical to (13). The normal vector to the surface changes its direction according to the +direction marked by the gradient of the recession rate in the plane tangent to the surface and in the +opposite direction. Equations (14) and (32) are a system equivalent to equation (10) that can be +integrated over time, using the information provided by the function ������������̇������������(������������, ������������, ������������), to obtain the evolution +of the combustion surface,. +The normal vector is tangent to the trajectory followed by the point ������������, so its variation with the length +traveled is the curvature, which will be proportional to the modulus of the gradient of the recession +rate referred to itself, as written in equation (13). Admitting that this quantity is constant, for small +values of the forward coordinate, the trajectory of this point describes an arc of radius ������������̇������������ �∇′������������̇������������� +⁄ +. + +10 + + +Figure 3: Schematic representation of the process that takes place when +the recession rate varies linearly on the surface of the propellant. The +trajectory of the ray is ������������������������′′′, although the point of tangency of the circle +envelope is ������������′′. +A schematic representation of the process that takes place with variable recession velocity has been +made in Figure 3. The recession rate is considered to vary linearly on the surface of the propellant, +������������̇������������ = ������������0 + ������������1������������, and it is assumed that ������������1������������������������ ≪ ������������0, so that the combustion front, for a time ������������������������, moves +from the point ������������ a distance ������������������������~������������0������������������������ that in the figure is used to draw a circle of center ������������ that locates +the possible points that the combustion surface could reach. Applying Piobert's principle directly, +equivalent to the exact result that the points on the surface move perpendicular to it, the ray would +describe the trajectory ������������������������′ and the new combustion surface would be built by joining the image points +������������′ of the entire surface. The above does not consider that applying Huygens' principle, each point ������������ of +the surface is the center of a circle of distinct radii that grows at a rate ������������������������~������������1������������������������, being ������������′′ the image +points in a position other than ������������′. However, considering what was shown in previous lines, the +trajectory of the point ������������ is an arc of radius ������������0 ������������1 +⁄ , and rotating an angle ������������������������~������������1������������������������~������������1 ������������������������ ������������0 +⁄ +, to the +point ������������′′′. The points ������������′ and ������������′′ are not correct and underestimate the position of the combustion +surface, situation which is relieved because the distance ������������������������′′′ must be smaller. +3.1. Uniform recession rate +If the recession rate is uniform, then, ������������������������̇������������ ������������������������ +⁄ += ������������������������̇������������ ������������������������ +⁄ += 0 and from equation (30): +�������������������������⃗ +������������������������ = 0 +(33) +Which indicates that the propagation directions remain unchanged, although the propagation velocity +may be a function of time. These circumstances have the consequence of the propagation problem +becoming decoupled from the temporal problem and being, therefore, purely geometric in nature. +Consequently, the temporal evolution of the combustion area satisfies that the normal directions to +the surface remain unchanged and the trajectories of the points on the surface are straight lines. The +centers of curvature of the surface are located above the normal lines in fixed positions and the shape +of the surface can be easily reconstructed. The surface retains its topology until the propellant +consumption reaches some center of curvature. At that moment the analysis ceases to be valid and if +the combustion front progresses there is an irreversible destruction of geometry. +Because ������������̇������������ = ������������������������ ������������������������ +⁄ +, the forward coordinate may be used in equation (14), instead of time: +������������������������⃗ +������������������������ = �������������⃗ +(34) + +11 + +Which is independent of the pace of recession and, therefore, the burnback problem is reduced to an +exercise in geometry. Effectively, it can be integrated using the initial geometry from ������������ = 0 +(corresponding to the initial time, ������������ = 0), obtaining a family of surfaces ������������⃗������������(������������, ������������, ������������). The recession rate +can present any sort of time dependency because, from the known family ������������⃗������������(������������, ������������, ������������) and the expression +������������������������ = ������������̇������������(������������)������������������������, its evolution with time can be calculated. +Depending on the nature of the initial surface, different methods may be used to obtain its evolution. +If the radii of curvature are defined at all points of interest, a possible procedure to obtain the surface +������������⃗������������(������������, ������������, ������������) is to evaluate the length of the radii of curvature, ������������1,2 = 1 ������������1,2 +⁄ +, and describe how they +change by means of equation (34). That is, solving ������������������������1,2 ������������������������ +⁄ += −1, expression that supports the +general solution +������������1,2(������������, ������������, ������������) = ������������1,2 +o (������������, ������������) − ������������ +(35) +Where the initial surface has the distribution ������������1,2 +o (������������, ������������) of radii of curvature. An immediate +consequence is that, when a radius of curvature cancels out (note that, for this to be possible, it is +necessary that the radius of curvature is strictly positive at ������������ = 0, which corresponds to an initially +convex geometry), there is an unavoidable discontinuity, since all the points of the combustion surface +collide in the center of curvature, without the integration being able to continue. This event partially +destroys geometry and requires a special analysis, since it is necessary to consider the evolution of a +surface that contains non-regular points or regions. +Note that, contrary to the usual definition of the radius of curvature as the absolute value of the +inverse of the curvature, here it has been given the sign of the curvature itself, to be able to generalize +the relations. In this way, those radii that extend behind the space traveled by the normal are +considered negative. When parameterizing the surface with the arc lengths, the absolute value of the +radius of curvature will be taken, so that the angular sectors traveled will be positive. +Calling back to relationship (29): +������������|������������⃗������������ × ������������⃗������������| +������������������������ += − � 1 +������������1 ++ 1 +������������2 +� |������������⃗������������ × ������������⃗������������| +(36) +Where the time variable has been replaced by the normal coordinate. If the surface is parameterized +by arc lengths following the main directions, ������������������������′ = |������������1|������������������������1 and ������������������������′ = |������������2|������������������������2(which are orthonormal, +|������������⃗������������′| = |������������⃗������������′| = 1, when considering the main curvatures), the variation of the combustion area is: +������������������������������������ +������������������������ = − � +� 1 +������������1 ++ 1 +������������2 +� |������������1| +������������(������������1,2) +|������������2|������������������������1������������������������2 +(37) +where ������������(������������1,2) expresses that integration variables now extend into a different domain than the +parameterization used before. Note that the initial radii of curvature, ������������1,2 +o += ������������1,2 +o (������������1, ������������2), that we are +going to use to calculate the area can be a function of the angles, (������������1, ������������2), so that there are no +restrictions on the combustion surface, other than the mere existence of the radii of curvature. In the +case of regions of null curvature, the original expression must be retrieved since the expression (37) +has been invalidated by using the inverse of the curvatures in this analysis. Without going into major +complications, what follows is useful to analyze the behavior of fixed angular sectors, since, for +calculation purposes, the area can be decomposed into an arbitrary number of portions. With the +intervention of (35) and some algebra, it can be obtained, +������������������������������������ +������������������������ = − � +(sgn(������������1)|������������2 +o − ������������| + sgn(������������2) |������������1 +o − ������������|) +������������(������������1,2) +������������������������1������������������������2 +(38) +To analyze the expression, it is necessary to separate the different cases according to the sign of the +curvatures, or the radii of curvature. If both are positive, the slope of the combustion area, depending + +12 + +on the forward coordinate, ������������������������������������ ������������������������ +⁄ +, is monotonically decreasing and, as both radii decrease, the +analysis is valid until the smaller one is canceled out. If both are negative, the slope is positive, with +no limits other than those of the surface itself or those imposed by the combustion chamber casing. If +the signs of the radii of curvature are different, it is necessary to elaborate the analysis with care. If +we consider the case of the negative radius being less in absolute value than the positive one, the +value of the initial slope is negative, and grows linearly with ������������ until it reaches the point where both +radii equal in absolute value (this coincides with zero variation of the slope, which corresponds to a +minimum of the local area enclosed in the angular sector considered). It then follows an upward slope +behavior, until the initially positive radius is canceled out, stopping the linear analysis. +Any of the situations considered above leads to a linear variation of the slope and, therefore, to a +quadratic variation of the combustion area with the forward coordinate of advance. Because of the +above considerations, expression (38) can be reordered as follows: +������������������������������������ +������������������������ = � +sgn(������������1������������2) [2������������ − (������������1 +o + ������������2 +o)] +������������(������������1,2) +������������������������1������������������������2 +(39) +The combustion area finally is +������������������������ = � +{sgn(������������1������������2) [������������2 − (������������1 +o + ������������2 +o)������������] + ������������1 +o������������2 +o} +������������(������������1,2) +������������������������1������������������������2 +(40) +Which is canceled out when ������������ = ������������1,2 +o and, in addition, it is fulfilled ������������1,2 +o +> 0, which corresponds to the +situation of zero radius of curvature when the combustion front destroys a rounded cusp, already +noted in the previous paragraph. +3.2. Cylindrical geometries +In line with the high slenderness of rocket-propelled aerospace vehicles, it is common to find +combustion surfaces where the longitudinal dimension predominates. If the vehicle is very slender, and +the thrust demand is high, the combustion surface must be greater than the cross-sectional area, and +the only way to achieve this is by longitudinal drilling. In this case, the combustion surface is of +cylindrical type, in which the characteristic dimension along the grain (~������������) is large compared to the +cross-sectional dimension (~������������), that is, ������������ ≫ ������������. Local curvatures (in the longitudinal and transverse +direction), necessarily, verify ������������1~1 ������������ +⁄ and ������������2~1 ������������ +⁄ , which results in ������������1 ≪ ������������2, so that equation (29) +may be simplified by ignoring ������������1 as compared with ������������2: +������������|������������⃗������������ × ������������⃗������������| +������������������������ +≈ −������������̇������������������������2|������������⃗������������ × ������������⃗������������| +(41) +The temporal variation of the combustion area is: +������������������������������������ +������������������������ = − � ������������̇������������������������2������������������������������������������������ +������������ + +(42) +where the surface differential element, |������������⃗������������ × ������������⃗������������|������������������������ ������������������������ = ������������������������������������������������, is expressed by the coordinate along the +cylinder ������������ and the arc length ������������. As the curvature of the cross section can be set as ������������2 = − ������������������������ ������������������������ +⁄ +, +being ������������ the angle formed by the tangent to the curve, the previous expression becomes: +������������������������������������ +������������������������ = � ������������̇������������������������������������������������������������ +������������ + +(43) +For a differential element of area, it is verified: +(44) + +13 + +������������ +������������������������ (������������������������������������) = ������������������������������������������������ +This is a very interesting expression. First, it shows again that if the recession rate is uniform then the +problem is exclusively geometric. Moreover, if the length of the cylinder remains unchanged in the +process then its influence is reduced to a constant factor. However, the most interesting property is +that the variation of the combustion area is independent of the shape of the cross section. All +reference to dimensions has disappeared from the expression. The variation of the combustion area is +proportional to the value of the angular sector traveled by the tangent when running around the +perimeter, and in the case of a straight cylinder of constant length (������������) and uniform recession rate, it +turns out to be: +������������������������������������ +������������������������ = 2������������������������ +(45) +This value is independent of the shape of the section and corresponds to a progressive combustion +process, identical to that which takes place for a cylinder of circular section. The combustion area is +obtained immediately, ������������������������ = ������������������������ +ο + 2������������������������������������, where ������������������������ +ο is the value of the initial area for ������������ = 0. Naturally, +these results are subject to the cross-section being regular, in the sense that the curvature is defined at +all points. Under these conditions, the variation of the area meets the following properties: i) it is +independent of the shape of the perimeter; ii) it has a constant value equal to the angle rotated by the +tangent to the curve; and iii) the sign (which marks the character of the combustion process) is the +contrary to that of the curvature, when the normal to the curve points in the direction of propagation. +Consequently, the expression of the perimeter is linear with the forward coordinate, and the process is +reversible, in the sense that, if the direction of propagation is reversed, the initial geometry is reached +uniquely. In the regressive regions of the perimeter, the propagation process decreases the radius of +curvature and when the depth of advance reaches the center of curvature a discontinuity is generated, +since a convex region disappears. At that point, the perimeter topology changes, and the surface +analysis must be restarted, probably considering the evolution of a cusp, as will be discussed later. +The process of combustion of slender channels can be adequately described by one-dimensional models +in which the geometry of the channel is determined by the distribution of port areas. Consider the +perimeter of each section ������������������������ = ∮ ������������������������ and the port area in each section ������������������������ = ∮ ������������������������������������. For calculation +purposes, the combustion area, ������������������������, can be defined at any given section as the area of combustion +exposed from ������������ = 0 to the considered section ������������. That is, +������������������������ = � ������������������������������������������������̅ +������������ +0 + +(46) +If the recession rate is uniform in the section, which is the most consistent simplification with the +slender cylinder approximation, the variation with time of the port area (invoking again ������������̇������������ = ������������������������ ������������������������ +⁄ += +������������������������ ������������������������ +⁄ +) is +������������������������������������ +������������������������ = ������������������������ +(47) +While the perimeter, in the assumption that it is a regular curve, complies with expression (45) and, +therefore, +������������������������������������ +������������������������ = 2������������ +(48) +The above expressions constitute a closed geometric system with which all geometric variables can be +calculated using very simple integrals. + +14 + +3.3. Non-regular geometries +The conclusions obtained in previous paragraphs can be generalized to contours in which the radius of +curvature may present discontinuities, but for which the tangent to the perimeters must be a +continuous function. In these circumstances, for each point of the combustion surface, an image point +can be defined as the surface evolves. That is, a bijective relationship can be established between the +points. This does not occur when: i) there are discontinuities in the tangent to the combustion +surface, ii) two combustion surfaces meet each other, or iii) the combustion front reaches the motor +case. In the first and second situations the trajectories of the surface points intersect. If the front +reaches the motor case or any other inert element, the points on the surface also disappear +irreversibly. All these situations are irreversible, in the sense that, if the sign of the recession rate is +changed, the succession of combustion areas produced is not the same, just reversed in time, but very +different, indeed. Next, a number of geometries that are commonly presented in solid propellant +engines and that do not have a regular behavior are analyzed. +3.3.1 Corners and cusp +When the perimeter of the section presents a break, which represents a discontinuity in the slope, a +non-regular situation is generated whose evolution is different depending on the direction of advance +of the front. Figure 4 depicts two different situations in which the gaseous and solid domains are +exchanged. In situation (a) the combustion process regularizes the geometry, the vertex of the corner +becomes a source point, origin of a rarefaction, and as the geometry generated presents a smooth +distribution of the angle (the tangent to the perimeter is continuous) the rate of increase of the +perimeter, as already seen in equation (44), is +������������������������������������ +������������������������ � +Corner += ∆������������ +(49) +The increase (decreases are also possible) of the perimeter is proportional to the angle rotated by the +tangent when following the curve. It is easy to imagine an algorithm that accumulates variations of +the angle associated. + +Figure 4: In configuration (a) the combustion front progresses from a +corner creating a cylindrical surface (rarefaction). In configuration (b) the +combustion front consumes a cusp destroying geometry and creating a +discontinuity (caustic). +The situation (b) is the opposite to (a). The combustion front destroys part of the geometry as it +advances. The collision of the two combustion fronts causes a discontinuity that is called a caustic. +The destruction of geometry is irreversible. In Figure 4 (b), a simple geometric analysis leads to the +relationship + +CORNER +CUSP +SP~yβ +SP~ - 2ytan △Φ/215 + +������������������������������������ +������������������������ � +Cusp += −2 ������������������������������������(∆������������ 2 +⁄ ) + +(50) +Which is similar to (49). Both relationships coincide if the angle rotated by the perimeter is very small +(∆������������ ≪ 1) but, in general, equation (50) has a nonlinear dependence on the angle. Fortunately, both +expressions have a linear dependence on the forward coordinate, and this allows combining different +geometries so that any target value of ������������������������������������ ������������������������ +⁄ + can be set. Specifically, an adequate combination of +valleys or corners (of a progressive nature, ������������������������������������ ������������������������ +⁄ +> 0) and vertices or cusps (of a regressive nature, +������������������������������������ ������������������������ +⁄ +< 0) can lead to a geometry in which the perimeter changes in a controlled way. For this +case, the most common solution is a star-shaped geometry, in which the angle and number of cusps +determines the progressive, regressive, or neutral character of the combustion. +3.3.2 Collisions +Figure 5 (a) shows a dendrite-like geometry in which, when the thickness is exhausted, the 2������������ arc of +circle at the end of the protuberance disappears and the two combustion fronts collide simultaneously, +producing an instantaneous drop (a discontinuity) of the combustion area of the form +������������������������ = −������������������������������������ℋ(������������ − ������������) +(51) +where, ������������������������ is the discrete variation of the perimeter, Δ������������������������ is the length of the dendrite, ℋ( ) is the +Heaviside function and ������������ is the semi-thickness of the dendrite (remember that ℋ(������������) = 0, ������������ < 0; and +ℋ(������������) = 1, ������������ ≥ 0). + +Figure 5: Sometimes there is the collision of two combustion front (a) or +with the engine casing (b), which causes a sudden destruction of the +combustion surface and a discontinuity in the evolution of the perimeter. +A similar situation occurs when cylindrical combustion surfaces collide with the motor case in the final +phase of propellant combustion. If the collision takes place sharing the center of curvature the +decrease in area will be sudden, +������������������������ = −������������������������������������ℋ(������������ − ������������) +(52) +Being here Δ������������������������ the arc length of the collision front. If the collision is not completely frontal, a very +rapid process of combustion area destruction occurs, which must be analyzed in each case. +In both these situations, the most relevant characteristic is that the processes are not linear. In fact, +the burning area presents discontinuity that originate an unsteady response of the chamber pressure. +Furthermore, the evolution is not reversible. + +DENDRITE +WALL COLLISION +SP~ - 2△P,H(y- ) +SP~-APH(y- w)16 + +4. Burnback analysis methods +Current methods can be classified into Analytical or Numerical. The analytical methods, essentially, +consist in using Piobert's aphorism and displacing the combustion surface, formed by simple geometric +figures, perpendicular to itself, incorporating the particular phenomenology imposed by cusps and +corners. This activity can be carried out for a simple geometry obtaining closed relationships, or by +automating operations through some algorithm, such as the SPP© program or other CAD-type +graphic programs. In contrast, numerical methods start from a discrete description of the combustion +surface, which allows them to be more flexible and general. Once the discretized surface is available, it +can act as in analytical methods using some specific property of the solution, or address the +propagation problem by integrating differential relationships. +Method +References +1. Analytical methods + +1.1. Simple/unique geometry +[2]–[7],[8], [9] +1.2. Combination of simple geometries +[1], [10]–[12], [13] +1.3. CAD based methods + +1.3.1. Parametrized geometry +[14]–[16] +1.3.2. Based in CAD in-house tools +[17]–[19] +2. Numerical methods + +2.1. Direct surface tracking +[20]–[23] +2.2. Minimum distance function (MDF) +[24]–[26], [27]–[30] +2.3. Theory of curve and surface evolution (PDE’s based) + +2.3.1. Set Level Methods (Hamilton-Jacobi equation) +[31]–[33] +2.3.1.1. +Standard (signed function evolution) +[34]–[41],[42]–[47] +2.3.1.2. +Narrow band + +2.3.2. Steady perspective (Eikonal equation) + +2.3.2.1. +Direct time marching +[48]–[53] +2.3.2.2. +Fast marching methods (FMM) +[54] +Table 1: Classification of the different methods of burnback analysis. +Table 1 lists all the categories of methods considered in this paper and indicates the most relevant +bibliographic sources. Applied to geometries accessible to the method, all those listed in the table +solve the problem satisfactorily, from the point of view of thrust curve calculations. Numerical +methods are usually able to deal with more general and complex problems than analytical methods, +though. Different arguments have been raised in the literature to evaluate the suitability of each +method. As the most versatile and powerful methods are numerical methods, the central argument is +usually efficiency, measured in terms of computational time requirements. However, the high power +achieved by computers today weakens the importance of this argument, because the computational +effort in the field of burnback analysis is moderate compared with that required for the study of, for +example, the rocket internal aerodynamics or the structural calculation of the propellant. Numerical +burnback analyses only need to obtain a single spatial function that determines the combustion +surface as the propellant is consumed. In addition, it is not necessary to use adapted meshes, but with +significantly uniform meshes that reasonably describe the geometry is sufficient to obtain satisfactory +results. From this perspective, other considerations, such as the flexibility in terms of the possibility of +carrying out complex three-dimensional geometric analyses, the possibility of analyzing cases with +variable recession velocity, and the economy of implementation, all make the methods based on the +Eikonal equation (2.3.2 in Table 1) the most attractive. This result contrasts with the very high +diffusion that LSM have reached in the analysis of the burnback problem in the last twenty years, +motivated by the evident generality of the method. However, the burnback analysis problem does not +need so much generality, and the LSM is oversized in this case. The integration of the Eikonal +equation is enough to obtain a completely satisfactory solution of the problem (that is, the calculation +of the thrust curve) and, eventually, allow the design of the initial combustion geometry. In the next + +17 + +section, the results obtained with method 2.3.2.1 of Table 1, which meets the above requirements, are +presented for a variety of grain geometries. +4.1. Analytical methods +Analytical methods make use of different properties of the solution that are incorporated into the +analytical calculation of the position of the combustion surface. These methods are fast, simple and +accurate. But they cannot address problems of arbitrary geometry, they have to solve complex +geometric situations with specially adapted procedures, and they cannot, in general, solve problems of +variable recession velocity. +4.1.1 Simple/unique geometry +It is the simplest approach and consists in the algebraic analysis of the evolution of a surface that +moves perpendicular to itself. During the second third of the twentieth century, in the early days of +the development of solid propellant engines, it was the only possible method. The work of Billheimer +and Wagner [2] contains an extensive bibliographical review of this period and the different +procedures with which the simple geometric calculation was enriched to achieve the determination of +the thrust curve of solid propellant engines with grain geometries that presented some complexity. For +example, the work of Thibodaux et al. [4] can be reviewed to verify the level of specialization achieved +in the analysis of, in this case, three-dimensional geometries in spherical chambers. Or the arduous +work of analyzing the interaction between the combustion front of a slotted-tube grain with the casing +of the engine, described in [5]. This type of method is still widely used, and the number of recent +citations, referring to burnback analysis with purely analytical methods, is very high (not all collected +in this review), because the immediacy of the method lends itself to its easy integration into internal +aerodynamics analysis systems [5], or its integration into all kinds of engine design optimization +algorithms [6][7]. +As already mentioned, the most interesting advantages of the method are its speed, simplicity, and +precision. Naturally, it is not possible to analyze arbitrary geometries and it is difficult to incorporate +realistic situations such as a non-constant recession velocity. In addition, the analyses must +incorporate a specific treatment of non-continuous geometries (such as cusps and corners) which, for +example, in three dimensions can significantly complicate the problem. However, it is possible to +address situations of industrial interest and others that initially would seem complex, such as the +analysis of two propellants burning simultaneously. In this sense, through analytical methods, it is +possible to address the problem of two propellants with two different recession rates, as for example, +to analyze the combustion of a bipropellant star geometry that does not present sliver mass +fraction[3][8] and that Krishnan and Bose [9] study with a high level of detail for various +configurations. +4.1.2 Combination of simple geometries +The simplicity of use of analytical methods facilitates a different strategy, combining elements of +simple geometry and automating the analysis of the evolution of the combustion surface. The best +known and most successful example is the burnback module of the SPP© software package, initially +presented by Coats et al. [1] in 1987, and continuously updated and improved since then (see [11]– +[13]). +The SPP© program has been a standard reference software in the United States for predicting the +performance of solid-propellant rocket engines. The methodology for evaluating the thrust coefficient, +starting from the chemical equilibrium value, which is corrected with individual efficiencies due to + +18 + +different effects, is an industry standard. The Grain Design and Ballistics module allows the design of +the initial combustion surface and calculates the thrust curve using a burnback analysis package, an +internal aerodynamics module, and calculations with finite chemical kinetics in a two-dimensional +nozzle flow. The SPP© program has been used in the past, and is still being used today, by major +agencies, institutions, and manufacturers of solid-propellant rocket engines in the United States and +other countries [13]. The grain design and analysis module construct the surface by extracting simple +geometric figures from an initial volume (the interior of the motor case). It is a Boolean operation that +can be repeated with the basic figures resized. In the calculation of the evolution of the combustion +surface, the dimensions of geometric figures are increased, emulating the advance of the front. +Operationally, the program is fed with symbolic commands, which are executed sequentially. It is a +flexible, versatile, and efficient tool, capable of modeling all the geometries that are usually presented +in solid propellant rocket engines, as long as they can be decomposed into simple volumes. Naturally, +it is an analytical methodology that retains the disadvantages already mentioned, but the product has +been adapted and consolidated to mitigate these disadvantages as much as possible. +4.1.3 CAD based methods +The increase in accessibility and power of computer-aided design (CAD) has meant that these +specialized programs have been used to conduct burnback analysis of realistic and overly complex +geometries. This is the main quality of the method, the ability to evaluate surfaces of complicated +shapes. Two strategies may be adopted for the calculation of the area evolution. +On the one hand, when modeling the initial combustion surface, the model can be parameterized so +that the recession process is considered, using the parameters that define the model itself (1.3.1. in +Table 1, see references [15]–[17]). For example, if a cylinder is parameterized by its radius, by varying +the radius a preset quantity, the process of recession is simulated. The next operation is to vary the +parameterized values and allow the graphic system to reconstruct the new combustion surface, +executing the corresponding symbolic operations. +The other possibility is to use CAD-specific capabilities that move the model surface with controlled +laws (1.3.2. in Table 1, see references [17]–[19]). That is, specific tools for translation, growth, or +projection of surfaces that the software makes available to the user. These procedures are quick and +versatile, can tackle complex geometries, and provide a fast and adequate response. However, the +information obtained must be extracted from within the CAD system. Furthermore, there is no +certainty of these geometric operations being able to capture the real problem physics, since many of +these operations are hidden from the user. Naturally, the user is forced to examine these operations +and, eventually, correct situations in which the graphical system fails because it is unable to +automatically generate rarefactions or caustics. +4.2. Numerical methods +Numerical methods approach the problem from a discrete description of a combustion surface. +Depending on the method, the initial combustion surface can be an external surface of a volumetric +mesh that represents the whole propellant, where other surfaces of interest can be easily identified as +well, such as the motor case or, for example, symmetries of the model. Alternatively, the combustion +surface is discretized as an isolated surface, whose movement is the objective of the calculation and +which, in one way or another, must incorporate an analysis of the interaction with other surfaces such +as the engine casing. The advantage of numerical methods is that they allow the description of the +evolution of complex combustion surfaces, and, with some exceptions, they allow variable recession +rate to be incorporated into the calculation. + +19 + +4.2.1 Direct surface tracking +This category includes methods that carry out local surface monitoring, combined with a position +identification that allows interaction with inert areas or with the engine casing. In principle, this type +of methods start from a discretization of the surface and obtain its evolution using displacement +algorithms that somehow consider properties exhibited by the propagation process. The most +commonly used of these properties is Piobert's postulate that the surface moves perpendicular to +itself. Typical methods of calculating free surfaces (Volume of fluid, VOF, method) are also used, +identifying the convection rate with the recession rate of the front. +Among these methods, one can mention the SLIC (Simple Line Interface Calculation) method devised +by Noh and Woodward [20]. The authors conceived it for use in one, two or three spatial dimensions. +The domain is discretized into enclosures, and fluid interfaces are represented locally for each +enclosure by lines, either perpendicular or parallel to the coordinate directions. Decision-making logic +is used in the propagation, depending on the arrangement of fluid regions. Due to the completely one- +dimensional nature of the interface description in SLIC, it is relatively easy to get correct results with +time. Another very similar method is FLAIR [55], which tries to increase the accuracy by +complicating a little the geometric description of the front within each control zone, and is used by +Mashayek et al. [21] for the analysis of two-dimensional combustion geometries. +Belonging to the surface tracking methods that use phenomenological algorithms, which basically +project the surface perpendicular to itself, the work of Hejl and Heister [22] carries out direct surface +tracking and incorporates locally the peculiarities that are presented in the form of rarefactions and +caustics. Also, in reference [56],[56][57],[57]Another work in this category is carried out by Ki et +al. [23] that present the PIT method (Partial Interface Tracking) in the analysis of combustion +surfaces of three-dimensional geometrics of type finocyl and conocyl. This method applies a +Lagrangian approach to the axisymmetric area of the transverse plane and the two-dimensional area +of the longitudinal plane separately, because the Lagrangian approach is an effective way to simulate +two-dimensional evolutions. In this way, a three-dimensional problem is solved with the computational +effort of two two-dimensional problems. The limitation is that geometries have to exhibit some +symmetry, which is usually common in solid propellant engines, such as finocyl and conocyl types. +However, it does not bring anything new in the spectrum of front-tracking methods, but it merely +solves with success three-dimensional problems approximately. + +4.2.2 Minimum distance function (MDF) +The method of calculating the minimum distance to the initial combustion surface, proposed by +Wilcox et al. [24], has been very fruitful in solving the burnback analysis problem and, in this case, is +used to allow internal ballistic calculation [25]. It is a very intuitive method, easy to implement, and +does not exhibit limitations in terms of the complexity of the geometry. Once the domain occupied by +the propellant has been discretized, the method consists in calculating the smallest distance from any +interior point to the initial surface. This calculation involves a search for the point of the initial +surface closest to the inner point, which is onerous from the computational standpoint. Usually, +methods of reducing this computational time are required, optimizing search algorithms using +standard techniques, such as Ren et al. [26] using a divide-and-conquer algorithm. Like other methods +already discussed, MDF employs a property of the solution, in this case, Fermat's Principle, and when +the propagation velocity is uniform, the minimum time condition is equivalent to the minimum +distance condition. Precisely, this is the disadvantage of the method, which cannot incorporate +variable recession rate without overcomplicating the algorithm. The reason the generalization of the +MDF method is not possible is that a global property is used, which leaves out of the calculation what + +20 + +is the path followed by each ray. However, the conceptual simplicity and the possibility of applying it +in realistic three-dimensional geometries, makes it a widely used method [28]–[30], see, for example, +how in [31] is concluded that it is superior to other surface monitoring methods. +4.2.3 Theory of curve and surface evolution (PDE’s based) +To describe the propagation of the combustion surface in solid propellant rockets, Saintout et al. [48] +implement an algorithm that incorporates all the characteristics that allow it to describe the physics +of the process properly, and identify the equation that they integrate numerically as of the Hamilton- +Jacobi type. This situation is reached from preliminary studies of the same research group on surface +tracking methods [50] and [49]. These works are part of the activity made by SNPE (Société Nationale +des Poudres et Explosifs, currently a subsidiary of Nexter) for the analysis and design of solid +propellant rocket engines in Europe that, in the case of burnback analysis, culminate with the work of +Dauch and Ribereau [51]. In this work, the general purpose tool called PIBAL© is presented, which +integrates an evolution of the IVOLINA© program (previously developed in the references [52] and +[53]) that addresses the integration of the Eikonal equation by a time marching method (2.3.2.2 in +Table 1). +However, in parallel to the developments described in the previous paragraph, for the treatment of +this type of problems (and other, more complex), the work of Osher and Sethian [31] initiates a +lineage of methods, based on the procedure called Level Set method (LSM). These methods have been +very fruitful and have been developed and employed on numerous occasions (see [32] for an overview). +In what follows, it is described how the problem has been solved by two different paths, the first +addresses the resolution of an equation of type Hamilton-Jacobi by means of the LSM that is capable +of solving problems of propagation of very general fronts, much more complex than the problem of +burnback. The second perspective addresses the steady problem that is circumscribed to the solution +of an equation of type Eikonal that is strictly the problem to be solved in the burnback analysis and, +in this sense, the modeling and computational effort made is more proportionate. A basic and +complete description of both approaches can be obtained in Sethian's text [33], which clearly identifies +and discusses both methods. +Consider the situation in Figure 6 in which a curve or surface, defined for example by the function +������������ = 0, spreads with velocity ṙp in the direction perpendicular to the surface itself. The problem is to +determine the evolution of the surface. In the most general situation, the propagation rate may +depend on local properties of the surface point, such as the direction of the normal, or curvature, or +on general properties of the curve, such as integral relations of all kinds, and, also, on properties +external to the problem itself, as would be the case of advection, due to a velocity field. + +Figure 6: Outline of the two approaches followed in the numerical +methods of solving the burnback problem. On the left, the LSM in which +the front is represented by the null value of a distance function, ������������ = 0. +On the right, the position of the front is represented by the values taken +by the solution of the Eikonal equation that corresponds to the travel +time. + +21 + +The problem can be approached from two points of view. The boundary value formulation calculates +the time ������������ = ������������(������������) it takes for the front to reach each point in the domain, and it is evident that the +definition of the velocity of the front leads to ṙ p = ������������������������ ������������������������ +⁄ + and, therefore, in several dimensions it is +fulfilled that, +������������̇������������|������������������������| = 1 +(53) +Already written before, with the condition ������������ = 0 on the initial combustion surface. This is the Eikonal +equation, which is a traditional problem in many physical systems. In this problem, it has been +implicitly assumed that the function ������������ is a single-value function, for which the propagation rate must +have a constant sign, either outwards from the domain, or inwards. This restriction, which for some +situations is very important, in the case of burnback analysis is fulfilled naturally and the unknown of +the Eikonal equation is strictly the function to be obtained to solve the evolution of the combustion +surface in a solid propellant rocket. +When propagation can take place in two directions, on both sides of the front, it is mandatory to +describe the movement of the front by a function ������������ with more spatial dimensions. To obtain an +equation of this evolution, consider the path ������������⃗(������������) that follows a particle of the front and how, without +loss of generality, one can assume the front defined by ������������(������������⃗(������������), ������������) = 0. Differentiating the function +yields, +������������������������ + ������������������������(������������⃗(������������), ������������) ∙ ������������⃗′(������������) = 0 +(54) +Which is the equation that allows us to obtain the function ������������. As the velocity of the front is ṙp = �������������⃗ ∙ +������������⃗′(������������) and the direction normal to the surface is �������������⃗ = ∇������������ |∇������������| +⁄ +, finally, the equation for ������������ is: +������������������������ + ������������̇������������|������������������������| = 0 +(55) +For which an initial value of the function must be supplied. This equation is of the Hamilton–Jacobi +type, for a wide spectrum of forms of ṙp. The problem of front propagation occurs in a wide variety of +configurations: from ocean waves, combustion fronts or interfaces in the movement of heterogeneous +substances; of course, in problems of light propagation or seismic wave propagation; but also, in +problems of character identification or image processing. +Equations (53) and (55) represent the two different approaches, and both of them provide fully +satisfactory results. The only difference is that solving the Eikonal equation involves an effort adjusted +to the problem. The method based on the Hamilton-Jacobi equation is designed for more complex +problems and needs further elaboration in the calculation, uses more memory and has to solve +numerical problems (such as the reinitialization of the distance function) typical of a more complex +method, but which are totally unnecessary in the burnback analysis problem. +4.2.3.1. Level Set Methods (Hamilton-Jacobi equation) +The evident generality of LSM has led the methodology to be used in the analysis of the burnback +problem on numerous occasion [34-47]. Usually, the initial function ������������(������������⃗, ������������ = 0) is fixed as a signed +distance function (SDF) containing the value of the minimum distance to the front from the initial +surface and which is calculated with some algorithm (2.3.1.1 in Table 1). The method is not exempt +from some problems, since the SDF can take poorly conditioned values as the integration progresses, +and it becomes necessary to reinitialize it periodically. [58] +It is evident that the method employs an implicit function defined throughout the propellant domain +of which the only useful information is the front defined by the null value of the function. For this +reason, some authors have used a strategy of limiting the calculated value of the SDF to the vicinity +of the front (2.3.1.2. in Table 1). However, it is necessary to incorporate a search and location +algorithm of the front to determine the narrow band. + +22 + +Notable is the contribution of Chiapolino et al. [47] that addresses the solution of a Hamilton-Jacobi +equation using a standard LSM but with a step function for the level function emulating the front +tracking methods commonly used in heterogeneous fluid problems. An instructive article describes a +numerical method on an unstructured mesh, in which it uses upwind techniques with limiters, for the +method stability, which have been developed in previous works. The examples that are included, +addressing three-dimensional burnback analysis, are very illustrative, and correspond to modern and +realistic grain geometries. +4.2.3.2. Steady perspective (Eikonal equation) +4.2.3.2.1. Direct time marching +The solution of equation (53) (also of the equation (6) using a Time Marching procedure), +������������������������ + ������������(������������������������) = 0 +(56) +where the Hamiltonian is +������������(������������������������) = 1 − ������������̇������������|������������������������| +(57) +As already mentioned, this type of equation belongs to the so-called Hamilton-Jacobi equations, which +arises as a problem of initial values with boundary conditions according to the situation to be +simulated. In the problem in hand, ������������ = 0 on the initial combustion surface. Usually, the rest of the +boundary conditions consist of boundaries at which the front extinguishes (for example, the engine +casing) in which, usually, no condition is necessary to be imposed due to the hyperbolic nature of the +equation; and contours of symmetry or periodicity, in which the implementation of the condition is +relatively simple. +References [48]–[53] pioneer the use of the Eikonal equation for the solution of the burnback problem. +These works constitute a frame of reference for the correct and adjusted solution. Since then, however, +the propagation problem has been addressed from different perspectives and for different problems, +though not necessarily in solid-propellant engine technology. The direct solution of the Eikonal has +been addressed on numerous occasions for the appropriate monitoring of surfaces, as in [59]. Singular +is the contribution of Gueyffier et al. [60], which addresses the solution of the Eikonal equation using +a spectral method for the description of the combustion surface with a philosophy similar to that +employed by surface tracking methods. +4.2.3.2.2. Fast marching methods (FMM) +The Eikonal equation in the form (53) can be solved by calling a method based on the traditional +alternate direction methods but using the propagation direction of the front to update the variables +and in this way obtain an additional advantage. These procedures are called Fast Marching methods +(FMM). It is possible to consult the book by S. Sethian [54] to have an overview, where an interesting +critical comparison between FMM and LSM is also established. The burnback problem has been +addressed by this method in unstructured mesh, for example, in [61] in a complete paper but there are +not many other contributions to the burnback problem using this procedure. +5. Burnback analytical solutions +Constant combustion surface area is the most common design condition for a solid propellant rocket +engine. This situation is generically optimal because it implies that the chamber structural design is +adjusted to the entire engine operation range. Otherwise, the thickness of the engine casing must be +sized for the most unfavorable load case, which corresponds to the maximum pressure reached and, +therefore, the combustion chamber is heavier than that of the engine that would provide the same + +23 + +total impulse with constant chamber pressure. To achieve constant pressure profiles with large +combustion areas, comparable to those of the chamber itself, it is necessary to resort to geometries +with a certain degree of complexity. The important variables are the web fraction, the volume +fraction, and the sliver fraction but, also, the Klemmung and ������������ (combustion to port area ratio). These +last two parameter are of interest because control the occurrence of the erosive combustion +phenomenon. +5.1. Classic star +Figure 7 shows the geometric description of the cross-section of a star-shaped propellant with ������������ tips, +as discussed in [62], [63]. For simplicity, only half of an angular sector, ������������/������������, is represented, taking +advantage of the symmetry properties of the section. The tip has an angle ������������ (the figure shows the +semi-angle ������������ 2 +⁄ ) while occupying a fraction ������������ of the entire angular sector, ������������(������������/������������). The depth of the +valley area has length ������������ from the center of the chamber and it is considered that the thickness of the +propellant is the necessary to finish the first phase of combustion at the moment in which the front +arrives for the first time to the engine casing. If the propellant web thickness is larger, a progressive +phase of linear perimeter growth begins at this time. This second phase of combustion would be +progressive, and in the design of the engine it will not be allowed to extend too much, as it raises the +chamber pressure. However, it can increase the web fraction or the combustion time, and it may be +necessary to satisfy design requirements. Nevertheless, the possibility of compensating this effect by +designing the star with a slightly regressive profile should be analyzed. + +Figure 7: Schematic of star geometry and definition of geometric +parameters. +A heuristic procedure to determine the variation of the perimeter of the section considered is to go +through the contour, measuring the rotation suffered by the normal to the surface and calculating, in +each case, the increase in perimeter that occurs. Performing this operation for the geometry in +Figure 7, the expression obtained is +∆������������ = 2������������ +� +(1) +������������ � ������������ +������������⏟ +(2) ++ ������������� +2 − ������������ +2� +����� +(3) +− +1 +������������������������������������[������������ 2 +⁄ ] +������� +(4) +� +(58) +The term (1) corresponds to the 2������������ half sectors, the term (2) is due to the turn suffered by the +normal in the half sector (if it were a cylinder, these first two factors would give rise to the simple +result already commented ∆������������ = 2������������������������). The term (3) is the one corresponding to orienting the normal +from the radial position, after the rotation ������������ ������������ +⁄ , to the surface of the cusp, which assumes a rotation +equal to the complementary angle of ������������ 2 +⁄ . Finally, the term (4) is the one corresponding to the + +0/2 +大24 + +destruction of part of the cusp. Indeed, the first thing to note is that ������������ 2 +⁄ is the complementary angle +of the angle ∆������������ 2 +⁄ in the Figure 4, and the tangent function of the complementary angle is the inverse +of the tangent of the angle and, in addition, in the generic expression (equation (50)) two slopes of the +cusp are taken into account, while in the sector in Figure 7 only one of them has to be accounted for. +This rapid assessment is delicate and subject to probable misinterpretation of the criteria under which +it is applied. However, it is a very interesting method to be used in combination with more elaborate +geometric evaluations. Because it provides a quick verification. In addition, it allows us to carry out +analyses that lead to the elaboration of optimal strategies for the design, merely using analytical +arguments. +To obtain a geometry in which the combustion area does not change (i.e. neutral combustion), it is +necessary that ∆������������ = 0 in equation (58) which is a simple nonlinear equation for ������������ as a function of ������������. +Table 2 shows the semi-angle of the tip, the web fraction, and the volumetric fraction, for different +values of the number of tips. Note that between 5 and 6 tips, the value of ������������ goes from being ������������ 2 +⁄ +< +������������ ������������ +⁄ to be ������������ 2 +⁄ +> ������������ ������������ +⁄ , showing that for less than 5 tips impossible geometries can arise in which the +tips collide with each other. It is especially interesting that if ������������ 2 +⁄ += ������������ ������������ +⁄ the channel is straight. This +is the condition for analyzing axial slots. The table indicates as well that for ������������ ≥ 6 (because ������������ 2 +⁄ +> +������������ ������������ +⁄ ) the combustion process should be regressive, which is very useful information when combining +combustion geometries. + +������������ +4 +5 +6 +7 +8 +������������ ������������ +⁄ +28.21 +31.12 +33.53 +35.55 +37.30 +������������ ������������ +⁄ +45.00 +36.00 +30.00 +25.70 +22.50 +������������ ������������������������ +⁄ + +0.200 +0.180 +0.164 +0.151 +0.140 +������������ +- +0.893 +0.804 +0.733 +0.674 +Table 2: Solution of equation (58) for neutral combustion ∆������������ = 0, and the +corresponding values of the web fraction and the volumetric fraction. +Combined propellant geometries are presented on many solid rockets. A common configuration is to +use simple cylindrical combustion and a slotted segment. The cylindrical section has a combustion +area that grows over time and the slotted segment can be configured so that the combustion area +decreases at the desired rate. The combination of both geometries can result in a thrust curve with a +specific profile. +5.1.5 Bipropellant star +The star configuration provides a constant combustion area curve for moderate values of the web +fraction. However, the mass of residual propellant after the neutral phase (sliver fraction) can be very +large, with a negative impact on the effective volumetric fraction. It is possible to design a sliverless +geometry using two propellants with different recession velocities. The idea is to fill the region of the +cusp with a high-speed recession propellant so that it reaches the engine casing at the same time as +the propellant, with a lower recession rate, that fills the web thickness. + +25 + + +Figure 8: Straight star loaded with two propellants of different recession +rates. On the left is a general scheme and on the right the notation used +in the analysis to determine the adequate interface. +Figure 8 shows a possible simple configuration for a straight star (similar to a slotted geometry) in +which two propellants of different combustion rate are used. Each propellant advances a different +amount at the same time due to the different rate of combustion. The propellant 1 advances ������������1 and +the propellant 2 advances ������������2. For the combustion front to reach the casing simultaneously at all +points, the combustion front in the propellant 1 has to be cylindrical with radius ������������1 from point ������������. +While, by construction, the combustion front in the propellant 2 is composed of a line and a circle arc +of radius ������������2, centered on point ������������’. The recession velocity in the propellant 2 must be such that it +reaches the point ������������ at the same time as the propellant 1. This imposes a geometric exception, +depending on which combustion front reaches the point ������������ in the propellant 2, whether it is the +straight front or the circular front. In what follows it is assumed that it is the circular combustion +front that reaches the point ������������. The rest of the parameters to be used are shown in Figure 8, in which +������������ is the depth of the slot, ������������������������ is the fillet radius in the slot, ������������������������ is the chamber radius and ������������1,2 and ������������1,2 +are the polar coordinates of the points on the two combustion fronts, respectively. The condition for +the fronts to progress simultaneously over the interface is expressed by +������������1 ������������������������������������ ������������1 = ������������2 ������������������������������������ ������������2 + ������������ +(59) +������������1 ������������������������������������ ������������1 = ������������2 ������������������������������������ ������������2 +(60) +Where ������������1 = ������������������������ + ������������ + ������������1 and ������������2 = ������������������������ + ������������2. Without loss of generality, it can be put ������������1 = ������������ and ������������2 = ������������������������ +with ������������ > 1. The regression rate of the propellant 2 is suitable so that, on the symmetry line, the front +reaches the housing at the point ������������ at the same time as in the propellant 1 reaches point ������������. The +propellant 2 induces a cylindrical combustion front on the propellant 1 with radius ������������1, while the front +in the propellant 2 is also cylindrical with radius ������������2 but with center at ������������’. The condition of reaching +the casing simultaneously at the point ������������ is expressed by removing ������������2 from expressions (59) and (60), +thus getting +(������������1 ������������������������������������ ������������1 − ������������)2 + (������������1 ������������������������������������ ������������1)2 = ������������2 +2 +(61) +And substituting ������������1 = ������������������������, ������������1 = ������������ ������������ +⁄ and ������������ = ������������, which is the value of the web thickness, result in: +������������������������2 − 2������������������������������������ ������������������������������������(������������ ������������ +⁄ ) + ������������2 = ������������������������� + ������������������������� +2 +(62) +Along with +������������������������ = ������������������������ + ������������ + ������������ +(63) +Once the geometry of the star is established (������������, ������������������������, ������������������������ and ������������ are known), equation (63) allows the +calculation of the web and equation (62) provides the needed value of ������������, that is, the ratio of recession +velocities of both propellants, ������������ = ������������������������2 ������������������������1 +⁄ +. + +1 +ri= yi+ d +r2= 2 + rf +2 +0226 + +The geometry of the interface can be obtained by taking as a parameter the depth of the forward +coordinate of propellant 1 (������������; 0 ≤ ������������ ≤ ������������) and explicitly resolving with +������������1 = ������������������������ + ������������ + ������������ +(64) +������������2 = ������������������������ + ������������2 +(65) +And using equation (61) to obtain ������������1 and equation (60) to obtain ������������2. Once the interface line has been +drawn, it is possible to calculate the burn perimeter on each propellant and, considering the different +recession velocities, calculate the mass released by each propellant. The length of each perimeter in +each propellant is no longer an intuitive measure of the mass burned by the entire surface or of the +chamber pressure reached at each moment. For this reason, in what follows, the geometric concept of +forward coordinate is momentarily abandoned in favor of a pseudotime, as an independent variable. + +Figure 9: Comparison of the equivalent burning surfaces of a +monopropellant and a bipropellant star-shaped geometry with 4 cusps, +with������������ = 0.4, ������������������������ = 0.1 and ������������ = 0.5; so that the ratio of recession velocities +takes the value ������������ = 1.592. +Figure 9 shows the simulation performed with a geometry of four slots. The results obtained in the +case of operating with a single propellant and in the case of operating with two sliverless propellants +are presented. To compare both situations, it is useful to represent the pseudotime lines that +correspond to ������������ = ������������1,2 ������������������������1,2 +⁄ +. Taking ������������������������1 = 1 in the case of a single propellant, the pseudotime is +equivalent to the forward coordinate. In the bipropellant case, an equivalent combustion area must be +defined in the form ������������������������,������������������������ = ������������������������,1 + ������������������������������������,2. The equivalent combustion area allows the calculation of the +mass released and the chamber pressure and thrust, using the combustion data of the propellant 1. In +this way, we can establish a reliable comparison with the operation of a single propellant. In the +monopropellant case, this geometry, with few slots, gives rise to an increasing combustion area profile, +until the combustion front reaches the engine casing for the first time. From that moment, the +combustion area decreases over time, giving rise to a long tail thrust phase as shown in the figure. In +the bipropellant case, however, the combustion process of the fast propellant generates at the +beginning more mass flow, compensating the initial deficit presented by the monopropellant. In this +way, as clearly shown in Figure 9, a near-neutral combustion area curve is provided. This remarkable +feature can be anticipated by designing the geometry so that the equivalent combustion areas are +similar at the initial and final times. As the combustion fronts reach the casing simultaneously, no +sliver fraction is produced and the combustion area curve drops sharply at that moment, forming an +optimal silverless geometry. +Figure 10 shows the comparison of operation with one and two propellants of an elliptical hole +geometry that initially presents a high volumetric filling. As a result, the ratio of recession rates is +also high, which translates, again, into a significant variation in the equivalent area of combustion. In +this case, the pseudotimes at the end of the combustion of both configurations are equal, highlighting + +0 0.1 0.2 0.3 0.4 0.5 0.6 +Equivalent burnning surface +0.9 +monopropellant +80 +bipropellant +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0.00 +0.20 +0.40 +0.60 +0.80 +Pseudo time27 + +the significant variation (up to 33%) in the equivalent combustion area. For the calculation of the +interface, the approximation of the combustion front in propellant 1 remaining elliptical has been +made. The numerical simulation shows how little importance this gross hypothesis has on the overall +result. + +Figure 10: Comparison of the equivalent burning surfaces of a +monopropellant and a bipropellant elliptical-hole geometry, with high +volumetric fraction and high ratio of recession velocities ������������. +The solution of these bipropellant cases has been approached without establishing any consideration +about how the combustion fronts interact with each other. As will be seen below, the interaction can +be complex and create rarefaction and caustic waves that significantly modify the combustion front +near the interface. This can lead to variations of some importance in the evaluation of the equivalent +combustion area and, therefore, in the prediction of the actions of the system. As will also be shown +later, the numerical analysis scheme proposed reliably captures these anomalies. For the examples +presented above, this anomaly does not occur, since it is a corner-type combustion situation in which +the design system guarantees that the interface is above the equilibrium point ������������ (the scheme ������������)) in +Figure 14, so that the combustion fronts do not present rarefactions or caustics of any kind. +5.2. Bipropellant burnback analysis +The combustion front in a bipropellant grain is determined by the difference between the recession +rates of each propellant and by the geometry of the front and of the interface. To approach a general +analysis with confidence, it is advisable to start with a simple situation, in which the combustion front +at the point of contact of both propellants is flat, as represented in Figure 11. The point ������������ separates +both propellants at the combustion surface, and the interface between them is straight and +perpendicular to said combustion surface. The recession rate of the propellant 1 (on the left in the +figure) is ������������1 and the combustion front of this propellant moves to the parallel line ������������1 a distance ������������1 = +������������1������������������������ after a time ������������������������, at points that are far enough from the point S. At the same time, the combustion +front for the propellant 2 moves ������������2 = ������������2������������������������ reaching the line ������������2. For the analysis it will be assumed +that ������������1 < ������������2 and, therefore, ������������1 < ������������2. To build the solution it is convenient to consider the point ������������ to +be the source of the propagation process in both propellants, then the combustion front will extend +into the propellant 1, at least, up to the cylinder ������������1; and, in the propellant 2, up to the cylinder ������������2. As +in time ������������������������ propellant 2 reaches the point ������������2, while propellant 1 only would reach point ������������1, faster +propellant acts as a source of ignition. Each of the points on the side of the propellant 2 over the line +������������������������2 +����� will be the center of a family of circles that consumes the propellant 1, and whose radius is +proportional to the distance remaining to travel to ������������2. Consequently, the combustion surface in the +propellant 1 will be the envelope of this family of cylinders, which is easily built by tracing the +tangent to the circle ������������1, from ������������2 to the point of tangency ������������. The segment ������������2������������ +����� intersect with line ������������1 at +the point ������������, separating the combustion surfaces obtained from the original surface (������������1), and that +obtained because of the phenomenon already described in the interface (������������2������������ +�����). On point ������������ two + +1.4 +Equivalent burnning surface +1.2 +0.8 +0.6 +0.4 +0.2 +monopropellant +bipropellant +0 +0.00 +0.20 +0.40 +0.60 +0.80 +1.00 +Pseudo time28 + +different combustion fronts converge, whose collision forms the caustic ������������, which is a straight line +starting from the point ������������ in the line ������������������������ +����. + +Figure 11: Diagram of the evolution of the combustion surface of a +bipropellant with flat front and interface perpendicular to the front. +The propagation of a combustion front, initially flat, along the interface between two propellants is a +relatively common situation that, for example, corresponds to that which occurs in the case of +thermally conductive wires, embedded in the propellant to increase the combustion area. In this case +the cable acts as an ignition source with a higher velocity than the propellant regression rate and the +combustion geometry obtained is conical with the axis on the cable, analogous to the construction +������������1������������������������2 +� in Figure 11. However, although the situation is simple, it allows the introduction of the basic +analysis mechanisms to be used in more complex situations. Thinking that the point S It is the origin +of the combustion front of each propellant, building the cylinders of influence ������������1 y ������������2, calculating the +intersection with the lines that establish the position of the fronts ������������1 and ������������2 far from ������������, and +determining the envelope of certain families of cylinders, leads to the construction of the combustion +surface at each time. +For the analysis of more complex situations, where the combustion front is not initially flat, it is +convenient to generalize the notation, as shown in Figure 12. Uppercase letters are used to name +points of interest and lowercase letters are used to name lines and circular arcs. The figure represents +the two possible situations for a non-flat combustion front, when ������������ is the vertex of a cusp, and when +it is the vertex of a corner. The bisector angle ������������ is used to represent the initial position of the fronts +and identify the angle ������������ (that lies between the lines ������������ and ������������2) as a measure of the difference in +burning rates, because if ������������ = ������������ the burning rates are equal. + + +Figure 12: Meaning of the different symbols used in the description of the +propagation process of a bipropellant for two initial configurations of the +initial combustion surface. +The lines ������������1 and ������������2 are parallel to the original surfaces and represent the position of each combustion +front if they were isolated (in the figure, ������������1 < ������������2). Unlike the simple flat-front case, lines ������������1 and ������������2 are +not parallel, but rather converge at the point ������������, that allows you to draw the equilibrium line ������������ from + +CE +F +/ F2 +e +S +业f +Ifi +/J2 +1 +1 +CUSPY1 +b, +2β +y2 +C1 +F +E +F. +K +e +1 +CORNER29 + +������������, towards ������������������������ +����. Once the position of the surfaces is known, the circles of influence, ������������1 and ������������2, can be +traced, tangents to the aforementioned lines at points ������������1 and ������������2, respectively. The perpendicular to +the initial surfaces, ������������1 and ������������2, are drawn from ������������ following the directions ������������������������1 +����� y ������������������������2 +�����. The conical zone +between ������������1 and ������������2 defines an interference region. The equilibrium line ������������ is a reference for the position +of the interface between the propellants, which will evolve differently if it is inside or outside the +interference region. Finally, if the angle formed by the initial surfaces of both propellants is 2������������, then +the angle of the equilibrium line ������������ can be calculated through the relationship: +������������1 +������������2 += ������������������������������������(2������������) − ������������������������������������(2������������) ������������������������������������(������������) +(66) +which shows that the structure of the study region depends on ������������ and the recession rate ratio, that is, +������������1 ������������2 +⁄ +. +Figure 13 shows the different results of the combustion surface if ������������ is the vertex of a cusp. Each +scheme corresponds to distinct positions of the interface, identified by the line ������������, determined by angle +������������������������ that the interface forms with the line ������������1 (Figure 12). The series starts with a sufficiently large value +of the angle ������������������������ (greater than ������������) and situations are analyzed for decreasing values of ������������������������. If ������������������������ > ������������, see +diagram ������������), the interface intercepts the line ������������1 at the point ������������ and the propagation process in the +propellant 1 produces the premature ignition of propellant 2 along the interface between them. The +combustion surface produced (segment ������������������������ +���) is generated by obtaining the line that starts from ������������ and is +tangent to the cylinder ������������2, which corresponds to the envelope of the family of circles generated by the +ignition points. The intersection of this line with the line ������������2 determines the position of the point ������������ +which is the vertex of caustic ������������ generated in this process which goes from ������������ towards ������������������������ +����. As ������������������������ +decreases, the point ������������ approaches point ������������, coinciding both when ������������������������ = ������������, and the caustic disappears, as +illustrated in the diagram ������������) of Figure 13. In this situation, the propellants are consumed at their own +rate without generating any additional structure, forming the fronts only by the lines ������������1 and ������������2. When +������������������������ < ������������, but before you get to ������������������������, the propellant 1 reaches point ������������ before propellant 2, but this time +the family of cylinders ends in the circle that passes through ������������ and is tangent to ������������2 and, as +represented in the scheme ������������), a partial rarefaction is created between lines ������������ and ������������. When ������������������������ reaches +the value of ������������������������ (scheme ������������)) the expansion is complete between the line ������������2 and the line ������������. Here, the +point B, that defines ������������������������, is obtained as the intersection of the circle ������������2 and the line ������������1. Until now, the +high inclination of the interface line causes the process to be dominated by the propellant 1 but when +������������������������ > ������������������������ the propellant 2 reaches the point ������������ earlier, producing the premature ignition of the propellant +1. In the scheme ������������) this situation is shown, in which the new surface ������������������������ +��� is obtained by drawing the +line that starts from ������������ and is tangent to ������������1, obtaining the position of ������������ as an intersection of this line +and ������������1. Under this, if ������������������������ > 0 remains the rarefaction between ������������ and ������������2 until ������������������������ is canceled (scheme ������������)) +and rarefaction disappears. Finally, as represented in the scheme ������������) for ������������������������ < 0 the structure of the +front is maintained. + +30 + + +Figure 13: Sequence of the different schemes for different positions of the +interface, in the case where the initial combustion surface has a cusp at +the vertex between the two propellants. +For the case where in cusp configuration the point ������������ is within the interference region (delimited by ������������1 +and ������������2) the different modes of propagation of the Figure 13 are simplified and only schemas ������������), ������������) and +������������) appear. +Figure 14 shows the different morphologies of the combustion surface when ������������ is the vertex of a corner +separating both propellants. Each scheme corresponds to distinct positions of the interface, identified +by the line ������������, determined by angle ������������������������ that the interface forms with the line ������������2 (Figure 12). When ������������������������ is +large enough, as depicted in the scheme ℎ), the propellant 2 runs through the entire interface to the +point ������������ faster than propellant 2. Therefore, the combustion surface ������������������������ is plotted by calculating the +envelope of the propagation cylinders in the propellant 1, that is, it is obtained from the line that +starts from ������������ and is tangent to ������������1 at the point ������������. As depicted in the scheme, between the lines ������������1 and +������������, a rarefaction is formed that is reduced as ������������������������ decreases. The rarefaction disappears when ������������������������ = ������������ +(scheme ������������)) which, as in the cusp, gives rise to a scenario without mutual interactions, each propellant +was consumed independently of the other. While 0 < ������������������������ < ������������ the process is, as presented in the scheme +������������), similar to the scheme ℎ), but in this case the point of tangency ������������ goes over the line ������������1 and the +envelope generates the caustic ������������ along the segment ������������������������ +����. + +Figure 14: Sequence of the different schemes for distinct positions of the +interface, in the case where the initial combustion surface has a corner at +the vertex between the two propellants. +The situation when ������������������������ < 0 is represented in scheme ������������) of Figure 14. A rarefaction is generated in the +propellant 2 between the lines ������������ and ������������2. The surface of the front coincides with ������������2 up to ������������2, and +between ������������2 and ������������ it coincides with ������������2. As the propellant 2 continues to dominate the process, the +combustion surface ������������������������ +��� is obtained as before, tracing the tangent to the circle ������������1 that goes through ������������. + +8;>8 +S +8; =8 +>8>B +s +S;=OB +S +s +I三E +I三B +b1 +C +b1 +F2 +b +i +b2 +c +i/ +f2 +S>0; > 0 +8; = 0 +0>1g +2 +C +L=F2 +i +c +iE +>0 +S +S +f2 +f2 +F2 +F2 +0>1g +F +8; <-4β -8 +; = -4β - 831 + +For negative angles (������������������������ < 0), but greater in absolute value, the circular arc ������������2������������ grows until it reaches +the line ������������1 in which, as represented in scheme ������������), caustic ������������ disappears, when reaching the interface +itself. If the interface tilts even more (scheme ������������)) is now the propellant 1 the one that dominates the +propagation process, causing the ignition of the propellant 2. The envelope ������������������������ is created from the +cylinder ������������2. +All the above situations correspond to the scheme of Figure 12 at which the equilibrium point ������������ is +outside the interference region bounded by ������������1 and ������������2. When point ������������ is situated within the interference +region, schemes ℎ), ������������) and ������������) are reproduced and a new configuration, not represented in the figures, +appears with two rarefactions, one in each propellant next to the lines ������������1 and ������������2. +Figure 15 shows the result of numerical analysis, with a code based on obtaining the solution of the +Eikonal equation, by simple time marching, which is described later in this work. As can be seen in +the figure, the structures of the schemes ������������), ������������) and ������������) are reproduced faithfully. The algorithm +efficiently captures rarefaction and caustic structures produced near the interface between the two +propellants. + +Figure 15: Result of the numerical simulation of three examples of +bipropellant interface corresponding to the same reference letters in +Figure 13 and Figure 14. +On the one hand, this shows that the previous analytical reasoning is correct, in general terms. On the +other, it shows the power and versatility of the numerical method proposed. As has been seen in the +review of the literature and the analysis of the different methods, the numerical integration of the +Eikonal equation is the best procedure to establish the combustion surfaces, even when the recession +rate is variable. Using a discrete representation and computation system, the kinematics of +geometrically complex burning fronts propagating with prescribed variable burning rates can be +efficiently described. +6. Burnback numerical solution +As shown in the previous sections, the solution of the burnback problem by numerical methods that +offers the best results passes, in the general case, through the solution of a Hamilton-Jacobi equation, +although, naturally, the direct solution of the Eikonal equation can be selected. Two lines of work can +be distinguished in the solution of this type of equations, one that addresses the mathematical +problem in a generic way (e.g. [64]–[66]) and another driven by the solution of front propagation +problems using LSM (e.g. [67] or, very recently, [47]). +The study of numerical approximations to the viscous solution was also initiated by Crandall and +Lions [65]. They introduced an important class of monotonic schemes for a simplified form of +equations and showed that these schemes converge to the viscous solution (for an in-depth review of +this matter from a general point of view, see [64]). However, it is known that monotonous schemes can +be at most first-order, so they are too dissipative for most practical applications, although they are + +32 + +used to build high-order algorithms. In reference [67], Osher and Sethian built a class of high-order +upwind-type schemes to, imitating ENO algorithm of high order developed by Harten et al. [68] and +Shu and Osher [69], approximate conservation laws. Its construction was based on the observation +that the Hamilton-Jacobi equations are closely related to conservation laws. In this sense, a wide +variety of algorithms have been proposed, such as those described in e.g. [69]–[71]. +In particular, in the problem of burnback analysis, this type of algorithms has been used on numerous +occasions, but the applications that are most interesting are those developed for unstructured meshes. +The nature of the initial combustion surfaces and the need to use complicated geometries that meet +the design requirements of solid-propellant rocket engines leads inexorably to the use of unstructured +meshes. In addition, this type of meshing allows noticeably short generation times, which has a +significant impact on the overall efficiency of the process. The solution of the unstructured Hamilton– +Jacobi equation composed of triangles was first proposed by Abgrall [72] by the approximate solution +of a classical Riemman problem, based on the work of Bardi and Evans [73]. These works have been +followed by others [74]–[77] in which the approximation order was increased or different schemes of +the same type were tested. Special mention should be made, in this category, of the schemes that +obtain the solution of the Eikonal equation by means of fast marching algorithms in unstructured +meshes, such as in [78] or [79]. +6.1. Time marching method +In the present work, the solution of the Eikonal equation is obtained by means of the simple time +marching procedure in an unstructured two-dimensional mesh composed of triangular elements. The +integration domain is the complete volume of propellant, delimited by the initial combustion surface +and the surfaces that remain inert (surfaces inhibited for combustion and the surfaces in contact with +insulating material or in contact with the case). The value of the unknown function ������������(������������⃗, ������������), which +represents the time of arrival of the front, is stored at the vertices of the mesh and, as already +indicated above, the problem to be solved is +������������������������ + ������������(������������������������) = 0 +(67) +In which the Hamiltonian is +������������(������������������������) = 1 − ������������̇������������|������������������������| +(68) +With the initial condition ������������(������������⃗, 0) = 0, which is also imposed as a boundary condition on the initial +propellant surface throughout the integration. The method used does not need to impose spatial +boundary conditions on inert surfaces, through which the combustion front passes without +disturbance. However, it is customary to select portions of the propellant volume delimited by surfaces +with symmetry conditions, which is easily implemented in the algorithm. + +Figure 16: In the diagram on the left, the main geometric elements used +in the basic discretization around the node ������������ are represented; and on the +right, the notation used in the edge-based algorithm to construct the +discrete solution is shown. + +U, +U, +0, +0j+1 +nj+1/2 +i +0j+1 +Uj+133 + +The solution of the equation can be obtained numerically efficiently, by means of a discretization +based on the work of Abgrall [72]. This requires a domain triangularization, using the variable values +������������������������ (������������ = 1. . ������������������������) at the vertices, to estimate the value of the gradients of the function at each triangle, +��������������⃗������������ = [������������������������]������������ (������������ = 1. . ������������������������). In Figure 16, the geometric configuration used is represented, in which the +angles around an edge connected to the node i are ������������������������ y ������������������������+1 and the unit vector in the direction of +the edge is �������������⃗������������+1 2 +⁄ . The value of the function over time ������������ = (������������ + 1)∆������������ is obtained from: +������������������������ +������������+1 = ������������������������ +������������ + ∆������������ℋ���������������⃗������������ +������������� +(69) +Where +ℋ���������������⃗������������ +������������� = ������������ � 1 +2������������ � ��������������������������������������⃗������������ +������������ +������������ +� − ������������������������ � ������������������������+1 2 +⁄ +��������������⃗������������ +������������ + ��������������⃗������������+1 +������������ +2 +∙ �������������⃗������������+1 2 +⁄ +������������ + +(70) +And +������������������������+1 2 +⁄ = ������������������������������������ ������������������������ +2 + ������������������������������������ ������������������������+1 +2 +(71) +Integration must be carried out under the stability condition ∆������������ ≤ ℎ ������������ +⁄ where ℎ is the minimum height +of the adjacent triangles and the diffusion factor is calculated by ������������i = ������������ π +⁄ , being ������������ = max +������������ ‖∇������������‖. +The algorithm is constructed by traversing the edges of the mesh and updating the value of the mean +gradient on each node (see the right diagram of Figure 16). The procedure is executed by using the +following relationships: +��������������⃗������������1 ← ��������������⃗������������1 + 1 +2 ���������������������������������������⃗������������ + ������������������������+1��������������⃗������������+1� +(72) +��������������⃗������������2 ← ��������������⃗������������2 + 1 +2 �������������′������������+1��������������⃗������������+1 + ������������′��������������������������⃗������������� +(73) +And calculating +������������������������+1 2 +⁄ = ������������������������������������ ������������������������ +2 + ������������������������������������ ������������������������+1 +2 +(74) +������������′������������+1 2 +⁄ = ������������������������������������ ������������′������������+1 +2 ++ ������������������������������������ ������������′������������ +2 +(75) +The diffusion terms of the equation are calculated by +������������������������1 ← ������������������������1 − ������������������������1������������������������+1 2 +⁄ +1 +2 ���������������⃗������������ + ��������������⃗������������+1� ∙ �������������⃗������������+1 2 +⁄ +(76) +������������������������2 ← ������������������������2 − ������������������������2������������′������������+1 2 +⁄ +1 +2 ���������������⃗������������+1 + ��������������⃗������������� ∙ �−�������������⃗������������+1 2 +⁄ � +(77) +Boundary conditions are applied for the nodes of each contour by modifying the values of ��������������⃗������������ and of ������������������������ +calculated on all nodes as follows: +a) Free contour +������������������������ ← ������������������������ +(78) +������������������������ ← 2������������������������ +(79) +b) Symmetry contour +��������������⃗������������ ← 1 +2 ���������������⃗������������ + ��������������⃗������������� +������������������������������������� +(80) +������������������������ ← 2������������������������ +(81) + +34 + +Where ��������������⃗������������� +������������������������������������ is the symmetric vector to ��������������⃗������������. The integration is advanced until reaching a steady state, +which is ensured by checking that the gradients of the variable within each triangle do not change +above a predetermined value. +6.2. Results and discussion +Previously, throughout this document, results of numerical simulations that employ the algorithm +described above have been presented in Figure 9, Figure 10 and Figure 15. These results clearly show +the method's ability to deal with situations in which the velocity of front propagation is not constant. +The cases of bipropellant, in star configuration and ellipse of high fill coefficient, are handled +efficiently. The presence of the interface that separates both propellants is undertaken with the single +implementation of assigning different values to the recession rate to each node within the domain. The +same technique is used in the three simulations presented in Figure 15, configuring the calculation +domain and the inclination of the interface properly. +Figure 17 shows the results of three representative cases. The results have been calculated with unit +recession rate in the system of units in which the geometry is represented, that is, advance coordinate +and pseudotime coincide. The three cases have been calculated with a modest number of elements not +exceeding 104 nodes. Even so, the results show reasonable precision in the absence of a more rigorous +error analysis that is carried out in the following section. The examples show the ability of the method +to describe all relevant phenomena in the analysis of these configurations. In the so-called anchor +geometry, the combustion front collides with the engine casing generating an abrupt change in the +combustion area, while, inside, a caustic is formed when the combustion fronts collide, coming from +the central slot and the circumferential groove. The second case corresponds to a star geometry +optimized to produce neutral combustion. Finally, an unoptimized case of dogbone geometry is +included in which it is observed that the condition of free contour in inert boundaries is treated +without visible reflections and disturbances. + +35 + + +Figure 17: From left to right: constant pseudotime lines, mesh utilized, +and curves of combustion surface area for three representative cases (top +to bottom: anchor geometry, optimized neutral-burn star, and +unoptimized dogbone). +Figure 18 shows the results obtained with a partially optimized axil-geometry. The constant +pseudotime line shows the full variety of situations; and the combustion area curve only needs a few +adjustments to present a properly flat profile. + +Figure 18: Constant pseudotime lines and combustion area curve for a +geometry corresponding to a low-slenderness engine with an axil-type +grain in the process of manual optimization. +6.2.7 Error analysis +A simple slotted geometry is chosen to perform error analysis. This geometry brings together two +aspects of interest: the expansion of a combustion front in which the combustion perimeter increases +and the collision of two combustion fronts with the consequent generation of a caustic. This is a +simple situation, and the error can be calculated by comparing it with the analytical solution of the +problem. + +0.7 +0.6 +0.5 +0.4 +Ab +0.3 +0.2 +0.1 +0 +0.00 +0.05 +0.10 +0.15 +y +0.20 +* +0.0 +0.1 +0.1 +0.2 +10 +Ab +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.00.15 +0.10 +WW +0.05 +0.00 +0.00 +0.05 +0.10 +0.15 +0.2036 + + +Figure 19 Level contours of pseudotime (left), 2500-node mesh (center), +and combustion area (right) on the problem used to evaluate the +discretization error. +The problem consists in the advance of a combustion front from a radial slot (only the right half of +the domain is considered using vertical symmetry) composed of a straight section ending in a +semicircle. As shown in Figure 19, the motor case would be located at the upper border and at the +right border where the combustion front leaves the domain. The combustion area that develops this +geometry is traced in the graph of the figure, and consists of a first section of neutral combustion, due +to the increase in perimeter caused by the circular expansion, combined with the destruction of +geometry caused by the caustic, followed by a process of a strong fall of the area, while the +combustion front leaves the asymmetrical upper part. + + +Figure 20: Constant normalized error contours in the single slot problem +for different number of nodes in the mesh. +Figure 20 displays the normalized error obtained in simulations with different number of nodes in the +mesh. The error has been calculated as the difference between the calculated value and the exact +analytically calculated one. The error is normalized with the maximum value of the penetration level +reached by the front, so the contours of the figure are representative of the relative error. This +procedure has been chosen because it is not possible to calculate the relative error in the initial +contours in which the value of the forward coordinate is very small. In all cases, it is observed that +the error incurred is extremely small in the advance of the straight fronts. However, on the +discontinuity the error is noticeable and in absolute value increases throughout the expansion range. +In the cases analyzed, the maximum error, corresponding to the coarsest mesh, is less than 1% and is +located where the discontinuity crosses the contour. By increasing the number of nodes of the mesh, +the error decreases significantly and as already mentioned, even with meshes of modest size, the +results obtained are very valuable. In the figure, the denser mesh provides a solution in which the +error in the front position is less than 0.3%. + +0.0 +1.0 +2.0 +3.0 +y +4.0error +0.01 +0.009 +0.008 +0.007 +0.006 +0.005 +0.004 +0.003 +0.002 +nodes=4000 +8000 +16000 +0.00137 + +7. Conclusions +Burnback analysis is a central issue in the calculation of the performances of solid propellant rocket +engines. Since the beginning of the development of these engines, a variety of methods have been used +to address this problem. The first methods used were purely analytical and could only be applied to +simple geometries, although the skill of some researchers led them to solve complex cases of industrial +interest. The use of the first digital calculators, to automate calculation, and numerical methods in +modern computers applied to differential equations, which adequately describe the kinematics of the +free surface, has put the problem of burnback in a state of remarkable technological maturity. Also, a +series of phenomenological methods have recently been developed, which use specific properties of the +solution, like the principle of minimum time or Piobert’s statement, which obtain interesting results +but are difficult to generalize to problems with non-uniform recession velocity. +The most general and fruitful methods lie in solving the Eikonal equation which, as shown in this +paper, is obtained from the detailed analysis of the process. Although the direct resolution of the +equation was addressed early, at the beginning of this century, giving rise to powerful and versatile +methods, during the last twenty years the developments have led to solving the burnback problem +using the so-called Level Set Method. LSM-based calculations solve a Hamilton-Jacobi equation, using +a signed level function, to get the solution robustly and reliably, without limitations in the geometries +to analyze nor in the recession velocity distributions. However, this strategy is oversized for the +burnback problem. LSM is a procedure that solves much more general problems than burnback but +enjoys great popularity because it is used in a very wide range of free-boundary problems and with +applications in very different fields. From a broad efficiency point of view, the burnback problem must +be solved using the Eikonal equation on an unstructured discretization of the propellant volume, so +that it is possible to address any geometric complication that the design problem of a solid-propellant +rocket engine requires. The method is computationally efficient, especially when compared with other +kinds of analyses that need to be addressed in the design of a solid-propellant rocket engine (e.g. +structural or internal aerodynamic calculations). The reason is that only one unknown needs to be +solved and the meshing does not need the sophistication of a CFD mesh. +This paper develops the basic theory of propagation of the combustion front, carries out a critical +review of the existing literature on burnback analysis, highlights the ability of analytical methods +solving very general problems of, for example, bipropellants, and shows the power and versatility of +the integration of the Eikonal equation, using simple time marching for the solution of any grain +design problem. +8. Acknowledgements +This study has been carried out as part of the PILUM project (Proyecto de Investigación de +tecnologías para Lanzador, Ubicado en plataforma aérea, de Micro y nano satélites) promoted by +INTA (Instituto Nacional de Tecnología Aeroespacial Esteban Terradas), an autonomous agency of +the Spanish public administration responsible for the aerospace and defense technologies research and +especially from the support received from Tcol. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='. 37 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Acknowledgements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='. 38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Abstract Burnback analysis is a geometric exercise, whose correct solution leads to obtaining the thrust curve of solid propellant rockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" Traditionally, Piobert's statement, which introduces a certain amount of intuition, is used as an argument to construct analytical and numerical algorithms, although it is also common to use numerical integration of differential equations, whose solution is free of ambiguities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This paper presents a detailed study of the process experienced by the combustion surface that allows enunciating the properties of the kinematics of the surface without the need to appeal to heuristic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To the author’s knowledge, although simple and usual in other disciplines, this kind of analysis has not been presented previously in the field of the combustion process of a solid propellant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' A formal development of the theory allows us to identify the Eikonal equation as representative of the physical process and the one that is necessary to solve to obtain a true problem description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Next, the methods used throughout the technological development of solid propellant rockets are reviewed, from 2 their beginnings, in which only analytical procedures and, at most, their automation were possible by means of the first calculators, to modern methods, which obtain solutions to complex problems, based on the numerical solution of PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Other methods are also reviewed, which are developed around some of the properties presented by the solution, that is, methods of heuristic or phenomenological foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As a result of the review, it becomes clear that the solution of the Eikonal equation for burnback analysis is undertaken in the early 2000’s, clarifying the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, all subsequent developments, systematically, employ techniques based on the Level Set Method developed in the late 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' But LSM is applied to much more general and complex problems, and its use adds nothing new to the problem solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Finally, several examples of the capabilities of the most relevant methods are provided, from the point of view of both efficiency and precision, presenting results in situations of interest, in the field of propulsion by solid-propellant rockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Introduction Solid-propellant rocket motors are the simplest high-performance propulsion system ever devised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It consists of a structural vessel filled with a mixture of energetic solid components, which react chemically at a high rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This reaction produces gases at high temperature and pressure, which are expelled at high speed through a nozzle, producing the consequent reaction force, that is, thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' When the solid propellant ignites and a combustion front is formed on its surface, it is gradually consumed layer by layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The combustion geometry determines the propulsive response of the system, as it directly controls the mass released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' By properly sizing the initially exposed area and anticipating what its variation will be, the thrust variation capacity is anticipated in the geometric design of the propellant (throttling by design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' From the economic point of view, solid propellant rocket engines are very effective propulsion systems due to the simplicity of their configuration and the ease and safety in the tasks of handling, transport, and use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' From the propulsive point of view, the specific impulse they provide is modest, but in many of the space and terrestrial applications this weakness is compensated by simplicity in design and manufacturing economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In addition, the solid propellant rocket motor has a very interesting impulse- density value that makes them the ideal system in applications where the volume is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To ensure the effective use of these systems and the fulfillment of the demanding requirements of the missions in which they are used, design and simulation tools with high degree of fidelity are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this sense, prediction of the thrust curve of the engine is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' And, for this task, one must have versatile, fast, reliable, and accurate tools for analyzing the evolution of the combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The calculation of the burning surface area as a function of time is an essential step in the analysis and design activities of solid propellant rocket engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It is relatively easy to establish a heuristic procedure, based on a set of simple rules, that determine the evolution of the combustion surface with time for simple geometries, but only by a rigorous procedure can realistic and complex problems be addressed: for any initial geometry, or when the combustion rate is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Towards the third decade of the nineteenth century the French general of artillery Guillaume Piobert (1793-1871), military engineer and scientist, enunciated a hypothesis about what was the process that followed the combustion of the substances used in the impulsion of projectiles: The combustion of the inner parts of the gunpowder grains takes place only when the layers that cover them are consumed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the speed with which the fire spreads from one cut to another, in the compound, has great influence on the effects of the explosion (in his own words: "Rapidité de combustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" - La combustion des parties intérieures des grains de poudre n'a lieu que lorsque les couches qui les recouvrent sont consumées;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' la rapidité avec laquelle le feu se propage de tranche en tranche, dans la composition, a la 3 plus grande influence sur les effets de l\'explosion", this quote is from the publication Mémoires sur les pouvoirs de guerre des différents procédés de fabrication: avec résumés des épreuves comparatives faites sur ces poudres à Esquerdes en 1831 et 1832 et à Metz en 1836 et 1837 , printer-bookseller Bachelier, 1844, Paris).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' That is, the propellant undergoes a local process, over the surface, and can be described by a combustion front that consumes it by layers, sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the rate of combustion is uniform, the layers have uniform thickness and the description of the evolution of the surface is reduced to a geometric calculation, in which the time variable is proportional to the depth advanced by the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In Figure 1, the photo corresponding to the geometry of a propellant in intermediate combustion times is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To obtain the images it is necessary to quench the motor (a procedure can be the sudden opening of the chamber, which causes a marked decrease in pressure that has as a consequence that the chemical reaction freezes, stopping the process of consumption of the solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The initial geometry is an eight-pointed star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the central photo the tips have been consumed, and the advance of the combustion front has also continued in the valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Finally, in the last photograph, taken close to the final moment, the combustion front is about to reach the engine casing, even though this will happen earlier at some points than at others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' All of these features are a direct consequence of the initial geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Many of the aspects discussed in the preceding description have a marked influence on the performance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The geometry with edges, the complete consumption of geometric entities (the tips) or the uneven consumption of the propellant that does not reach the casing simultaneously are indicators that determine the efficiency of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 1: Situation of the combustion surface in three instants, the initial one, an intermediate state, and shortly before finishing the combustion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In practice, with uniform surface recession rate (idealized situation in which the pressure of the chamber must be uniform and the erosive combustion effects non-existent) the calculation of the evolution of the combustion surface involves its displacement perpendicular to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' That is, each point on the surface is projected to a point on the new surface along the line perpendicular to the original surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The normal distance traveled by the combustion front at each point will be called forward coordinate (symbol ������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the situation of constant burning rate, the value of the forward coordinate is proportional to the burning time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this text, the term pseudotime is used (symbol ������������) when calculations are made with recession velocity equal to unity in the system of units in which the geometry of the propellant has been stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This has led to burnback analysis being approached on many occasions through analytical procedures with heuristic foundations, as in the well-known SPP© program [1], in which the initial surface is formed by extracting simple geometric elements from the volume of the chamber, such as parallelepipeds, spheres or cones whose combination and evolution reproduces the movement of a complex surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, the complexity of some combustion surfaces and the possibility of the process not taking place with constant recession velocity, make it advisable to establish a well-founded general analysis that allows the problem to be addressed in any situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4 Discrete methods should be used to assess the evolution of combustion surfaces in a general and automatic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Although analytical methods can be very quick and immediate, their application to complex geometries becomes complicated and laborious, or even unapproachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Discrete methods offer the possibility of representing arbitrary combustion surfaces and delivering results automatically and repetitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Usually, in the relevant literature, emphasis is placed on whether the methods solve the problem quickly or not, that is, whether they are computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This interest is motivated because some methods employ search algorithms, which can slow them down if special precautions are not taken, and others involve the numerical integration of differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Today, this aspect is of less relevance, because the power of computers has suffered a spectacular increase in recent years and because the impact of the method used in burnback analysis is small, on a global calculation of the design tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For application in the current context, the algorithms used to calculate the combustion surface at different times must be flexible, reliable, robust, and accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Flexible in the sense of allowing discretization of any surface and treatment of variable recession velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Effective and robust when calculating solutions in which interference effects may appear, such as caustics and rarefactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' And finally, accurate, which in principle could be regarded as a consequence of the previous but is also achieved by using adequate algorithms and well-founded mesh studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In problems closely coupled with the resolution of the internal aerodynamics of the engine, the calculation time of the combustion area can be a non-negligible fraction of the total time, but an inert scalar in a domain of similar size should not exceed the fraction corresponding to the advection calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In addition, the calculation of the combustion surface must not contain many mesh points, when compared with those required in the detailed solution of a fluid field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Combustion front kinematics Mathematically, the problem is to determine the function ������������(������������, ������������, ������������) − ������������ = 0 that describes the combustion surface at each point in time, in the domain initially occupied by the propellant, ������������, ������������, ������������ ∈ ������������, where ������������ ≥ 0 is the time elapsed since ignition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It can also be said that the expression allows us to calculate the time (������������) it takes for the combustion front to reach the point (������������, ������������, ������������) at which, naturally, ������������(������������, ������������, ������������) = 0 defines the initial surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For the correct approach to the problem, it is necessary to provide sufficient information about the value of the burning rate at each point, and that means knowing the recession velocity at all points of the volume initially occupied by the propellant, although its calculation is a consequence of the geometry at each instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" Piobert's statement establishes that the combustion surface moves in the normal direction and suggests that each point on the surface moves perpendicular to the surface itself, but what happens is that the points disappear." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The intuition of the scientist was correct, but it is worth developing a procedure that can be followed with confidence in any situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To do this, imagine that we can refer to each point of the combustion surface ������������(������������, ������������, ������������) = ������������ by means of a position vector, ������������⃗������������(������������, ������������, ������������) where ������������ and ������������ are two parameters, without specific physical dimensions, whose variation defines the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Now, it is assumed that both parameters define the surface in the region of interest with values of order unity, ������������~������������~1, although sometimes it may be convenient to parameterize the surface using the arc lengths, which will be expressly indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' All points on the surface are subjected to the combustion process simultaneously and the geometry obtained is a consequence of this on the region occupied by the propellant (for example, ������������ ≥ ������������ ∩ ������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To correctly analyze the problem, we will use the Huygens–Fresnel principle, which states that each point of a wavefront acts as a source point of a spherical wavefront, and that the interaction of all of them forms the propagation of the original front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Consider that the combustion process will affect only one point, ������������, at which the combustion process 5 begins, as shown in Figure 2, and that the burning rate is constant and of value ������������̇������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' After a time ������������������������ the material consumed will be the one inside the intersection between the propellant and the sphere of center ������������ and radius ������������̇������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If it is now considered that all points on the surface of the propellant participate in the combustion process, each of them will be the center of a sphere that will have consumed the propellant inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Over time the propellant contained inside all spheres will have been consumed and the combustion surface will be the envelope of the family of spheres internal to the propellant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This is a generalized algorithm for the determination of the new position of the combustion surface that can be applied whatever the shape of the combustion surface and that helps to solve any complicated configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 2: Diagram of the application of the Huygens–Fresnel principle to the determination of the motion of the combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The family of spheres that have their center at a point on the surface ������������ and radio ������������̇������������������������������������ is: (������������⃗ − ������������⃗������������) ∙ (������������⃗ − ������������⃗������������) = �������������̇������������������������������������� 2 (1) The envelope of the family is obtained by canceling out the derivative of the equation of the surface with respect to the parameters ������������ and ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and solving the generated system together with the equation of the family itself (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' it is assumed that the burning rate depends on the position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='differentiating yields ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ∙ (������������⃗ − ������������⃗������������) = −�������������̇������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ∙ (������������⃗ − ������������⃗������������) = −�������������̇������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Replacing the parameters ������������ and ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' equations (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' (2) and (3) provide the expression of the new surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Note that the evolution of the combustion surface must be smooth, at least, in this development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As the combustion surface in time ������������ + ������������������������ is arbitrarily close to the original, using ������������������������⃗ = ������������⃗ − ������������⃗������������, it is obtained that |������������������������⃗|~������������̇������������������������������������, according to equation (1), which is small compared to the characteristic size of the combustion surface ������������ ≫ |������������������������⃗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the left-hand side of equations (2) and (3) is of the order of ������������|������������������������⃗|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' while the right-hand side is of the order of |������������������������⃗|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and since |������������������������⃗|2 ≪ ������������|������������������������⃗|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' equations (2) y (3) must be replaced by ������������������������⃗������������ ������������������������ ⁄ (������������⃗ − ������������⃗������������) = 0 (4) ������������������������⃗������������ ������������������������ ⁄ (������������⃗ − ������������⃗������������) = 0 (5) Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the equations to be solved are (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Vectors ������������������������⃗������������ ������������������������ ⁄ and ������������������������⃗������������ ������������������������ ⁄ are tangent to the surface ������������ and it is concluded that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' both ������������������������⃗������������ ������������������������ ⁄ (������������⃗ − ������������⃗������������) = 0 and ������������������������⃗������������ ������������������������ ⁄ (������������⃗ − ������������⃗������������) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' are the equations of planes perpendicular to the tangent vectors at the point ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The above result cannot be applied on a combustion surface where the normal direction is not defined, but the algorithm of the sphere family does not require the surfaces to be smooth and is very useful when analyzing the evolution of the combustion surface in non-regular situations, with geometric elements such as cusps or corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Also, it is possible to easily analyze complex situations, for 6 example, conductive cables embedded in the propellant or bipropellant situations, with different burning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The direction of advance of the surface is perpendicular to the surface ������������ and therefore parallel to the gradient vector, ∇������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The modulus of the vector is related to the speed of advance of the front since, by the expression chosen at the beginning of this section, ������������(������������, ������������, ������������) − ������������ = 0, and in this way ������������������������ = ������������������������ or, what is the same, |������������������������| = 1 ������������̇������������ ⁄ (6) Which is known as the Eikonal equation (word that, in Greek, means "image").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" This equation is basic in Geometric Optics because it allows the calculation of the trajectories of light rays, perpendicular to the surfaces of the same optical path and, therefore, the calculation of the trajectories that reverse a minimum time (Fermat's principle)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this case, the inverse of the burning rate plays the role of the refractive index (ratio between the light speed in vacuum and that of the medium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This equation is used not only in geometric optic applications, but also in other wave propagation problems, such as electromagnetism or seismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The solutions of the equation can exhibit geometric singularities called caustic ("causticus" in Latin means “burnt”) or the well-known mirage phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The vector ������������������������⃗ has the direction of ∇������������ and the modulus is the variation of the forward normal coordinate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������ = ������������̇������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' with which equation (6) can be written as ������������������������ = 1 ������������̇������������ ������������������������⃗ ������������������������ (7) Differentiating with respect to ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ ������������������������ [������������������������] = ������������ ������������������������ �1 ������������̇������������ ������������������������⃗ ������������������������� (8) The left-hand side can be transformed by the chain rule as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ [������������������������] = ������������������������⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ∙ ������������[������������������������] = ������������̇������������������������������������ ∙ ������������[������������������������] = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ������������̇������������������������[������������������������ ∙ ������������������������] = ������������ �1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='And equation (8) becomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ �1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������� = ������������ �1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='The equation with which the trajectory of the surface points can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Developing the derivatives yields 1 ������������̇������������ ������������������������̇������������ ������������������������ ������������������������⃗ ������������������������ − ������������2������������⃗ ������������������������2 = ������������������������̇������������ ������������̇������������ (11) From which certain interesting properties can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The first one is that, if the burning rate is constant, the surface points move along straight lines since the solution of ������������2������������⃗ ������������������������2 ⁄ = 0, is ������������⃗ = ������������⃗0 + ������������(������������������������ |������������������������| ⁄ ) (12) Where ������������⃗0 is the starting position and it has been used that ������������̇������������|∇������������| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On the other hand, by construction, ������������������������⃗ ������������������������ ⁄ is a vector in the direction of the normal to the surface, whereas ������������2������������⃗ ������������������������2 ⁄ is perpendicular to it, so that the recession rate gradient can be broken down into a normal component ∇⊥������������̇������������ = ������������������������̇������������ ������������������������ ⁄ and a parallel component ∇∥������������̇������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Equation (11) can therefore be projected in the directions perpendicular and parallel to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the direction perpendicular to the surface the result is trivial (equation (12)) while in the parallel direction 7 ������������2������������⃗ ������������������������2 = − ������������∥������������̇������������ ������������̇������������ (13) Which expresses that the trajectories only turn if there is a non-zero parallel gradient of the recession rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' When the recession rate is constant the combustion surface can be reconstructed by simple translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For this reason, numerous heuristic algorithms have been developed over time to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Some general results, related to geometric optics, of interest for the performances of rocket engines have been reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' But the relevant thing is to calculate the combustion area at each moment, because it allows us to determine the thrust curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To have a means of assessing the area of combustion, the surface must be parameterized with ������������⃗������������(������������, ������������, ������������) (note that the subscript will be ignored hereafter), assuming that the values of ������������ y ������������ identify a point on the surface and, as long as the value of the parameters is maintained, the point follows the trajectory described by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' parameterization complies with ������������������������⃗ ������������������������ = ������������̇�������������������������⃗ (14) Where the normal to the surface �������������⃗ is calculated as usual,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' �������������⃗ = ������������⃗������������ × ������������⃗������������ |������������⃗������������ × ������������⃗������������| (15) And ������������⃗������������ = ������������������������⃗ ������������������������ ⁄ and ������������⃗������������ = ������������������������⃗ ������������������������ ⁄ are used to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Equation (14) is equivalent to equation (7), precursor of equation (10) that describes the trajectory, but, in this case, to express the normal it is necessary to reconstruct the surface with the values of ������������⃗ near the considered ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On the other hand, the direction of the normal has been chosen in the direction of advance of the front, that is, the same as ∇������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The combustion area, ������������������������(������������), at any given moment, is calculated by ������������������������ = � |������������⃗������������ × ������������⃗������������| ������������(������������,������������) ������������������������ ������������������������ (16) traversing the set of parameters ������������(������������, ������������) that defines the combustion surface at each instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The temporal variation of the area is therefore ������������ ������������������������ (������������������������) = � ������������|������������⃗������������ × ������������⃗������������| ������������������������ ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Differentiating the cross product yields ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ (������������⃗������������ × ������������⃗������������) = ������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ × ������������⃗������������ + ������������⃗������������ × ������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='The time derivatives of the position vector with respect to the parameters are obtained from equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(14): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ = ������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ �������������⃗ + ������������̇�������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(19) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ = ������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ �������������⃗ + ������������̇�������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Where the nomenclature is �������������⃗������������ = �������������������������⃗ ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='and �������������⃗������������ = �������������������������⃗ ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='for the derivatives of the normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Substituting expressions (19) and (20) into (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ (������������⃗������������ × ������������⃗������������) = ������������̇������������(�������������⃗������������ × ������������⃗������������ − �������������⃗������������ × ������������⃗������������) − �������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ������������⃗������������ − ������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ������������⃗������������� × �������������⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Note that the first term in the right-hand side of equation (21) is a vector perpendicular to the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='tangent plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' parallel to the normal direction) since both �������������⃗������������ and �������������⃗������������ are vectors contained in the tangent plane defined by ������������⃗������������ and ������������⃗������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, the second term is a vector perpendicular to the previous one, contained in the tangent plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Considering that ������������⃗������������ × ������������⃗������������ = |������������⃗������������ × ������������⃗������������|�������������⃗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' it can also be written,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ (������������⃗������������ × ������������⃗������������) = ������������|������������⃗������������ × ������������⃗������������| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='�������������⃗ + |������������⃗������������ × ������������⃗������������| �������������������������⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='and the comparison of equations (21) and (22) yields: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������|������������⃗������������ × ������������⃗������������| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='= ������������̇������������(�������������⃗������������ × ������������⃗������������ − �������������⃗������������ × ������������⃗������������) ∙ �������������⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(23) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='�������������������������⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='|������������⃗������������ × ������������⃗������������| �������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ������������⃗������������ − ������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ������������⃗������������� × �������������⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(24) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Expression (23) evaluates the temporal evolution of the combustion area element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' while expression (24) determines whether the direction of propagation changes or not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which is a result that had already been advanced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' making use of the typical developments of geometric optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These two expressions summarize the behavior of the combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the recession rate is uniform, the direction of the normal at each point remains unchanged and the surface points move in a fixed direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Conversely, if the recession rate changes from one point to another on the surface, the direction of the normal vector changes and the surface is distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To further analyze expression (23), it is convenient to use some concepts of differential geometry of surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The vectors �������������⃗������������ and �������������⃗������������ are contained in the tangent plane and can be expressed as a linear combination of the vectors ������������⃗������������ and ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in the form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='�������������⃗������������ = ������������11������������⃗������������ + ������������21������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='�������������⃗������������ = ������������12������������⃗������������ + ������������22������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='The matrix of coefficients is calculated by: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='�������������11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������22� = − ������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������� ������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(27) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='where the coefficients of the First Fundamental Form (which corresponds to the inner product ������������������������⃗ ∙ ������������������������⃗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='are ������������ = ������������⃗������������ ∙ ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = ������������⃗������������ ∙ ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = ������������⃗������������ ∙ ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and are related to the area element by |������������⃗������������ × ������������⃗������������| = √������������������������ − ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The coefficients of the Second Fundamental Form (which corresponds to the inner product ������������������������⃗ ∙ �������������������������⃗) are ������������ = −�������������⃗������������ ∙ ������������⃗������������ = �������������⃗ ∙ ������������⃗������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = −�������������⃗������������ ∙ ������������⃗������������ = �������������⃗ ∙ ������������⃗������������������������ = �������������⃗ ∙ ������������⃗������������������������ = −�������������⃗������������ ∙ ������������⃗������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = −�������������⃗������������ ∙ ������������⃗������������ = �������������⃗ ∙ ������������⃗������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and are related to the curvature of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The normal curvature of the surface is the ratio of both fundamental forms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������ = (������������������������⃗ ∙ �������������������������⃗) (������������������������⃗ ∙ ������������������������⃗) ⁄ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which is the component of the curvature vector ������������⃗ = ������������������������⃗ ������������������������ ⁄ in the direction of the normal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' where ������������⃗ = ������������������������⃗ ������������������������ ⁄ is the tangent vector (in this case the parameter ������������ describes any curve contained in ������������ that passes through the point in question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The normal curvature is independent of the curve on which it is defined and depends only on the orientation of the tangent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Principal curvatures are the maximum and minimum values of the normal curvatures of a given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the main curvatures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������1 and ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' of the surface are the eigenvalues of the matrix��������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the average curvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = 1 2 (������������1 + ������������2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' is half of the trace of the matrix with the sign changed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = − 1 2 (������������11 + ������������22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and 9 Gaussian curvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = ������������1������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' coincides with the determinant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = det��������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which corresponds to the intrinsic curvature of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Naturally, all these values do not depend on the chosen parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Substituting expressions (25) and (26) into (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������|������������⃗������������ × ������������⃗������������| ������������������������ = ������������̇������������(������������11������������⃗������������ × ������������⃗������������ − ������������22������������⃗������������ × ������������⃗������������) ∙ �������������⃗ (28) That is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������|������������⃗������������ × ������������⃗������������| ������������������������ = −������������̇������������(������������1 + ������������2)|������������⃗������������ × ������������⃗������������| (29) Expression (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which can be rewritten as ������������(������������������������) ������������������������ ⁄ = 2������������(������������������������),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' is a classical result in differential geometry when one intends to obtain the variation of the area,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' of a family of surfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and is directly related to very interesting topics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' such as the plotting of surfaces of constant average curvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' or obtaining surfaces of minimum area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the current context, it provides a direct geometric interpretation of how the combustion area evolves over time, due to the local value of the recession rate and as a function of surface curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' At a symmetrical saddle point, ������������1 = −������������2, the net variation of the combustion area is zero, whereas, if the surface concavity prevails at the point, ������������1 + ������������2 > 0, the area decreases, but if the surface is globally convex, ������������1 + ������������2 < 0, the combustion area increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' equation (24) can be rewritten as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='�������������������������⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='|������������⃗������������ × ������������⃗������������|2 �������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ������������⃗������������ × (������������⃗������������ × ������������⃗������������) − ������������������������̇������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ������������⃗������������ × (������������⃗������������ × ������������⃗������������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='The expression is apparently complicated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' but if a new parameterization of the surface is used,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' being (������������′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������′) the arc lengths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' it is then verified that |������������⃗������������′| = |������������⃗������������′| = 1 and if,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' orthogonality is required,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������⃗������������′ ∙ ������������⃗������������′ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' this yields �������������������������⃗ ������������������������ = − �������������������������̇������������ ������������������������′ ������������⃗������������′ + ������������������������̇������������ ������������������������′ ������������⃗������������′� ≡ −������������∥������������̇������������ (31) Where the vector identity ������������⃗ × ��������������⃗ × ������������⃗� = �������������⃗(������������⃗ ∙ ������������⃗) − ������������⃗�������������⃗ ∙ �������������⃗� has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To interpret the expression more easily one can use ������������������������ = ������������̇������������������������������������ and write �������������������������⃗ ������������������������ = − ������������∥������������̇������������ ������������̇������������ (32) which is identical to (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The normal vector to the surface changes its direction according to the direction marked by the gradient of the recession rate in the plane tangent to the surface and in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Equations (14) and (32) are a system equivalent to equation (10) that can be integrated over time, using the information provided by the function ������������̇������������(������������, ������������, ������������), to obtain the evolution of the combustion surface,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The normal vector is tangent to the trajectory followed by the point ������������, so its variation with the length traveled is the curvature, which will be proportional to the modulus of the gradient of the recession rate referred to itself, as written in equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Admitting that this quantity is constant, for small values of the forward coordinate, the trajectory of this point describes an arc of radius ������������̇������������ �∇′������������̇������������� ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 10 Figure 3: Schematic representation of the process that takes place when the recession rate varies linearly on the surface of the propellant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The trajectory of the ray is ������������������������′′′, although the point of tangency of the circle envelope is ������������′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' A schematic representation of the process that takes place with variable recession velocity has been made in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The recession rate is considered to vary linearly on the surface of the propellant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������̇������������ = ������������0 + ������������1������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and it is assumed that ������������1������������������������ ≪ ������������0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' so that the combustion front,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' for a time ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' moves from the point ������������ a distance ������������������������~������������0������������������������ that in the figure is used to draw a circle of center ������������ that locates the possible points that the combustion surface could reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" Applying Piobert's principle directly, equivalent to the exact result that the points on the surface move perpendicular to it, the ray would describe the trajectory ������������������������′ and the new combustion surface would be built by joining the image points ������������′ of the entire surface." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" The above does not consider that applying Huygens' principle, each point ������������ of the surface is the center of a circle of distinct radii that grows at a rate ������������������������~������������1������������������������, being ������������′′ the image points in a position other than ������������′." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, considering what was shown in previous lines, the trajectory of the point ������������ is an arc of radius ������������0 ������������1 ⁄ , and rotating an angle ������������������������~������������1������������������������~������������1 ������������������������ ������������0 ⁄ , to the point ������������′′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The points ������������′ and ������������′′ are not correct and underestimate the position of the combustion surface, situation which is relieved because the distance ������������������������′′′ must be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Uniform recession rate If the recession rate is uniform, then, ������������������������̇������������ ������������������������ ⁄ = ������������������������̇������������ ������������������������ ⁄ = 0 and from equation (30): �������������������������⃗ ������������������������ = 0 (33) Which indicates that the propagation directions remain unchanged, although the propagation velocity may be a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These circumstances have the consequence of the propagation problem becoming decoupled from the temporal problem and being, therefore, purely geometric in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Consequently, the temporal evolution of the combustion area satisfies that the normal directions to the surface remain unchanged and the trajectories of the points on the surface are straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The centers of curvature of the surface are located above the normal lines in fixed positions and the shape of the surface can be easily reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The surface retains its topology until the propellant consumption reaches some center of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' At that moment the analysis ceases to be valid and if the combustion front progresses there is an irreversible destruction of geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Because ������������̇������������ = ������������������������ ������������������������ ⁄ , the forward coordinate may be used in equation (14), instead of time: ������������������������⃗ ������������������������ = �������������⃗ (34) 11 Which is independent of the pace of recession and, therefore, the burnback problem is reduced to an exercise in geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Effectively, it can be integrated using the initial geometry from ������������ = 0 (corresponding to the initial time, ������������ = 0), obtaining a family of surfaces ������������⃗������������(������������, ������������, ������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The recession rate can present any sort of time dependency because, from the known family ������������⃗������������(������������, ������������, ������������) and the expression ������������������������ = ������������̇������������(������������)������������������������, its evolution with time can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Depending on the nature of the initial surface, different methods may be used to obtain its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the radii of curvature are defined at all points of interest, a possible procedure to obtain the surface ������������⃗������������(������������, ������������, ������������) is to evaluate the length of the radii of curvature, ������������1,2 = 1 ������������1,2 ⁄ , and describe how they change by means of equation (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' That is, solving ������������������������1,2 ������������������������ ⁄ = −1, expression that supports the general solution ������������1,2(������������, ������������, ������������) = ������������1,2 o (������������, ������������) − ������������ (35) Where the initial surface has the distribution ������������1,2 o (������������, ������������) of radii of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' An immediate consequence is that, when a radius of curvature cancels out (note that, for this to be possible, it is necessary that the radius of curvature is strictly positive at ������������ = 0, which corresponds to an initially convex geometry), there is an unavoidable discontinuity, since all the points of the combustion surface collide in the center of curvature, without the integration being able to continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This event partially destroys geometry and requires a special analysis, since it is necessary to consider the evolution of a surface that contains non-regular points or regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Note that, contrary to the usual definition of the radius of curvature as the absolute value of the inverse of the curvature, here it has been given the sign of the curvature itself, to be able to generalize the relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this way, those radii that extend behind the space traveled by the normal are considered negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' When parameterizing the surface with the arc lengths, the absolute value of the radius of curvature will be taken, so that the angular sectors traveled will be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Calling back to relationship (29): ������������|������������⃗������������ × ������������⃗������������| ������������������������ = − � 1 ������������1 + 1 ������������2 � |������������⃗������������ × ������������⃗������������| (36) Where the time variable has been replaced by the normal coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the surface is parameterized by arc lengths following the main directions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������′ = |������������1|������������������������1 and ������������������������′ = |������������2|������������������������2(which are orthonormal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' |������������⃗������������′| = |������������⃗������������′| = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' when considering the main curvatures),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the variation of the combustion area is: ������������������������������������ ������������������������ = − � � 1 ������������1 + 1 ������������2 � |������������1| ������������(������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2) |������������2|������������������������1������������������������2 (37) where ������������(������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2) expresses that integration variables now extend into a different domain than the parameterization used before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Note that the initial radii of curvature, ������������1,2 o = ������������1,2 o (������������1, ������������2), that we are going to use to calculate the area can be a function of the angles, (������������1, ������������2), so that there are no restrictions on the combustion surface, other than the mere existence of the radii of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the case of regions of null curvature, the original expression must be retrieved since the expression (37) has been invalidated by using the inverse of the curvatures in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Without going into major complications, what follows is useful to analyze the behavior of fixed angular sectors, since, for calculation purposes, the area can be decomposed into an arbitrary number of portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' With the intervention of (35) and some algebra, it can be obtained, ������������������������������������ ������������������������ = − � (sgn(������������1)|������������2 o − ������������| + sgn(������������2) |������������1 o − ������������|) ������������(������������1,2) ������������������������1������������������������2 (38) To analyze the expression, it is necessary to separate the different cases according to the sign of the curvatures, or the radii of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If both are positive, the slope of the combustion area, depending 12 on the forward coordinate, ������������������������������������ ������������������������ ⁄ , is monotonically decreasing and, as both radii decrease, the analysis is valid until the smaller one is canceled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If both are negative, the slope is positive, with no limits other than those of the surface itself or those imposed by the combustion chamber casing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the signs of the radii of curvature are different, it is necessary to elaborate the analysis with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If we consider the case of the negative radius being less in absolute value than the positive one, the value of the initial slope is negative, and grows linearly with ������������ until it reaches the point where both radii equal in absolute value (this coincides with zero variation of the slope, which corresponds to a minimum of the local area enclosed in the angular sector considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It then follows an upward slope behavior, until the initially positive radius is canceled out, stopping the linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Any of the situations considered above leads to a linear variation of the slope and, therefore, to a quadratic variation of the combustion area with the forward coordinate of advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Because of the above considerations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' expression (38) can be reordered as follows: ������������������������������������ ������������������������ = � sgn(������������1������������2) [2������������ − (������������1 o + ������������2 o)] ������������(������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2) ������������������������1������������������������2 (39) The combustion area finally is ������������������������ = � {sgn(������������1������������2) [������������2 − (������������1 o + ������������2 o)������������] + ������������1 o������������2 o} ������������(������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2) ������������������������1������������������������2 (40) Which is canceled out when ������������ = ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 o and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' it is fulfilled ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 o > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which corresponds to the situation of zero radius of curvature when the combustion front destroys a rounded cusp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' already noted in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Cylindrical geometries In line with the high slenderness of rocket-propelled aerospace vehicles, it is common to find combustion surfaces where the longitudinal dimension predominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the vehicle is very slender, and the thrust demand is high, the combustion surface must be greater than the cross-sectional area, and the only way to achieve this is by longitudinal drilling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this case, the combustion surface is of cylindrical type, in which the characteristic dimension along the grain (~������������) is large compared to the cross-sectional dimension (~������������), that is, ������������ ≫ ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Local curvatures (in the longitudinal and transverse direction),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' necessarily,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' verify ������������1~1 ������������ ⁄ and ������������2~1 ������������ ⁄ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which results in ������������1 ≪ ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' so that equation (29) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='may be simplified by ignoring ������������1 as compared with ������������2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������|������������⃗������������ × ������������⃗������������| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='≈ −������������̇������������������������2|������������⃗������������ × ������������⃗������������| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(41) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='The temporal variation of the combustion area is: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ = − � ������������̇������������������������2������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(42) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='where the surface differential element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' |������������⃗������������ × ������������⃗������������|������������������������ ������������������������ = ������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' is expressed by the coordinate along the cylinder ������������ and the arc length ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As the curvature of the cross section can be set as ������������2 = − ������������������������ ������������������������ ⁄ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' being ������������ the angle formed by the tangent to the curve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the previous expression becomes: ������������������������������������ ������������������������ = � ������������̇������������������������������������������������������������ ������������ (43) For a differential element of area,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' it is verified: (44) 13 ������������ ������������������������ (������������������������������������) = ������������������������������������������������ This is a very interesting expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' First, it shows again that if the recession rate is uniform then the problem is exclusively geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Moreover, if the length of the cylinder remains unchanged in the process then its influence is reduced to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, the most interesting property is that the variation of the combustion area is independent of the shape of the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' All reference to dimensions has disappeared from the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The variation of the combustion area is proportional to the value of the angular sector traveled by the tangent when running around the perimeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and in the case of a straight cylinder of constant length (������������) and uniform recession rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' it turns out to be: ������������������������������������ ������������������������ = 2������������������������ (45) This value is independent of the shape of the section and corresponds to a progressive combustion process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' identical to that which takes place for a cylinder of circular section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The combustion area is obtained immediately, ������������������������ = ������������������������ ο + 2������������������������������������, where ������������������������ ο is the value of the initial area for ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Naturally, these results are subject to the cross-section being regular, in the sense that the curvature is defined at all points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Under these conditions, the variation of the area meets the following properties: i) it is independent of the shape of the perimeter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ii) it has a constant value equal to the angle rotated by the tangent to the curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and iii) the sign (which marks the character of the combustion process) is the contrary to that of the curvature, when the normal to the curve points in the direction of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Consequently, the expression of the perimeter is linear with the forward coordinate, and the process is reversible, in the sense that, if the direction of propagation is reversed, the initial geometry is reached uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the regressive regions of the perimeter, the propagation process decreases the radius of curvature and when the depth of advance reaches the center of curvature a discontinuity is generated, since a convex region disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' At that point, the perimeter topology changes, and the surface analysis must be restarted, probably considering the evolution of a cusp, as will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The process of combustion of slender channels can be adequately described by one-dimensional models in which the geometry of the channel is determined by the distribution of port areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Consider the perimeter of each section ������������������������ = ∮ ������������������������ and the port area in each section ������������������������ = ∮ ������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For calculation purposes, the combustion area, ������������������������, can be defined at any given section as the area of combustion exposed from ������������ = 0 to the considered section ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' That is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������ = � ������������������������������������������������̅ ������������ 0 (46) If the recession rate is uniform in the section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which is the most consistent simplification with the slender cylinder approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the variation with time of the port area (invoking again ������������̇������������ = ������������������������ ������������������������ ⁄ = ������������������������ ������������������������ ⁄ ) is ������������������������������������ ������������������������ = ������������������������ (47) While the perimeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in the assumption that it is a regular curve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' complies with expression (45) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������������������ ������������������������ = 2������������ (48) The above expressions constitute a closed geometric system with which all geometric variables can be calculated using very simple integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Non-regular geometries The conclusions obtained in previous paragraphs can be generalized to contours in which the radius of curvature may present discontinuities, but for which the tangent to the perimeters must be a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In these circumstances, for each point of the combustion surface, an image point can be defined as the surface evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' That is, a bijective relationship can be established between the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This does not occur when: i) there are discontinuities in the tangent to the combustion surface, ii) two combustion surfaces meet each other, or iii) the combustion front reaches the motor case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the first and second situations the trajectories of the surface points intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the front reaches the motor case or any other inert element, the points on the surface also disappear irreversibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' All these situations are irreversible, in the sense that, if the sign of the recession rate is changed, the succession of combustion areas produced is not the same, just reversed in time, but very different, indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Next, a number of geometries that are commonly presented in solid propellant engines and that do not have a regular behavior are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 Corners and cusp When the perimeter of the section presents a break, which represents a discontinuity in the slope, a non-regular situation is generated whose evolution is different depending on the direction of advance of the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 4 depicts two different situations in which the gaseous and solid domains are exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In situation (a) the combustion process regularizes the geometry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the vertex of the corner becomes a source point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' origin of a rarefaction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and as the geometry generated presents a smooth distribution of the angle (the tangent to the perimeter is continuous) the rate of increase of the perimeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' as already seen in equation (44),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' is ������������������������������������ ������������������������ � Corner = ∆������������ (49) The increase (decreases are also possible) of the perimeter is proportional to the angle rotated by the tangent when following the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It is easy to imagine an algorithm that accumulates variations of the angle associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 4: In configuration (a) the combustion front progresses from a corner creating a cylindrical surface (rarefaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In configuration (b) the combustion front consumes a cusp destroying geometry and creating a discontinuity (caustic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The situation (b) is the opposite to (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The combustion front destroys part of the geometry as it advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The collision of the two combustion fronts causes a discontinuity that is called a caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The destruction of geometry is irreversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In Figure 4 (b), a simple geometric analysis leads to the relationship CORNER CUSP SP~yβ SP~ - 2ytan △Φ/215 ������������������������������������ ������������������������ � Cusp = −2 ������������������������������������(∆������������ 2 ⁄ ) (50) Which is similar to (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Both relationships coincide if the angle rotated by the perimeter is very small (∆������������ ≪ 1) but, in general, equation (50) has a nonlinear dependence on the angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Fortunately, both expressions have a linear dependence on the forward coordinate, and this allows combining different geometries so that any target value of ������������������������������������ ������������������������ ⁄ can be set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Specifically, an adequate combination of valleys or corners (of a progressive nature, ������������������������������������ ������������������������ ⁄ > 0) and vertices or cusps (of a regressive nature, ������������������������������������ ������������������������ ⁄ < 0) can lead to a geometry in which the perimeter changes in a controlled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For this case, the most common solution is a star-shaped geometry, in which the angle and number of cusps determines the progressive, regressive, or neutral character of the combustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 Collisions Figure 5 (a) shows a dendrite-like geometry in which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' when the thickness is exhausted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the 2������������ arc of circle at the end of the protuberance disappears and the two combustion fronts collide simultaneously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' producing an instantaneous drop (a discontinuity) of the combustion area of the form ������������������������ = −������������������������������������ℋ(������������ − ������������) (51) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������ is the discrete variation of the perimeter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Δ������������������������ is the length of the dendrite,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ℋ( ) is the Heaviside function and ������������ is the semi-thickness of the dendrite (remember that ℋ(������������) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and ℋ(������������) = 1, ������������ ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 5: Sometimes there is the collision of two combustion front (a) or with the engine casing (b), which causes a sudden destruction of the combustion surface and a discontinuity in the evolution of the perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' A similar situation occurs when cylindrical combustion surfaces collide with the motor case in the final phase of propellant combustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the collision takes place sharing the center of curvature the decrease in area will be sudden, ������������������������ = −������������������������������������ℋ(������������ − ������������) (52) Being here Δ������������������������ the arc length of the collision front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the collision is not completely frontal, a very rapid process of combustion area destruction occurs, which must be analyzed in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In both these situations, the most relevant characteristic is that the processes are not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In fact, the burning area presents discontinuity that originate an unsteady response of the chamber pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Furthermore, the evolution is not reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' DENDRITE WALL COLLISION SP~ - 2△P,H(y- ) SP~-APH(y- w)16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Burnback analysis methods Current methods can be classified into Analytical or Numerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" The analytical methods, essentially, consist in using Piobert's aphorism and displacing the combustion surface, formed by simple geometric figures, perpendicular to itself, incorporating the particular phenomenology imposed by cusps and corners." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This activity can be carried out for a simple geometry obtaining closed relationships, or by automating operations through some algorithm, such as the SPP© program or other CAD-type graphic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In contrast, numerical methods start from a discrete description of the combustion surface, which allows them to be more flexible and general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Once the discretized surface is available, it can act as in analytical methods using some specific property of the solution, or address the propagation problem by integrating differential relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Method References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Analytical methods 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Simple/unique geometry [2]–[7],[8], [9] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Combination of simple geometries [1], [10]–[12], [13] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' CAD based methods 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Parametrized geometry [14]–[16] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Based in CAD in-house tools [17]–[19] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Numerical methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Direct surface tracking [20]–[23] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Minimum distance function (MDF) [24]–[26], [27]–[30] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Theory of curve and surface evolution (PDE’s based) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Set Level Methods (Hamilton-Jacobi equation) [31]–[33] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Standard (signed function evolution) [34]–[41],[42]–[47] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Narrow band 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Steady perspective (Eikonal equation) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Direct time marching [48]–[53] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Fast marching methods (FMM) [54] Table 1: Classification of the different methods of burnback analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Table 1 lists all the categories of methods considered in this paper and indicates the most relevant bibliographic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Applied to geometries accessible to the method, all those listed in the table solve the problem satisfactorily, from the point of view of thrust curve calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Numerical methods are usually able to deal with more general and complex problems than analytical methods, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Different arguments have been raised in the literature to evaluate the suitability of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As the most versatile and powerful methods are numerical methods, the central argument is usually efficiency, measured in terms of computational time requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, the high power achieved by computers today weakens the importance of this argument, because the computational effort in the field of burnback analysis is moderate compared with that required for the study of, for example, the rocket internal aerodynamics or the structural calculation of the propellant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Numerical burnback analyses only need to obtain a single spatial function that determines the combustion surface as the propellant is consumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In addition, it is not necessary to use adapted meshes, but with significantly uniform meshes that reasonably describe the geometry is sufficient to obtain satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' From this perspective, other considerations, such as the flexibility in terms of the possibility of carrying out complex three-dimensional geometric analyses, the possibility of analyzing cases with variable recession velocity, and the economy of implementation, all make the methods based on the Eikonal equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 in Table 1) the most attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This result contrasts with the very high diffusion that LSM have reached in the analysis of the burnback problem in the last twenty years, motivated by the evident generality of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, the burnback analysis problem does not need so much generality, and the LSM is oversized in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The integration of the Eikonal equation is enough to obtain a completely satisfactory solution of the problem (that is, the calculation of the thrust curve) and, eventually, allow the design of the initial combustion geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the next 17 section, the results obtained with method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 of Table 1, which meets the above requirements, are presented for a variety of grain geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Analytical methods Analytical methods make use of different properties of the solution that are incorporated into the analytical calculation of the position of the combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These methods are fast, simple and accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' But they cannot address problems of arbitrary geometry, they have to solve complex geometric situations with specially adapted procedures, and they cannot, in general, solve problems of variable recession velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 Simple/unique geometry It is the simplest approach and consists in the algebraic analysis of the evolution of a surface that moves perpendicular to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' During the second third of the twentieth century, in the early days of the development of solid propellant engines, it was the only possible method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The work of Billheimer and Wagner [2] contains an extensive bibliographical review of this period and the different procedures with which the simple geometric calculation was enriched to achieve the determination of the thrust curve of solid propellant engines with grain geometries that presented some complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For example, the work of Thibodaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [4] can be reviewed to verify the level of specialization achieved in the analysis of, in this case, three-dimensional geometries in spherical chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Or the arduous work of analyzing the interaction between the combustion front of a slotted-tube grain with the casing of the engine, described in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This type of method is still widely used, and the number of recent citations, referring to burnback analysis with purely analytical methods, is very high (not all collected in this review), because the immediacy of the method lends itself to its easy integration into internal aerodynamics analysis systems [5], or its integration into all kinds of engine design optimization algorithms [6][7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As already mentioned, the most interesting advantages of the method are its speed, simplicity, and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Naturally, it is not possible to analyze arbitrary geometries and it is difficult to incorporate realistic situations such as a non-constant recession velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In addition, the analyses must incorporate a specific treatment of non-continuous geometries (such as cusps and corners) which, for example, in three dimensions can significantly complicate the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, it is possible to address situations of industrial interest and others that initially would seem complex, such as the analysis of two propellants burning simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this sense, through analytical methods, it is possible to address the problem of two propellants with two different recession rates, as for example, to analyze the combustion of a bipropellant star geometry that does not present sliver mass fraction[3][8] and that Krishnan and Bose [9] study with a high level of detail for various configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 Combination of simple geometries The simplicity of use of analytical methods facilitates a different strategy, combining elements of simple geometry and automating the analysis of the evolution of the combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The best known and most successful example is the burnback module of the SPP© software package, initially presented by Coats et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [1] in 1987, and continuously updated and improved since then (see [11]– [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The SPP© program has been a standard reference software in the United States for predicting the performance of solid-propellant rocket engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The methodology for evaluating the thrust coefficient, starting from the chemical equilibrium value, which is corrected with individual efficiencies due to 18 different effects, is an industry standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The Grain Design and Ballistics module allows the design of the initial combustion surface and calculates the thrust curve using a burnback analysis package, an internal aerodynamics module, and calculations with finite chemical kinetics in a two-dimensional nozzle flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The SPP© program has been used in the past, and is still being used today, by major agencies, institutions, and manufacturers of solid-propellant rocket engines in the United States and other countries [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The grain design and analysis module construct the surface by extracting simple geometric figures from an initial volume (the interior of the motor case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It is a Boolean operation that can be repeated with the basic figures resized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the calculation of the evolution of the combustion surface, the dimensions of geometric figures are increased, emulating the advance of the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Operationally, the program is fed with symbolic commands, which are executed sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It is a flexible, versatile, and efficient tool, capable of modeling all the geometries that are usually presented in solid propellant rocket engines, as long as they can be decomposed into simple volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Naturally, it is an analytical methodology that retains the disadvantages already mentioned, but the product has been adapted and consolidated to mitigate these disadvantages as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3 CAD based methods The increase in accessibility and power of computer-aided design (CAD) has meant that these specialized programs have been used to conduct burnback analysis of realistic and overly complex geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This is the main quality of the method, the ability to evaluate surfaces of complicated shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Two strategies may be adopted for the calculation of the area evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On the one hand, when modeling the initial combustion surface, the model can be parameterized so that the recession process is considered, using the parameters that define the model itself (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in Table 1, see references [15]–[17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For example, if a cylinder is parameterized by its radius, by varying the radius a preset quantity, the process of recession is simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The next operation is to vary the parameterized values and allow the graphic system to reconstruct the new combustion surface, executing the corresponding symbolic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The other possibility is to use CAD-specific capabilities that move the model surface with controlled laws (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in Table 1, see references [17]–[19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' That is, specific tools for translation, growth, or projection of surfaces that the software makes available to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These procedures are quick and versatile, can tackle complex geometries, and provide a fast and adequate response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, the information obtained must be extracted from within the CAD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Furthermore, there is no certainty of these geometric operations being able to capture the real problem physics, since many of these operations are hidden from the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Naturally, the user is forced to examine these operations and, eventually, correct situations in which the graphical system fails because it is unable to automatically generate rarefactions or caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Numerical methods Numerical methods approach the problem from a discrete description of a combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Depending on the method, the initial combustion surface can be an external surface of a volumetric mesh that represents the whole propellant, where other surfaces of interest can be easily identified as well, such as the motor case or, for example, symmetries of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Alternatively, the combustion surface is discretized as an isolated surface, whose movement is the objective of the calculation and which, in one way or another, must incorporate an analysis of the interaction with other surfaces such as the engine casing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The advantage of numerical methods is that they allow the description of the evolution of complex combustion surfaces, and, with some exceptions, they allow variable recession rate to be incorporated into the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 Direct surface tracking This category includes methods that carry out local surface monitoring, combined with a position identification that allows interaction with inert areas or with the engine casing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In principle, this type of methods start from a discretization of the surface and obtain its evolution using displacement algorithms that somehow consider properties exhibited by the propagation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" The most commonly used of these properties is Piobert's postulate that the surface moves perpendicular to itself." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Typical methods of calculating free surfaces (Volume of fluid, VOF, method) are also used, identifying the convection rate with the recession rate of the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Among these methods, one can mention the SLIC (Simple Line Interface Calculation) method devised by Noh and Woodward [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The authors conceived it for use in one, two or three spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The domain is discretized into enclosures, and fluid interfaces are represented locally for each enclosure by lines, either perpendicular or parallel to the coordinate directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Decision-making logic is used in the propagation, depending on the arrangement of fluid regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Due to the completely one- dimensional nature of the interface description in SLIC, it is relatively easy to get correct results with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Another very similar method is FLAIR [55], which tries to increase the accuracy by complicating a little the geometric description of the front within each control zone, and is used by Mashayek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [21] for the analysis of two-dimensional combustion geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Belonging to the surface tracking methods that use phenomenological algorithms, which basically project the surface perpendicular to itself, the work of Hejl and Heister [22] carries out direct surface tracking and incorporates locally the peculiarities that are presented in the form of rarefactions and caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Also, in reference [56],[56][57],[57]Another work in this category is carried out by Ki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [23] that present the PIT method (Partial Interface Tracking) in the analysis of combustion surfaces of three-dimensional geometrics of type finocyl and conocyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This method applies a Lagrangian approach to the axisymmetric area of the transverse plane and the two-dimensional area of the longitudinal plane separately, because the Lagrangian approach is an effective way to simulate two-dimensional evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this way, a three-dimensional problem is solved with the computational effort of two two-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The limitation is that geometries have to exhibit some symmetry, which is usually common in solid propellant engines, such as finocyl and conocyl types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, it does not bring anything new in the spectrum of front-tracking methods, but it merely solves with success three-dimensional problems approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 Minimum distance function (MDF) The method of calculating the minimum distance to the initial combustion surface, proposed by Wilcox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [24], has been very fruitful in solving the burnback analysis problem and, in this case, is used to allow internal ballistic calculation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It is a very intuitive method, easy to implement, and does not exhibit limitations in terms of the complexity of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Once the domain occupied by the propellant has been discretized, the method consists in calculating the smallest distance from any interior point to the initial surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This calculation involves a search for the point of the initial surface closest to the inner point, which is onerous from the computational standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Usually, methods of reducing this computational time are required, optimizing search algorithms using standard techniques, such as Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [26] using a divide-and-conquer algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" Like other methods already discussed, MDF employs a property of the solution, in this case, Fermat's Principle, and when the propagation velocity is uniform, the minimum time condition is equivalent to the minimum distance condition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Precisely, this is the disadvantage of the method, which cannot incorporate variable recession rate without overcomplicating the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The reason the generalization of the MDF method is not possible is that a global property is used, which leaves out of the calculation what 20 is the path followed by each ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, the conceptual simplicity and the possibility of applying it in realistic three-dimensional geometries, makes it a widely used method [28]–[30], see, for example, how in [31] is concluded that it is superior to other surface monitoring methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3 Theory of curve and surface evolution (PDE’s based) To describe the propagation of the combustion surface in solid propellant rockets, Saintout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [48] implement an algorithm that incorporates all the characteristics that allow it to describe the physics of the process properly, and identify the equation that they integrate numerically as of the Hamilton- Jacobi type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This situation is reached from preliminary studies of the same research group on surface tracking methods [50] and [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These works are part of the activity made by SNPE (Société Nationale des Poudres et Explosifs, currently a subsidiary of Nexter) for the analysis and design of solid propellant rocket engines in Europe that, in the case of burnback analysis, culminate with the work of Dauch and Ribereau [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this work, the general purpose tool called PIBAL© is presented, which integrates an evolution of the IVOLINA© program (previously developed in the references [52] and [53]) that addresses the integration of the Eikonal equation by a time marching method (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, in parallel to the developments described in the previous paragraph, for the treatment of this type of problems (and other, more complex), the work of Osher and Sethian [31] initiates a lineage of methods, based on the procedure called Level Set method (LSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These methods have been very fruitful and have been developed and employed on numerous occasions (see [32] for an overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In what follows, it is described how the problem has been solved by two different paths, the first addresses the resolution of an equation of type Hamilton-Jacobi by means of the LSM that is capable of solving problems of propagation of very general fronts, much more complex than the problem of burnback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The second perspective addresses the steady problem that is circumscribed to the solution of an equation of type Eikonal that is strictly the problem to be solved in the burnback analysis and, in this sense, the modeling and computational effort made is more proportionate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" A basic and complete description of both approaches can be obtained in Sethian's text [33], which clearly identifies and discusses both methods." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Consider the situation in Figure 6 in which a curve or surface, defined for example by the function ������������ = 0, spreads with velocity ṙp in the direction perpendicular to the surface itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The problem is to determine the evolution of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the most general situation, the propagation rate may depend on local properties of the surface point, such as the direction of the normal, or curvature, or on general properties of the curve, such as integral relations of all kinds, and, also, on properties external to the problem itself, as would be the case of advection, due to a velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 6: Outline of the two approaches followed in the numerical methods of solving the burnback problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On the left, the LSM in which the front is represented by the null value of a distance function, ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On the right, the position of the front is represented by the values taken by the solution of the Eikonal equation that corresponds to the travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 21 The problem can be approached from two points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The boundary value formulation calculates the time ������������ = ������������(������������) it takes for the front to reach each point in the domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and it is evident that the definition of the velocity of the front leads to ṙ p = ������������������������ ������������������������ ⁄ and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in several dimensions it is fulfilled that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������̇������������|������������������������| = 1 (53) Already written before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' with the condition ������������ = 0 on the initial combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This is the Eikonal equation, which is a traditional problem in many physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this problem, it has been implicitly assumed that the function ������������ is a single-value function, for which the propagation rate must have a constant sign, either outwards from the domain, or inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This restriction, which for some situations is very important, in the case of burnback analysis is fulfilled naturally and the unknown of the Eikonal equation is strictly the function to be obtained to solve the evolution of the combustion surface in a solid propellant rocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' When propagation can take place in two directions, on both sides of the front, it is mandatory to describe the movement of the front by a function ������������ with more spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To obtain an equation of this evolution, consider the path ������������⃗(������������) that follows a particle of the front and how, without loss of generality, one can assume the front defined by ������������(������������⃗(������������), ������������) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Differentiating the function yields, ������������������������ + ������������������������(������������⃗(������������), ������������) ∙ ������������⃗′(������������) = 0 (54) Which is the equation that allows us to obtain the function ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As the velocity of the front is ṙp = �������������⃗ ∙ ������������⃗′(������������) and the direction normal to the surface is �������������⃗ = ∇������������ |∇������������| ⁄ , finally, the equation for ������������ is: ������������������������ + ������������̇������������|������������������������| = 0 (55) For which an initial value of the function must be supplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This equation is of the Hamilton–Jacobi type, for a wide spectrum of forms of ṙp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The problem of front propagation occurs in a wide variety of configurations: from ocean waves, combustion fronts or interfaces in the movement of heterogeneous substances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' of course, in problems of light propagation or seismic wave propagation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' but also, in problems of character identification or image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Equations (53) and (55) represent the two different approaches, and both of them provide fully satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The only difference is that solving the Eikonal equation involves an effort adjusted to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The method based on the Hamilton-Jacobi equation is designed for more complex problems and needs further elaboration in the calculation, uses more memory and has to solve numerical problems (such as the reinitialization of the distance function) typical of a more complex method, but which are totally unnecessary in the burnback analysis problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Level Set Methods (Hamilton-Jacobi equation) The evident generality of LSM has led the methodology to be used in the analysis of the burnback problem on numerous occasion [34-47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Usually, the initial function ������������(������������⃗, ������������ = 0) is fixed as a signed distance function (SDF) containing the value of the minimum distance to the front from the initial surface and which is calculated with some algorithm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The method is not exempt from some problems, since the SDF can take poorly conditioned values as the integration progresses, and it becomes necessary to reinitialize it periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [58] It is evident that the method employs an implicit function defined throughout the propellant domain of which the only useful information is the front defined by the null value of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For this reason, some authors have used a strategy of limiting the calculated value of the SDF to the vicinity of the front (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, it is necessary to incorporate a search and location algorithm of the front to determine the narrow band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 22 Notable is the contribution of Chiapolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [47] that addresses the solution of a Hamilton-Jacobi equation using a standard LSM but with a step function for the level function emulating the front tracking methods commonly used in heterogeneous fluid problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' An instructive article describes a numerical method on an unstructured mesh, in which it uses upwind techniques with limiters, for the method stability, which have been developed in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The examples that are included, addressing three-dimensional burnback analysis, are very illustrative, and correspond to modern and realistic grain geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Steady perspective (Eikonal equation) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Direct time marching The solution of equation (53) (also of the equation (6) using a Time Marching procedure),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������ + ������������(������������������������) = 0 (56) where the Hamiltonian is ������������(������������������������) = 1 − ������������̇������������|������������������������| (57) As already mentioned,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' this type of equation belongs to the so-called Hamilton-Jacobi equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which arises as a problem of initial values with boundary conditions according to the situation to be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the problem in hand, ������������ = 0 on the initial combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Usually, the rest of the boundary conditions consist of boundaries at which the front extinguishes (for example, the engine casing) in which, usually, no condition is necessary to be imposed due to the hyperbolic nature of the equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and contours of symmetry or periodicity, in which the implementation of the condition is relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' References [48]–[53] pioneer the use of the Eikonal equation for the solution of the burnback problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These works constitute a frame of reference for the correct and adjusted solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Since then, however, the propagation problem has been addressed from different perspectives and for different problems, though not necessarily in solid-propellant engine technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The direct solution of the Eikonal has been addressed on numerous occasions for the appropriate monitoring of surfaces, as in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Singular is the contribution of Gueyffier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [60], which addresses the solution of the Eikonal equation using a spectral method for the description of the combustion surface with a philosophy similar to that employed by surface tracking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Fast marching methods (FMM) The Eikonal equation in the form (53) can be solved by calling a method based on the traditional alternate direction methods but using the propagation direction of the front to update the variables and in this way obtain an additional advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These procedures are called Fast Marching methods (FMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It is possible to consult the book by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Sethian [54] to have an overview, where an interesting critical comparison between FMM and LSM is also established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The burnback problem has been addressed by this method in unstructured mesh, for example, in [61] in a complete paper but there are not many other contributions to the burnback problem using this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Burnback analytical solutions Constant combustion surface area is the most common design condition for a solid propellant rocket engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This situation is generically optimal because it implies that the chamber structural design is adjusted to the entire engine operation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Otherwise, the thickness of the engine casing must be sized for the most unfavorable load case, which corresponds to the maximum pressure reached and, therefore, the combustion chamber is heavier than that of the engine that would provide the same 23 total impulse with constant chamber pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To achieve constant pressure profiles with large combustion areas, comparable to those of the chamber itself, it is necessary to resort to geometries with a certain degree of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The important variables are the web fraction, the volume fraction, and the sliver fraction but, also, the Klemmung and ������������ (combustion to port area ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These last two parameter are of interest because control the occurrence of the erosive combustion phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Classic star Figure 7 shows the geometric description of the cross-section of a star-shaped propellant with ������������ tips, as discussed in [62], [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For simplicity, only half of an angular sector, ������������/������������, is represented, taking advantage of the symmetry properties of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The tip has an angle ������������ (the figure shows the semi-angle ������������ 2 ⁄ ) while occupying a fraction ������������ of the entire angular sector, ������������(������������/������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The depth of the valley area has length ������������ from the center of the chamber and it is considered that the thickness of the propellant is the necessary to finish the first phase of combustion at the moment in which the front arrives for the first time to the engine casing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the propellant web thickness is larger, a progressive phase of linear perimeter growth begins at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This second phase of combustion would be progressive, and in the design of the engine it will not be allowed to extend too much, as it raises the chamber pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, it can increase the web fraction or the combustion time, and it may be necessary to satisfy design requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Nevertheless, the possibility of compensating this effect by designing the star with a slightly regressive profile should be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 7: Schematic of star geometry and definition of geometric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' A heuristic procedure to determine the variation of the perimeter of the section considered is to go through the contour, measuring the rotation suffered by the normal to the surface and calculating, in each case, the increase in perimeter that occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Performing this operation for the geometry in Figure 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the expression obtained is ∆������������ = 2������������ � (1) ������������ � ������������ ������������⏟ (2) + ������������� 2 − ������������ 2� ����� (3) − 1 ������������������������������������[������������ 2 ⁄ ] ������� (4) � (58) The term (1) corresponds to the 2������������ half sectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the term (2) is due to the turn suffered by the normal in the half sector (if it were a cylinder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' these first two factors would give rise to the simple result already commented ∆������������ = 2������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The term (3) is the one corresponding to orienting the normal from the radial position, after the rotation ������������ ������������ ⁄ , to the surface of the cusp, which assumes a rotation equal to the complementary angle of ������������ 2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Finally, the term (4) is the one corresponding to the 0/2 大24 destruction of part of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Indeed, the first thing to note is that ������������ 2 ⁄ is the complementary angle of the angle ∆������������ 2 ⁄ in the Figure 4, and the tangent function of the complementary angle is the inverse of the tangent of the angle and, in addition, in the generic expression (equation (50)) two slopes of the cusp are taken into account, while in the sector in Figure 7 only one of them has to be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This rapid assessment is delicate and subject to probable misinterpretation of the criteria under which it is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, it is a very interesting method to be used in combination with more elaborate geometric evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Because it provides a quick verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In addition, it allows us to carry out analyses that lead to the elaboration of optimal strategies for the design, merely using analytical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To obtain a geometry in which the combustion area does not change (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' neutral combustion), it is necessary that ∆������������ = 0 in equation (58) which is a simple nonlinear equation for ������������ as a function of ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Table 2 shows the semi-angle of the tip, the web fraction, and the volumetric fraction, for different values of the number of tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Note that between 5 and 6 tips, the value of ������������ goes from being ������������ 2 ⁄ < ������������ ������������ ⁄ to be ������������ 2 ⁄ > ������������ ������������ ⁄ , showing that for less than 5 tips impossible geometries can arise in which the tips collide with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It is especially interesting that if ������������ 2 ⁄ = ������������ ������������ ⁄ the channel is straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This is the condition for analyzing axial slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The table indicates as well that for ������������ ≥ 6 (because ������������ 2 ⁄ > ������������ ������������ ⁄ ) the combustion process should be regressive, which is very useful information when combining combustion geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ 4 5 6 7 8 ������������ ������������ ⁄ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='21 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='12 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='53 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='55 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='30 ������������ ������������ ⁄ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='70 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='50 ������������ ������������������������ ⁄ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='140 ������������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='674 Table 2: Solution of equation (58) for neutral combustion ∆������������ = 0, and the corresponding values of the web fraction and the volumetric fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Combined propellant geometries are presented on many solid rockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' A common configuration is to use simple cylindrical combustion and a slotted segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The cylindrical section has a combustion area that grows over time and the slotted segment can be configured so that the combustion area decreases at the desired rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The combination of both geometries can result in a thrust curve with a specific profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='5 Bipropellant star The star configuration provides a constant combustion area curve for moderate values of the web fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, the mass of residual propellant after the neutral phase (sliver fraction) can be very large, with a negative impact on the effective volumetric fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It is possible to design a sliverless geometry using two propellants with different recession velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The idea is to fill the region of the cusp with a high-speed recession propellant so that it reaches the engine casing at the same time as the propellant, with a lower recession rate, that fills the web thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 25 Figure 8: Straight star loaded with two propellants of different recession rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On the left is a general scheme and on the right the notation used in the analysis to determine the adequate interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 8 shows a possible simple configuration for a straight star (similar to a slotted geometry) in which two propellants of different combustion rate are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Each propellant advances a different amount at the same time due to the different rate of combustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The propellant 1 advances ������������1 and the propellant 2 advances ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For the combustion front to reach the casing simultaneously at all points, the combustion front in the propellant 1 has to be cylindrical with radius ������������1 from point ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' While, by construction, the combustion front in the propellant 2 is composed of a line and a circle arc of radius ������������2, centered on point ������������’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The recession velocity in the propellant 2 must be such that it reaches the point ������������ at the same time as the propellant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This imposes a geometric exception, depending on which combustion front reaches the point ������������ in the propellant 2, whether it is the straight front or the circular front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In what follows it is assumed that it is the circular combustion front that reaches the point ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The rest of the parameters to be used are shown in Figure 8, in which ������������ is the depth of the slot, ������������������������ is the fillet radius in the slot, ������������������������ is the chamber radius and ������������1,2 and ������������1,2 are the polar coordinates of the points on the two combustion fronts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The condition for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='the fronts to progress simultaneously over the interface is expressed by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������1 ������������������������������������ ������������1 = ������������2 ������������������������������������ ������������2 + ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(59) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������1 ������������������������������������ ������������1 = ������������2 ������������������������������������ ������������2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(60) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Where ������������1 = ������������������������ + ������������ + ������������1 and ������������2 = ������������������������ + ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Without loss of generality, it can be put ������������1 = ������������ and ������������2 = ������������������������ with ������������ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The regression rate of the propellant 2 is suitable so that, on the symmetry line, the front reaches the housing at the point ������������ at the same time as in the propellant 1 reaches point ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The propellant 2 induces a cylindrical combustion front on the propellant 1 with radius ������������1, while the front in the propellant 2 is also cylindrical with radius ������������2 but with center at ������������’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The condition of reaching the casing simultaneously at the point ������������ is expressed by removing ������������2 from expressions (59) and (60),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' thus getting (������������1 ������������������������������������ ������������1 − ������������)2 + (������������1 ������������������������������������ ������������1)2 = ������������2 2 (61) And substituting ������������1 = ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������1 = ������������ ������������ ⁄ and ������������ = ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which is the value of the web thickness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' result in: ������������������������2 − 2������������������������������������ ������������������������������������(������������ ������������ ⁄ ) + ������������2 = ������������������������� + ������������������������� 2 (62) Along with ������������������������ = ������������������������ + ������������ + ������������ (63) Once the geometry of the star is established (������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������ and ������������ are known),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' equation (63) allows the calculation of the web and equation (62) provides the needed value of ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the ratio of recession velocities of both propellants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������ = ������������������������2 ������������������������1 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 1 ri= yi+ d r2= 2 + rf 2 0226 The geometry of the interface can be obtained by taking as a parameter the depth of the forward coordinate of propellant 1 (������������;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 0 ≤ ������������ ≤ ������������) and explicitly resolving with ������������1 = ������������������������ + ������������ + ������������ (64) ������������2 = ������������������������ + ������������2 (65) And using equation (61) to obtain ������������1 and equation (60) to obtain ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Once the interface line has been drawn, it is possible to calculate the burn perimeter on each propellant and, considering the different recession velocities, calculate the mass released by each propellant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The length of each perimeter in each propellant is no longer an intuitive measure of the mass burned by the entire surface or of the chamber pressure reached at each moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For this reason, in what follows, the geometric concept of forward coordinate is momentarily abandoned in favor of a pseudotime, as an independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 9: Comparison of the equivalent burning surfaces of a monopropellant and a bipropellant star-shaped geometry with 4 cusps, with������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='4, ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 and ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' so that the ratio of recession velocities takes the value ������������ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 9 shows the simulation performed with a geometry of four slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The results obtained in the case of operating with a single propellant and in the case of operating with two sliverless propellants are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To compare both situations, it is useful to represent the pseudotime lines that correspond to ������������ = ������������1,2 ������������������������1,2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Taking ������������������������1 = 1 in the case of a single propellant, the pseudotime is equivalent to the forward coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the bipropellant case, an equivalent combustion area must be defined in the form ������������������������,������������������������ = ������������������������,1 + ������������������������������������,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The equivalent combustion area allows the calculation of the mass released and the chamber pressure and thrust, using the combustion data of the propellant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this way, we can establish a reliable comparison with the operation of a single propellant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the monopropellant case, this geometry, with few slots, gives rise to an increasing combustion area profile, until the combustion front reaches the engine casing for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' From that moment, the combustion area decreases over time, giving rise to a long tail thrust phase as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the bipropellant case, however, the combustion process of the fast propellant generates at the beginning more mass flow, compensating the initial deficit presented by the monopropellant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this way, as clearly shown in Figure 9, a near-neutral combustion area curve is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This remarkable feature can be anticipated by designing the geometry so that the equivalent combustion areas are similar at the initial and final times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As the combustion fronts reach the casing simultaneously, no sliver fraction is produced and the combustion area curve drops sharply at that moment, forming an optimal silverless geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 10 shows the comparison of operation with one and two propellants of an elliptical hole geometry that initially presents a high volumetric filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As a result, the ratio of recession rates is also high, which translates, again, into a significant variation in the equivalent area of combustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this case, the pseudotimes at the end of the combustion of both configurations are equal, highlighting 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='6 Equivalent burnning surface 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='9 monopropellant 80 bipropellant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='80 Pseudo time27 the significant variation (up to 33%) in the equivalent combustion area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For the calculation of the interface, the approximation of the combustion front in propellant 1 remaining elliptical has been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The numerical simulation shows how little importance this gross hypothesis has on the overall result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 10: Comparison of the equivalent burning surfaces of a monopropellant and a bipropellant elliptical-hole geometry, with high volumetric fraction and high ratio of recession velocities ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The solution of these bipropellant cases has been approached without establishing any consideration about how the combustion fronts interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As will be seen below, the interaction can be complex and create rarefaction and caustic waves that significantly modify the combustion front near the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This can lead to variations of some importance in the evaluation of the equivalent combustion area and, therefore, in the prediction of the actions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As will also be shown later, the numerical analysis scheme proposed reliably captures these anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For the examples presented above, this anomaly does not occur, since it is a corner-type combustion situation in which the design system guarantees that the interface is above the equilibrium point ������������ (the scheme ������������)) in Figure 14, so that the combustion fronts do not present rarefactions or caustics of any kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Bipropellant burnback analysis The combustion front in a bipropellant grain is determined by the difference between the recession rates of each propellant and by the geometry of the front and of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To approach a general analysis with confidence, it is advisable to start with a simple situation, in which the combustion front at the point of contact of both propellants is flat, as represented in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The point ������������ separates both propellants at the combustion surface, and the interface between them is straight and perpendicular to said combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The recession rate of the propellant 1 (on the left in the figure) is ������������1 and the combustion front of this propellant moves to the parallel line ������������1 a distance ������������1 = ������������1������������������������ after a time ������������������������, at points that are far enough from the point S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' At the same time, the combustion front for the propellant 2 moves ������������2 = ������������2������������������������ reaching the line ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For the analysis it will be assumed that ������������1 < ������������2 and, therefore, ������������1 < ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' To build the solution it is convenient to consider the point ������������ to be the source of the propagation process in both propellants, then the combustion front will extend into the propellant 1, at least, up to the cylinder ������������1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and, in the propellant 2, up to the cylinder ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As in time ������������������������ propellant 2 reaches the point ������������2, while propellant 1 only would reach point ������������1, faster propellant acts as a source of ignition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Each of the points on the side of the propellant 2 over the line ������������������������2 ����� will be the center of a family of circles that consumes the propellant 1, and whose radius is proportional to the distance remaining to travel to ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Consequently, the combustion surface in the propellant 1 will be the envelope of this family of cylinders, which is easily built by tracing the tangent to the circle ������������1, from ������������2 to the point of tangency ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The segment ������������2������������ ����� intersect with line ������������1 at the point ������������, separating the combustion surfaces obtained from the original surface (������������1), and that obtained because of the phenomenon already described in the interface (������������2������������ �����).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On point ������������ two 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='4 Equivalent burnning surface 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 monopropellant bipropellant 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00 Pseudo time28 different combustion fronts converge, whose collision forms the caustic ������������, which is a straight line starting from the point ������������ in the line ������������������������ ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 11: Diagram of the evolution of the combustion surface of a bipropellant with flat front and interface perpendicular to the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The propagation of a combustion front, initially flat, along the interface between two propellants is a relatively common situation that, for example, corresponds to that which occurs in the case of thermally conductive wires, embedded in the propellant to increase the combustion area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this case the cable acts as an ignition source with a higher velocity than the propellant regression rate and the combustion geometry obtained is conical with the axis on the cable, analogous to the construction ������������1������������������������2 � in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, although the situation is simple, it allows the introduction of the basic analysis mechanisms to be used in more complex situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Thinking that the point S It is the origin of the combustion front of each propellant, building the cylinders of influence ������������1 y ������������2, calculating the intersection with the lines that establish the position of the fronts ������������1 and ������������2 far from ������������, and determining the envelope of certain families of cylinders, leads to the construction of the combustion surface at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For the analysis of more complex situations, where the combustion front is not initially flat, it is convenient to generalize the notation, as shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Uppercase letters are used to name points of interest and lowercase letters are used to name lines and circular arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The figure represents the two possible situations for a non-flat combustion front, when ������������ is the vertex of a cusp, and when it is the vertex of a corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The bisector angle ������������ is used to represent the initial position of the fronts and identify the angle ������������ (that lies between the lines ������������ and ������������2) as a measure of the difference in burning rates, because if ������������ = ������������ the burning rates are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 12: Meaning of the different symbols used in the description of the propagation process of a bipropellant for two initial configurations of the initial combustion surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The lines ������������1 and ������������2 are parallel to the original surfaces and represent the position of each combustion front if they were isolated (in the figure, ������������1 < ������������2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Unlike the simple flat-front case, lines ������������1 and ������������2 are not parallel, but rather converge at the point ������������, that allows you to draw the equilibrium line ������������ from CE F / F2 e S 业f Ifi /J2 1 1 CUSPY1 b, 2β y2 C1 F E F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' K e 1 CORNER29 ������������, towards ������������������������ ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Once the position of the surfaces is known, the circles of influence, ������������1 and ������������2, can be traced, tangents to the aforementioned lines at points ������������1 and ������������2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The perpendicular to the initial surfaces, ������������1 and ������������2, are drawn from ������������ following the directions ������������������������1 ����� y ������������������������2 �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The conical zone between ������������1 and ������������2 defines an interference region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The equilibrium line ������������ is a reference for the position of the interface between the propellants, which will evolve differently if it is inside or outside the interference region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' if the angle formed by the initial surfaces of both propellants is 2������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' then the angle of the equilibrium line ������������ can be calculated through the relationship: ������������1 ������������2 = ������������������������������������(2������������) − ������������������������������������(2������������) ������������������������������������(������������) (66) which shows that the structure of the study region depends on ������������ and the recession rate ratio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������1 ������������2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 13 shows the different results of the combustion surface if ������������ is the vertex of a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Each scheme corresponds to distinct positions of the interface, identified by the line ������������, determined by angle ������������������������ that the interface forms with the line ������������1 (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The series starts with a sufficiently large value of the angle ������������������������ (greater than ������������) and situations are analyzed for decreasing values of ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If ������������������������ > ������������, see diagram ������������), the interface intercepts the line ������������1 at the point ������������ and the propagation process in the propellant 1 produces the premature ignition of propellant 2 along the interface between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The combustion surface produced (segment ������������������������ ���) is generated by obtaining the line that starts from ������������ and is tangent to the cylinder ������������2, which corresponds to the envelope of the family of circles generated by the ignition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The intersection of this line with the line ������������2 determines the position of the point ������������ which is the vertex of caustic ������������ generated in this process which goes from ������������ towards ������������������������ ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As ������������������������ decreases, the point ������������ approaches point ������������, coinciding both when ������������������������ = ������������, and the caustic disappears, as illustrated in the diagram ������������) of Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this situation, the propellants are consumed at their own rate without generating any additional structure, forming the fronts only by the lines ������������1 and ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' When ������������������������ < ������������, but before you get to ������������������������, the propellant 1 reaches point ������������ before propellant 2, but this time the family of cylinders ends in the circle that passes through ������������ and is tangent to ������������2 and, as represented in the scheme ������������), a partial rarefaction is created between lines ������������ and ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' When ������������������������ reaches the value of ������������������������ (scheme ������������)) the expansion is complete between the line ������������2 and the line ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Here, the point B, that defines ������������������������, is obtained as the intersection of the circle ������������2 and the line ������������1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Until now, the high inclination of the interface line causes the process to be dominated by the propellant 1 but when ������������������������ > ������������������������ the propellant 2 reaches the point ������������ earlier, producing the premature ignition of the propellant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the scheme ������������) this situation is shown, in which the new surface ������������������������ ��� is obtained by drawing the line that starts from ������������ and is tangent to ������������1, obtaining the position of ������������ as an intersection of this line and ������������1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Under this, if ������������������������ > 0 remains the rarefaction between ������������ and ������������2 until ������������������������ is canceled (scheme ������������)) and rarefaction disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Finally, as represented in the scheme ������������) for ������������������������ < 0 the structure of the front is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 30 Figure 13: Sequence of the different schemes for different positions of the interface, in the case where the initial combustion surface has a cusp at the vertex between the two propellants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' For the case where in cusp configuration the point ������������ is within the interference region (delimited by ������������1 and ������������2) the different modes of propagation of the Figure 13 are simplified and only schemas ������������), ������������) and ������������) appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 14 shows the different morphologies of the combustion surface when ������������ is the vertex of a corner separating both propellants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Each scheme corresponds to distinct positions of the interface, identified by the line ������������, determined by angle ������������������������ that the interface forms with the line ������������2 (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' When ������������������������ is large enough, as depicted in the scheme ℎ), the propellant 2 runs through the entire interface to the point ������������ faster than propellant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Therefore, the combustion surface ������������������������ is plotted by calculating the envelope of the propagation cylinders in the propellant 1, that is, it is obtained from the line that starts from ������������ and is tangent to ������������1 at the point ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As depicted in the scheme, between the lines ������������1 and ������������, a rarefaction is formed that is reduced as ������������������������ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The rarefaction disappears when ������������������������ = ������������ (scheme ������������)) which, as in the cusp, gives rise to a scenario without mutual interactions, each propellant was consumed independently of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' While 0 < ������������������������ < ������������ the process is, as presented in the scheme ������������), similar to the scheme ℎ), but in this case the point of tangency ������������ goes over the line ������������1 and the envelope generates the caustic ������������ along the segment ������������������������ ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 14: Sequence of the different schemes for distinct positions of the interface, in the case where the initial combustion surface has a corner at the vertex between the two propellants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The situation when ������������������������ < 0 is represented in scheme ������������) of Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' A rarefaction is generated in the propellant 2 between the lines ������������ and ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The surface of the front coincides with ������������2 up to ������������2, and between ������������2 and ������������ it coincides with ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As the propellant 2 continues to dominate the process, the combustion surface ������������������������ ��� is obtained as before, tracing the tangent to the circle ������������1 that goes through ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='>8 S 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' =8 >8>B s S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='=OB S s I三E I三B b1 C b1 F2 b i b2 c i/ f2 S>0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' > 0 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' = 0 0>1g 2 C L=F2 i c iE >0 S S f2 f2 F2 F2 0>1g F 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' <-4β -8 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' = -4β - 831 For negative angles (������������������������ < 0), but greater in absolute value, the circular arc ������������2������������ grows until it reaches the line ������������1 in which, as represented in scheme ������������), caustic ������������ disappears, when reaching the interface itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' If the interface tilts even more (scheme ������������)) is now the propellant 1 the one that dominates the propagation process, causing the ignition of the propellant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The envelope ������������������������ is created from the cylinder ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' All the above situations correspond to the scheme of Figure 12 at which the equilibrium point ������������ is outside the interference region bounded by ������������1 and ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' When point ������������ is situated within the interference region, schemes ℎ), ������������) and ������������) are reproduced and a new configuration, not represented in the figures, appears with two rarefactions, one in each propellant next to the lines ������������1 and ������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 15 shows the result of numerical analysis, with a code based on obtaining the solution of the Eikonal equation, by simple time marching, which is described later in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As can be seen in the figure, the structures of the schemes ������������), ������������) and ������������) are reproduced faithfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The algorithm efficiently captures rarefaction and caustic structures produced near the interface between the two propellants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 15: Result of the numerical simulation of three examples of bipropellant interface corresponding to the same reference letters in Figure 13 and Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On the one hand, this shows that the previous analytical reasoning is correct, in general terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' On the other, it shows the power and versatility of the numerical method proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As has been seen in the review of the literature and the analysis of the different methods, the numerical integration of the Eikonal equation is the best procedure to establish the combustion surfaces, even when the recession rate is variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Using a discrete representation and computation system, the kinematics of geometrically complex burning fronts propagating with prescribed variable burning rates can be efficiently described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Burnback numerical solution As shown in the previous sections, the solution of the burnback problem by numerical methods that offers the best results passes, in the general case, through the solution of a Hamilton-Jacobi equation, although, naturally, the direct solution of the Eikonal equation can be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Two lines of work can be distinguished in the solution of this type of equations, one that addresses the mathematical problem in a generic way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [64]–[66]) and another driven by the solution of front propagation problems using LSM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [67] or, very recently, [47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The study of numerical approximations to the viscous solution was also initiated by Crandall and Lions [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' They introduced an important class of monotonic schemes for a simplified form of equations and showed that these schemes converge to the viscous solution (for an in-depth review of this matter from a general point of view, see [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, it is known that monotonous schemes can be at most first-order, so they are too dissipative for most practical applications, although they are 32 used to build high-order algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In reference [67], Osher and Sethian built a class of high-order upwind-type schemes to, imitating ENO algorithm of high order developed by Harten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [68] and Shu and Osher [69], approximate conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Its construction was based on the observation that the Hamilton-Jacobi equations are closely related to conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In this sense, a wide variety of algorithms have been proposed, such as those described in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' [69]–[71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In particular, in the problem of burnback analysis, this type of algorithms has been used on numerous occasions, but the applications that are most interesting are those developed for unstructured meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The nature of the initial combustion surfaces and the need to use complicated geometries that meet the design requirements of solid-propellant rocket engines leads inexorably to the use of unstructured meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In addition, this type of meshing allows noticeably short generation times, which has a significant impact on the overall efficiency of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The solution of the unstructured Hamilton– Jacobi equation composed of triangles was first proposed by Abgrall [72] by the approximate solution of a classical Riemman problem, based on the work of Bardi and Evans [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' These works have been followed by others [74]–[77] in which the approximation order was increased or different schemes of the same type were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Special mention should be made, in this category, of the schemes that obtain the solution of the Eikonal equation by means of fast marching algorithms in unstructured meshes, such as in [78] or [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Time marching method In the present work, the solution of the Eikonal equation is obtained by means of the simple time marching procedure in an unstructured two-dimensional mesh composed of triangular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The integration domain is the complete volume of propellant, delimited by the initial combustion surface and the surfaces that remain inert (surfaces inhibited for combustion and the surfaces in contact with insulating material or in contact with the case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The value of the unknown function ������������(������������⃗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which represents the time of arrival of the front,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' is stored at the vertices of the mesh and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' as already indicated above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' the problem to be solved is ������������������������ + ������������(������������������������) = 0 (67) In which the Hamiltonian is ������������(������������������������) = 1 − ������������̇������������|������������������������| (68) With the initial condition ������������(������������⃗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 0) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' which is also imposed as a boundary condition on the initial propellant surface throughout the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The method used does not need to impose spatial boundary conditions on inert surfaces, through which the combustion front passes without disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, it is customary to select portions of the propellant volume delimited by surfaces with symmetry conditions, which is easily implemented in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 16: In the diagram on the left, the main geometric elements used in the basic discretization around the node ������������ are represented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and on the right, the notation used in the edge-based algorithm to construct the discrete solution is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' U, U, 0, 0j+1 nj+1/2 i 0j+1 Uj+133 The solution of the equation can be obtained numerically efficiently, by means of a discretization based on the work of Abgrall [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This requires a domain triangularization, using the variable values ������������������������ (������������ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������) at the vertices, to estimate the value of the gradients of the function at each triangle, ��������������⃗������������ = [������������������������]������������ (������������ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' ������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In Figure 16, the geometric configuration used is represented, in which the angles around an edge connected to the node i are ������������������������ y ������������������������+1 and the unit vector in the direction of the edge is �������������⃗������������+1 2 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The value of the function over time ������������ = (������������ + 1)∆������������ is obtained from: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������+1 = ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ + ∆������������ℋ���������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(69) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='ℋ���������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������� = ������������ � 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2������������ � ��������������������������������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='� − ������������������������ � ������������������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='��������������⃗������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ + ��������������⃗������������+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='�������������⃗������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(70) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='And ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ = ������������������������������������ ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 + ������������������������������������ ������������������������+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(71) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Integration must be carried out under the stability condition ∆������������ ≤ ℎ ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='where ℎ is the minimum height ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='of the adjacent triangles and the diffusion factor is calculated by ������������i = ������������ π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' being ������������ = max ������������ ‖∇������������‖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The algorithm is constructed by traversing the edges of the mesh and updating the value of the mean gradient on each node (see the right diagram of Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The procedure is executed by using the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='following relationships: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='��������������⃗������������1 ← ��������������⃗������������1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ���������������������������������������⃗������������ + ������������������������+1��������������⃗������������+1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(72) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='��������������⃗������������2 ← ��������������⃗������������2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 �������������′������������+1��������������⃗������������+1 + ������������′��������������������������⃗������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(73) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='And calculating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ = ������������������������������������ ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 + ������������������������������������ ������������������������+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(74) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������′������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ = ������������������������������������ ������������′������������+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='+ ������������������������������������ ������������′������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(75) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='The diffusion terms of the equation are calculated by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������1 ← ������������������������1 − ������������������������1������������������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ���������������⃗������������ + ��������������⃗������������+1� ∙ �������������⃗������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(76) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������2 ← ������������������������2 − ������������������������2������������′������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ���������������⃗������������+1 + ��������������⃗������������� ∙ �−�������������⃗������������+1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='⁄ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(77) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Boundary conditions are applied for the nodes of each contour by modifying the values of ��������������⃗������������ and of ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='calculated on all nodes as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='a) Free contour ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ← ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(78) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ← 2������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(79) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='b) Symmetry contour ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='��������������⃗������������ ← 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 ���������������⃗������������ + ��������������⃗������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(80) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������ ← 2������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='(81) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='Where ��������������⃗������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='������������������������������������ is the symmetric vector to ��������������⃗������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The integration is advanced until reaching a steady state, which is ensured by checking that the gradients of the variable within each triangle do not change above a predetermined value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Results and discussion Previously, throughout this document, results of numerical simulations that employ the algorithm described above have been presented in Figure 9, Figure 10 and Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=" These results clearly show the method's ability to deal with situations in which the velocity of front propagation is not constant." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The cases of bipropellant, in star configuration and ellipse of high fill coefficient, are handled efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The presence of the interface that separates both propellants is undertaken with the single implementation of assigning different values to the recession rate to each node within the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The same technique is used in the three simulations presented in Figure 15, configuring the calculation domain and the inclination of the interface properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 17 shows the results of three representative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The results have been calculated with unit recession rate in the system of units in which the geometry is represented, that is, advance coordinate and pseudotime coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The three cases have been calculated with a modest number of elements not exceeding 104 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Even so, the results show reasonable precision in the absence of a more rigorous error analysis that is carried out in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The examples show the ability of the method to describe all relevant phenomena in the analysis of these configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the so-called anchor geometry, the combustion front collides with the engine casing generating an abrupt change in the combustion area, while, inside, a caustic is formed when the combustion fronts collide, coming from the central slot and the circumferential groove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The second case corresponds to a star geometry optimized to produce neutral combustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Finally, an unoptimized case of dogbone geometry is included in which it is observed that the condition of free contour in inert boundaries is treated without visible reflections and disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 35 Figure 17: From left to right: constant pseudotime lines, mesh utilized, and curves of combustion surface area for three representative cases (top to bottom: anchor geometry, optimized neutral-burn star, and unoptimized dogbone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 18 shows the results obtained with a partially optimized axil-geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The constant pseudotime line shows the full variety of situations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' and the combustion area curve only needs a few adjustments to present a properly flat profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 18: Constant pseudotime lines and combustion area curve for a geometry corresponding to a low-slenderness engine with an axil-type grain in the process of manual optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='7 Error analysis A simple slotted geometry is chosen to perform error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This geometry brings together two aspects of interest: the expansion of a combustion front in which the combustion perimeter increases and the collision of two combustion fronts with the consequent generation of a caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This is a simple situation, and the error can be calculated by comparing it with the analytical solution of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='4 Ab 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2 10 Ab 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='10 WW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='2036 Figure 19 Level contours of pseudotime (left), 2500-node mesh (center), and combustion area (right) on the problem used to evaluate the discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The problem consists in the advance of a combustion front from a radial slot (only the right half of the domain is considered using vertical symmetry) composed of a straight section ending in a semicircle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' As shown in Figure 19, the motor case would be located at the upper border and at the right border where the combustion front leaves the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The combustion area that develops this geometry is traced in the graph of the figure, and consists of a first section of neutral combustion, due to the increase in perimeter caused by the circular expansion, combined with the destruction of geometry caused by the caustic, followed by a process of a strong fall of the area, while the combustion front leaves the asymmetrical upper part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 20: Constant normalized error contours in the single slot problem for different number of nodes in the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Figure 20 displays the normalized error obtained in simulations with different number of nodes in the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The error has been calculated as the difference between the calculated value and the exact analytically calculated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The error is normalized with the maximum value of the penetration level reached by the front, so the contours of the figure are representative of the relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This procedure has been chosen because it is not possible to calculate the relative error in the initial contours in which the value of the forward coordinate is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In all cases, it is observed that the error incurred is extremely small in the advance of the straight fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, on the discontinuity the error is noticeable and in absolute value increases throughout the expansion range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the cases analyzed, the maximum error, corresponding to the coarsest mesh, is less than 1% and is located where the discontinuity crosses the contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' By increasing the number of nodes of the mesh, the error decreases significantly and as already mentioned, even with meshes of modest size, the results obtained are very valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' In the figure, the denser mesh provides a solution in which the error in the front position is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0 y 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='0error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='002 nodes=4000 8000 16000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='00137 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Conclusions Burnback analysis is a central issue in the calculation of the performances of solid propellant rocket engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Since the beginning of the development of these engines, a variety of methods have been used to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The first methods used were purely analytical and could only be applied to simple geometries, although the skill of some researchers led them to solve complex cases of industrial interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The use of the first digital calculators, to automate calculation, and numerical methods in modern computers applied to differential equations, which adequately describe the kinematics of the free surface, has put the problem of burnback in a state of remarkable technological maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Also, a series of phenomenological methods have recently been developed, which use specific properties of the solution, like the principle of minimum time or Piobert’s statement, which obtain interesting results but are difficult to generalize to problems with non-uniform recession velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The most general and fruitful methods lie in solving the Eikonal equation which, as shown in this paper, is obtained from the detailed analysis of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Although the direct resolution of the equation was addressed early, at the beginning of this century, giving rise to powerful and versatile methods, during the last twenty years the developments have led to solving the burnback problem using the so-called Level Set Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' LSM-based calculations solve a Hamilton-Jacobi equation, using a signed level function, to get the solution robustly and reliably, without limitations in the geometries to analyze nor in the recession velocity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' However, this strategy is oversized for the burnback problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' LSM is a procedure that solves much more general problems than burnback but enjoys great popularity because it is used in a very wide range of free-boundary problems and with applications in very different fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' From a broad efficiency point of view, the burnback problem must be solved using the Eikonal equation on an unstructured discretization of the propellant volume, so that it is possible to address any geometric complication that the design problem of a solid-propellant rocket engine requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The method is computationally efficient, especially when compared with other kinds of analyses that need to be addressed in the design of a solid-propellant rocket engine (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' structural or internal aerodynamic calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' The reason is that only one unknown needs to be solved and the meshing does not need the sophistication of a CFD mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' This paper develops the basic theory of propagation of the combustion front, carries out a critical review of the existing literature on burnback analysis, highlights the ability of analytical methods solving very general problems of, for example, bipropellants, and shows the power and versatility of the integration of the Eikonal equation, using simple time marching for the solution of any grain design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Acknowledgements This study has been carried out as part of the PILUM project (Proyecto de Investigación de tecnologías para Lanzador, Ubicado en plataforma aérea, de Micro y nano satélites) promoted by INTA (Instituto Nacional de Tecnología Aeroespacial Esteban Terradas), an autonomous agency of the Spanish public administration responsible for the aerospace and defense technologies research and especially from the support received from Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Jesús Sánchez, head of the Department of Rockets and Orthotronics at INTA-Marañosa Campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' It has also received partial support from the Scholarship- Collaboration program of the Spanish Ministry of Education and Science and in part from a similar program sponsored by Universidad Politécnica de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' 38 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Coats, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Nickerson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Bao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Liu, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Wu, “Efficient simulation of grain burning surface regression,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE3T4oBgHgl3EQfgQo6/content/2301.04559v1.pdf'} +page_content=', vol.' metadata={'source': 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0000000000000000000000000000000000000000..21147b27da070649a0d57d032814c425d11ae94a --- /dev/null +++ b/idE1T4oBgHgl3EQfzwUn/content/tmp_files/2301.03447v1.pdf.txt @@ -0,0 +1,1618 @@ +Proceeding of International Conference on Ummah 2022 +4th – 5th December 2022 | Universiti Malaysia Kelantan +1013 +Automatic Standardization of Arabic Dialects for Machine Translation + +Abidrabbo Alnassan +Jean Moulin Lyon 3 University +Linguistics Research Center - Corpus, Discourse and Societies +ILCEA4-CREO (Grenoble Alpes University) +abidrabbo.alnassan@univ-lyon3.fr + +ABSTRACT +Based on an annotated multimedia corpus, television series Marāyā 2013, we dig into the +question of “automatic standardization” of Arabic dialects for machine translation. Here we +distinguish between rule-based machine translation and statistical machine translation. +Machine translation from Arabic most of the time takes standard or modern Arabic as the +source language and produces quite satisfactory translations thanks to the availability of the +translation memories necessary for training the models. The case is different for the translation +of Arabic dialects. The productions are much less efficient. In our research we try to apply +machine translation methods to a dialect/standard (or modern) Arabic pair to automatically +produce a standard Arabic text from a dialect input, a process we call “automatic +standardization”. we opt here for the application of "statistical models" because "automatic +standardization" based on rules is more hard with the lack of "diglossic" dictionaries on the +one hand and the difficulty of creating linguistic rules for each dialect on the other. Carrying +out this research could then lead to combining "automatic standardization" software and +automatic translation software so that we take the output of the first software and introduce it +as input into the second one to obtain at the end a quality machine translation. This approach +may also have educational applications such as the development of applications to help +understand different Arabic dialects by transforming dialectal texts into standard Arabic. +Keywords: Arabic dialect; Syrian dialect; Automatic standardization; Modern Standard +Arabic; Machine translation + +INTRODUCTION + +The Arabic language is a collection of varieties: Standard Arabic (SA) or Classical Arabic +(CA), the language used in the Quran as well as in numerous literary texts; Modern Standard +Arabic (MSA), the formal and official language of the Arab World; and Arabic dialects (AD), +the commonly used informal native varieties. SA differs significantly in its grammatical +properties from ADs. ADs have no standard orthographies and rules, they differ from each +other and currently have an increasing presence on the web. + +Arabic NLP researches, which focused mostly on SA and MSA, is now dealing more +with ADs, especially when the DARPA73 launched, in October 2011, the Broad Operational +Language Translation (BOLT) program to attempt to create new techniques for automated +translation and linguistic analysis that can be applied to the informal genres of text and speech +common in online and in-person communication. Machine translation of ADs is a real +challenge because of the lack of dialectal linguistic resources while SA and MSA has a wealth +of resources in terms of morphological analyzers, disambiguation systems, annotated data, and +parallel corpora. + + +73 Defense Advanced Research Projects Agency + +Proceeding of International Conference on Ummah 2022 +e ISBN 978-967-0021-48-5 +1014 +Based on an annotated multimedia corpus, television series Marāyā 2013, we dig into +the question of “automatic standardization” of Arabic dialects for machine translation. Here +we distinguish between rule-based machine translation and statistical and neural machine +translation. Rule-based machine translation software relies on the use of many linguistic rules +and large volumes of dictionary entries for each language pair. The software iterates through +the text to be translated and creates an intermediate representation from which the translation +is generated. This process requires the use of voluminous dictionaries, syntactic, morphological +and semantic data, and numerous linguistic rules. + +Statistical machine translation software translates using auto-constructed “statistical +models” from monolingual and bilingual corpora. The construction of these “statistical +models” requires the prior existence and availability of large volumes of translated texts +(translation memories) to train the model to generate the translation. + +Neural machine translation, on the other hand, is processed through a neural network +where each neuron is a mathematical function that processes data. The initial translation +training is done by feeding examples into the neural network and making adjustments based +on how much error in the output there was. The network is continually used and continue to +fine-tune itself to provide better results. + +Machine translation from Arabic most of the time takes standard or modern Arabic as +the source language and produces quite satisfactory translations, thanks to the availability of +the translation memories necessary for training the models. The case is different for the +translation of Arabic dialects. The productions are much less efficient. + +In our research we try to apply machine translation methods to a dialect/standard (or +modern) Arabic pair to automatically produce a standard Arabic text from a dialect input, a +process we call “automatic standardization” versus “automatic translation” or “machine +translation”. "automatic standardization" is done in one direction: informal or non-standard +variety to formal or standard variety of the same language while “automatic translation” is done +between different languages regardless of the direction of the process. The other process which +goes from the standard variety to the non-standard variety of the same language may be called +"automatic destandardization". + +We aim through our research to develop a strategy to enrich linguistic resources and +parallel ADs-MSA corpora by involving a huge number of human resources not yet involved +in this process. We opt here for the application of "statistical models" because "automatic +standardization" based on rules is more hard with the lack of standard orthographies for ADs, +their numerous varieties, the absence of "diglossic" dictionaries and the difficulty of creating +linguistic rules and dedicated tools for each dialect. + +PREVIOUS WORK + +Previous research on ADs machine translation has focused on mapping AD input words into +MSA equivalents before translating. Researchers have used different techniques to do that. +Chiang et al. (2006) built a parser for spoken Levantine Arabic (LA) transcripts using an MSA +treebank. They used an LA-MSA lexicon in addition to morphological and syntactic rules to +map the LA sentences to MSA. Riesa and Yarowsky (2006) built a statistical morphological +segmenter for Iraqi and Levantine speech transcripts, and showed that they outperformed rule- +based segmentation with small amounts of training. Abo Bakr et al. (2008) suggested a hybrid, + +Proceeding of International Conference on Ummah 2022 +4th – 5th December 2022 | Universiti Malaysia Kelantan +1015 +rule-based and statistical, system to map Egyptian Arabic to MSA, using morphological +analysis on the input and an Egyptian-MSA lexicon. Sawaf (2010) normalized the dialectal +words in a hybrid machine translation system. Salloum and Habash (2011) also mapped AD to +MSA. + +Some tools exist for preprocessing and tokenizing Arabic text with a focus on ADs. +MAGEAD (Habash and Rambow, 2006) is a morphological analyzer and generator that can +analyze words into their root/pattern and affixed morphemes, or generate a word in the opposite +direction. Amazon’s Mechanical Turk (MTurk) help creating annotated resources for +computational linguistics. It is an online marketplace that allows “Requesters” to create simple +tasks requiring human knowledge, and have them completed by “Workers” from all over the +world. Zaidan and Callison-Burch (2011) created the Arabic Online Commentary (AOC) +dataset by crawling the websites of three Arabic newspapers74, and extracting online articles +and readers' comments75. Over 100k sentences from the AOC were annotated by native Arabic +speakers on MTurk to identify ADs and dialect level in each one. The collected labels were +used to train automatic dialect identification systems. Laith H. Baniata et al (2018) study the +problem of employing a neural machine translation model to translate ADs to MSA. They +propose the development of a multitask learning model which shares one decoder among +language pairs, and every source language has a separate encoder. + +SPOKEN ARABIC VS STANDARD ARABIC + +In spontaneous oral communication, the letter ( َق: qa) of SA is pronounced ( َأ: ’a) in some ADs, +which can make it difficult, out of context, to determine the word we hear. This difference in +pronunciation between spoken Arabic and standard Arabic mobilizes on the one hand the +writing of the word produced orally and on the other its meaning in SA. + َق(qa) → َأ(’a) + ﻢَﻠَﻗ(qalam: pen) → ﻢَﻟَأ(’alam: pen “in dialect” or pain “in SA”)76 + +In some cases, the new word heard is even non-existent in SA. + َق(qa) → َأ(’a) + راَﺮَﻗ(qarār: decision) → راَرَأ(’arār: decision “in dialect” but there is no meaning to +this pronunciation in SA) + +A Syrian speaker and a Lebanese or Jordanian speaker can understand each other if +both express themselves in their dialectal Arabic, because they are from neighboring linguistic +cultures. However, a Syrian and a Moroccan cannot easily communicate through their dialect. +There, MSA is essential (Alnassan, 2017). “Automatic standardization” therefore is not only +important for the translation of ADs into other foreign languages, but also for creating a passage +between the different ADs themselves. + +Research on ADs machine translation is mainly based on written productions. The +enrichment of machine translation tools therefore also presupposes the use of oral productions. + +74 The three newspapers are: 1) Al-Ghad (ﺪﻐﻟا), a Jordanian newspaper (www.alghad.com), 2) Al-Riyadh (ضﺎﯾﺮﻟا), +a Saudi newspaper (www.alriyadh.com), 3) Al-Youm Al-Sabe' (ﻊﺑﺎﺴﻟا مﻮﯿﻟا), an Egyptian newspaper +(www.youm7.com). +75 The commentary data consists of 3.1M segments, corresponding to 52.1M words. +76 We follow for the transliteration the system of the Arabica journal : (lettees : ء: ’ , ب : b, ت: t, ث: ṯ, ج: j, ح: +ḥ, خ: ḫ, د: d, ذ: ḏ, ر: r, ز: z, س: s, ش: š, ص: ṣ, ض: ḍ, ط: ṭ, ظ: ẓ, ع : ‘ , غ: ġ, ف: f, ق: q, ك: k, ل: l, م: m, ن: n, + ـھ: h, و: w, ي: y. Short vowels : ـَــ: a, ـُــ: u, ـِــ: i. Long vowels : ا: ā, و: ū, ي: ī). + +Proceeding of International Conference on Ummah 2022 +e ISBN 978-967-0021-48-5 +1016 +It is for this reason that our study relied on a television corpus (Marāyā 2013) in which most +of the speech is in the Syrian dialect of Damascus. + +MARĀYĀ 2013 + +Marāyā is a Syrian television series whose first season was broadcast in 1982. Very popular in +Syria and the Arab world, this series deals ironically, and sometimes satirically, with themes +relating to Syrian daily, social and political life. Marāyā 2013 is the last season of this series. +The corpus includes all the thirty episodes broadcast on the Algerian channel Aš-šurūq TV. +The average length of each episode is eighteen minutes, while it was around forty minutes in +previous seasons. Although the Damascus dialect is the main language used, dialects from other +Syrian regions may appear depending on the characters. Similarly, SA and MSA arises from +time to time and exceptionally occupies the entirety of episode 25 in which the text is narrative. + +Analysis of the lexicon of Marāyā 2013 + +The construction and analysis of Marāyā 2013 were initially carried out within the framework +of previous research in Arabic language didactics (Alnassan, 2016). The speeches of the +characters in the videos were transcribed and annotated using the ELAN (EUDICO Linguistic +Annotator)77 tool. The analysis of the transcripts was implemented in two steps: +· The first consisted of a statistical and descriptive analysis to define the nature of the +lexical elements of the corpus and their distribution according to categories (according +to lexicon proximity to SA/MSA) and groups (nouns, verbs, adjectives, ..., etc.); +· The second was based on a linguistic analysis (morphological, phonetic and semantic) +which aimed to identify useful elements for the improvement of Arabic language +teaching manuals. +The statistical analysis showed that 60% of the lexical elements of the corpus +studied are common in SA/MSA and in the Syrian dialect (SD). However, at the level +of the construction of the sentence, the phonetic aspect and the semantics in context, +the two systems will be even more differentiated. +Through morphological and semantic analysis, we have been able to distinguish: +· Lexical elements that have the same meaning and the same form in SA/MSA and in +dialect (ﻲﺧأ: ’ḫī: my brother); +· Lexical elements that have undergone slight modifications between SA/MSA and the +dialect (letters or/and vowels), retaining the same meaning (ﺮﯿِﺘْﻛ/ﺮﯿِﺜَﻛ: kaṯīr/ktīr: many); +· Lexical elements which have the same form in SA/MSA and in dialect but whose +meaning is different between the two registers (ةَﺮﻜُﺑ: bukrah: “the time just before +sunrise” in SA/MSA / “tomorrow” in Syrian dialect). +We also distinguish at the sentence level: +· Constructions which have the same components in SA/MSA and in dialect and which +produce the same meaning (ﷲ َءﺎﺷ ْنإ: ’in šā’ Allāh: God willing); +· Constructions of lexical elements belonging to SA/MSA, but which are used only in +dialectal context (ﻚَﻧﻮُﯿﻋﺮْﻣأ: ’amr ‘yūnak: “as you wish!/ at your service”, in the sense of +obeying an order or responding to a request kindly.); +· Dialectal constructions containing lexical items related to SA/MSA, retaining the same +meaning as the original SA/MSA construction (ﺎﻣ ﻞِﺑَأ ﺔّﯿِﻤﻠﻟ ّﺪِﻌﺗ: t'idd lal-miyyih 'bil mā…: + +77 ELAN is an annotation tool that allows you to create, edit, view and search complex annotations for audio or +video data. https://archive.mpi.nl/tla/elan. + +Proceeding of International Conference on Ummah 2022 +4th – 5th December 2022 | Universiti Malaysia Kelantan +1017 +count to a hundred before doing something, the equivalent standard expression is ﱡﺪُﻌَﺗ + ْنأ َﻞْﺒَﻗ ﺔَﺌِﻤﻠِﻟ…: ta'uddu lil-mi'ah qabla 'an…); +· Constructions containing lexical elements related to SA/MSA, but which only exist and +have a meaning in dialectal context (ﻲﻄﯿِﺣ ﻲِطﻮَﺘﺴِﺗ: tistawṭī ḥiyṭī: "you see it low, my +wall/roof"78, in the sense of challenging someone's contemptuous look at you). +Phonetic analysis allowed us to identify seven letters of SA/MSA that can be +pronounced differently in the regional dialect of Damascus. The following table shows, with +examples, how to pronounce these letters in different contexts. (Alnassan, 2016b) +Abbreviations: +SA: Standard Arabic +L: letter in Arabic script +RdD: Regional dialect of Damascus +T: Transliteration of the Arabic letter +Tra: Translation of the example in English + +Table 1. Arabic letters pronounced differently in RdD + + +From these observations, we can imagine the difficulty of carrying out rule-based +machine translation or “automatic standardization” of ADs. For this reason, we think that +creating and continuously enriching AD-MSA parallel corpora can, with the help of computer +tools, considerably advance research in the machine translation of ADs. Audio-visual resources + +78 This expression is typically metaphorical. It represents two very distinct situations according to the meanings +of the word “ﻲﻄﯿﺣ: ḥiyṭī: my wall/my roof” in the Syrian dialect. By the first meaning “ ﻂﯿﺣ: ḥiyṭ: wall”, we imagine +two neighbors for whom there is a wall separating their house. If one of the two is intrusive and the wall is high, +he cannot do anything to disturb his neighbor. On the other hand, if this wall is low, he can easily overcome it and +interfere in the affairs of the other. Here, the latter, angry, can use the expression “ﻲﻄﯿِﺣ ﻲِطﻮَﺘﺴِﺗ: tistawṭī ḥiyṭī” +while they are arguing. In the second case where the meaning is " ﻂﯿﺣ: ḥiyṭ: roof", the expression refers to a +situation where the roof of someone's house is too low so that anyone can climb it and then gain access to the +inner courtyard. The metaphor of this expression most often refers to the first image. + +Pronunciation +SA letter +Example +Original word +inRdD +L +T +L +T +Word : T :Tra +Word : T +t + : Mutallat : triangle + : Mutallat +t +t +Cio : mitil : like +Jie : mitil +s +Shu : masalan : for example + : matalan +p +JIoM : q.p : . +q.p : +p +Z +O!:izin:permission +:"idn +d +Ol: "idin : hear +si: 'udun +CD +s +ann : ynyps : r +yyps : ? +s +: saddiqni +p +y: marid : ill +y: marid +d +Z +Lj:mazbut:absolutely +-oisa : madbut +L +z +:muzaharah:demonstration + y-: muzaharah +L +z +d +e- : dahr : back +e : zahr +Z +J:galiz:heavy/ annoying +Lule : galiz +b + : hadiqah : garden / park +2 : hadiqah +b +1 +st: 'adim : ancient + : qadim +g +rl l : 'abu gasim : the father of Gasim + :'abi gasimProceeding of International Conference on Ummah 2022 +e ISBN 978-967-0021-48-5 +1018 +must also be taken into account in this process. Most current research focuses on textual +resources. + +ADs-MSA PARALLEL CORPORA CREATION + +Researchers in machine translation of ADs most often use methods that pivot through MSA. +Harrath et al. (2017) for example show in a survey eight research works out of thirteen pivoting +through MSA to translate ADs into English. Some works are based on open source parallel +corpora like what can be found on OPUS79 (the Open Parallel Corpus). Others build their own +corpus. + +For our work, we have tried to see if we can find on OPUS an ADs-MSA parallel +corpora. By searching on OPUS, we were able to find voluminous resources dealing with the +Arabic language. However, almost all the resources found are related to the MSA. The +following table shows some of the information obtained by running a query to find Arabic- +English parallel corpora. The complete result of our query can be found in the appendix. + +Table 2. Some Arabic-English/English-Arabic corpora on OPUS +Corpus +Arabic-English +English-Arabic +Sentence pairs +Words +Sentence pairs +Words +United Nations Parallel Corpus +16,637,291 +832.98M +20,044,653 +904.08M +OpenSubtitles v2018 +25,855,525 +339.10M +29,823,188 +356.14M +Tanzil +184,894 +13.02M +187,052 +13.07M +TED2020 v1 +397,962 +12.52M +407,595 +12.54M +tico-19 v2020-10-28 +3,070 +0.14M +3,071 +0.14M +WikiMatrix v1 +999,763 +41.98M +999,763 +41.98M +wikimedia v20210402 +374,437 +31.49M +407,543 +31.84M +Wikipedia +146,131 +5.34M +151,136 +5.38M + +By searching for corpora containing ADs, we were able to identify two dialects listed +among the source languages; the Syrian dialect “ar-SY(Arabic)” and the Tunisian dialect “ar- +TN(Arabic)”. No dialect has been listed in the target languages. We then made the request to +obtain the corpus containing the translations of the Syrian dialect “ar-SY(Arabic)” into MSA +(which is represented in OPUS as “ar(Arabic)”) and then the corpus containing the +translations of the Tunisian dialect “ar-TN(Arabic)” into MSA. The results were as follows: + +Table 3. ar-SY(Arabic)-MSA corpora on OPUS + + + + + + + + + + +79 OPUS is a growing collection of translated texts from the web. : https://opus.nlpl.eu/ + +Search & download resources:ar_ SY (Arabic) +ar (Arabic) +all +show all versions +Language resources: click on [ tmx moses xces | lang-id J to download the data! (raw = untokenized, ud = parsed with universal dependencies, alg = word alignments and phrase tables) +corpus +doc's sent's ar tokens ar_SY tokens +XCES/XMI +raw +TMX +Moses +mono +raw +pn +alg + dic +freq +other files +Ubuntu v14.10 +xces ar ar_SYar ar_SY +tmx +moses +AsAs +dic +ar ar_Sy +sample +total +0 +0 +0 +0 +0 +0 +color: +size (src+trg): 16.4k 32.8k 65.5k 0.1M 0.3M 0.5M 1.0M 2.1M 4.2M 8.4M 16.8M 33.6M 67.1M 134.2MProceeding of International Conference on Ummah 2022 +4th – 5th December 2022 | Universiti Malaysia Kelantan +1019 +Table 4. ar-TN(Arabic)-MSA corpora on OPUS + + +By consulting the tmx version of the two corpora obtained, we were able to discover +that the corpus for the Syrian dialect was an empty corpus, while the corpus which was +supposed to contain the Tunisian dialect actually contained only words and sentences in MSA +accompanied by their equivalent in MSA too. It was therefore not a Tunisian dialectal source +language translated into MSA as a target language but rather a source in MSA produced by a +Tunisian or in Tunisia and its equivalent in MSA as the target language. Here is an example of +the content of this corpus : + +Table 5. ar-TN(Arabic)-MSA tmx content + + +We also noticed the existence of an "ara (arabic)" in the list of source languages and +which does not exist in the list of target languages. Looking also at the tmx version of the +parallel corpus for the language pair ara(Arabic)-MSA, we found that these two varieties are +only MSA. + + +64 + +65 +mi +66 +67 + +68 + +69 +aLZumb. +70 +ssLiaii +71 + +72 + +73 +Li +74 +LLigl +75 + +76 + +77 +r4hyi/tuv> +78 +wiihiyr +80 + +81 +slyiJsiLL +82 +seg>pLyyaLabisi-si +83 + +84 + +85 +86 +seg>uLaJwyLaauLi ypLyy. +87 + +88 + +89 +Liaglil/tuv> +90 +seg>Liagl1 +91 +Search & download resources: ar_ TN (Arabic) +var (Arabic) +all + show all versions +Language resources: click on [ tmx [ moses I xces I lang-id J to download the data! (raw = untokenized, ud = parsed with universal dependencies, alg = word alignments and phrase tables) +corpus +doc's +sent's +ar tokens +ar_TN tokens +XCES/XML +raw +TMX +Moses +mono +raw +ud +alg +dic +freq +other files +GNOME v1 +1 +0.9k +3.7k +7.2k +xces ar ar_TN + ar ar TN +tmxmoses +NIe JNe Je +alg smt +ar ar_TN +sample +total +1 +0.9k +3.7k +7.2k +0.9k +0.7k +0.9k +color: +size (src+trg): 16.4k 32.8k 65.5k 0.1M 0.3M 0.5M 1.0M 2.1M 4.2M 8.4M 16.8M 33.6M 67.1M 134.2MProceeding of International Conference on Ummah 2022 +e ISBN 978-967-0021-48-5 +1020 +Table 6. ara(Arabe)-MSA corpora on OPUS + + +To create their own corpus, some researchers use MTurk (Zaidan and Callison-Burch: 2011a,b; +Zbib et al.: 2012). The idea is to create a parallel corpus by hiring non-professional translators +and annotators to translate or annotate the sentences that were labeled as being ADs or MSA +in documents collected from the web. +This method, from our point of view, is limited because: +· It is based on the work of a small number of contributors (translators or +annotators); +· It is costly in terms of financial investment. +· The work is not durable and the enrichment of the corpus is not continuous. +· It is often based on the analysis of written documents. +Our work on Marāyā 2013 pushed us to reflect on methods that may allow us to build +big textual corpora based essentially on audio-visual elements. Collecting subtitle texts from +films, for example, does not provide this opportunity because in this case the text does not +accurately represent the language content of the video. The transcription of the dialogues of +Marāyā 2013 series was done manually and took a long time despite the fact that the average +duration of each episode was around eighteen minutes. The other seasons of Marāyā had an +average duration of forty-five minutes for each episode. We can so imagine how much time +and money an individual researcher must spend to manually transcribe the remaining eighteen +seasons of Marāyā, where each season contains at least thirty episodes. Automatic video +transcription tools (speech to text tools) are not efficient enough for the Arabic language, +especially when it comes to ADs. +The creation of textual resources from audio or audio-visual resources cannot therefore +be carried out within the framework of individual work, which is our current case. In the same +way, creating large ADs-MSA corpora requires the contribution of a very large number of +contributors who are able to bring their help to advance this work. + +Development of applications and platforms for the massive transcription and +standardization of ADs. + +During our computer-assisted translation (CAT) courses, we invite our students to practice +using open source CAT applications and platforms. This practice allows the student to become +familiar with these CAT tools. It also contributes to the continuous enrichment of translation +memories (TM) which eventually become parallel corpora. The only problem is that these +parallel corpora are not accessible to users. In other words, the service is provided free of +charge, an individual user can retrieve the TM of his present work, but the parallel corpus +produced by all users is not accessible. +To circumvent this problem we imagine the following scenario: +· IT developers or web developers build a web application or a platform allowing the +entry of a word, an expression or a sentence in AD, define which AD it belongs to, then +standardize it into MSA; +· This web application or platform must be unique and centralized to avoid duplication +of data collected by users. + +Search & dowynload resources: +ara (Arabic) +ar (Arabic) +all +show all versions +Language resources: click on [ tmx I moses I xces I lang-id J to download the data! (raw = untokenized, ud = parsed with universal dependencies, alg = word alignments and phrase tables) +corpus +s,sop +sent's +ar tokens +ara tokens +XCES/XIMIL +raw +TMX +Moses +mono +raw +pn +alg +dic +freq +other files +GNOME v1 +1 +0.6k +1.7k +1.6k +xces ar ara +ar ara +tmx +moses +ar ara +ar ara +alg smt +ar ara +sample +[D10] +1 +0.6k +1.7k +1.6k +0.6k +0.4k +0.6k +color: +size (src+trg):16.4k 32.8k 65.5k 0.1M 0.3M 0.5M 1.0M 2.1M 4.2M 8.4M 16.8M 33.6M 67.1M 134.2MProceeding of International Conference on Ummah 2022 +4th – 5th December 2022 | Universiti Malaysia Kelantan +1021 +· The application must be accessible for free +· In educational and higher education institutions, we develop introductory and practical +courses for the standardization of ADs into MSA. +· The practical part is done using the above-mentioned application or platform. +· Each user can retrieve the result of his current work to be able to develop his own +resources; +· Each user can also download the global parallel corpus produced and enriched +continuously by all users. +· This possibility of downloading the global parallel corpus may also be available to +researchers in DAs and MSA. +· Researchers may also contribute to the development of the application or the platform, +or to the development of output evaluation tools, for example. +Through this approach, researchers in dialectology, translation and machine translation +of ADs will have an additional resource to those that already exist. A large ADs-MSA parallel +corpus, built by a large number of contributors who will not necessarily be translators80. If only +students from the Arabic language departments of all the universities in the Arab world +participate in this work, we will very quickly have the ADs-MSA parallel corpus which we +hope to obtain. Such a corpus can help very considerably in the development of the statistical +and neural models for ADs machine translation, and before that, the development of statistical +and neural models for the automatic standardization of ADs. +In the same way, we can also develop a unique application or platform for the massive +transcription of audio and audio-visual resources where the speeches are in ADs. The textual +resources obtained by such an approach can be the basis on which the users of the manual +standardization application will work. + +CONCLUSION + +Through this brief presentation of the difficulties related to the ADs machine translation, and +of the work carried out and in progress in this field, we can come back to the idea that we really +need to develop methods and tools to fill the lack of resources for ADs. We have seen that there +is currently a significant lack of monolingual dialect corpora based on audio or audio-visual +resources. There is also a significant lack in ADs-MSA parallel corpora necessary for training +statistical or neural models in ADs machine translation systems. + +The solutions we propose: developing a unique application or platform for the massive +transcription of audio or audio-visual data which are in ADs, then another application or +platform for the standardization of ADs, can significantly help to create and enrich +continuously textual resources and large parallel ADs-MSA corpora. We can thus involve a +very large number of participants who are not yet involved in this kind of practice while they +can help without it being expensive in terms of time and money. + +Carrying out this project could then lead to combining "automatic standardization" +software and automatic translation software to obtain at the end a quality ADs machine +translation. + + +80 Native speakers of the different ADs, even if they do not know any other foreign language, can participate in +this standardization work because SA and MSA are learned in school from childhood. + +Proceeding of International Conference on Ummah 2022 +e ISBN 978-967-0021-48-5 +1022 +This approach may also have educational applications such as the development of +applications to help understanding different ADs by transforming dialectal texts into standard +Arabic. + +REFERENCES + +Abo Bakr, H., Shaalan, K., & Ziedan, I. (2008). A Hybrid Approach for Converting Written +Egyptian Colloquial Dialect into Diacritized Arabic. In The 6th International +Conference on Informatics and Systems, INFOS2008. Cairo University. +Alnassan, A. (2016a). Les compétences lexicales en arabe langue étrangère/seconde : analyse +d'un corpus télévisuel syrien. Thèse de doctorat en sciences du langage préparée à +l’Université Lumière Lyon 2 et soutenue publiquement le 02 juillet 2016. +Alnassan, A. (2016b). Written and Spoken Arabic in Syria : towards the development of teaching Arabic +as a foreign language at the Higher Language Institute of Damascus. In : The 4th FLLT +Conference Proceedings by LITU, 4 (1), pp.33 – 45. The 4the FLLT Conference "Foreign +Language Learning and Teaching", 24 – 25 June 2016, The Ambassador Hotel, Bangkok, +Thailand. +Alnassan, A. (2017). Didactique de l’arabe et problématique de la polyglossie : approche +comparative entre l’arabe littéraire et le dialecte syrien en vue d’améliorer la qualité de +l’enseignement de l’arabe, langue étrangère. In Les Carnets du Cediscor, 13. Presses +Sorbonne Nouvelle, Paris, pages 46 – 59. +Attia, M., Pecina, P., Toral, A., & Genabith, J. V. (2013). A corpusbased finite-state +morphological toolkit for contemporary arabic. Journal of Logic and Computation (pp. +exs070). +Baniata, L. H., Park, S., & Park, S. B. (2018). A Neural Machine Translation Model for Arabic +Dialects That Utilises Multitask Learning (MTL). In Computational Intelligence and +Neuroscience. +Retrieved +October +31, +2022, +from +https://www.hindawi.com/journals/cin/2018/7534712/ +Brown, P. F., Cocke, J., Della Pietra, S. A., Della Pietra, V. J., Jelinek, F., Lafferty, J. D., +Mercer, R. L., & Roossin, P. S. (1990). A statistical approach to machine translation. +Computational Linguistics, 16(2), pages 79–85. +Callison-Burch, C., Koehn, P., & Osborne, M. (2006). Improved statistical machine translation +using paraphrases. In Proceedings of the Human Language Technology Conference of +the NAACL, Main Conference (pp. 17–24). +Chiang, D., Diab, M., Habash, N., Rambow, O., & Shareef, S. (2006). Parsing Arabic Dialects. +In Proceedings of the European Chapter of ACL (EACL). Retrieved October 31, 2022, +from https://aclanthology.org/events/eacl-2006/ +Du, J., Jiang, J., & Way, A. (2010). Facilitating translation using source language paraphrase +lattices. In Proceedings of the 2010 Conference on Empirical Methods in Natural +Language Processing, EMNLP’10 (pp. 420–429). Cambridge, Massachusetts. +Habash, N. & Rambow, O. (2006). MAGEAD: A Morphological Analyzer and Generator for +the Arabic Dialects. In Proceedings of the 21st International Conference on +Computational Linguistics and 44th Annual Meeting of the Association for +Computational Linguistics (pp. 681–688). Sydney, Australia. +Harrat, S., Meftouh, K., & Smaïli. K. (2017). Machine translation for Arabic dialects (survey). +Information Processing and Management (56 (2), pp. 262-273). Elsevier. +Riesa, J., & Yarowsky. D. (2006). Minimally Supervised Morphological Segmentation with +Applications to Machine Translation. In Proceedings of the 7th Conference of the +Association for Machine Translation in the Americas (AMTA06) (pp. 185–192). +Cambridge,MA. + +Proceeding of International Conference on Ummah 2022 +4th – 5th December 2022 | Universiti Malaysia Kelantan +1023 +Salloum, W. & Habash, N. (2011). Dialectal to Standard Arabic Paraphrasing to Improve +Arabic-English Statistical Machine Translation. In Proceedings of the First Workshop +on Algorithms and Resources for Modelling of Dialects and Language Varieties (pp. +10–21). Edinburgh, Scotland. +Sawaf, H. (2010). Arabic dialect handling in hybrid machine translation. In Proceedings of the +Conference of the Association for Machine Translation in the Americas (AMTA), +Denver, Colorado. +Zaidan, O. F., & Callison-Burch, C. (2011). Crowdsourcing translation: Professional quality +from non-professionals. In Proceedings of the 49th Annual Meeting of the Association +for Computational Linguistics: Human Language Technologies (pp. 1220–1229). +Portland, Oregon. +Zbib, R., Malchiodi, E., Devlin, J., Stallard, D., Matsoukas, S., Schwartz, R., Makhoul, J., +Zaidan†, O. F., & Callison-Burch‡, C. (2012). Machine Translation of Arabic Dialects. +Conference of the North American Chapter of the Association for Computational +Linguistics: Human Language Technologies (pp. 49–59). Montréal, Canada, June 3-8, +2012. + + + +Proceeding of International Conference on Ummah 2022 +e ISBN 978-967-0021-48-5 +1024 +Appendix A +Transliteration of the Arabic alphabet +Appendix B +Arabic-English corpora on OPUS + +Search&downloadresources: +ar(Arabic) +en(English) +all +showall versions +Language resources: click on [tmx moses xces lang-id Jto download the data! (raw =untokenized, ud = parsed with universal dependencies,alg =word alignments and phrase tables) +corpus +doc's +sent's +artokens +en tokens +XCES/XML +raw +TMX +Moses +mono +raw +pn +alg +dic +freq +other files +CCMatrix l +1 +49.7M +805.8M +900.9M +xces ar en +aren +tmx +moses +ar en ar en +aren +sample +WikiMatrix vl +1 +2.0M +79.6M +1.0G +xces aren +aren +tmx +moses +ar en ar en +aren +sample +UNPCv1.0 +114067 +16.6M +394.7M +445.4M +xces ar en +aren +tmx +moses +ar en aren +alg +aren +sample +MultiUNvl +67617 +8.2M +201.7M +228.2M +xces ar en +aren +tmx +moses +aren +aren +alg +ar en query sample +CCAligned vl +507 +13.0M +188.7M +200.6M +xces ar en +aren +tmx +moses +ar en +aren +ar en +sample +wikimediav20210402 +1 +0.4M +24.7M +349.2M +xces ar en +aren +tmx +moses +aren +aren +aren +sample +OpenSubtitlesv2018 +8256 +4.6M +26.8M +29.9M +xces ar en +aren +xuum +moses +aren +aren +alg smt +dic +arenquerysamplexces/alt +XLEnt v1.1 +1 +5.6M +19.1M +18.7M +xces ar en +aren +tmx +moses +aren +aren +aren +sample +QEDv2.0a +5033 +0.7M +6.6M +9.5M +xces ar en +aren +tmx +moses +aren +aren +alg smt +dic +ar en +sample +TED2020v1 +3879 +0.4M +6.4M +8.1M +xces aren +aren +tmx +moses +aren +aren +aren +sample +Tanzil vi +30 +0.2M +7.9M +5.6M +xces ar en +aren +tmx +moses +aren +aren +alg smt +dic +aren query sample +News-Commentaryv16 +7185 +83.2k +5.0M +3.8M +xcesar en +aren +tmx +moses +aren +aren +algsmt +dic +aren +sample +UNv20090831 +1 +74.1k +3.3M +3.7M +xcesaren +aren +tmx +moses +aren +aren +alg smt +aren query +sample +Wikipedia v1.0 +1 +0.2M +3.2M +3.5M +xcesaren +aren +tmx +moses +aren: +aren +alg smt +dic +aren query +sample +TED2013 v1.1 +1 +0.2M +2.4M +3.0M +xces ar en +aren +tmx +moses +ar en +aren +alg smt +dic +ar en query +sample +GNOME v1 +1313 +0.5M +2.4M +2.6M +xces aren +aren +tmx +moses +aren +aren +algsmt +aren +sample +bible-uedin vl +2 +61.5k +0.9M +1.5M +xces ar en +aren +tmx +moses +ar en +aren +alg smt +dic +ar en +sample +GlobalVoicesv2018q4 +3875 +58.3k +1.0M +1.3M +xcesaren +aren +tmx +moses +aren +aren +algsmt +dic +aren +sample +KDE4v2 +784 +0.1M +0.7M +0.8M +xces ar en +aren +tmx +moses +aren +aren +algsmt +dic +ar en query sample +Mozilla-I110n vl +1 +51.7k +0.2M +0.7M +xces ar en +aren +aren aren +aren +sample +ELRC_2922v1 +1 +15.1k +0.3M +0.3M +xces ar en +aren +tmx +moses +ar en +aren +alg smt +dic +aren +sample +EUbookshopv2 +30 +1.7k +80.0k +0.4M +xcesar en +aren +tmx +moses +aren +aren +algsmt +dic +ar en query +samplemoses/strict +infopankkivi +290 +16.0k +0.2M +0.2M +xces ar en +aren +tmx +moses +ar en ar en +alg smt +dic +aren +sample +Tatoebav2022-03-03 +1 +27.3k +0.1M +0.2M +xces ar en +aren +tmx +moses +aren +aren +aren +sample +tico-19 v2020-10-28 +1 +3.1k +67.9k +70.4k +xces ar en +aren +tmx +moses +aren +aren +alg smt +dic +aren +sample +Ubuntu v14.10 +xces ar en +aren +tmx +moses +ar en ar en +dic +aren +sample +212879 +102.8M +1.8G +3.3G +102.8M +101.0M +115.6M +color: +size(src+trg):16.4k32.8k65.5k0.1M0.3M0.5M1.0M2.1M4.2M8.4M16.8M33.6M67.1M134.2MShort vowels +Long vowels +二 +a +- +iad +二 +u +9 +n +- +i +1Arabic letter +Symbol +Arabic letter +Symbol +Arabic letter +Symbol +Arabic letter +Symbol +L +d +d +F +k +b +d +L +t +J +1 +t +r +上 +Z +2 +m +t +z +3 +Q +n +j +Cw +S +g +h +2 +h +3 +> +f +9 +W +2 +h +CD +s +q +yProceeding of International Conference on Ummah 2022 +4th – 5th December 2022 | Universiti Malaysia Kelantan +1025 +English-Arabic corpora on OPUS + + + + +Search&downloadresources:en (English) +V +ar (Arabic) +all +V +show all versions +Language resources: click on[tmx moses xceslang-id J to download the data! (raw=untokenized,ud =parsed with universal dependencies,alg=word alignments and phrasetables) +corpus +doc's +sent's +artokens +en tokens +XCES/XML +raw +TMX +Moses +mono +raw +pn +alg +dic +freq +otherfilles +CCMatrix vl +1 +49.7M +805.8M +900.9M +xces ar en +aren +tmx +moses +ar en +aren +aren +sample +WikiMatrix vl +1 +2.0M +79.6M +1.0G +xces aren +aren +tmx +moses +ar en ar en +aren +sample +UNPCv1.0 +114067 +16.6M +394.7M +445.4M +xces ar en +aren +tmx +moses +ar en aren +alg +aren +sample +MultiUNvl +67617 +8.2M +201.7M +228.2M +xces ar en +aren +tmx +moses +ar en ar en +alg +ar en query sample +CCAligned vl +507 +13.0M +188.7M +200.6M +xces ar en +aren +tmx +moses +ar en +aren +aren +sample +wikimediav20210402 +1 +0.4M +24.7M +349.2M +xces ar en +aren +tmx +moses +ar en +aren +aren +sample +OpenSubtitlesv2018 +8256 +4.6M +26.8M +29.9M +xces ar en +aren +tmx +moses +aren +aren +algsmt +dic +ar en query samplexces/alt +XLEnt v1.1 +1 +5.6M +19.1M +18.7M +xces ar en +aren +tmx 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newline at end of file diff --git a/idE1T4oBgHgl3EQfzwUn/content/tmp_files/load_file.txt b/idE1T4oBgHgl3EQfzwUn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f05f239b251873af77399830f30f3351692b0d1c --- /dev/null +++ b/idE1T4oBgHgl3EQfzwUn/content/tmp_files/load_file.txt @@ -0,0 +1,681 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf,len=680 +page_content='Proceeding of International Conference on Ummah 2022 4th – 5th December 2022 | Universiti Malaysia Kelantan 1013 Automatic Standardization of Arabic Dialects for Machine Translation Abidrabbo Alnassan Jean Moulin Lyon 3 University Linguistics Research Center - Corpus, Discourse and Societies ILCEA4-CREO (Grenoble Alpes University) abidrabbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='alnassan@univ-lyon3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='fr ABSTRACT Based on an annotated multimedia corpus, television series Marāyā 2013, we dig into the question of “automatic standardization” of Arabic dialects for machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Here we distinguish between rule-based machine translation and statistical machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Machine translation from Arabic most of the time takes standard or modern Arabic as the source language and produces quite satisfactory translations thanks to the availability of the translation memories necessary for training the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The case is different for the translation of Arabic dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The productions are much less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' In our research we try to apply machine translation methods to a dialect/standard (or modern) Arabic pair to automatically produce a standard Arabic text from a dialect input, a process we call “automatic standardization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' we opt here for the application of "statistical models" because "automatic standardization" based on rules is more hard with the lack of "diglossic" dictionaries on the one hand and the difficulty of creating linguistic rules for each dialect on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Carrying out this research could then lead to combining "automatic standardization" software and automatic translation software so that we take the output of the first software and introduce it as input into the second one to obtain at the end a quality machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' This approach may also have educational applications such as the development of applications to help understand different Arabic dialects by transforming dialectal texts into standard Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Keywords: Arabic dialect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Syrian dialect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Automatic standardization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Modern Standard Arabic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Machine translation INTRODUCTION The Arabic language is a collection of varieties: Standard Arabic (SA) or Classical Arabic (CA), the language used in the Quran as well as in numerous literary texts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Modern Standard Arabic (MSA), the formal and official language of the Arab World;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' and Arabic dialects (AD), the commonly used informal native varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' SA differs significantly in its grammatical properties from ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' ADs have no standard orthographies and rules, they differ from each other and currently have an increasing presence on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Arabic NLP researches, which focused mostly on SA and MSA, is now dealing more with ADs, especially when the DARPA73 launched, in October 2011, the Broad Operational Language Translation (BOLT) program to attempt to create new techniques for automated translation and linguistic analysis that can be applied to the informal genres of text and speech common in online and in-person communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Machine translation of ADs is a real challenge because of the lack of dialectal linguistic resources while SA and MSA has a wealth of resources in terms of morphological analyzers, disambiguation systems, annotated data, and parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 73 Defense Advanced Research Projects Agency Proceeding of International Conference on Ummah 2022 e ISBN 978-967-0021-48-5 1014 Based on an annotated multimedia corpus, television series Marāyā 2013, we dig into the question of “automatic standardization” of Arabic dialects for machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Here we distinguish between rule-based machine translation and statistical and neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Rule-based machine translation software relies on the use of many linguistic rules and large volumes of dictionary entries for each language pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The software iterates through the text to be translated and creates an intermediate representation from which the translation is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' This process requires the use of voluminous dictionaries, syntactic, morphological and semantic data, and numerous linguistic rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Statistical machine translation software translates using auto-constructed “statistical models” from monolingual and bilingual corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The construction of these “statistical models” requires the prior existence and availability of large volumes of translated texts (translation memories) to train the model to generate the translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Neural machine translation, on the other hand, is processed through a neural network where each neuron is a mathematical function that processes data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The initial translation training is done by feeding examples into the neural network and making adjustments based on how much error in the output there was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The network is continually used and continue to fine-tune itself to provide better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Machine translation from Arabic most of the time takes standard or modern Arabic as the source language and produces quite satisfactory translations, thanks to the availability of the translation memories necessary for training the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The case is different for the translation of Arabic dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The productions are much less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' In our research we try to apply machine translation methods to a dialect/standard (or modern) Arabic pair to automatically produce a standard Arabic text from a dialect input, a process we call “automatic standardization” versus “automatic translation” or “machine translation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' "automatic standardization" is done in one direction: informal or non-standard variety to formal or standard variety of the same language while “automatic translation” is done between different languages regardless of the direction of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The other process which goes from the standard variety to the non-standard variety of the same language may be called "automatic destandardization".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' We aim through our research to develop a strategy to enrich linguistic resources and parallel ADs-MSA corpora by involving a huge number of human resources not yet involved in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' We opt here for the application of "statistical models" because "automatic standardization" based on rules is more hard with the lack of standard orthographies for ADs, their numerous varieties, the absence of "diglossic" dictionaries and the difficulty of creating linguistic rules and dedicated tools for each dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' PREVIOUS WORK Previous research on ADs machine translation has focused on mapping AD input words into MSA equivalents before translating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Researchers have used different techniques to do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Chiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' (2006) built a parser for spoken Levantine Arabic (LA) transcripts using an MSA treebank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' They used an LA-MSA lexicon in addition to morphological and syntactic rules to map the LA sentences to MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Riesa and Yarowsky (2006) built a statistical morphological segmenter for Iraqi and Levantine speech transcripts, and showed that they outperformed rule- based segmentation with small amounts of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Abo Bakr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' (2008) suggested a hybrid, Proceeding of International Conference on Ummah 2022 4th – 5th December 2022 | Universiti Malaysia Kelantan 1015 rule-based and statistical, system to map Egyptian Arabic to MSA, using morphological analysis on the input and an Egyptian-MSA lexicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Sawaf (2010) normalized the dialectal words in a hybrid machine translation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Salloum and Habash (2011) also mapped AD to MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Some tools exist for preprocessing and tokenizing Arabic text with a focus on ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' MAGEAD (Habash and Rambow, 2006) is a morphological analyzer and generator that can analyze words into their root/pattern and affixed morphemes, or generate a word in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Amazon’s Mechanical Turk (MTurk) help creating annotated resources for computational linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' It is an online marketplace that allows “Requesters” to create simple tasks requiring human knowledge, and have them completed by “Workers” from all over the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=" Zaidan and Callison-Burch (2011) created the Arabic Online Commentary (AOC) dataset by crawling the websites of three Arabic newspapers74, and extracting online articles and readers' comments75." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Over 100k sentences from the AOC were annotated by native Arabic speakers on MTurk to identify ADs and dialect level in each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The collected labels were used to train automatic dialect identification systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Laith H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Baniata et al (2018) study the problem of employing a neural machine translation model to translate ADs to MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' They propose the development of a multitask learning model which shares one decoder among language pairs, and every source language has a separate encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' SPOKEN ARABIC VS STANDARD ARABIC In spontaneous oral communication, the letter ( َق: qa) of SA is pronounced ( َأ: ’a) in some ADs, which can make it difficult, out of context, to determine the word we hear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' This difference in pronunciation between spoken Arabic and standard Arabic mobilizes on the one hand the writing of the word produced orally and on the other its meaning in SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' َق(qa) → َأ(’a) ﻢَﻠَﻗ(qalam: pen) → ﻢَﻟَأ(’alam: pen “in dialect” or pain “in SA”)76 In some cases, the new word heard is even non-existent in SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' َق(qa) → َأ(’a) راَﺮَﻗ(qarār: decision) → راَرَأ(’arār: decision “in dialect” but there is no meaning to this pronunciation in SA) A Syrian speaker and a Lebanese or Jordanian speaker can understand each other if both express themselves in their dialectal Arabic, because they are from neighboring linguistic cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' However, a Syrian and a Moroccan cannot easily communicate through their dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' There, MSA is essential (Alnassan, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' “Automatic standardization” therefore is not only important for the translation of ADs into other foreign languages, but also for creating a passage between the different ADs themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Research on ADs machine translation is mainly based on written productions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The enrichment of machine translation tools therefore also presupposes the use of oral productions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 74 The three newspapers are: 1) Al-Ghad (ﺪﻐﻟا), a Jordanian newspaper (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='alghad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='com), 2) Al-Riyadh (ضﺎﯾﺮﻟا), a Saudi newspaper (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='alriyadh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content="com), 3) Al-Youm Al-Sabe' (ﻊﺑﺎﺴﻟا مﻮﯿﻟا), an Egyptian newspaper (www." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='youm7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 75 The commentary data consists of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='1M segments, corresponding to 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='1M words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 76 We follow for the transliteration the system of the Arabica journal : (lettees : ء: ’ , ب : b, ت: t, ث: ṯ, ج: j, ح: ḥ, خ: ḫ, د: d, ذ: ḏ, ر: r, ز: z, س: s, ش: š, ص: ṣ, ض: ḍ, ط: ṭ, ظ: ẓ, ع : ‘ , غ: ġ, ف: f, ق: q, ك: k, ل: l, م: m, ن: n, ـھ: h, و: w, ي: y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Short vowels : ـَــ: a, ـُــ: u, ـِــ: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Long vowels : ا: ā, و: ū, ي: ī).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Proceeding of International Conference on Ummah 2022 e ISBN 978-967-0021-48-5 1016 It is for this reason that our study relied on a television corpus (Marāyā 2013) in which most of the speech is in the Syrian dialect of Damascus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' MARĀYĀ 2013 Marāyā is a Syrian television series whose first season was broadcast in 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Very popular in Syria and the Arab world, this series deals ironically, and sometimes satirically, with themes relating to Syrian daily, social and political life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Marāyā 2013 is the last season of this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The corpus includes all the thirty episodes broadcast on the Algerian channel Aš-šurūq TV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The average length of each episode is eighteen minutes, while it was around forty minutes in previous seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Although the Damascus dialect is the main language used, dialects from other Syrian regions may appear depending on the characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Similarly, SA and MSA arises from time to time and exceptionally occupies the entirety of episode 25 in which the text is narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Analysis of the lexicon of Marāyā 2013 The construction and analysis of Marāyā 2013 were initially carried out within the framework of previous research in Arabic language didactics (Alnassan, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The speeches of the characters in the videos were transcribed and annotated using the ELAN (EUDICO Linguistic Annotator)77 tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The analysis of the transcripts was implemented in two steps: The first consisted of a statistical and descriptive analysis to define the nature of the lexical elements of the corpus and their distribution according to categories (according to lexicon proximity to SA/MSA) and groups (nouns, verbs, adjectives, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=', etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The second was based on a linguistic analysis (morphological, phonetic and semantic) which aimed to identify useful elements for the improvement of Arabic language teaching manuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The statistical analysis showed that 60% of the lexical elements of the corpus studied are common in SA/MSA and in the Syrian dialect (SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' However, at the level of the construction of the sentence, the phonetic aspect and the semantics in context, the two systems will be even more differentiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Through morphological and semantic analysis, we have been able to distinguish: Lexical elements that have the same meaning and the same form in SA/MSA and in dialect (ﻲﺧأ: ’ḫī: my brother);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Lexical elements that have undergone slight modifications between SA/MSA and the dialect (letters or/and vowels), retaining the same meaning (ﺮﯿِﺘْﻛ/ﺮﯿِﺜَﻛ: kaṯīr/ktīr: many);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Lexical elements which have the same form in SA/MSA and in dialect but whose meaning is different between the two registers (ةَﺮﻜُﺑ: bukrah: “the time just before sunrise” in SA/MSA / “tomorrow” in Syrian dialect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' We also distinguish at the sentence level: Constructions which have the same components in SA/MSA and in dialect and which produce the same meaning (ﷲ َءﺎﺷ ْنإ: ’in šā’ Allāh: God willing);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Constructions of lexical elements belonging to SA/MSA, but which are used only in dialectal context (ﻚَﻧﻮُﯿﻋﺮْﻣأ: ’amr ‘yūnak: “as you wish!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='/ at your service”, in the sense of obeying an order or responding to a request kindly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=" Dialectal constructions containing lexical items related to SA/MSA, retaining the same meaning as the original SA/MSA construction (ﺎﻣ ﻞِﺑَأ ﺔّﯿِﻤﻠﻟ ّﺪِﻌﺗ: t'idd lal-miyyih 'bil mā…: 77 ELAN is an annotation tool that allows you to create, edit, view and search complex annotations for audio or video data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='mpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='nl/tla/elan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=" Proceeding of International Conference on Ummah 2022 4th – 5th December 2022 | Universiti Malaysia Kelantan 1017 count to a hundred before doing something, the equivalent standard expression is ﱡﺪُﻌَﺗ ْنأ َﻞْﺒَﻗ ﺔَﺌِﻤﻠِﻟ…: ta'uddu lil-mi'ah qabla 'an…);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Constructions containing lexical elements related to SA/MSA, but which only exist and have a meaning in dialectal context (ﻲﻄﯿِﺣ ﻲِطﻮَﺘﺴِﺗ: tistawṭī ḥiyṭī: "you see it low, my wall/roof"78, in the sense of challenging someone\'s contemptuous look at you).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Phonetic analysis allowed us to identify seven letters of SA/MSA that can be pronounced differently in the regional dialect of Damascus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The following table shows, with examples, how to pronounce these letters in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' (Alnassan, 2016b) Abbreviations: SA: Standard Arabic L: letter in Arabic script RdD: Regional dialect of Damascus T: Transliteration of the Arabic letter Tra: Translation of the example in English Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Arabic letters pronounced differently in RdD From these observations, we can imagine the difficulty of carrying out rule-based machine translation or “automatic standardization” of ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' For this reason, we think that creating and continuously enriching AD-MSA parallel corpora can, with the help of computer tools, considerably advance research in the machine translation of ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Audio-visual resources 78 This expression is typically metaphorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' It represents two very distinct situations according to the meanings of the word “ﻲﻄﯿﺣ: ḥiyṭī: my wall/my roof” in the Syrian dialect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' By the first meaning “ ﻂﯿﺣ: ḥiyṭ: wall”, we imagine two neighbors for whom there is a wall separating their house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' If one of the two is intrusive and the wall is high, he cannot do anything to disturb his neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' On the other hand, if this wall is low, he can easily overcome it and interfere in the affairs of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Here, the latter, angry, can use the expression “ﻲﻄﯿِﺣ ﻲِطﻮَﺘﺴِﺗ: tistawṭī ḥiyṭī” while they are arguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' In the second case where the meaning is " ﻂﯿﺣ: ḥiyṭ: roof", the expression refers to a situation where the roof of someone\'s house is too low so that anyone can climb it and then gain access to the inner courtyard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The metaphor of this expression most often refers to the first image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Pronunciation SA letter Example Original word inRdD L T L T Word : T :Tra Word : T t : Mutallat : triangle : Mutallat t t Cio : mitil : like Jie : mitil s Shu : masalan : for example : matalan p JIoM : q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='p : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='p : p Z O!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' :izin:permission :"idn d Ol: "idin : hear si: \'udun CD s ann : ynyps : r yyps : ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=" s : saddiqni p y: marid : ill y: marid d Z Lj:mazbut:absolutely oisa : madbut L z :muzaharah:demonstration y-: muzaharah L z d e- : dahr : back e : zahr Z J:galiz:heavy/ annoying Lule : galiz b : hadiqah : garden / park 2 : hadiqah b 1 st: 'adim : ancient : qadim g rl l : 'abu gasim : the father of Gasim :'abi gasimProceeding of International Conference on Ummah 2022 e ISBN 978-967-0021-48-5 1018 must also be taken into account in this process." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Most current research focuses on textual resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' ADs-MSA PARALLEL CORPORA CREATION Researchers in machine translation of ADs most often use methods that pivot through MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Harrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' (2017) for example show in a survey eight research works out of thirteen pivoting through MSA to translate ADs into English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Some works are based on open source parallel corpora like what can be found on OPUS79 (the Open Parallel Corpus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Others build their own corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' For our work, we have tried to see if we can find on OPUS an ADs-MSA parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' By searching on OPUS, we were able to find voluminous resources dealing with the Arabic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' However, almost all the resources found are related to the MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The following table shows some of the information obtained by running a query to find Arabic- English parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The complete result of our query can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Some Arabic-English/English-Arabic corpora on OPUS Corpus Arabic-English English-Arabic Sentence pairs Words Sentence pairs Words United Nations Parallel Corpus 16,637,291 832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='98M 20,044,653 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='08M OpenSubtitles v2018 25,855,525 339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='10M 29,823,188 356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='14M Tanzil 184,894 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='02M 187,052 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='07M TED2020 v1 397,962 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='52M 407,595 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='54M tico-19 v2020-10-28 3,070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='14M 3,071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='14M WikiMatrix v1 999,763 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='98M 999,763 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='98M wikimedia v20210402 374,437 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='49M 407,543 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='84M Wikipedia 146,131 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='34M 151,136 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='38M By searching for corpora containing ADs, we were able to identify two dialects listed among the source languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' the Syrian dialect “ar-SY(Arabic)” and the Tunisian dialect “ar- TN(Arabic)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' No dialect has been listed in the target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' We then made the request to obtain the corpus containing the translations of the Syrian dialect “ar-SY(Arabic)” into MSA (which is represented in OPUS as “ar(Arabic)”) and then the corpus containing the translations of the Tunisian dialect “ar-TN(Arabic)” into MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The results were as follows: Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' ar-SY(Arabic)-MSA corpora on OPUS 79 OPUS is a growing collection of translated texts from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' : https://opus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='nlpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='eu/ Search & download resources:ar_ SY (Arabic) ar (Arabic) all show all versions Language resources: click on [ tmx moses xces | lang-id J to download the data!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=" (raw = untokenized, ud = parsed with universal dependencies, alg = word alignments and phrase tables) corpus doc's sent's ar tokens ar_SY tokens XCES/XMI raw TMX Moses mono raw pn alg dic freq other files Ubuntu v14." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='10 xces ar ar_SYar ar_SY tmx moses AsAs dic ar ar_Sy sample total 0 0 0 0 0 0 color: size (src+trg): 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='4k 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='8k 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='5k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='1M 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6M 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='1M 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='2MProceeding of International Conference on Ummah 2022 4th – 5th December 2022 | Universiti Malaysia Kelantan 1019 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' ar-TN(Arabic)-MSA corpora on OPUS By consulting the tmx version of the two corpora obtained, we were able to discover that the corpus for the Syrian dialect was an empty corpus, while the corpus which was supposed to contain the Tunisian dialect actually contained only words and sentences in MSA accompanied by their equivalent in MSA too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' It was therefore not a Tunisian dialectal source language translated into MSA as a target language but rather a source in MSA produced by a Tunisian or in Tunisia and its equivalent in MSA as the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Here is an example of the content of this corpus : Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' ar-TN(Arabic)-MSA tmx content We also noticed the existence of an "ara (arabic)" in the list of source languages and which does not exist in the list of target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Looking also at the tmx version of the parallel corpus for the language pair ara(Arabic)-MSA, we found that these two varieties are only MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 64 65 mi 66 67 68 69 aLZumb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 70 ssLiaii 71 72 73 Li 74 LLigl 75 76 77 r4hyi/tuv> 78 wiihiyr 80 81 slyiJsiLL 82 seg>pLyyaLabisi-si 83 84 85 86 seg>uLaJwyLaauLi ypLyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 87 88 89 Liaglil/tuv> 90 seg>Liagl1 91 Search & download resources: ar_ TN (Arabic) var (Arabic) all show all versions Language resources: click on [ tmx [ moses I xces I lang-id J to download the data!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=" (raw = untokenized, ud = parsed with universal dependencies, alg = word alignments and phrase tables) corpus doc's sent's ar tokens ar_TN tokens XCES/XML raw TMX Moses mono raw ud alg dic freq other files GNOME v1 1 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='9k 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='7k 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='2k xces ar ar_TN ar ar TN tmxmoses NIe JNe Je alg smt ar ar_TN sample total 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='9k 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='4M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='8M 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6M 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='1M 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='2MProceeding of International Conference on Ummah 2022 e ISBN 978-967-0021-48-5 1020 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' ara(Arabe)-MSA corpora on OPUS To create their own corpus, some researchers use MTurk (Zaidan and Callison-Burch: 2011a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Zbib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' : 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The idea is to create a parallel corpus by hiring non-professional translators and annotators to translate or annotate the sentences that were labeled as being ADs or MSA in documents collected from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' This method, from our point of view, is limited because: It is based on the work of a small number of contributors (translators or annotators);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' It is costly in terms of financial investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The work is not durable and the enrichment of the corpus is not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' It is often based on the analysis of written documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Our work on Marāyā 2013 pushed us to reflect on methods that may allow us to build big textual corpora based essentially on audio-visual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Collecting subtitle texts from films, for example, does not provide this opportunity because in this case the text does not accurately represent the language content of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The transcription of the dialogues of Marāyā 2013 series was done manually and took a long time despite the fact that the average duration of each episode was around eighteen minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The other seasons of Marāyā had an average duration of forty-five minutes for each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' We can so imagine how much time and money an individual researcher must spend to manually transcribe the remaining eighteen seasons of Marāyā, where each season contains at least thirty episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Automatic video transcription tools (speech to text tools) are not efficient enough for the Arabic language, especially when it comes to ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The creation of textual resources from audio or audio-visual resources cannot therefore be carried out within the framework of individual work, which is our current case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' In the same way, creating large ADs-MSA corpora requires the contribution of a very large number of contributors who are able to bring their help to advance this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Development of applications and platforms for the massive transcription and standardization of ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' During our computer-assisted translation (CAT) courses, we invite our students to practice using open source CAT applications and platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' This practice allows the student to become familiar with these CAT tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' It also contributes to the continuous enrichment of translation memories (TM) which eventually become parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The only problem is that these parallel corpora are not accessible to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' In other words, the service is provided free of charge, an individual user can retrieve the TM of his present work, but the parallel corpus produced by all users is not accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' To circumvent this problem we imagine the following scenario: IT developers or web developers build a web application or a platform allowing the entry of a word, an expression or a sentence in AD, define which AD it belongs to, then standardize it into MSA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' This web application or platform must be unique and centralized to avoid duplication of data collected by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Search & dowynload resources: ara (Arabic) ar (Arabic) all show all versions Language resources: click on [ tmx I moses I xces I lang-id J to download the data!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=" (raw = untokenized, ud = parsed with universal dependencies, alg = word alignments and phrase tables) corpus s,sop sent's ar tokens ara tokens XCES/XIMIL raw TMX Moses mono raw pn alg dic freq other files GNOME v1 1 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='7k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6k xces ar ara ar ara tmx moses ar ara ar ara alg smt ar ara sample [D10] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='7k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='4k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6k color: size (src+trg):16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='4k 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='8k 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='5k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='1M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='3M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='5M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='0M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='1M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='2M 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='4M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='8M 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6M 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='1M 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='2MProceeding of International Conference on Ummah 2022 4th – 5th December 2022 | Universiti Malaysia Kelantan 1021 The application must be accessible for free In educational and higher education institutions, we develop introductory and practical courses for the standardization of ADs into MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The practical part is done using the above-mentioned application or platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Each user can retrieve the result of his current work to be able to develop his own resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Each user can also download the global parallel corpus produced and enriched continuously by all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' This possibility of downloading the global parallel corpus may also be available to researchers in DAs and MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Researchers may also contribute to the development of the application or the platform, or to the development of output evaluation tools, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Through this approach, researchers in dialectology, translation and machine translation of ADs will have an additional resource to those that already exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' A large ADs-MSA parallel corpus, built by a large number of contributors who will not necessarily be translators80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' If only students from the Arabic language departments of all the universities in the Arab world participate in this work, we will very quickly have the ADs-MSA parallel corpus which we hope to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Such a corpus can help very considerably in the development of the statistical and neural models for ADs machine translation, and before that, the development of statistical and neural models for the automatic standardization of ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' In the same way, we can also develop a unique application or platform for the massive transcription of audio and audio-visual resources where the speeches are in ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The textual resources obtained by such an approach can be the basis on which the users of the manual standardization application will work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' CONCLUSION Through this brief presentation of the difficulties related to the ADs machine translation, and of the work carried out and in progress in this field, we can come back to the idea that we really need to develop methods and tools to fill the lack of resources for ADs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' We have seen that there is currently a significant lack of monolingual dialect corpora based on audio or audio-visual resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' There is also a significant lack in ADs-MSA parallel corpora necessary for training statistical or neural models in ADs machine translation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' The solutions we propose: developing a unique application or platform for the massive transcription of audio or audio-visual data which are in ADs, then another application or platform for the standardization of ADs, can significantly help to create and enrich continuously textual resources and large parallel ADs-MSA corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' We can thus involve a very large number of participants who are not yet involved in this kind of practice while they can help without it being expensive in terms of time and money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Carrying out this project could then lead to combining "automatic standardization" software and automatic translation software to obtain at the end a quality ADs machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 80 Native speakers of the different ADs, even if they do not know any other foreign language, can participate in this standardization work because SA and MSA are learned in school from childhood.' 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F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=', & Callison-Burch‡, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Machine Translation of Arabic Dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' 49–59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Montréal, Canada, June 3-8, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=' Proceeding of International Conference on Ummah 2022 e ISBN 978-967-0021-48-5 1024 Appendix A Transliteration of the Arabic alphabet Appendix B Arabic-English corpora on OPUS Search&downloadresources: ar(Arabic) en(English) all showall versions Language resources: click on [tmx moses xces lang-id Jto download the data!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content=" (raw =untokenized, ud = parsed with universal dependencies,alg =word alignments and phrase tables) corpus doc's sent's artokens en tokens XCES/XML raw TMX Moses mono raw pn alg dic freq other files CCMatrix l 1 49." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='7M 805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='8M 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='9M xces ar en aren tmx moses ar en ar en aren sample WikiMatrix vl 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='0M 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='6M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='0G xces aren aren tmx moses ar en ar en aren sample UNPCv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE1T4oBgHgl3EQfzwUn/content/2301.03447v1.pdf'} +page_content='0 114067 16.' metadata={'source': 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processing is an increasingly important domain of computer science, with applications in data and +network analysis, among others. Target graphs in these applications are often large, leading to the creation of +“big data” systems designed to provide the scalability needed to analyze these graphs using parallel processing. +However, researchers have shown that while these systems often provide scalability, they also often introduce +overheads that exceed the benefits they provide, sometimes lower absolute performance than even simple serial +implementations. +This report studies the viability and performance of actor model to implement scalable concurrent programs +to perform common graph computations. We show that relatively simple actor-based implementations +outperform both dedicated graph processing systems and the benchmark serial implementations. +I. +Introduction +Scalability is often heralded as the most im- +portant property of a “big data” software sys- +tem. +In [1], McSherry et al. show that this +metric is frequently misleading when not cou- +pled with performance comparisons against +an appropriate baseline. Providing decreased +runtime as core count increases is certainly a +desirable property of a well-engineered parallel +system, but, in the end, absolute performance +is paramount from the perspective of a user, +trumping scalability. +McSherry et al. introduce a metric they term +COST, or Configuration that Outperforms a +Single Thread. +This metric measures the +amount of hardware resources required before +a particular system beats the runtime of a rea- +sonably designed single threaded solution. In +particular, they found that several parallel data +processing systems from the literature were +outperformed by a simple serial program for +various common graph processing tasks, even +when the parallel systems were given over 100x +the computational resources. This analysis sug- +gests that these big data systems achieve their +scalability by parallelizing overhead that they +themselves have introduced rather than provid- +ing a useful speedup relative to a fair baseline. +While some problem domains have inher- +ent algorithmic or dataset related impediments +to achieving performance improvements with +parallel execution, such issues are not present +here. Datasets are large, providing the scale +to decompose across several processors, and +algorithms are generally data parallel without +fine-grained global synchronization or depen- +dencies, only requiring interactions within a +local neighborhood. Given this potential, well- +engineered parallel implementations should be +able to provide speedups over the serial ver- +sions. +Taking inspiration from the findings of the +COST work, in this report, we test this sup- +position. We implement parallel versions of +two common graph algorithms using an actor- +based framework and compare performance +and programmabilty against a well-established +existing parallel graph processing system and +the baseline serial implementations. +1 +arXiv:2301.01395v1 [cs.DC] 4 Jan 2023 + +COST of Graph Processing Using Actors +II. +Approach +In this section, we introduce and describe the +ideology and principles of the actor model and +the specific framework used in our implemen- +tations, Charm++. We also discuss the graph +computations performed by our implementa- +tions. +i. +Actor Model +The actor model [2] provides a conceptual +framework to reason about and design con- +current computation. In this model, the com- +putation is decomposed into several entities +called actors. Each individual actor operates +independently from the others, with its own +private state that no other object can read or +modify. Actors pass data and coordinate exe- +cution via sending messages to each other. No- +tably, this messaging is entity-centric (i.e. an +actor is specified as the destination of a mes- +sage) as opposed to processor-centric, as seen +in the bulk synchronous parallel model or mes- +sage passing à la MPI. When an actor receives +a message, it gains ownership over the payload +data. In response to a message, an actor can +perform some computation, send additional +messages, spawn new actors, etc. +This encapsulation of state and restriction +of interaction to only messages reduces the +complexity of implementing concurrent appli- +cations. +Rather than having to think about +the global state of the application or multi- +ple non-deterministic threads of execution and +their interaction with each other, the program- +mer merely has to consider the received mes- +sages and local state on a given individual actor +when designing the logic of the program. +Using actors improves the safety of parallel +applications as compared to shared memory +concurrency. Since state is private and cannot +be concurrently accessed or altered, the actor +model prevents memory access order race con- +ditions and obviates the need for lock-based +synchronization. +Programs expressed in the actor model can +naturally be executed in parallel due to the +inherently concurrent semantics of actor com- +putation. +Further, the messaging semantics +allow execution to be fully distributed, since +there is no need for shared state. +i.1 +Concrete Implementations +Due to these properties, concrete implementa- +tions of the actor model are popular in the field +of distributed systems, with examples such as +ActorFoundry, Akka, and Erlang, among oth- +ers. These implementations often exploit the +properties of actor-based computation to add +additional features such as fault tolerance and +migratability. +However, while these languages and libraries +provide the many benefits of the actor model +to users, a common concern is that they are +focused more on expressivity and programma- +bility than on pure raw performance. The need +to do location management, message delivery, +garbage collection, etc. adds overhead rela- +tive to more primitive, lower-level forms of +writing parallel programs. Given that we are +interested primarily in a performance compar- +ison, it is important that our actor-based im- +plementations utilize a platform designed with +performance in mind, such as Charm++. +i.2 +Charm++ +Charm++ [3] [4] is an adaptive, asynchronous +task-based parallel programming framework +based on the actor model and focused on +high performance computing applications. The +core entities in a Charm++ program are ob- +jects called chares, which are essentially actors. +Chares hold private state and generally com- +municate with each other using messages. The +Charm++ runtime system automatically mea- +sures the load of chares, and coupled with the +message delivery semantics of actors, can use +these measurements to migrate chares to auto- +matically balance load between processors. +Both the runtime of Charm++ and applica- +tions that use it are written in C++, meaning +there is no garbage collection and the over- +heads are that of a low-level systems program- +ming language. +2 + +COST of Graph Processing Using Actors +While one may use Charm++ to develop +applications that completely adhere to the se- +mantics of the actor model, it allows users to +break some of these constraints in the interest +of performing performance optimizations. For +example, users can create shared buffers be- +tween chares to more efficiently share data as +compared to messaging. These boundary vio- +lations must be done carefully and can often +lead to degraded performance if not done well, +but they are often needed in the HPC domain +where performance is sacrosanct. +Additionally, Charm++ offers some program- +ming niceties such as quiescence detection, au- +tomatic message aggregation, and the ability to +do bulk creation of actors in collections called +“chare arrays”, which allow actors to also be +referenced via an index rather than just an ad- +dress. +ii. +Computations +We implement two common graph computa- +tions, PageRank and connected component de- +tection via label propagation. +ii.1 +PageRank +PageRank [5] is a method for ranking vertices +in a directed graph by “link popularity.” Fa- +mously, it was developed by the founders of +Google and served as the original algorithm +underpinning the results of their eponymous +web search engine. +PageRank is an iterative algorithm that main- +tains a rank for each vertex in the graph. Dur- +ing each iteration, the rank of a vertex is damp- +ened by a given factor α, divided by its out- +degree d, and sent to each of its outgoing neigh- +bors. The new rank of a vertex is computed by +adding all of the contributions it receives from +its incoming neighbors to 1 − α. +The original implementation from the COST +paper is shown in Listing 1 and the Charm++ +implementation in Listing 2 (declarations and +initialization have been condensed or omitted +for space). +1 fn +PageRank20 (graph: GraphIter , alpha: f32) { +2 +let mut a = vec! [0 f32; graph.nodes ()]; +3 +let mut b = vec! [0 f32; graph.nodes ()]; +4 +let mut d = vec! [0 f32; graph.nodes ()]; +5 +6 +graph.map_edges (|x, y| { d[x] += 1; }); +7 +for +iter in 0..20 { +8 +for i in 0.. graph.nodes () { +9 +b[i] = alpha * a[i] / d[i]; +10 +a[i] = 1f32 - alpha; +11 +} +12 +graph.map_edges (|x, y| { a[y] += b[x]; }) +13 +} +14 } +Listing 1: Serial PageRank +1 // Driver +function , only +runs on 0th chare +2 void +runpagerank (float +alpha) { +3 +for (int i = 0; i < 20; i++) { +4 +// Call +update on all +chares in array +5 +thisProxy.update(alpha); +6 +CkWaitQD (); // Sleep +until +quiescence +7 +// Call +iterate +on all +chares in array +8 +thisProxy.iterate (); +9 +CkWaitQD (); +10 +} +11 } +12 +13 // Set up values +for new +iteration +14 void +update(float +alpha) { +15 +for (int i = 0; i < d.size (); i++) { +16 +b[i] = alpha * a[i] / d[i]; +17 +a[i] = 1 - alpha; +18 +} +19 } +20 +21 // Run +PageRank +iteration , send to +neighbors +22 void +iterate () { +23 +vector >> outgoing; +24 +auto +edgeIt = edges.begin (); +25 +for (int i = 0; i < degs.size (); i++) { +26 +for (int j = 0; j < degs[i]; j++) { +27 +const +auto +dest = *edgeIt ++ +28 +outgoing[CHUNKINDEX(dest)]. emplace_back +(dest , b[i]); +29 +} +30 +} +31 +for (int i = 0; i < outgoing.size (); i++) { +32 +// Call +addB on chare +with +index i +33 +thisProxy[i]. addB(outgoing[i]); +34 +} +35 } +36 +37 // +Receive +values +from +neighbor +38 void +addB(vector > b_in) { +39 +for (const +auto& entry : b_in) { +40 +const +auto +dest = entry.first; +41 +const +auto +value = entry.second; +42 +// base is first +index on this +chunk +43 +a[dest - base] += value; +44 +} +45 } +Listing 2: Charm++ PageRank +3 + +COST of Graph Processing Using Actors +ii.2 +Connected Components +A connected component of a graph is a con- +nected subgraph that is not part of any larger +connected subgraph, or, more formally, a sub- +graph C of an undirected graph G such that +every vertex in V(C) is reachable from all ver- +tices in V(C) and not reachable from any vertex +in V(G) − V(C). +There are many known algorithms to find +connected components, but here we use the la- +bel propagation method [6] due to the suitability +of its neighborhood communication pattern for +distributed computation. Label propagation +is an iterative algorithm that maintains a label +for each vertex in the graph, initially set to the +unique index of the vertex. In each iteration, +the current label of each vertex is sent to each +of its neighbors. Upon receiving a candidate +label from a neighbor, a vertex changes its la- +bel to the candidate if the candidate is strictly +less than its current label. This process contin- +ues until an iteration occurs where no vertex +changes its label. At termination, the label of +each vertex is equal to the smallest index of the +vertices in its component. +III. +Methodology +We compare our implementations to two refer- +ences, the same serial versions1 implemented +in Rust as used in the original COST paper, and +the implementations provided by GraphX [7], +the graph processing component of the Apache +Spark data analysis engine. +GraphX was one of the “big data” systems +compared in the COST paper, and it either +approximately matched in performance or out- +performed the other big data systems used +in that comparison. +Here, we take it as a +representative for the landscape of data pro- +cessing systems due to its popularity and past +performance. Additionally, on their website2, +GraphX claims to have “Comparable perfor- +mance to the fastest specialized graph process- +ing systems.” +1https://github.com/frankmcsherry/COST +2https://spark.apache.org/graphx/ +Name +Vertices +Edges +soc-LiveJournal1 +4, 847, 571 +68, 993, 773 +twitter_rv +61, 578, 415 +1, 468, 365, 182 +uk-2007-05 +105, 896, 555 +3, 738, 733, 648 +Table 1: Properties of Selected Graphs +We evaluate the performance of the imple- +mentations on three different real-world input +graphs: +• soc-LiveJournal1 [8] - Friendship network +graph of the social media service LiveJour- +nal +• twitter_rv [9] - Following network graph +of the social media service Twitter +• uk-2007-05 [10] [11] - Hyperlink graph of +web pages in the .uk domain +The properties of these graphs are given +in Table 1. +twitter_rv and uk-2007-05 are +the same datasets used for evaluation in the +original COST paper and widely used in the +benchmarking of big data systems, and soc- +LiveJournal1 is a smaller dataset added to ease +testing during development of the new actor- +based versions. +Note that all of the chosen graphs are di- +rected graphs. +For use with label propaga- +tion, input graphs are converted to undirected +versions by adding a reverse edge for each +edge if it does not already exist in the edge set, +meaning that the graphs have more edges than +shown in Table 1 during label propagation. +Experiments were conducted on the CPU +partition of Delta3 at the National Center for +Supercomputing Applications. Each node of +Delta in the CPU partition has two AMD EPYC +7763 processors, with 128 cores across two sock- +ets and 256 GB of memory. +Finally, note that all provided timings are +only of the actual computation, the time taken +to ingest the graph from storage and perform +other upfront preparation and initialization are +not included. +3https://delta.ncsa.illinois.edu/ +4 + +COST of Graph Processing Using Actors +1 +2 +4 +8 +16 +32 +64 +128 +PEs +0 +50 +100 +150 +200 +250 +300 +Runtime (s) +soc-LiveJournal1 PageRank +Serial +GraphX +Figure 1: GraphX PageRank on soc-LiveJournal1 +IV. +Results & Discussion +i. +Serial and GraphX Baselines +In order to establish a baseline for comparison, +we first examine the GraphX and serial imple- +mentations. Figure 1 shows the performance +of these baseline runs for PageRank with soc- +LiveJournal1. GraphX performance scales as +the number of processors (PEs) increases, but +even with its fastest configuration of 128 PEs, +it takes 27.9 s for 20 iterations of PageRank, +while the serial version takes merely 3.18 s on +a single core. +Figure 2 tells a similar story for label propa- +gation: GraphX improves with scale, but even +its fastest time of 16.31 s at 64 PEs is over an +order of magnitude slower than the 1.05 s of +the serial version. +These +GraphX +implementations +use +GraphX’s Pregel API, which claims to avoid +excessive storage of intermediate results for +iterative computations like these, but there +appears to be a bug in practice, as even small +graphs can cause out of memory conditions +(e.g. a small 11.1 MB test graph running out +of memory during GraphX label propagation +given a 16 GB allocation). +Running PageRank with GraphX on twit- +ter_rv also resulted in an OOM crash; the +graph is 5.8 GB and GraphX was given 128 PEs +and 180 GB of memory. Running label propa- +gation on twitter_rv resulted in a timeout after +1 +2 +4 +8 +16 +32 +64 +128 +PEs +0 +50 +100 +150 +200 +Runtime (s) +soc-LiveJournal1 label prop +Serial +GraphX +Figure 2: GraphX Label Prop. on soc-LiveJournal1 +30 minutes of processing on 128 PEs. We did +not attempt runs with uk-2007-05 due to these +failures and the larger size of that graph. +GraphX was generally outperformed by the +serial implementations in the original COST +paper, but not to the extent seen in our ex- +periments. We are using a newer version of +GraphX, which may account for some of the +difference, but even so, the performance gulf is +stark. We implemented several different vari- +ants of the GraphX client application in an +attempt to improve the performance, but the +results shown were the best we were able to +obtain. +These GraphX results correspond to a COST +of ∞, as performance never matches the sin- +gle threaded version, regardless of how much +hardware we provide. We did not perform +runs beyond 128 PEs, but scaling improve- +ments appear to end at or before that point +in our results. +ii. +Charm++ +Our basic Charm++ implementation assigns +contiguous chunks of vertices to chares. Local +vertices are stored in index order, and the des- +tinations of outgoing edges are stored for each +vertex. During each iteration, a chare loops +over its local vertices and their outgoing edges, +performing any necessary preparation and ag- +gregating outgoing message data in per-chare +buffers. Messages are sent to their destination +5 + +COST of Graph Processing Using Actors +chares after the conclusion of this loop. Upon +reception of one of these messages, the chare +performs the required computation to apply +the payload of the message to its local vertices. +To analyze their performance impact, we im- +plemented several different variants with opti- +mizations on top of this basic implementation. +Some of these optimizations violate the seman- +tics of the actor model, namely those of private +state and only exchanging data via messages. +Atomic Passes data using a global vertex +buffer updated concurrently via atomic +operations instead of using messages. +Pairs Passes data using a collection of global +buffers, one for each ordered pair of chares +instead of using messages. No locks or +atomics are needed, synchronized via a +message telling consumer that the pro- +ducer is finished. +Reduction Passes data and computes results +using a parallel reduction tree instead of +using point to point messages. Each chare +contributes a buffer containing data for all +vertices with the updates coming from its +local vertices applied, the buffers are then +reduced in parallel, and finally each chare +does local updates using the correspond- +ing portion of the reduced buffer. +Sort Destination Reorders how edges are +stored on a chare; instead of ordering by +and storing (local source, {destinations}), +this orders and stores by (destination, {list +of local sources}). This orders local iter- +ations by destination, meaning that mes- +sages can be sent earlier than in the basic +version: after the edges incident to a sin- +gle destination chunk are done rather than +waiting until all edges are done. +ii.1 +PageRank +Figures 3, 4, and 5 show PageRank perfor- +mance for the Charm++ variants and the se- +rial implementation on soc-LiveJournal1, twit- +ter_rv, and uk-2007-05, respectively. Table 2 +1 +2 +4 +8 +16 +32 +64 +128 +PEs +0 +2 +4 +6 +8 +10 +12 +Runtime (s) +soc-LiveJournal1 PageRank +Serial +Charm +Charm Atomic +Charm Pairs +Charm Reduction +Charm Sortdest +Figure 3: Charm++ PageRank on soc-LiveJournal1 +1 +2 +4 +8 +16 +32 +64 +128 +PEs +0 +100 +200 +300 +400 +Runtime (s) +twitter rv PageRank +Serial +Charm +Charm Atomic +Charm Pairs +Charm Reduction +Charm Sortdest +Figure 4: Charm++ PageRank on twitter_rv +1 +2 +4 +8 +16 +32 +64 +128 +PEs +0 +100 +200 +300 +400 +Runtime (s) +uk-2007-05 PageRank +Serial +Charm +Charm Atomic +Charm Pairs +Charm Reduction +Charm Sortdest +Figure 5: Charm++ PageRank on uk-2007-05 +6 + +COST of Graph Processing Using Actors +Graph +soc-LJ1 +twitter_rv +uk-2007-05 +Serial +3.18 +180.69 +83.62 +PEs +1 +2.33 +119.28 +58.65 +2 +2.22 +73.93 +47.83 +4 +1.90 +56.01 +31.36 +8 +1.90 +52.84 +19.94 +16 +1.35 +39.02 +11.19 +32 +1.09 +28.55 +8.61 +64 +1.13 +21.48 +5.11 +128 +1.08 +25.00 +4.56 +Table 2: Best Charm++ PageRank Runtime (s) Across +All Variants +shows the runtime of the best performing vari- +ant at every scale. +Significantly, the COST with all input graphs +is 1, as the performance of the Charm++ vari- +ants matches or exceeds the performance of the +serial version on a single processor. However, +the performance and scalability of the different +variants varies greatly. +The basic variant performs well on 1 PE, but +then spikes in runtime when moving to 2 PEs, +before regaining performance with scale. This +is due to the allocation and serialization over- +head of messaging, which diminishes in rela- +tive importance as buffers become smaller as +the computation is scaled. +The atomic variant generally performs well +at all scales and for all graphs, since it avoids +the overheads of messaging and uses a highly +performant technique to operate on shared +data. +While the use of shared state breaks +a tenant of the actor model and does not work +for distributed execution, from a programma- +bilty perspective, using atomics is simpler and +less error-prone than using locks while also +being suitable for fine-grained parallelism. +The pairs variant works fairly well at the +medium to large scale on twitter_rv but poorly +on the other graphs and at smaller scale. As +with the basic variant, this is likely due to allo- +cation overheads from needing to dynamically +manage multiple large buffers, possibly exacer- +bated by NUMA effects. Note that the buffers +here are proportional in size to the number +of edges, rather than the much smaller num- +ber of vertices, as in the atomic variant. This +variant shows one of the risks of using shared +state: getting a pointer to it may be cheap, but +managing it may be costly. +The reduction variant is very reasonable at +small to medium scale, but is the worst per- +forming variant for every graph at large scale. +Its poor performance is due to two factors: load +imbalance and memory utilization. +1. In the other variants (with the exception +of atomic, which has the same global syn- +chronization), a chare can progressively +proceed with applying updates as neigh- +boring chares send data to it, whereas the +reduction variant requires all chares to +have finished their local loop, as the re- +duction can only complete after all objects +have contributed their data. +2. Each chare must allocate a buffer of size +equal to the number of total vertices in +the graph to contribute to the reduction +since sparse contributions are not sup- +ported. The other variants communicate +using space proportional to the number +of edges (with the exception of atomic, +which uses a single global vertex buffer). +While the number of edges is larger than +the number of vertices for all of the input +graphs, it is only 14-35x the number of ver- +tices, meaning the reduction variant will +be using much more memory at 128 PEs, +leading to allocation overhead and more +cache evictions. +The sort destination variant is the best per- +forming overall, due to several different rea- +sons: +1. Better locality and more efficient memory +utilization, as it only needs to maintain +a single buffer for message data since it +computes all of the outgoing data for a +single destination chunk before moving +onto the next chunk. +Similarly, on the +receive side, the payload of the message +7 + +COST of Graph Processing Using Actors +is arranged in the same order as the local +vertex data are stored. +2. Reduced load imbalance by sending mes- +sages earlier than other variants, allowing +otherwise idle chares to move onto the +next phase of the iteration instead of wait- +ing with no work to do. +3. Sending less data than other variants by +locally reducing the data bound for an ex- +ternal vertex before sending, which is also +enabled by its access pattern. As an ex- +ample, suppose two vertices A and B on a +chare both have outgoing edges to an ex- +ternal vertex C. In the basic variant, the up- +date from A is added to a message buffer, +and later the update from B is added to +the same message buffer. We could search +this buffer to combine the two updates, +but doing so is relatively costly, requiring +either a linear search or maintaining some +sorted or hash-based data structure and +the associated memory and computational +overheads, so we send the update from A +and B separately. On the other hand, due +to the arrangement of edges in the sort +destination variant, we process all vertices +with an outgoing edge to C consecutively, +so combining the update from A with the +update from B before sending is trivial. +ii.2 +Label Propagation +Figures 6, 7, and 8 show label propagation +performance for the Charm++ variants and +the serial implementation on soc-LiveJournal1, +twitter_rv, and uk-2007-05, respectively. Ta- +ble 3 shows the runtime of the best performing +variant at every scale. +Scaling performance for the various variants +with label propagation are similar to those of +PageRank. At least one variant is faster than +the serial baseline version for every graph and +at every scale, so the COST is again 1. +One notable difference between the PageR- +ank and label propagation implementations is +that PageRank sends the same volume of data +at every iteration, whereas label propagation +1 +2 +4 +8 +16 +32 +64 +128 +PEs +0 +1 +2 +3 +4 +5 +6 +Runtime (s) +soc-LiveJournal1 label prop +Serial +Charm +Charm Atomic +Charm Pairs +Charm Reduction +Charm Sortdest +Figure 6: Charm++ Label Prop. on soc-LiveJournal1 +1 +2 +4 +8 +16 +32 +64 +128 +PEs +0 +20 +40 +60 +80 +100 +120 +140 +Runtime (s) +twitter rv label prop +Serial +Charm +Charm Atomic +Charm Pairs +Charm Reduction +Charm Sortdest +Figure 7: Charm++ Label Prop. on twitter_rv +1 +2 +4 +8 +16 +32 +64 +128 +PEs +0 +200 +400 +600 +800 +1000 +1200 +Runtime (s) +uk-2007-05 label prop +Serial +Charm +Charm Atomic +Charm Pairs +Charm Reduction +Charm Sortdest +Figure 8: Charm++ Label Prop. on uk-2007-05 +8 + +COST of Graph Processing Using Actors +Graph +soc-LJ1 +twitter_rv +uk-2007-05 +Serial +1.05 +71.85 +83.59 +PEs +1 +0.67 +50.32 +43.83 +2 +0.72 +31.07 +33.45 +4 +0.80 +36.68 +21.56 +8 +0.69 +26.69 +13.81 +16 +0.45 +18.68 +10.24 +32 +0.33 +12.97 +7.15 +64 +0.45 +8.24 +5.69 +128 +0.25 +8.66 +6.11 +Table 3: Best Charm++ Label Propagation Runtime (s) +Across All Variants +sends a variable amount of data, only sending +updates on edges coming from a vertex that +has updated its label since the previous itera- +tion (does not apply to the reduction variant, +which continues sending a buffer of global ver- +tices). The serial implementation does not have +this optimization +The reduction variant is even slower at the +large scale here than it was for PageRank. +Structurally, the computation is not very differ- +ent, so the performance delta is likely due to +increased cache or memory pressure because +of the larger number of edges used for label +propagation. +V. +Conclusions +Using COST is a simple, practical way to com- +paratively assess the performance of concur- +rent systems. Too often, benchmarks are given +in isolation without providing a reasonable +baseline, highlighting specious scalability over +practical performance. +Developing performant, scalable software +is difficult and fraught with the complexities +of managing threads, messaging, synchroniza- +tion, scheduling, and more. The actor model +provides an elegant way to design concurrent +applications, ameliorating many of the tra- +ditional difficulties of parallel programming +without adding constraining restrictions or sac- +rificing performance. +It is a testament to the actor model that with +only minor changes to convert a simple se- +rial code into a message driven, actor-based, +concurrent implementation, we were able to +achieve scalability in parallel execution while +maintaining absolute performance, matching +the serial version when executing on a single +processor. Furthermore, our implementations +greatly outperformed a purportedly scalable +“big data” system, providing scalable perfor- +mance up to 128 PEs. +Optimizations that violated the semantics +of the pure actor model by using shared state +were helpful in some cases. However, empir- +ically, the most beneficial optimization, sort +destination, merely involved altering the order +and organization of private data within a chare +to improve cache performance, reduce load im- +balance, and shrink messages sizes by doing +local reductions. +VI. +Acknowledgements +This research used the Delta advanced comput- +ing and data resource which is supported by +the National Science Foundation (award OAC +2005572) and the State of Illinois. Delta is a +joint effort of the University of Illinois Urbana- +Champaign and its National Center for Super- +computing Applications. +9 + +COST of Graph Processing Using Actors +References +[1] +Frank McSherry, Michael Isard, and +Derek G. Murray. “Scalability! But at +What Cost?” In: Proceedings of the 15th +USENIX Conference on Hot Topics in Op- +erating Systems. HOTOS’15. Switzerland: +USENIX Association, 2015, p. 14. +[2] +Carl Hewitt, Peter Bishop, and Richard +Steiger. “A Universal Modular ACTOR +Formalism for Artificial Intelligence”. +In: Proceedings of the 3rd International +Joint Conference on Artificial Intelligence. +IJCAI’73. Stanford, USA: Morgan Kauf- +mann Publishers Inc., 1973, pp. 235–245. +[3] +Laxmikant Kale et al. The Charm++ Par- +allel Programming System. Aug. 2019. doi: +10.5281/zenodo.3370873. url: https: +//charm.cs.illinois.edu. +[4] +Laxmikant V. Kale and Gengbin Zheng. +“Chapter 1: The Charm++ Programming +Model”. In: Parallel Science and Engineer- +ing Applications: The Charm++ Approach. +Ed. by Laxmikant V. Kale and Abhi- +nav Bhatele. 1st. Boca Raton, FL, USA: +CRC Press, Inc., 2013. Chap. 1, pp. 1– +16. isbn: 1466504129, 9781466504127. doi: +10.1201/b16251. +[5] +Lawrence Page et al. The PageRank Cita- +tion Ranking: Bringing Order to the Web. +Technical Report 1999-66. Previous num- +ber = SIDL-WP-1999-0120. Stanford In- +foLab, Nov. 1999. url: http://ilpubs. +stanford.edu:8090/422/. +[6] +Da Yan et al. “Pregel Algorithms for +Graph Connectivity Problems with Per- +formance Guarantees”. In: Proc. VLDB +Endow. 7.14 (Oct. 2014), pp. 1821–1832. +issn: 2150-8097. doi: 10.14778/2733085. +2733089. url: https : / / doi . org / 10 . +14778/2733085.2733089. +[7] +Joseph E. Gonzalez et al. “GraphX: +Graph +Processing +in +a +Distributed +Dataflow Framework”. In: Proceedings +of the 11th USENIX Conference on Op- +erating Systems Design and Implementa- +tion. OSDI’14. Broomfield, CO: USENIX +Association, 2014, pp. 599–613. isbn: +9781931971164. +[8] +Jure Leskovec and Andrej Krevl. SNAP +Datasets: Stanford Large Network Dataset +Collection. http://snap.stanford.edu/ +data. June 2014. +[9] +Haewoon Kwak et al. “What is Twit- +ter, a social network or a news media?” +In: WWW ’10: Proceedings of the 19th in- +ternational conference on World wide web. +Raleigh, North Carolina, USA: ACM, +2010, pp. 591–600. isbn: 978-1-60558-799- +8. doi: http://doi.acm.org/10.1145/ +1772690.1772751. +[10] +Paolo Boldi and Sebastiano Vigna. “The +WebGraph Framework I: Compression +Techniques”. In: Proc. of the Thirteenth +International World Wide Web Conference +(WWW 2004). Manhattan, USA: ACM +Press, 2004, pp. 595–601. +[11] +Paolo Boldi et al. “Layered Label Prop- +agation: A MultiResolution Coordinate- +Free Ordering for Compressing Social +Networks”. In: Proceedings of the 20th in- +ternational conference on World Wide Web. +Ed. by Sadagopan Srinivasan et al. ACM +Press, 2011, pp. 587–596. +10 + diff --git a/pdAzT4oBgHgl3EQfb_x4/content/tmp_files/load_file.txt b/pdAzT4oBgHgl3EQfb_x4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb174cd30e0750b250a22da5fe6ee1af866f633f --- /dev/null +++ b/pdAzT4oBgHgl3EQfb_x4/content/tmp_files/load_file.txt @@ -0,0 +1,498 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf,len=497 +page_content='COST of Graph Processing Using Actors Ronak Buch University of Illinois at Urbana-Champaign rabuch2@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='edu Abstract Graph processing is an increasingly important domain of computer science, with applications in data and network analysis, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Target graphs in these applications are often large, leading to the creation of “big data” systems designed to provide the scalability needed to analyze these graphs using parallel processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' However, researchers have shown that while these systems often provide scalability, they also often introduce overheads that exceed the benefits they provide, sometimes lower absolute performance than even simple serial implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' This report studies the viability and performance of actor model to implement scalable concurrent programs to perform common graph computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' We show that relatively simple actor-based implementations outperform both dedicated graph processing systems and the benchmark serial implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Introduction Scalability is often heralded as the most im- portant property of a “big data” software sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In [1], McSherry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' show that this metric is frequently misleading when not cou- pled with performance comparisons against an appropriate baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Providing decreased runtime as core count increases is certainly a desirable property of a well-engineered parallel system, but, in the end, absolute performance is paramount from the perspective of a user, trumping scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' McSherry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' introduce a metric they term COST, or Configuration that Outperforms a Single Thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' This metric measures the amount of hardware resources required before a particular system beats the runtime of a rea- sonably designed single threaded solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In particular, they found that several parallel data processing systems from the literature were outperformed by a simple serial program for various common graph processing tasks, even when the parallel systems were given over 100x the computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' This analysis sug- gests that these big data systems achieve their scalability by parallelizing overhead that they themselves have introduced rather than provid- ing a useful speedup relative to a fair baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' While some problem domains have inher- ent algorithmic or dataset related impediments to achieving performance improvements with parallel execution, such issues are not present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Datasets are large, providing the scale to decompose across several processors, and algorithms are generally data parallel without fine-grained global synchronization or depen- dencies, only requiring interactions within a local neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Given this potential, well- engineered parallel implementations should be able to provide speedups over the serial ver- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Taking inspiration from the findings of the COST work, in this report, we test this sup- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' We implement parallel versions of two common graph algorithms using an actor- based framework and compare performance and programmabilty against a well-established existing parallel graph processing system and the baseline serial implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='01395v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='DC] 4 Jan 2023 COST of Graph Processing Using Actors II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Approach In this section, we introduce and describe the ideology and principles of the actor model and the specific framework used in our implemen- tations, Charm++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' We also discuss the graph computations performed by our implementa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Actor Model The actor model [2] provides a conceptual framework to reason about and design con- current computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In this model, the com- putation is decomposed into several entities called actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Each individual actor operates independently from the others, with its own private state that no other object can read or modify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Actors pass data and coordinate exe- cution via sending messages to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' No- tably, this messaging is entity-centric (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' an actor is specified as the destination of a mes- sage) as opposed to processor-centric, as seen in the bulk synchronous parallel model or mes- sage passing à la MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' When an actor receives a message, it gains ownership over the payload data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In response to a message, an actor can perform some computation, send additional messages, spawn new actors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' This encapsulation of state and restriction of interaction to only messages reduces the complexity of implementing concurrent appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Rather than having to think about the global state of the application or multi- ple non-deterministic threads of execution and their interaction with each other, the program- mer merely has to consider the received mes- sages and local state on a given individual actor when designing the logic of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Using actors improves the safety of parallel applications as compared to shared memory concurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Since state is private and cannot be concurrently accessed or altered, the actor model prevents memory access order race con- ditions and obviates the need for lock-based synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Programs expressed in the actor model can naturally be executed in parallel due to the inherently concurrent semantics of actor com- putation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Further, the messaging semantics allow execution to be fully distributed, since there is no need for shared state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1 Concrete Implementations Due to these properties, concrete implementa- tions of the actor model are popular in the field of distributed systems, with examples such as ActorFoundry, Akka, and Erlang, among oth- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' These implementations often exploit the properties of actor-based computation to add additional features such as fault tolerance and migratability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' However, while these languages and libraries provide the many benefits of the actor model to users, a common concern is that they are focused more on expressivity and programma- bility than on pure raw performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The need to do location management, message delivery, garbage collection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' adds overhead rela- tive to more primitive, lower-level forms of writing parallel programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Given that we are interested primarily in a performance compar- ison, it is important that our actor-based im- plementations utilize a platform designed with performance in mind, such as Charm++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='2 Charm++ Charm++ [3] [4] is an adaptive, asynchronous task-based parallel programming framework based on the actor model and focused on high performance computing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The core entities in a Charm++ program are ob- jects called chares, which are essentially actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Chares hold private state and generally com- municate with each other using messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The Charm++ runtime system automatically mea- sures the load of chares, and coupled with the message delivery semantics of actors, can use these measurements to migrate chares to auto- matically balance load between processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Both the runtime of Charm++ and applica- tions that use it are written in C++, meaning there is no garbage collection and the over- heads are that of a low-level systems program- ming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 2 COST of Graph Processing Using Actors While one may use Charm++ to develop applications that completely adhere to the se- mantics of the actor model, it allows users to break some of these constraints in the interest of performing performance optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' For example, users can create shared buffers be- tween chares to more efficiently share data as compared to messaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' These boundary vio- lations must be done carefully and can often lead to degraded performance if not done well, but they are often needed in the HPC domain where performance is sacrosanct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Additionally, Charm++ offers some program- ming niceties such as quiescence detection, au- tomatic message aggregation, and the ability to do bulk creation of actors in collections called “chare arrays”, which allow actors to also be referenced via an index rather than just an ad- dress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Computations We implement two common graph computa- tions, PageRank and connected component de- tection via label propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1 PageRank PageRank [5] is a method for ranking vertices in a directed graph by “link popularity.” Fa- mously, it was developed by the founders of Google and served as the original algorithm underpinning the results of their eponymous web search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' PageRank is an iterative algorithm that main- tains a rank for each vertex in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Dur- ing each iteration, the rank of a vertex is damp- ened by a given factor α, divided by its out- degree d, and sent to each of its outgoing neigh- bors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The new rank of a vertex is computed by adding all of the contributions it receives from its incoming neighbors to 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The original implementation from the COST paper is shown in Listing 1 and the Charm++ implementation in Listing 2 (declarations and initialization have been condensed or omitted for space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 1 fn PageRank20 (graph: GraphIter , alpha: f32) { 2 let mut a = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' [0 f32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='nodes ()];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 3 let mut b = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' [0 f32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='nodes ()];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 4 let mut d = vec!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' [0 f32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='nodes ()];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 5 6 graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='map_edges (|x, y| { d[x] += 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' });' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 7 for iter in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='.20 { 8 for i in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='. graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='nodes () { 9 b[i] = alpha * a[i] / d[i];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 10 a[i] = 1f32 - alpha;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 11 } 12 graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='map_edges (|x, y| { a[y] += b[x];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' }) 13 } 14 } Listing 1: Serial PageRank 1 // Driver function , only runs on 0th chare 2 void runpagerank (float alpha) { 3 for (int i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i < 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i++) { 4 // Call update on all chares in array 5 thisProxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='update(alpha);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 6 CkWaitQD ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' // Sleep until quiescence 7 // Call iterate on all chares in array 8 thisProxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='iterate ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 9 CkWaitQD ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 10 } 11 } 12 13 // Set up values for new iteration 14 void update(float alpha) { 15 for (int i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='size ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i++) { 16 b[i] = alpha * a[i] / d[i];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 17 a[i] = 1 - alpha;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 18 } 19 } 20 21 // Run PageRank iteration , send to neighbors 22 void iterate () { 23 vector >> outgoing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 24 auto edgeIt = edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='begin ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 25 for (int i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i < degs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='size ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i++) { 26 for (int j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' j < degs[i];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' j++) { 27 const auto dest = *edgeIt ++ 28 outgoing[CHUNKINDEX(dest)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' emplace_back (dest , b[i]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 29 } 30 } 31 for (int i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i < outgoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='size ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' i++) { 32 // Call addB on chare with index i 33 thisProxy[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' addB(outgoing[i]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 34 } 35 } 36 37 // Receive values from neighbor 38 void addB(vector > b_in) { 39 for (const auto& entry : b_in) { 40 const auto dest = entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='first;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 41 const auto value = entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='second;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 42 // base is first index on this chunk 43 a[dest - base] += value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 44 } 45 } Listing 2: Charm++ PageRank 3 COST of Graph Processing Using Actors ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='2 Connected Components A connected component of a graph is a con- nected subgraph that is not part of any larger connected subgraph, or, more formally, a sub- graph C of an undirected graph G such that every vertex in V(C) is reachable from all ver- tices in V(C) and not reachable from any vertex in V(G) − V(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' There are many known algorithms to find connected components, but here we use the la- bel propagation method [6] due to the suitability of its neighborhood communication pattern for distributed computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Label propagation is an iterative algorithm that maintains a label for each vertex in the graph, initially set to the unique index of the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In each iteration, the current label of each vertex is sent to each of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Upon receiving a candidate label from a neighbor, a vertex changes its la- bel to the candidate if the candidate is strictly less than its current label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' This process contin- ues until an iteration occurs where no vertex changes its label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' At termination, the label of each vertex is equal to the smallest index of the vertices in its component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Methodology We compare our implementations to two refer- ences, the same serial versions1 implemented in Rust as used in the original COST paper, and the implementations provided by GraphX [7], the graph processing component of the Apache Spark data analysis engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' GraphX was one of the “big data” systems compared in the COST paper, and it either approximately matched in performance or out- performed the other big data systems used in that comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Here, we take it as a representative for the landscape of data pro- cessing systems due to its popularity and past performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Additionally, on their website2, GraphX claims to have “Comparable perfor- mance to the fastest specialized graph process- ing systems.” 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='com/frankmcsherry/COST 2https://spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='org/graphx/ Name Vertices Edges soc-LiveJournal1 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 847,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 571 68,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 993,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 773 twitter_rv 61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 578,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 415 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 468,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 365,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 182 uk-2007-05 105,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 896,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 555 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 738,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 733,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 648 Table 1: Properties of Selected Graphs We evaluate the performance of the imple- mentations on three different real-world input graphs: soc-LiveJournal1 [8] - Friendship network graph of the social media service LiveJour- nal twitter_rv [9] - Following network graph of the social media service Twitter uk-2007-05 [10] [11] - Hyperlink graph of web pages in the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='uk domain The properties of these graphs are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' twitter_rv and uk-2007-05 are the same datasets used for evaluation in the original COST paper and widely used in the benchmarking of big data systems, and soc- LiveJournal1 is a smaller dataset added to ease testing during development of the new actor- based versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Note that all of the chosen graphs are di- rected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' For use with label propaga- tion, input graphs are converted to undirected versions by adding a reverse edge for each edge if it does not already exist in the edge set, meaning that the graphs have more edges than shown in Table 1 during label propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Experiments were conducted on the CPU partition of Delta3 at the National Center for Supercomputing Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Each node of Delta in the CPU partition has two AMD EPYC 7763 processors, with 128 cores across two sock- ets and 256 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Finally, note that all provided timings are only of the actual computation, the time taken to ingest the graph from storage and perform other upfront preparation and initialization are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 3https://delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='ncsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='edu/ 4 COST of Graph Processing Using Actors 1 2 4 8 16 32 64 128 PEs 0 50 100 150 200 250 300 Runtime (s) soc-LiveJournal1 PageRank Serial GraphX Figure 1: GraphX PageRank on soc-LiveJournal1 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Results & Discussion i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Serial and GraphX Baselines In order to establish a baseline for comparison, we first examine the GraphX and serial imple- mentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Figure 1 shows the performance of these baseline runs for PageRank with soc- LiveJournal1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' GraphX performance scales as the number of processors (PEs) increases, but even with its fastest configuration of 128 PEs, it takes 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='9 s for 20 iterations of PageRank, while the serial version takes merely 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='18 s on a single core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Figure 2 tells a similar story for label propa- gation: GraphX improves with scale, but even its fastest time of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='31 s at 64 PEs is over an order of magnitude slower than the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='05 s of the serial version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' These GraphX implementations use GraphX’s Pregel API, which claims to avoid excessive storage of intermediate results for iterative computations like these, but there appears to be a bug in practice, as even small graphs can cause out of memory conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' a small 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1 MB test graph running out of memory during GraphX label propagation given a 16 GB allocation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Running PageRank with GraphX on twit- ter_rv also resulted in an OOM crash;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' the graph is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='8 GB and GraphX was given 128 PEs and 180 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Running label propa- gation on twitter_rv resulted in a timeout after 1 2 4 8 16 32 64 128 PEs 0 50 100 150 200 Runtime (s) soc-LiveJournal1 label prop Serial GraphX Figure 2: GraphX Label Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' on soc-LiveJournal1 30 minutes of processing on 128 PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' We did not attempt runs with uk-2007-05 due to these failures and the larger size of that graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' GraphX was generally outperformed by the serial implementations in the original COST paper, but not to the extent seen in our ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' We are using a newer version of GraphX, which may account for some of the difference, but even so, the performance gulf is stark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' We implemented several different vari- ants of the GraphX client application in an attempt to improve the performance, but the results shown were the best we were able to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' These GraphX results correspond to a COST of ∞, as performance never matches the sin- gle threaded version, regardless of how much hardware we provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' We did not perform runs beyond 128 PEs, but scaling improve- ments appear to end at or before that point in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Charm++ Our basic Charm++ implementation assigns contiguous chunks of vertices to chares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Local vertices are stored in index order, and the des- tinations of outgoing edges are stored for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' During each iteration, a chare loops over its local vertices and their outgoing edges, performing any necessary preparation and ag- gregating outgoing message data in per-chare buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Messages are sent to their destination 5 COST of Graph Processing Using Actors chares after the conclusion of this loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Upon reception of one of these messages, the chare performs the required computation to apply the payload of the message to its local vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' To analyze their performance impact, we im- plemented several different variants with opti- mizations on top of this basic implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Some of these optimizations violate the seman- tics of the actor model, namely those of private state and only exchanging data via messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Atomic Passes data using a global vertex buffer updated concurrently via atomic operations instead of using messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Pairs Passes data using a collection of global buffers, one for each ordered pair of chares instead of using messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' No locks or atomics are needed, synchronized via a message telling consumer that the pro- ducer is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Reduction Passes data and computes results using a parallel reduction tree instead of using point to point messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Each chare contributes a buffer containing data for all vertices with the updates coming from its local vertices applied, the buffers are then reduced in parallel, and finally each chare does local updates using the correspond- ing portion of the reduced buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Sort Destination Reorders how edges are stored on a chare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' instead of ordering by and storing (local source, {destinations}), this orders and stores by (destination, {list of local sources}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' This orders local iter- ations by destination, meaning that mes- sages can be sent earlier than in the basic version: after the edges incident to a sin- gle destination chunk are done rather than waiting until all edges are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1 PageRank Figures 3, 4, and 5 show PageRank perfor- mance for the Charm++ variants and the se- rial implementation on soc-LiveJournal1, twit- ter_rv, and uk-2007-05, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Table 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='PEs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Runtime (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='soc-LiveJournal1 PageRank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Serial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Atomic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Pairs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Reduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Sortdest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Figure 3: Charm++ PageRank on soc-LiveJournal1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='PEs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Runtime (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='twitter rv PageRank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Serial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Atomic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Pairs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Reduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Sortdest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Figure 4: Charm++ PageRank on twitter_rv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='PEs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Runtime (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='uk-2007-05 PageRank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Serial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Atomic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Pairs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Reduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Charm Sortdest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Figure 5: Charm++ PageRank on uk-2007-05 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='COST of Graph Processing Using Actors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='soc-LJ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='twitter_rv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='uk-2007-05 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='Serial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='18 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='69 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='62 PEs 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='33 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='28 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='65 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='22 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='93 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='83 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='90 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='01 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='36 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='90 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='84 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='94 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='35 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='02 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='19 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='09 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='61 64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='13 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='11 128 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='08 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='56 Table 2: Best Charm++ PageRank Runtime (s) Across All Variants shows the runtime of the best performing vari- ant at every scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Significantly, the COST with all input graphs is 1, as the performance of the Charm++ vari- ants matches or exceeds the performance of the serial version on a single processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' However, the performance and scalability of the different variants varies greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The basic variant performs well on 1 PE, but then spikes in runtime when moving to 2 PEs, before regaining performance with scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' This is due to the allocation and serialization over- head of messaging, which diminishes in rela- tive importance as buffers become smaller as the computation is scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The atomic variant generally performs well at all scales and for all graphs, since it avoids the overheads of messaging and uses a highly performant technique to operate on shared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' While the use of shared state breaks a tenant of the actor model and does not work for distributed execution, from a programma- bilty perspective, using atomics is simpler and less error-prone than using locks while also being suitable for fine-grained parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The pairs variant works fairly well at the medium to large scale on twitter_rv but poorly on the other graphs and at smaller scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' As with the basic variant, this is likely due to allo- cation overheads from needing to dynamically manage multiple large buffers, possibly exacer- bated by NUMA effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Note that the buffers here are proportional in size to the number of edges, rather than the much smaller num- ber of vertices, as in the atomic variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' This variant shows one of the risks of using shared state: getting a pointer to it may be cheap, but managing it may be costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The reduction variant is very reasonable at small to medium scale, but is the worst per- forming variant for every graph at large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Its poor performance is due to two factors: load imbalance and memory utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In the other variants (with the exception of atomic, which has the same global syn- chronization), a chare can progressively proceed with applying updates as neigh- boring chares send data to it, whereas the reduction variant requires all chares to have finished their local loop, as the re- duction can only complete after all objects have contributed their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Each chare must allocate a buffer of size equal to the number of total vertices in the graph to contribute to the reduction since sparse contributions are not sup- ported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The other variants communicate using space proportional to the number of edges (with the exception of atomic, which uses a single global vertex buffer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' While the number of edges is larger than the number of vertices for all of the input graphs, it is only 14-35x the number of ver- tices, meaning the reduction variant will be using much more memory at 128 PEs, leading to allocation overhead and more cache evictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The sort destination variant is the best per- forming overall, due to several different rea- sons: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Better locality and more efficient memory utilization, as it only needs to maintain a single buffer for message data since it computes all of the outgoing data for a single destination chunk before moving onto the next chunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Similarly, on the receive side, the payload of the message 7 COST of Graph Processing Using Actors is arranged in the same order as the local vertex data are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Reduced load imbalance by sending mes- sages earlier than other variants, allowing otherwise idle chares to move onto the next phase of the iteration instead of wait- ing with no work to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Sending less data than other variants by locally reducing the data bound for an ex- ternal vertex before sending, which is also enabled by its access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' As an ex- ample, suppose two vertices A and B on a chare both have outgoing edges to an ex- ternal vertex C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In the basic variant, the up- date from A is added to a message buffer, and later the update from B is added to the same message buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' We could search this buffer to combine the two updates, but doing so is relatively costly, requiring either a linear search or maintaining some sorted or hash-based data structure and the associated memory and computational overheads, so we send the update from A and B separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' On the other hand, due to the arrangement of edges in the sort destination variant, we process all vertices with an outgoing edge to C consecutively, so combining the update from A with the update from B before sending is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='2 Label Propagation Figures 6, 7, and 8 show label propagation performance for the Charm++ variants and the serial implementation on soc-LiveJournal1, twitter_rv, and uk-2007-05, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Ta- ble 3 shows the runtime of the best performing variant at every scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Scaling performance for the various variants with label propagation are similar to those of PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' At least one variant is faster than the serial baseline version for every graph and at every scale, so the COST is again 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' One notable difference between the PageR- ank and label propagation implementations is that PageRank sends the same volume of data at every iteration, whereas label propagation 1 2 4 8 16 32 64 128 PEs 0 1 2 3 4 5 6 Runtime (s) soc-LiveJournal1 label prop Serial Charm Charm Atomic Charm Pairs Charm Reduction Charm Sortdest Figure 6: Charm++ Label Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' on soc-LiveJournal1 1 2 4 8 16 32 64 128 PEs 0 20 40 60 80 100 120 140 Runtime (s) twitter rv label prop Serial Charm Charm Atomic Charm Pairs Charm Reduction Charm Sortdest Figure 7: Charm++ Label Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' on twitter_rv 1 2 4 8 16 32 64 128 PEs 0 200 400 600 800 1000 1200 Runtime (s) uk-2007-05 label prop Serial Charm Charm Atomic Charm Pairs Charm Reduction Charm Sortdest Figure 8: Charm++ Label Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' on uk-2007-05 8 COST of Graph Processing Using Actors Graph soc-LJ1 twitter_rv uk-2007-05 Serial 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='05 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='85 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='59 PEs 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='67 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='32 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='83 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='72 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='07 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='45 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='80 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='68 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='56 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='69 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='69 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='81 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='45 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='68 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='24 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='33 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='97 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='15 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='45 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='69 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='66 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='11 Table 3: Best Charm++ Label Propagation Runtime (s) Across All Variants sends a variable amount of data, only sending updates on edges coming from a vertex that has updated its label since the previous itera- tion (does not apply to the reduction variant, which continues sending a buffer of global ver- tices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The serial implementation does not have this optimization The reduction variant is even slower at the large scale here than it was for PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Structurally, the computation is not very differ- ent, so the performance delta is likely due to increased cache or memory pressure because of the larger number of edges used for label propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Conclusions Using COST is a simple, practical way to com- paratively assess the performance of concur- rent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Too often, benchmarks are given in isolation without providing a reasonable baseline, highlighting specious scalability over practical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Developing performant, scalable software is difficult and fraught with the complexities of managing threads, messaging, synchroniza- tion, scheduling, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' The actor model provides an elegant way to design concurrent applications, ameliorating many of the tra- ditional difficulties of parallel programming without adding constraining restrictions or sac- rificing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' It is a testament to the actor model that with only minor changes to convert a simple se- rial code into a message driven, actor-based, concurrent implementation, we were able to achieve scalability in parallel execution while maintaining absolute performance, matching the serial version when executing on a single processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Furthermore, our implementations greatly outperformed a purportedly scalable “big data” system, providing scalable perfor- mance up to 128 PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Optimizations that violated the semantics of the pure actor model by using shared state were helpful in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' However, empir- ically, the most beneficial optimization, sort destination, merely involved altering the order and organization of private data within a chare to improve cache performance, reduce load im- balance, and shrink messages sizes by doing local reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Acknowledgements This research used the Delta advanced comput- ing and data resource which is supported by the National Science Foundation (award OAC 2005572) and the State of Illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Delta is a joint effort of the University of Illinois Urbana- Champaign and its National Center for Super- computing Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 9 COST of Graph Processing Using Actors References [1] Frank McSherry, Michael Isard, and Derek G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Murray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' “Scalability!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' But at What Cost?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In: Proceedings of the 15th USENIX Conference on Hot Topics in Op- erating Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' HOTOS’15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Switzerland: USENIX Association, 2015, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 14.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' [4] Laxmikant V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Kale and Gengbin Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' “Chapter 1: The Charm++ Programming Model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In: Parallel Science and Engineer- ing Applications: The Charm++ Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' by Laxmikant V.' metadata={'source': 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doi: http://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1145/ 1772690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content='1772751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' [10] Paolo Boldi and Sebastiano Vigna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' “The WebGraph Framework I: Compression Techniques”.' metadata={'source': 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Ordering for Compressing Social Networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' In: Proceedings of the 20th in- ternational conference on World Wide Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' by Sadagopan Srinivasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' ACM Press, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 587–596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfb_x4/content/2301.01395v1.pdf'} diff --git a/q9A0T4oBgHgl3EQfKv-o/content/tmp_files/2301.02109v1.pdf.txt b/q9A0T4oBgHgl3EQfKv-o/content/tmp_files/2301.02109v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f5575e5c965c99ce29e641348aeae97aca12ee54 --- /dev/null +++ b/q9A0T4oBgHgl3EQfKv-o/content/tmp_files/2301.02109v1.pdf.txt @@ -0,0 +1,690 @@ +arXiv:2301.02109v1 [gr-qc] 5 Jan 2023 +January 5th, 2023 +Theories of gravity with nonminimal matter-curvature +coupling and the de Sitter swampland conjectures +Orfeu Bertolami1,2, Cl´audio Gomes2 and Paulo M. S´a3,4 +1 Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias, Universidade do Porto, +Rua do Campo Alegre s/n, 4169-007 Porto, Portugal +2 Centro de F´ısica das Universidades do Minho e do Porto, Rua do Campo Alegre s/n, +4169-007 Porto, Portugal +3 Departamento de F´ısica, Faculdade de Ciˆencias e Tecnologia, Universidade do Algarve, +Campus de Gambelas, 8005-139 Faro, Portugal +4 Instituto de Astrof´ısica e Ciˆencias do Espa¸co, Faculdade de Ciˆencias, Universidade de +Lisboa, Campo Grande, 1749-016 Lisboa, Portugal +Abstract +We discuss, in the context of alternative theories of gravity with nonminimal coupling +between matter and curvature, if inflationary solutions driven by a single scalar field can +be reconciled with the swampland conjectures about the emergence of de Sitter solutions in +string theory. We find that the slow-roll conditions are incompatible with the swampland +conjectures for a fairly generic inflationary solution in such alternative theories of gravity. +E-mail addresses: orfeu.bertolami@fc.up.pt; claudio.gomes@fc.up.pt; pmsa@ualg.pt +1 + +1 +Introduction +Swampland conjectures have been proposed in order to distinguish consistent-looking low- +energy effective field theories that do not admit a suitable ultraviolet completion in string +theory — and, therefore, are said to be in the swampland — from those that lie in the string +theory landscape. This is particularly relevant as it is notoriously difficult to obtain inflation +from the fundamental fields that naturally arise in string theory. +This difficulty is somewhat surprising as in N = 1 supergravity — which, under certain +conditions, can be thought to be a low-energy limit of string theory — inflation can be rather +easily setup (see, for instance, Ref. [1]). In fact, alternative routes to obtain inflation in string +theory have been discussed, but they tend to be more involved (see, for instance, Ref. [2]). +It is relevant to point out that some phenomenologically viable string models, the ones with +an intermediate Grand Unified Theory energy scale, ask for a period of inflation for its full +implementation [3]. +The above-mentioned swampland conjectures are concretely a broad range of assumptions +about the conditions required to admit local gauge symmetries and at least one Planck mass +particle so to account for the weakness of gravity. One must also require that high-order terms +in the effective action do not admit superluminal propagation (see Ref. [4] for a review). To +our knowledge, there is no assumption, among this set of requirements, concerning the Strong +Equivalence Principle and implying that the gravity theory is necessarily General Relativity. +Thus, it is natural to ask if the swampland conjectures hold for alternative theories of gravity +in the context of which single-field inflation can take place. This is the case of gravity theories +with nonminimal coupling between matter and curvature [5] where inflationary solutions can +be found [6]. +In order to be more specific about the conditions to be met, let us review the swampland +conjectures relevant for our discussion. These conjectures impose some constraints on scalar +fields emerging at low energy, generically denoted by φ [7,8], namely: +∆φ +MP +< c1, +(1) +MP +|∂φV | +V +> c2, +(2) +where ∆φ is the range of variation of the field, MP ≡ MPl/ +√ +8π is the reduced Planck’s mass, +V (φ) is the scalar field potential, c1 and c2 are constants of order one, and we have used the +notation ∂φV ≡ ∂V/∂φ. It has been further argued that one should consider the more refined +condition [9–11] +M2 +P +∂2 +φφV +V +< −c3, +(3) +where c3 is also a constant of order one and ∂2 +φφV ≡ ∂2V/∂φ2. +Conditions given by Eqs. (2) and (3) can, in principle, be compared with the onset conditions +of single-field inflation which require that the parameters for the inflaton field [12] +ǫ = M2 +P +2 +�∂φV +V +�2 +(4) +2 + +and +η = M2 +P +∂2 +φφV +V +(5) +satisfy the slow-roll requirements ǫ ≪ 1 and |η| ≪ 1 at the onset of inflation, so that at the +end of inflation ǫ ∼ |η| ∼ 1. +These last requirements are consistent with constraints arising from the CMB data [12] (see +Ref. [13] for a detailed discussion), +ǫ < 0.0044 +(6) +and +η = −0.015 ± 0.006, +(7) +whose values, clearly, do not match the requirements on c2 and c3. +Actually, it can be shown that the incompatibility remains for whatever number of scalar +fields drives inflation provided their kinetic energy terms are canonical [13]. However, it is +possible to reconcile the swampland conjectures with observations in the context of warm +inflationary models [14, 15] in the regime of strong dissipation for one [16, 17] or more scalar +fields [13]. +In what follows we shall consider the situation in the context of a nonminimally coupled +matter-curvature gravity theory in a single-field inflationary setup to be specified below. Thus, +in the next section, we shall detail the alternative gravity theory in consideration and the +associated inflationary model. We shall see that despite the similarities between the slow-roll +parameters in the nonminimal coupled model and warm inflation, it is not possible, in the +context of the former, to satisfy the swampland conjectures. Finally, in section 3, we present +our conclusions. +2 +Theories of gravity with nonminimal matter-curvature +coupling +String theory itself does give origin to more complex gravitational theories than General Rela- +tivity. Effective models of string theory exhibit corrections to General Relativity that include, +for instance, high-order curvature terms and curvature terms coupled with derivatives of the +dilaton field (see, for instance, Refs. [18–22]). +However, independently from string and quantum gravity considerations, alternative theo- +ries of gravity are motivated as possible routes for addressing cosmological and astrophysical +phenomena, such as the accelerated expansion of the Universe and the flattening of the ro- +tation curves of galaxies, instead of resorting to dark energy and dark matter. Well studied +models include f(R) gravity [23,24], where the scalar curvature R in the Einstein-Hilbert ac- +tion is replaced by a more general function, f(R). A further possibility to generalize General +Relativity is to nonminimally couple matter and curvature, substituting the Einstein-Hilbert +action by a more general form involving two functions of curvature f1(R) and f2(R) [5]. The +function f1(R) has a role analogous to f(R) gravity theory, and the function f2(R) multiplies +3 + +the matter Lagrangian density giving rise to a nonminimal coupling between matter and ge- +ometry. This possibility has been extensively studied in the context of dark matter [25], dark +energy [26], inflation [6], energy density fluctuations [27], gravitational waves [28], and the +cosmic virial theorem [29]. This model has also been examined with the Newton-Schr¨odinger +approach [30,31]. +Analytic extensions at R = 0 of functions f1(R), f2(R) were also considered and constraints +to the resulting nonminimally coupled gravity model have been computed through perturbations +to the perihelion precession of Mercury’s orbit [32]. +It turns out that nonminimally coupled gravity modifies the gravitational attraction by +introducing both a fifth force of the Yukawa type and an extra force which depends on the +spatial gradient of the Ricci scalar R. While the Yukawa force is typical also of f(R) gravity, +the existence of the extra force is specific to nonminimally coupled gravity [5, 33], and it is +an effect of the nonminimal coupling that induces a non-vanishing covariant derivative of the +energy-momentum tensor. The arising Yukawa contribution can give origin to static solutions +even though in the absence of pressure [31]. The Yukawa contribution was also examined in the +context of experiments in deep ocean [34] and through the Cassini radiometric experiment [35]. +2.1 +Action, field equations and main features +In the present work we consider theories of gravity with an action functional of the form [5] +S = +� +d4x√−g +�M2 +P +2 f1(R) + f2(R)L +� +, +(8) +where fi(R) (with i = 1, 2) are functions of the Ricci scalar curvature R, L is the Lagrangian +density of matter, and g is the metric determinant. The Einstein-Hilbert action of General +Relativity is recovered by taking f1(R) = R and f2(R) = 1. +The variation of the action functional with respect to the metric gµν yields the field equations +� +F1 + 2F2L +M2 +P +� +Gµν = f2 +M2 +P +Tµν + ∆µν +� +F1 + 2F2L +M2 +P +� ++ 1 +2gµν +� +f1 − F1R − 2F2L +M2 +P +R +� +, +(9) +where Gµν is the Einstein tensor, Fi = ∂fi/∂R (i = 1, 2), and ∆µν ≡ ∇µ∇µ − gµν∇α∇α. +A relevant feature of nonminimally coupled gravity is that the energy-momentum tensor +of matter is not covariantly conserved. Indeed, applying the Bianchi identities to Eq. (9), one +obtains that +∇µTµν = F2 +f2 +(Lgµν − Tµν) ∇µR, +(10) +meaning that the nonminimal coupling cannot be “gauged away” by a convenient conformal +transformation, being thus a distinctive feature of the nonminimal model discussed here. +2.2 +Inflation in the nonminimally coupled theory +As widely discussed, within General Relativity the swampland conjectures and the slow-roll +conditions cannot be matched for single-field cold inflation (see, for instance, Refs. [36,37]). +4 + +Given that the incompatibility of the swampland conjectures with the observations has +been an object of critique from the authors of Ref. [38] and that multi-field inflationary models +show no contradiction with the CMB features [39], it was logical to ask whether the swampland +conjectures would hold for many fields. In fact, multi-field cosmological models open interesting +perspectives, for instance, for unification of dark matter and dark energy [40–43]. Two-field +inflationary models were first considered in the context of N = 1 supergravity [44] and their +dynamics was scrutinized in Refs. [45,46] for a broader class of models. In a broad context and +in string theory, two-field inflationary with different mass scales and an interaction term were +considered in Refs. [47–49]. In the context of the swampland conjectures, two-field inflationary +models were examined in Refs. [50, 51], where in Ref. [50] non-canonical kinetic energy terms +have been considered. More recently, it has been shown that multi-field inflation cannot be made +compatible with the swampland conjectures without a significant amount of dissipation [13]. +In this paper, we will consider single-field cold inflation within the nonminimally coupled +theory of gravity defined by the action functional (8). +Assuming the Friedmann-Lemaˆıtre-Robertson-Walker metric +ds2 = −dt2 + a2(t)dx2, +(11) +from Eqs. (9) and (10) we obtain +H2 = 1 +6F +�2f2ρ +M2 +P +− 6H ˙F − f1 + FR +� +, +(12) +− 2 +� +2 ˙H + 3H2� +F = 2f2p +M2 +P ++ 2 ¨F + 6H ˙F + f1 − FR, +(13) +˙ρ + 3H(ρ + p) = −F2 +f2 +(ρ + p) ˙R, +(14) +where a(t) is the scalar factor, H = ˙a/a is the Hubble parameter, an overdot denote a derivative +with respect to time t, and we have introduced the notation F ≡ F1 + 2F2p/M2 +P. +In the +above equations, we have also assumed that matter is represented by an homogeneous scalar +field φ with a Lagrangian density L = p, for which pressure and energy density are given by +p = ˙φ2/2 − V (φ) and ρ = ˙φ2/2 + V (φ), respectively. +In what follows, we consider theories for which the pure gravitational sector of the action +has the Einstein-Hilbert form; more specifically, we choose f1(R) = R, implying F1 = 1. +With this assumption, Eqs. (12) and (13) become +H2 = +1 +3(M4 +P − 4G2) +� +ρf2(M2 +P − G) − 3pf2G − 6H(M2 +P + 2G) ˙G − 6G ¨G +� +, +(15) +and +˙H = −f2(ρ + p) + 2 ¨G +2(M2 +P + 2G) , +(16) +where the notation G ≡ pF2 was introduced. +5 + +Let us now choose +f2(R) = 1 + α +� R +6M2 +P +�3 +, +(17) +where α is a positive dimensionless parameter that sets the scale of the nonminimal coupling, +which is not necessarily the Planck scale. The cubic choice is the simplest power-law type +function which renders non-trivial solutions for the Friedmann equation. In fact, for a linear +monomial no real solutions for that equation are found, and for the quadratic scenario the stan- +dard solution in General Relativity is surprisingly retrieved; as for cubic and higher monomials +the behaviour is similar among each choice, up to small numerical factors [6]. Furthermore, we +assume that inflation is quasi-exponential, i.e., V ≫ ˙φ2, implying ρ ≃ −p ≃ V . +Under these assumptions, and taking into account that +R = 6( ˙H + 2H2), +(18) +we obtain +f2 = 1 + α +M6 +P +� +8H6 + 12H4 ˙H + 6H2 ˙H2 + ˙H3� +, +(19) +G = α +M6 +P +� +2H4p + 2H2 ˙Hp + +˙H2 +2 p +� +, +(20) +˙G = +α +M6 +P +� +8H3 ˙Hp + 4H ˙H2p + 2H4 ˙p + 2H2 ˙H ˙p + +˙H2 +2 ˙p + 2H2 ¨Hp + ˙H ¨Hp +� +, +(21) +¨G = +α +M6 +P +� +24H2 ˙H2p + 4 ˙H3p + 16H3 ˙H ˙p + 8H ˙H2 ˙p + 8H3 ¨Hp + 12H ˙H ¨Hp + 4H2 ¨H ˙p ++ 2 ˙H ¨H ˙p + ¨H2p + 2H4¨p + 2H2 ˙H ¨p + +˙H2 +2 ¨p + 2H2H(3)p + ˙HH(3)p +� +, +(22) +where H(3) denotes the third derivative of H with respect to time t. +Equation (15) can now be written as +3 +� +M2 +P − 4αH4 +M6 +P +p +� � +M2 +P + 4αH4 +M6 +P +p +� +≃ ρ +� +1 + 8αH6 +M6 +P +� � +M2 +P − 2αH4 +M6 +P +p +� +− 3p2 +� +1 + 8αH6 +M6 +P +� 2αH4 +M6 +P +(23) +or, for M8 +P − 4αH4V ̸= 0, +4αV H6 + 3M8 +PH2 − M6 +PV ≃ 0, +(24) +where we have taken only the first two terms of f2 and the first term of G in Eqs. (19) and +(20), respectively; all the other terms in these equations, as well as the terms of ˙G and ¨G in +Eqs. (21) and (22), were neglected since they contain time derivatives of H and p. +For the sake of simplicity, let us now introduce the dimensionless variable +V = V +M4 +P +. +(25) +6 + +Then, equation (24) becomes +4αV H6 + 3M4 +PH2 − M6 +PV ≃ 0, +(26) +yielding the solution +H2 ≃ +−1 + +�� +αV +3 + +� +1 + αV +3�2/3 +2 +√ +αV +�� +αV +3 + +� +1 + αV +3�1/3M2 +P. +(27) +Taking into account that the energy scale of inflation, defined as Einf = V 1/4, is much +smaller than the reduced Planck mass, the right-hand side of Eq. (27) can be expanded in +power series of V ≪ 1, yielding +H2 ≃ M2 +P +3 V +� +1 − 4 +27αV +3� +, +(28) +where the first term on the right-hand side is the General Relativity term and the second one +is a correction due the presence of a nonminimal coupling between matter and curvature. +Let us now turn to Eq. (16). It can be written as +2 +� +M2 +P + 4αH4 +M6 +P +p +� +˙H ≃ − +� +1 + 8αH6 +M6 +P +� +(ρ + p) +(29) +or, equivalently, +˙H ≃ − +(M6 +P + 8αH6) ˙φ2 +2(M8 +P − 4αM4 +PH4V ), +(30) +where we have used ρ + p = ˙φ2 and taken only the first two terms of f2 and the first term of G +in Eqs. (19) and (20), respectively; all the other terms in these equations, as well as the terms +of ¨G in Eq. (22), were neglected since they contain time derivatives of H and p. +Using H2 given by Eq. (28) and expanding in power series of V , Eq. (30) yields +˙H ≃ − +˙φ2 +2M2 +P +� +1 + 20 +27αV +3� +, +(31) +where, again, the first term on the right-hand side is the General Relativity term and the second +is a correction due to the direct matter-curvature coupling. +Note that, if in Eq. (29) we had also taken for f2 the term proportional to ˙H and for ¨G the +term proportional to ˙H ˙p, we would have obtained in the right-hand side of Eq. (31) an extra +term proportional to ˙φ4V +2; however, since ˙φ2/M4 +P ≪ V , this extra term can be neglected in +comparison with the term proportional to ˙φ2V +3. +Now, taking the time derivative of Eq. (28) and using Eq. (31) to eliminate ˙H, we obtain +∂φV ≃ −3H ˙φ +� +1 + 20 +27αV +3� � +1 − 16 +27αV +3�−1 +(32) +7 + +or, expanding in power series of V , +∂φV ≃ −3H ˙φ +� +1 + 4 +3αV +3� +. +(33) +Taking a second time derivative, Eq. (33) becomes: +∂2 +φφV ≃ 3H2 +� +1 + 4 +3αV +3� � +− +˙H +H2 − +¨φ +˙φH +� +, +(34) +where we have neglected terms proportional to ˙φ2V +2. +Using Eqs. (28), (31), (33), and (34), the quantities ˙H/H2 and ¨φ/( ˙φH) can be expressed as +˙H +H2 ≃ ǫ +� +1 − 44 +27αV +3� +(35) +and +¨φ +˙φH +≃ ǫ +� +1 − 44 +27αV +3� +− η +� +1 − 32 +27αV +3� +, +(36) +where the slow-roll parameters ǫ and η are given by Eqs. (4) and (5). +Now, taking into account that in the slow-roll inflationary regime | ˙H|/H2 ≪ 1 and |¨φ/( ˙φH)| +≪ 1 we conclude that +ǫ ≪ 1 + 44 +27αV +3 +and +|η| ≪ 1 + 32 +27αV +3. +(37) +Since V ≡ V/M4 +P ≪ 1 and assuming for naturalness that α = O(1), we conclude that in the +nonminimally coupled theory of gravity under consideration, the slow-roll parameters satisfy +the conditions ǫ ≪ 1 and |η| ≪ 1. Because these parameters are related to the constants c2 +and c3 arising within the de Sitter swampland conjectures [see Eqs. (1) and (2)] through the +relations +c2 +2 < 2ǫ +and +c3 < |η|, +(38) +we arrive at the conclusion that c2 ≪ 1 and c3 ≪ 1 during a quasi-exponential inflationary +period. +Thus, we clearly see that the swampland conjectures cannot be met for inflation in the +context of theories of gravity with nonminimally coupled matter and curvature. +3 +Discussion and Conclusions +In this work, in the context of the nonminimally coupled matter-curvature theory of gravity, +we have considered the compatibility of the slow-roll conditions of inflation and the de Sitter +swampland conjectures. +8 + +Despite the specificities of the nonminimally coupled theory and the fact that it can lead +to an inflationary regime which differs from the one in General Relativity for the choice of the +f2(R)-function such as in Eq. (17), we find that under quite general conditions the requirements +for the inflaton potential are still very much controlled by the slow-roll conditions. 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Noori Gashti, “Two-Field Inflationary Model and Swampland de Sitter Conjecture”, +arxiv: 2111.06421 [gr-qc] (2021). +12 + diff --git a/q9A0T4oBgHgl3EQfKv-o/content/tmp_files/load_file.txt b/q9A0T4oBgHgl3EQfKv-o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b26d43294f7b292d068491458f81c037b3d69c7 --- /dev/null +++ b/q9A0T4oBgHgl3EQfKv-o/content/tmp_files/load_file.txt @@ -0,0 +1,509 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf,len=508 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='02109v1 [gr-qc] 5 Jan 2023 January 5th, 2023 Theories of gravity with nonminimal matter-curvature coupling and the de Sitter swampland conjectures Orfeu Bertolami1,2, Cl´audio Gomes2 and Paulo M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Campus de Gambelas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 8005-139 Faro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Portugal 4 Instituto de Astrof´ısica e Ciˆencias do Espa¸co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Faculdade de Ciˆencias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Universidade de Lisboa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Campo Grande,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 1749-016 Lisboa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Portugal Abstract We discuss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' in the context of alternative theories of gravity with nonminimal coupling between matter and curvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' if inflationary solutions driven by a single scalar field can be reconciled with the swampland conjectures about the emergence of de Sitter solutions in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' We find that the slow-roll conditions are incompatible with the swampland conjectures for a fairly generic inflationary solution in such alternative theories of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' E-mail addresses: orfeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='bertolami@fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' claudio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='gomes@fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' pmsa@ualg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='pt 1 1 Introduction Swampland conjectures have been proposed in order to distinguish consistent-looking low- energy effective field theories that do not admit a suitable ultraviolet completion in string theory — and, therefore, are said to be in the swampland — from those that lie in the string theory landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' This is particularly relevant as it is notoriously difficult to obtain inflation from the fundamental fields that naturally arise in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' This difficulty is somewhat surprising as in N = 1 supergravity — which, under certain conditions, can be thought to be a low-energy limit of string theory — inflation can be rather easily setup (see, for instance, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In fact, alternative routes to obtain inflation in string theory have been discussed, but they tend to be more involved (see, for instance, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' It is relevant to point out that some phenomenologically viable string models, the ones with an intermediate Grand Unified Theory energy scale, ask for a period of inflation for its full implementation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' The above-mentioned swampland conjectures are concretely a broad range of assumptions about the conditions required to admit local gauge symmetries and at least one Planck mass particle so to account for the weakness of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' One must also require that high-order terms in the effective action do not admit superluminal propagation (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [4] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' To our knowledge, there is no assumption, among this set of requirements, concerning the Strong Equivalence Principle and implying that the gravity theory is necessarily General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Thus, it is natural to ask if the swampland conjectures hold for alternative theories of gravity in the context of which single-field inflation can take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' This is the case of gravity theories with nonminimal coupling between matter and curvature [5] where inflationary solutions can be found [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In order to be more specific about the conditions to be met, let us review the swampland conjectures relevant for our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' These conjectures impose some constraints on scalar fields emerging at low energy, generically denoted by φ [7,8], namely: ∆φ MP < c1, (1) MP |∂φV | V > c2, (2) where ∆φ is the range of variation of the field, MP ≡ MPl/ √ 8π is the reduced Planck’s mass, V (φ) is the scalar field potential, c1 and c2 are constants of order one, and we have used the notation ∂φV ≡ ∂V/∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' It has been further argued that one should consider the more refined condition [9–11] M2 P ∂2 φφV V < −c3, (3) where c3 is also a constant of order one and ∂2 φφV ≡ ∂2V/∂φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Conditions given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (2) and (3) can, in principle, be compared with the onset conditions of single-field inflation which require that the parameters for the inflaton field [12] ǫ = M2 P 2 �∂φV V �2 (4) 2 and η = M2 P ∂2 φφV V (5) satisfy the slow-roll requirements ǫ ≪ 1 and |η| ≪ 1 at the onset of inflation, so that at the end of inflation ǫ ∼ |η| ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' These last requirements are consistent with constraints arising from the CMB data [12] (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [13] for a detailed discussion), ǫ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='0044 (6) and η = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='015 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='006, (7) whose values, clearly, do not match the requirements on c2 and c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Actually, it can be shown that the incompatibility remains for whatever number of scalar fields drives inflation provided their kinetic energy terms are canonical [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' However, it is possible to reconcile the swampland conjectures with observations in the context of warm inflationary models [14, 15] in the regime of strong dissipation for one [16, 17] or more scalar fields [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In what follows we shall consider the situation in the context of a nonminimally coupled matter-curvature gravity theory in a single-field inflationary setup to be specified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Thus, in the next section, we shall detail the alternative gravity theory in consideration and the associated inflationary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' We shall see that despite the similarities between the slow-roll parameters in the nonminimal coupled model and warm inflation, it is not possible, in the context of the former, to satisfy the swampland conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Finally, in section 3, we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 2 Theories of gravity with nonminimal matter-curvature coupling String theory itself does give origin to more complex gravitational theories than General Rela- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Effective models of string theory exhibit corrections to General Relativity that include, for instance, high-order curvature terms and curvature terms coupled with derivatives of the dilaton field (see, for instance, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [18–22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' However, independently from string and quantum gravity considerations, alternative theo- ries of gravity are motivated as possible routes for addressing cosmological and astrophysical phenomena, such as the accelerated expansion of the Universe and the flattening of the ro- tation curves of galaxies, instead of resorting to dark energy and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Well studied models include f(R) gravity [23,24], where the scalar curvature R in the Einstein-Hilbert ac- tion is replaced by a more general function, f(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' A further possibility to generalize General Relativity is to nonminimally couple matter and curvature, substituting the Einstein-Hilbert action by a more general form involving two functions of curvature f1(R) and f2(R) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' The function f1(R) has a role analogous to f(R) gravity theory, and the function f2(R) multiplies 3 the matter Lagrangian density giving rise to a nonminimal coupling between matter and ge- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' This possibility has been extensively studied in the context of dark matter [25], dark energy [26], inflation [6], energy density fluctuations [27], gravitational waves [28], and the cosmic virial theorem [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' This model has also been examined with the Newton-Schr¨odinger approach [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Analytic extensions at R = 0 of functions f1(R), f2(R) were also considered and constraints to the resulting nonminimally coupled gravity model have been computed through perturbations to the perihelion precession of Mercury’s orbit [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' It turns out that nonminimally coupled gravity modifies the gravitational attraction by introducing both a fifth force of the Yukawa type and an extra force which depends on the spatial gradient of the Ricci scalar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' While the Yukawa force is typical also of f(R) gravity, the existence of the extra force is specific to nonminimally coupled gravity [5, 33], and it is an effect of the nonminimal coupling that induces a non-vanishing covariant derivative of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' The arising Yukawa contribution can give origin to static solutions even though in the absence of pressure [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' The Yukawa contribution was also examined in the context of experiments in deep ocean [34] and through the Cassini radiometric experiment [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='1 Action, field equations and main features In the present work we consider theories of gravity with an action functional of the form [5] S = � d4x√−g �M2 P 2 f1(R) + f2(R)L � , (8) where fi(R) (with i = 1, 2) are functions of the Ricci scalar curvature R, L is the Lagrangian density of matter, and g is the metric determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' The Einstein-Hilbert action of General Relativity is recovered by taking f1(R) = R and f2(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' The variation of the action functional with respect to the metric gµν yields the field equations � F1 + 2F2L M2 P � Gµν = f2 M2 P Tµν + ∆µν � F1 + 2F2L M2 P � + 1 2gµν � f1 − F1R − 2F2L M2 P R � , (9) where Gµν is the Einstein tensor, Fi = ∂fi/∂R (i = 1, 2), and ∆µν ≡ ∇µ∇µ − gµν∇α∇α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' A relevant feature of nonminimally coupled gravity is that the energy-momentum tensor of matter is not covariantly conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Indeed, applying the Bianchi identities to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (9), one obtains that ∇µTµν = F2 f2 (Lgµν − Tµν) ∇µR, (10) meaning that the nonminimal coupling cannot be “gauged away” by a convenient conformal transformation, being thus a distinctive feature of the nonminimal model discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='2 Inflation in the nonminimally coupled theory As widely discussed, within General Relativity the swampland conjectures and the slow-roll conditions cannot be matched for single-field cold inflation (see, for instance, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [36,37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 4 Given that the incompatibility of the swampland conjectures with the observations has been an object of critique from the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [38] and that multi-field inflationary models show no contradiction with the CMB features [39], it was logical to ask whether the swampland conjectures would hold for many fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In fact, multi-field cosmological models open interesting perspectives, for instance, for unification of dark matter and dark energy [40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Two-field inflationary models were first considered in the context of N = 1 supergravity [44] and their dynamics was scrutinized in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [45,46] for a broader class of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In a broad context and in string theory, two-field inflationary with different mass scales and an interaction term were considered in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In the context of the swampland conjectures, two-field inflationary models were examined in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [50, 51], where in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' [50] non-canonical kinetic energy terms have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' More recently, it has been shown that multi-field inflation cannot be made compatible with the swampland conjectures without a significant amount of dissipation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In this paper, we will consider single-field cold inflation within the nonminimally coupled theory of gravity defined by the action functional (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Assuming the Friedmann-Lemaˆıtre-Robertson-Walker metric ds2 = −dt2 + a2(t)dx2, (11) from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (9) and (10) we obtain H2 = 1 6F �2f2ρ M2 P − 6H ˙F − f1 + FR � , (12) − 2 � 2 ˙H + 3H2� F = 2f2p M2 P + 2 ¨F + 6H ˙F + f1 − FR, (13) ˙ρ + 3H(ρ + p) = −F2 f2 (ρ + p) ˙R, (14) where a(t) is the scalar factor, H = ˙a/a is the Hubble parameter, an overdot denote a derivative with respect to time t, and we have introduced the notation F ≡ F1 + 2F2p/M2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In the above equations, we have also assumed that matter is represented by an homogeneous scalar field φ with a Lagrangian density L = p, for which pressure and energy density are given by p = ˙φ2/2 − V (φ) and ρ = ˙φ2/2 + V (φ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In what follows, we consider theories for which the pure gravitational sector of the action has the Einstein-Hilbert form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' more specifically, we choose f1(R) = R, implying F1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' With this assumption, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (12) and (13) become H2 = 1 3(M4 P − 4G2) � ρf2(M2 P − G) − 3pf2G − 6H(M2 P + 2G) ˙G − 6G ¨G � , (15) and ˙H = −f2(ρ + p) + 2 ¨G 2(M2 P + 2G) , (16) where the notation G ≡ pF2 was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 5 Let us now choose f2(R) = 1 + α � R 6M2 P �3 , (17) where α is a positive dimensionless parameter that sets the scale of the nonminimal coupling, which is not necessarily the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' The cubic choice is the simplest power-law type function which renders non-trivial solutions for the Friedmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' In fact, for a linear monomial no real solutions for that equation are found, and for the quadratic scenario the stan- dard solution in General Relativity is surprisingly retrieved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' as for cubic and higher monomials the behaviour is similar among each choice, up to small numerical factors [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Furthermore, we assume that inflation is quasi-exponential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=', V ≫ ˙φ2, implying ρ ≃ −p ≃ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Under these assumptions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' and taking into account that R = 6( ˙H + 2H2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (18) we obtain f2 = 1 + α M6 P � 8H6 + 12H4 ˙H + 6H2 ˙H2 + ˙H3� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (19) G = α M6 P � 2H4p + 2H2 ˙Hp + ˙H2 2 p � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (20) ˙G = α M6 P � 8H3 ˙Hp + 4H ˙H2p + 2H4 ˙p + 2H2 ˙H ˙p + ˙H2 2 ˙p + 2H2 ¨Hp + ˙H ¨Hp � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (21) ¨G = α M6 P � 24H2 ˙H2p + 4 ˙H3p + 16H3 ˙H ˙p + 8H ˙H2 ˙p + 8H3 ¨Hp + 12H ˙H ¨Hp + 4H2 ¨H ˙p + 2 ˙H ¨H ˙p + ¨H2p + 2H4¨p + 2H2 ˙H ¨p + ˙H2 2 ¨p + 2H2H(3)p + ˙HH(3)p � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (22) where H(3) denotes the third derivative of H with respect to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Equation (15) can now be written as 3 � M2 P − 4αH4 M6 P p � � M2 P + 4αH4 M6 P p � ≃ ρ � 1 + 8αH6 M6 P � � M2 P − 2αH4 M6 P p � − 3p2 � 1 + 8αH6 M6 P � 2αH4 M6 P (23) or, for M8 P − 4αH4V ̸= 0, 4αV H6 + 3M8 PH2 − M6 PV ≃ 0, (24) where we have taken only the first two terms of f2 and the first term of G in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (19) and (20), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' all the other terms in these equations, as well as the terms of ˙G and ¨G in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (21) and (22), were neglected since they contain time derivatives of H and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' For the sake of simplicity, let us now introduce the dimensionless variable V = V M4 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (25) 6 Then, equation (24) becomes 4αV H6 + 3M4 PH2 − M6 PV ≃ 0, (26) yielding the solution H2 ≃ −1 + �� αV 3 + � 1 + αV 3�2/3 2 √ αV �� αV 3 + � 1 + αV 3�1/3M2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (27) Taking into account that the energy scale of inflation, defined as Einf = V 1/4, is much smaller than the reduced Planck mass, the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (27) can be expanded in power series of V ≪ 1, yielding H2 ≃ M2 P 3 V � 1 − 4 27αV 3� , (28) where the first term on the right-hand side is the General Relativity term and the second one is a correction due the presence of a nonminimal coupling between matter and curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Let us now turn to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' It can be written as 2 � M2 P + 4αH4 M6 P p � ˙H ≃ − � 1 + 8αH6 M6 P � (ρ + p) (29) or, equivalently, ˙H ≃ − (M6 P + 8αH6) ˙φ2 2(M8 P − 4αM4 PH4V ), (30) where we have used ρ + p = ˙φ2 and taken only the first two terms of f2 and the first term of G in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (19) and (20), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' all the other terms in these equations, as well as the terms of ¨G in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (22), were neglected since they contain time derivatives of H and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Using H2 given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (28) and expanding in power series of V , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (30) yields ˙H ≃ − ˙φ2 2M2 P � 1 + 20 27αV 3� , (31) where, again, the first term on the right-hand side is the General Relativity term and the second is a correction due to the direct matter-curvature coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Note that, if in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (29) we had also taken for f2 the term proportional to ˙H and for ¨G the term proportional to ˙H ˙p, we would have obtained in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (31) an extra term proportional to ˙φ4V 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' however, since ˙φ2/M4 P ≪ V , this extra term can be neglected in comparison with the term proportional to ˙φ2V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Now, taking the time derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (28) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (31) to eliminate ˙H, we obtain ∂φV ≃ −3H ˙φ � 1 + 20 27αV 3� � 1 − 16 27αV 3�−1 (32) 7 or, expanding in power series of V , ∂φV ≃ −3H ˙φ � 1 + 4 3αV 3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (33) Taking a second time derivative, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (33) becomes: ∂2 φφV ≃ 3H2 � 1 + 4 3αV 3� � − ˙H H2 − ¨φ ˙φH � , (34) where we have neglected terms proportional to ˙φ2V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (28), (31), (33), and (34), the quantities ˙H/H2 and ¨φ/( ˙φH) can be expressed as ˙H H2 ≃ ǫ � 1 − 44 27αV 3� (35) and ¨φ ˙φH ≃ ǫ � 1 − 44 27αV 3� − η � 1 − 32 27αV 3� , (36) where the slow-roll parameters ǫ and η are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Now, taking into account that in the slow-roll inflationary regime | ˙H|/H2 ≪ 1 and |¨φ/( ˙φH)| ≪ 1 we conclude that ǫ ≪ 1 + 44 27αV 3 and |η| ≪ 1 + 32 27αV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (37) Since V ≡ V/M4 P ≪ 1 and assuming for naturalness that α = O(1), we conclude that in the nonminimally coupled theory of gravity under consideration, the slow-roll parameters satisfy the conditions ǫ ≪ 1 and |η| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Because these parameters are related to the constants c2 and c3 arising within the de Sitter swampland conjectures [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (1) and (2)] through the relations c2 2 < 2ǫ and c3 < |η|, (38) we arrive at the conclusion that c2 ≪ 1 and c3 ≪ 1 during a quasi-exponential inflationary period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Thus, we clearly see that the swampland conjectures cannot be met for inflation in the context of theories of gravity with nonminimally coupled matter and curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 3 Discussion and Conclusions In this work, in the context of the nonminimally coupled matter-curvature theory of gravity, we have considered the compatibility of the slow-roll conditions of inflation and the de Sitter swampland conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' 8 Despite the specificities of the nonminimally coupled theory and the fact that it can lead to an inflationary regime which differs from the one in General Relativity for the choice of the f2(R)-function such as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (17), we find that under quite general conditions the requirements for the inflaton potential are still very much controlled by the slow-roll conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Even though it is conceivable that the free parameter α introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' (17), which sets the impact of the nonminimal coupling, could be greater than one, it cannot overcome the typical scale of the inflaton potential and its smallness in comparison with the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Of course, for naturalness reasons, we assume that α = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Thus, we conclude that the de Sitter swampland conditions cannot be met in the context of gravity theories with nonminimal coupling between matter and curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' We expect these conclusions to hold for any number of inflation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' Acknowledgments PMS acknowledges support from Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portugal) through the research grants UIDB/04434/2020 and UIDP/04434/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' OB and CG acknowledge sup- port from Funda¸c˜ao para a Ciˆencia e a Tecnologia (Portugal) through the research project CERN/FIS-PAR/0027/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9A0T4oBgHgl3EQfKv-o/content/2301.02109v1.pdf'} +page_content=' References [1] J.' metadata={'source': 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0000000000000000000000000000000000000000..e0f9bacab514f7cc7ed7278c565e60ee72a2987a --- /dev/null +++ b/qdE2T4oBgHgl3EQfKwYZ/content/tmp_files/2301.03705v1.pdf.txt @@ -0,0 +1,3314 @@ +arXiv:2301.03705v1 [stat.ME] 9 Jan 2023 +Locally sparse quantile estimation for a partially functional interaction model +Weijuan Lianga, Qingzhao Zhangb,∗, Shuangge Mac,∗ +aSchool of Statistics, Renmin University of China, Beijing, China +bDepartment of Statistics and Data Science, School of Economics, The Wang Yanan Institute for Studies in Economics, and Fujian Key Lab of +Statistics, Xiamen University, Xiamen, China +cDepartment of Biostatistics, Yale School of Public Health, New Haven, Connecticut, USA +Abstract +Functional data analysis has been extensively conducted. In this study, we consider a partially functional model, +under which some covariates are scalars and have linear effects, while some other variables are functional and have +unspecified nonlinear effects. Significantly advancing from the existing literature, we consider a model with inter- +actions between the functional and scalar covariates. To accommodate long-tailed error distributions which are not +uncommon in data analysis, we adopt the quantile technique for estimation. To achieve more interpretable estimation, +and to accommodate many practical settings, we assume that the functional covariate effects are locally sparse (that +is, there exist subregions on which the effects are exactly zero), which naturally leads to a variable/model selection +problem. We propose respecting the “main effect, interaction” hierarchy, which postulates that if a subregion has a +nonzero effect in an interaction term, then its effect has to be nonzero in the corresponding main functional effect. +For estimation, identification of local sparsity, and respect of the hierarchy, we propose a penalization approach. An +effective computational algorithm is developed, and the consistency properties are rigorously established under mild +regularity conditions. Simulation shows the practical effectiveness of the proposed approach. The analysis of the +Tecator data further demonstrates its practical applicability. Overall, this study can deliver a novel and practically +useful model and a statistically and numerically satisfactory estimation approach. +Keywords: Partially functional model, interaction analysis, locally sparse estimation, robust estimation +1. Introduction +Functional data analysis has become routine in statistics. A popular regression setting has a scalar response and +functional covariates. In practice, we may directly observe the functional covariates or their realizations at discrete +observational (usually time or space) points. In the latter case, estimation of the functional covariates may be first +needed. For this regression setting, there have been extensive methodological, computational, and theoretical devel- +opments as well as data analyses [1–4]. In particular, both mean and robust estimations have been developed [5–7]. +As a natural extension of the aforementioned model, in a partially functional model, there are two types of co- +variates. The first type of covariates is functional, as described above. In addition, there are also scalar covariates +with linear effects. Such a model shares some similar spirit with the partially linear regression [8] but may be more +complicated in multiple aspects. As a “natural next step”, we further consider the model with interactions between the +functional and scalar covariates. Interaction is a “basic” concept in data analysis. However, most of the existing inter- +action analyses are limited to parametric covariate effects. In the literature, there are a handful of studies that examine +interactions in the partially linear models [9, 10], and statistical and computational analysis of such interactions has +been shown to be highly nontrivial. To the best of our knowledge, there has been no interaction analysis with partially +functional models that consist of two distinct types of covariate effects. +∗Corresponding author +Email addresses: weijuanliang@yeah.net (Weijuan Liang), qzzhang@xmu.edu.cn (Qingzhao Zhang), shuangge.ma@yale.edu +(Shuangge Ma) +Preprint submitted to Elsevier +January 11, 2023 + +For the estimation of functional models, both mean and quantile regression methods have been developed, accom- +modating “regular” and long-tailed error distributions. In this article, we consider data with long-tailed errors and +quantile estimation, which can be technically more challenging than mean estimation. In the existing (both quantile +and mean regression) studies, it is commonly assumed that the functional covariate effects are smooth. Without addi- +tional assumptions/constraints, the estimates are nonzero everywhere. In the past few years, there has been a strong +advocacy on locally sparse estimation. Under such an estimation, there exist continual subregions, on which the +estimates are exactly zero. In terms of both concept and statistical techniques, this has a strong tie with the sparse esti- +mation for parametric covariate effects. It has been argued that sparse estimation in general can be more interpretable +and more reliable. Sparse estimation is “naturally equivalent to” variable/model selection, for which regularization +especially penalization techniques have been extensively developed in the past decades. Examples of penalized sparse +estimation for functionals include [11–13]. +If there are no interactions in the model, conceptually, some of the existing penalized sparse methods for function- +als can be adapted to the partially functional models, although we note that there has been very limited research in this +aspect [14–16]. When interactions are present, however, these methods may lead to a violation of the “main effect, +interaction” variable selection hierarchy. This hierarchy has been strongly stressed in the recent parametric interaction +analysis studies. Under this hierarchy, if an interaction effect is identified, then one or both of the corresponding main +effects have to be identified, corresponding to the weak and strong hierarchy, respectively. It has been argued that +in interaction analysis, this hierarchy is statistically sensible and necessary. For the specific model we are interested +in, this hierarchy means that, for any specific subregion, if a functional effect is nonzero in an interaction term, then +the corresponding main functional effect must be nonzero in this subregion. This brings additional constraints and +complexity to estimation. To the best of our knowledge, there is no existing estimation technique that can respect this +hierarchy in estimation for our proposed model. +This study may complement and advance the existing literature in multiple important ways. First, a novel model +is developed, which can accommodate not only two distinct types of covariate effects but more importantly their +interactions. Such extensions are natural and strongly motivated by practical data analysis. This model includes +multiple existing models as special cases. Second, we consider quantile estimation, which is also motivated by +many practical data settings and can be more challenging than mean estimation. It is noted that the proposed model +and penalized estimation can also be coupled with mean squares loss function. Third, locally sparse estimation is +conducted, which can lead to more interpretable and more reliable results than those without sparsity. Fourth, as a +major advancement, we develop an estimation approach that respects the “main effect, interaction” hierarchy, making +this study more aligned with parametric interaction analysis. Last but not least, this study delivers a useful tool for data +considered in Section 4 and those alike. Overall, with the significant statistical developments and strong application +potential, this study is warranted beyond the existing literature. +The rest of the article is organized as follows. In Section 2, we first describe the data setting, proposed model, and +estimation approach. An effective computational algorithm is developed, and statistical properties are then rigorously +established. Practical performance of the proposed approach is examined using simulation (Section 3) and data +analysis (Section 4). The article concludes with brief discussions in Section 5. Additional theoretical developments +and numerical results are presented in the Appendix and Supplemental Materials. +2. Methods +2.1. Data and model settings +Consider a random sample of size n: {Xi(t), zi, yi}n +i=1, where Xi(t) is a functional covariate, zi = (zi1, · · · , ziq)⊤ is a +q-dimensional vector of scalar covariates, and yi is a scalar response. The proposed model, estimation approach, and +statistical and computational properties can be easily extended to data with multiple functional covariates. Assume +that Xi(t), i = 1, · · · , n are independent realizations of an unknown smooth and square-integrable function X(t) on the +domain [0, T]. Without loss of generality, assume that the functional covariate, scalar covariates, and scalar response +have been centered to mean zero. +Consider the partially functional interaction model: +yi = +� T +0 +Xi(t)β∗ +0(t)dt + +q +� +k=1 +zik +� T +0 +Xi(t)β∗ +k(t)dt + +q +� +k=1 +zikγ∗ +k + ǫi, +(1) +2 + +where β∗ +k(t)’s for k = 0, 1, · · · , q are smooth and square-integrable coefficient functions, γ∗ +k’s are scalar coefficients of +zi, and the error terms ǫi’s are independent of (Xi(t), zi) and satisfy Pr(ǫi ≤ 0|Xi(t), zi) = τ for τ ∈ (0, 1). Note that this +assumption accommodates long-tailed error distributions. +As described above, we consider the setting with local sparsity. Take β∗ +0(t) as an example. We say that β∗ +0(t) is +locally sparse if there exists a subregion I ⊂ [0, T], and β∗ +0(t) = 0 for all t ∈ I. Accordingly, X(t) has no contribution +to the response for t ∈ I. Here, we note that there can be more than one region with zero effects, and the region +location information is not known a priori. In addition, the proposed approach is flexible enough to also accommodate +the case with functional covariate effects being nonzero everywhere. Local sparsity can assist in distinguishing regions +with and without effects. And it is easy to see the natural connection with variable selection for parametric models. +The proposed model is more complicated than some existing alternatives as local sparsity may apply to both the +main effect and interactions. With the connection between local sparsity and variable selection, we naturally encounter +the “main effect, interaction” variable selection hierarchy. In our analysis, we are not interested in the sparsity in γ∗ +k’s +(although we note that extending to accommodate potential sparsity in γ∗ +k’s is relatively easy with the parametric +nature). As such, the hierarchy boils down to the relationship between the subregions of β∗ +k’s (k = 1, . . ., q) with +zero/nonzero effects and those of β∗ +0. More specifically, we say that the hierarchy is satisfied if and only if for any +subregion I ⊂ [0, T], if β∗ +0(t) = 0 for all t ∈ I, then β∗ +k(t) = 0. We note that a more rigorous definition should rule out +any measure zero set. +2.2. Estimation +For estimating the unknown parameters, we propose minimizing the penalized objective function: +Q(β(t), γ) =1 +n +n +� +i=1 +ρτ +yi − +� T +0 +Xi(t)β0(t)dt − +q +� +k=1 +zik +� T +0 +Xi(t)βk(t)dt − +q +� +k=1 +zikγk + ++ +q +� +k=1 +κ +T +� T +0 +pλ1(|βk(t)|)dt + κ +T +� T +0 +pλ2(∥β(t)∥2)dt + η +q +� +k=0 +� T +0 +β′′2 +k (t)dt, +(2) +where ρτ(u) = u(τ−I(u < 0)) is the quantile loss function, ∥β(t)∥2 = (�q +k=0 β2 +k(t))1/2, pλj(·)’s are penalty functions with +tuning parameters λ j’s (for j = 1, 2), κ is a modifier and will be discussed below, η is a tuning parameter, and β′′ +k (t) +is the second-order derivative of βk(t) with respect to t. Various penalty functions can be adopted here, and pλ1(·) and +pλ2(·) do not need to be the same. In our theoretical and numerical developments, we adopt MCP [17] for both pλ1(·) +and pλ2(·), where pλj(t) = λ j +� |t| +0 +� +1 − +x +λjξ +� ++ dt, λ j ≥ 0, and ξ > 0 is a regularization parameter. It is expected that, with +SCAD and some other penalties, properties will be similar. +In (2), the first term is a “standard” lack-of-fit based on the quantile technique. Under the smoothness assumption, +the last penalty on derivative has been routinely adopted. Here we note that a stronger smoothness assumption and +correspondingly a higher order derivative can also be adopted. The most significant and innovative advancement is the +first and second penalty terms. In “ordinary” locally sparse estimation, penalties similar to +� T +0 pλ1(|βk(t)|)dt have been +adopted [12]. In our estimation, new challenges are brought by the hierarchy. Motivated by the sparse group penaliza- +tion for parametric models [18], we treat (β0, β1, . . ., βq) as a “group”. For a subregion, κ +T +� T +0 pλ2(∥β(t)∥2)dt determines +whether this group of functionals has overall zero effect. If not, then �q +k=1 +κ +T +� T +0 pλ1(|βk(t)|)dt determines which of the +q interaction effects are nonzero. Note that here no penalty is applied to β0(t), ensuring that the corresponding estimate +is nonzero, and hence the hierarchy is guaranteed. +Directly optimizing (2) is challenging with the infinite dimension of the unknown functionals. Here we adopt a +popular B-spline expansion-based technique, which can be preferred with its compact support property, computational +efficiency, and satisfactory performance with capturing local sparsity. Denote HdMn as the linear space spanned by +a set of order d + 1 B-spline basis functions B1(t), · · · , BMn+d(t), each with Mn + 1 equally spaced knots 0 = t0 < +t1 < · · · < tMn = T in the domain [0, T]. In (2), second-order derivatives are taken, corresponding to d = 2. We +refer to [19] for the construction of B-spline basis functions and related. Denote B(t) = (B1(t), · · · , BMn+d(t))⊤. +Then we parameterize coefficient functions βk(t) = B(t)⊤bk for k = 0, · · · , q, where bk = (bk,1, · · · , bk,Mn+d)⊤. Let +Z = (z1, · · · , zn)⊤, X = (x1, . . ., xn)⊤ be the n × (Mn + d) matrix with the (i, j)th entry being xi j = +� T +0 Xi(t)B j(t)dt, and +3 + +U = (u1, . . ., un)⊤ be the n × q(Mn + d) matrix with ui = zi ⊗ xi, where ⊗ is the Kronecker product. Further denote +Ψ = (X, U), which is n × qn with qn = (q + 1) × (Mn + d). +The first term of (2) can be rewritten as: +1 +n +n +� +i=1 +ρτ(yi − ψ⊤ +i b − z⊤ +i γ), +(3) +where b = (b⊤ +0 , · · · , b⊤ +q )⊤ and γ = (γ1, · · · , γq)⊤. In Lemma 1 (Appendix), we examine approximating the sparse +group penalty under this basis expansion. In particular, setting the modifier κ = Mn, we have: +q +� +k=1 +Mn +T +� T +0 +pλ1(|βk(t)|)dt + Mn +T +� T +0 +pλ2(∥β(t)∥2)dt +≈ +q +� +k=1 +Mn +� +l=1 +pλ1 +�∥bk∥Wl +� + +Mn +� +l=1 +pλ2 +�∥b∥Wl +� , +(4) +where Wl is the (Mn + d) × (Mn + d) matrix with the (i, j)th entry wli j = Mn +T +� tl +tl−1 Bi(t)B j(t)dt if l ≤ i, j ≤ l + d, and +wli j = 0 otherwise, ∥bk∥Wl = (b⊤ +k Wlbk)1/2, and ∥b∥Wl = (�q +k=0 b⊤ +k Wlbk)1/2. Let V be the (Mn + d) × (Mn + d) matrix +with the (i, j)th entry vi j = +� T +0 +d2Bi(t) +dt2 +d2B j(t) +dt2 dt. Then, +q +� +k=0 +� T +0 +β′′2 +k (t)dt = +q +� +k=0 +b⊤ +k Vbk. +(5) +With (3), (4) and (5), we propose estimating (b∗, γ∗) by minimizing the following objective function: +Q(b, γ) =1 +n +n +� +i=1 +ρτ +� +yi − ψ⊤ +i b − z⊤ +i γ +� ++ +q +� +k=1 +Mn +� +l=1 +pλ1 +�∥bk∥Wl +� + +Mn +� +l=1 +pλ2 +�∥b∥Wl +� + η +q +� +k=0 +b⊤ +k Vbk. +(6) +Denote (ˆb, ˆγ) as the minimizer. Then the estimate of β∗ +k(t) is ˆβk(t) = B⊤(t)ˆbk. +2.3. Computation +To accommodate the non-differentiable quantile loss function, we resort to the majorize-minimization (MM) tech- +nique. In addition, we adopt the local quadratic approximation (LQA) technique for the sparse group penalty. +The proposed algorithm is iterative. At the (m + 1)th iteration, with estimate b(m) +k +from the mth iteration, we have: +pλ1 +�∥bk∥Wl +� ≈ pλ1(∥b(m) +k ∥Wl) + 1 +2 +p′ +λ1(∥b(m) +k ∥Wl) +∥b(m) +k ∥Wl +(∥bk∥2 +Wl − ∥b(m) +k ∥2 +Wl) += 1 +2 +p′ +λ1(∥b(m) +k ∥Wl) +∥b(m) +k ∥Wl +∥bk∥2 +Wl + G0(b(m) +k ), +where p′ +λ1(t) = λ1(1 − |t|/(λ1ξ))+ is the first-order derivative of pλ1(t), and G0(b(m) +k ) is a function of b(m) +k +and does not +depend on bk. As such, we can obtain the LQA approximation of the sparse group penalty as: +q +� +k=1 +Mn +� +l=1 +pλ1 +�∥bk∥Wl +� + +Mn +� +l=1 +pλ2 +�∥b∥Wl +� ≈ b⊤ ˘W(m)b + G1(b(m)), +(7) +4 + +where ˘W(m) = diag( ˘W(m) +0 , · · · , ˘W(m) +q ) is the qn × qn block diagonal matrix with: +˘W(m) +0 += 1 +2 +Mn +� +l=1 +p′ +λ2 +� +∥b(m)∥Wl +� +∥b(m)∥Wl +Wl, +and +˘W(m) +k += 1 +2 +Mn +� +l=1 + +p′ +λ1(∥b(m) +k ∥Wl) +∥b(m) +k ∥Wl ++ +p′ +λ2 +� +∥b(m)∥Wl +� +∥b(m)∥Wl +Wl, +for k = 1, · · · , q, and G1(b(m)) is free of b. +Let Φ be the n × dn matrix with the ith row being φi = (x⊤ +i , u⊤ +i , z⊤ +i )⊤ ∈ Rdn and dn = qn + q. Denote the coefficient +vector as ω = (b⊤, γ⊤)⊤ ∈ Rdn. Let ˜V = diag(V, · · · , V, 0q) and ˜W(m) = diag( ˘W(m), 0q) be dn × dn block-diagonal +matrices, where 0q is the q × q matrix with all entries being 0. +With the MM algorithm, at the (m + 1)th iteration, given the residual value r(m) = y − Φω(m), the quantile loss is +majorized at r(m) = (r(m) +1 , · · · , r(m) +n )⊤ by the quadratic function: +ξ(r|r(m)) = 1 +n +n +� +i=1 +1 +4 + +r2 +i +̺ + |r(m) +i +| ++ (4τ − 2)ri + c + , +where r = (r1, · · · , rn)⊤, ̺ is a small perturbation, and c is a constant. +The overall objective function at the (m + 1)th iteration is: +˜Q(ω|ω(m)) = ξ(r|r(m)) + ω⊤ ˜W(m) ˜ω + ηω⊤ ˜Vω. The +first-order derivative of ˜Q(ω|ω(m)) with respect to ω is: +˜Q′(ω|ω(m)) = 1 +2n +n +� +i=1 +φi +� +1 − 2τ − ri/(̺ + |r(m) +i +|) +� ++ 2 ˜W(m)ω + 2η ˜Vω += 1 +2nΦ⊤v̺(ω|ω(m)) + 2 ˜W(m)ω + 2η ˜Vω, +where +v̺(ω|ω(m)) = +1 − 2τ − +r1 +̺ + |r(m) +1 | +, · · · , 1 − 2τ − +rn +̺ + |r(m) +n | + +⊤ +is a length-n column vector. The second-order derivative of ˜Q(ω|ω(m)) with respect to ω is: +˜Q′′(ω|ω(m)) = 1 +2n +n +� +i=1 +φiφ⊤ +i +̺ + |r(m) +i +| ++ 2 ˜W(m) + 2η ˜V = 1 +2nΦ⊤R(m)Φ + 2 ˜W(m) + 2η ˜V, +where +R(m) = diag + +1 +̺ + |r(m) +1 | +, · · · , +1 +̺ + |r(m) +n | + +is an n × n diagonal matrix. Then the Gauss-Newton step direction is: +∆(m) +̺ (ω|ω(m)) = − +� ˜Q′′(ω|ω(m)) +�−1 ˜Q′(ω|ω(m)) += +− +� +Φ⊤R(m)Φ + 4n ˜W(m) + 4nη ˜V +�−1 � +Φ⊤v̺(ω|ω(m)) + 4n ˜W(m)ω + 4nη ˜Vω +� +. +(8) +Overall, the proposed computational algorithm proceeds as follows: +Step 1. Initialize ˆω as ˆω(0) = (Φ⊤Φ + nη ˜V)−1Φ⊤y and m = 0. +Step 2. Given ˆω(m), compute ˜W(m), R(m), and v̺( ˆω(m)). Update: +ˆω(m+1) = ˆω(m) + ∆(m) +̺ ( ˆω(m)), and m = m + 1. +5 + +Step 3. Repeat 2 until convergence, which is concluded if the norm of the difference between the estimates from two +consecutive iterations is smaller than a prespecified cutoff. The final estimate of ω is obtained by further +setting the elements of ˆω with absolute values smaller than a prespecified threshold to zero. +This algorithm is built on the MM and LQA techniques, both of which have been well examined in published +literature. Convergence of the algorithm can be established following the literature and is achieved in all of our +numerical studies. With the LQA, finite iterations cannot lead to sparse estimation. Following published studies, a +cutoff (whose value is not crucial) is imposed in Step 3. In our numerical study, we use 10−3. As in the literature, the +value of Mn is also not crucial since the smoothness of estimation is controlled by the roughness penalty, as opposed to +the number of knots. In our numerical study, we use cubic B-splines with 71 equally spaced knots to estimate β(t)’s, +following [12]. Note that other knot placement strategies can be considered such as some data-driven methods putting +knots at certain quantiles of covariates. Following [20, 21], we set λ2 = +� +q + 1λ1 and ξ = 6 and perform a grid search +for the optimal (η, λ1) based on prediction performance. More details are provided in the numerical studies below. The +R code implementing the proposed algorithm is publicly available at https://github.com/weijuanliang12138/SHLoS- +R-Code. +2.4. Theoretical properties +Let fi(·) and Fi(·) be the probability density function and distribution function of ǫi given (Xi(t), zi), respectively. +Denote Bn = diag{f1(0), · · · , fn(0)}. We assume the following conditions. +Condition 1. For i = 1, · · · , n, in a neighborhood of zero, fi is continuous and satisfies 0 < c ≤ fi ≤ C < ∞, where c +and C are constants. In addition, the first-order derivative f ′ +i has a uniform upper bound. +Condition 2. For k = 0, · · · , q, β∗ +k(t) belongs to the H¨older space Cα,ν([0, 1]). Specifically, |β∗(α) +k +(x1) − β∗(α) +k +(x2)| ≤ +C1|x1 − x2|ν for a constant C1, positive integer α, and ν ∈ (0, 1], and for all 0 ≤ x1, x2 ≤ 1, where β∗(α) +k +(·) is the +αth-order derivative of β∗ +k(·). Let r = a + ν. Assume r > 1.5. +Condition 3. There exist positive constants C2 and C3, such that ( +� T +0 X(t)2dt)1/2 ≤ C2 < ∞ and |zk| ≤ C3 < ∞ for +k = 1, · · · , q. In addition, there exist positive constants C4, C5, C6, and C7, such that +C4M−1 +n +≤ λmin(n−1ΨΨ⊤) ≤ λmax(n−1ΨΨ⊤) ≤ C5M−1 +n , +C6 ≤ λmin(n−1 ˇZ ˇZ⊤) ≤ λmax(n−1 ˇZ ˇZ⊤) ≤ C7, +where ˇZ = (In − Ψ(Ψ⊤BnΨ)−1Ψ⊤Bn)Z. +Condition 4. Mn = O(n +1 +2r+1 ). +Condition 1 is common in the quantile regression literature and weaker than those assumed with mean estimations. +Condition 2 ensures that there exists b∗ +k ∈ RMn+d such that supt∈[0,T] |β∗ +k(t) − B(t)⊤b∗ +k| = O(M−r +n ) for k = 0, · · · , q [22]. +Condition 3 is on the covariates and design matrices, which is analogous to those in [13] and [23]. Condition 4 is also +common in the spline literature. +Denote the null region of β∗ +k(t) as Nk = {t ∈ [0, T] : β∗ +k(t) = 0}. The asymptotic properties of the proposed +estimator can be summarized as follows. +Theorem 1. Under Conditions 1-4, if n− +r +2r+1 / min(λ1, λ2) = o(1), max(λ1, λ2) = o(1) and η = o(n−1/2), then there +exists a local minimizer (ˆb, ˆγ) of (6), such that for all k = 0, · · · , q with ˆβk(t) = B⊤(t)ˆbk, +(1) +� T +0 (ˆβk(t) − β∗ +k(t))2dt = Op(n−2r/(2r+1)) and ∥ˆγ − γ∗∥2 = Op(n−1/2), +(2) ˆβk(t) = 0 for all t ∈ Nk with probability tending to one. +Proof is provided in the Appendix. This theorem establishes the estimation and selection consistency properties. +It is observed that the convergence rate of ˆβk(t) is n−r/(2r+1), which is optimal [24]. The convergence rate of ˆγ is free of +Mn – the optimal root-n rate is achieved. The selection consistency holds by result (2). With the design of the penalty, +the “main effect, interaction” hierarchy is automatically satisfied. +6 + +3. Simulation +Data is generated from the following model: +yi = +� 1 +0 +Xi(t)β∗ +0(t)dt + +2 +� +k=1 +zik +� 1 +0 +Xi(t)β∗ +k(t)dt + +2 +� +k=1 +zikγ∗ +k + ǫi. +(9) +We consider three different scenarios of coefficient functions corresponding to various levels of sparsity and number +of null regions. All of these functions in each scenario satisfy the “main effect, interaction” hierarchy. +Scenario I: 60% regions of β∗ +10(t) have contribution to the response, and there is a null region in β∗ +10(t). The +functional main effect is: +β∗ +10(t) = + +2(1 − t) sin(2π(t + 0.2)) +0 ≤ t ≤ 0.3, +0 +0.3 < t < 0.7, +2t sin(2π(t − 0.2)) +0.7 ≤ t ≤ 1. +For the functional interaction effects, we consider: (1) β∗ +11(t) = β∗ +10(t) for t ∈ [0, 0.3], and β∗ +11(t) = 0 otherwise, (2) +β∗ +12(t) = β∗ +10(t) for t ∈ [0.7, 1], and β∗ +12(t) = 0 otherwise. These functions are demonstrated in Figure 1 by black solid +lines. +Scenario II: 30% regions of the main effect are nonnull regions, and there are four null regions on the entire domain +of β∗ +20(t). The functional main effect and interactions are defined as: +β∗ +20(t) = + +5 sin(10π(t − 0.2)) +0.2 < t ≤ 0.3, +−3 sin(10π(t − 0.5)) +0.5 < t ≤ 0.6, +3.5 sin(10π(t − 0.7)) +0.7 < t ≤ 0.8, +0 +otherwise, +β∗ +21(t) = + +2(t − 0.25)2/0.052 − 2 +0.2 < t ≤ 0.3, +5 sin(10π(t − 0.5)) +0.5 < t ≤ 0.6, +0 +otherwise, +and +β∗ +22(t) = + +2.5 sin(10π(t − 0.5)) +0.5 < t ≤ 0.6, +4(t − 0.75)2/0.052 − 4 +0.7 < t ≤ 0.8, +0 +otherwise, +respectively. These functions are presented in Figure 2 by black solid lines. +Scenario III: 17.5% regions of β∗ +30(t) have nonzero effects on the response, and there are eight null regions on the +entire domain of the main effect. The functional main effect and interactions are defined as: +β∗ +30(t) = + +4(t − 0.1375)2/0.01252 − 4 +0.125 < t ≤ 0.15, +7 sin(40π(t − 0.175)) +0.175 < t ≤ 0.2, +−6 sin(40π(t − 0.325)) +0.325 < t ≤ 0.35, +8 sin(40π(t − 0.6)) +0.6 < t ≤ 0.625, +−10 sin(40π(t − 0.7)) +0.7 < t ≤ 0.725, +5 sin(40π(t − 0.8)) +0.8 < t ≤ 0.825, +−7 sin(40π(t − 0.875)) +0.875 < t ≤ 0.9, +0 +otherwise, +β∗ +31(t) = + +10 sin(40π(t − 0.125)) +0.125 < t ≤ 0.15, +6 sin(40π(t − 0.325)) +0.325 < t ≤ 0.35, +8(t − 0.7125)2/0.01252 − 8 +0.7 < t ≤ 0.725, +9 sin(40π(t − 0.875)) +0.875 < t ≤ 0.9, +0 +otherwise, +7 + +and +β∗ +32(t) = + +5 sin(40π(t − 0.175)) +0.175 < t ≤ 0.2, +10(t − 0.6125)2/0.01252 − 10 +0.6 < t ≤ 0.625, +7 sin(40π(t − 0.8)) +0.8 < t ≤ 0.825, +0 +otherwise, +respectively. These functions are demonstrated in Figure 3 by black solid lines. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−2 +−1 +0 +1 +2 +ε ~ N(0, σ2) +t +β0(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−2 +−1 +0 +1 +2 +ε ~ N(0, σ2) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−2 +−1 +0 +1 +2 +ε ~ N(0, σ2) +t +β2(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−2 +−1 +0 +1 +2 +ε ~ t(3) +t +β0(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−2 +−1 +0 +1 +2 +ε ~ t(3) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−2 +−1 +0 +1 +2 +ε ~ t(3) +t +β2(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +Figure 1: Average of ˆβ(t)’s in Scenario I with n = 300 based on 100 replicates under Case 1 (top) and Case 2 (bottom), respectively. +Left/middle/right: β0(t)/β1(t)/β2(t). +The scalar covariates z·k, k = 1, 2 (where z·k is the k-th column of Z) are generated independently from the +standard normal distribution, and the corresponding coefficient vector is γ∗ = (0.5, 0.8)⊤. The functional covariate +Xi(t) is generated as Xi(t) = � ai jB j(t), where ai j’s are generated from a normal distribution with mean zero and +standard deviation 5, and each B j(t) is a B-spline basis function with order 5 and 71 equally spaced knots. Consider +three distributions for ǫi: +Case 1: (homoscedasticity) ǫi follows a normal distribution N(0, σ2), and σ is chosen so that the signal-to-noise ratio +equals 4. +Case 2: (homoscedasticity) ǫi follows a t(3) distribution. +Case 3: (heteroscedasticity) ǫi = ( 3 +2|zi1 +� 1 +0 Xi(t)β∗ +1(t)dt|)˜ǫi, where ˜ǫi ∼ N(0, 1) − QN(τ), and QN(τ) denotes the τth +quantile of a standard norm distribution. Note that here the model is misspecified. +For comparison, we consider the following alternatives: (a) Alt.1 adopts the mean squares lack-of-fit and a smooth- +ness penalty (which is the last term of the proposed approach). As such, it can control smoothness as in many published +studies but cannot conduct selection; (b) Alt.2 adopts the mean squares lack-of-fit and the “functional MCP penalty ++ smoothness penalty”. It computes conditional mean of the response and does not have a mechanism to respect the +“main effect, interaction” hierarchy; (c) Alt.3 adopts the mean squares lack-of-fit and the same penalty as the pro- +posed. As such, the only difference lies in the measured conditional quantity; (d) Alt.4 adopts the same quantile-based +loss as the proposed approach and the penalty in Alt.1; (e) Alt.5 adopts the same quantile-based loss as the proposed +approach and the penalty in Alt.2. For the quantile-based approaches, we set τ = 0.5 for homoscedasticity errors +8 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−4 +0 +2 +4 +6 +8 +ε ~ N(0, σ2) +t +β0(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−4 +0 +2 +4 +6 +8 +ε ~ N(0, σ2) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−4 +0 +2 +4 +6 +8 +ε ~ N(0, σ2) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−4 +0 +2 +4 +6 +8 +ε ~ t(3) +t +β0(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−4 +0 +2 +4 +6 +8 +ε ~ t(3) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−4 +0 +2 +4 +6 +8 +ε ~ t(3) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +Figure 2: Average of ˆβ(t)’s in Scenario II with n = 300 based on 100 replicates under Case 1 (top) and Case 2 (bottom), respectively. +Left/middle/right: β0(t)/β1(t)/β2(t). +(Cases 1 and 2) and τ = 0.3, 0.5, 0.7 for heteroscedasticity errors (Case 3). We consider sample size n = 300, 500. For +each simulation replicate, we generate an independent dataset under the same setting with sample size 500 and select +the optimal tunings corresponding to the best prediction. Summary statistics are computed based on 100 independent +replicates. +Performance is evaluated using the following criteria: (a) Average integrated squared errors on null region (ISE0): +ISE0k = +1 +l0k +� +Nk(ˆβk(t) − β∗ +k(t))2dt, where l0k is the length of null region Nk of β∗ +k(t), k = 0, 1, 2. (b) Average integrated +squared errors on nonnull region (ISE1): ISE1k = +1 +l1k +� +Nc +k (ˆβk(t) − β∗ +k(t))2dt, where l1k is the length of nonnull region +Nc +k of β∗ +k(t), k = 0, 1, 2. (c) Root mean squared errors of γ∗ (RMSEγ): RMSEγ = ∥ˆγ − γ∗∥2. (d) Average proportion +of nonnull regions that are correctly identified (fTPR), which is the functional counterpart of true positive rate in +parametric variable selection. (e) Average proportion of null regions that are correctly identified (fTNR), which is the +functional counterpart of true negative rate in parametric variable selection. +The results for Scenario I under Case 1 are provided in Table 1, and those for Scenario I under Cases 2 and 3 +are provided in the Appendix. In addition, the results for all cases under Scenarios II and III are provided in the +supplemental materials. Figures 1-3 present the estimated β(t)’s for Scenario I-III under Cases 1 and 2 with n = 300. +Overall, the findings are highly “as expected”. In particular, when the errors are normally distributed, the mean-based +methods can be advantageous. However, with Cases 2 and 3, the superiority of the quantile-based methods is obvious. +In addition, it is observed that introducing local sparsity can improve estimation, and that respecting the hierarchy +can further improve selection. As a representative example, consider Scenario 1 under Case 2 (Table 3, Appendix) +and n = 300. The ISE0(×102) for β2 are 10.785, 4.577, 3.505, 6.868, 1.400, and 1.053 for the five alternative and +proposed approaches, respectively. The corresponding fTNR values are 0.003 (Alt.1), 0.715 (Alt.2), 0.802 (Alt.3), +0.003 (Alt.4), 0.840 (Alt.5), and 0.891 (proposed), and the fTPR values are similar. Furthermore, it is observed that, +as the proportion of signal regions increases, it gets easier to identify sparsity. +9 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−10 +−5 +0 +5 +10 +ε ~ N(0, σ2) +t +β0(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−10 +−5 +0 +5 +10 +ε ~ N(0, σ2) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−10 +−5 +0 +5 +10 +ε ~ N(0, σ2) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−10 +−5 +0 +5 +10 +ε ~ t(3) +t +β0(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−10 +−5 +0 +5 +10 +ε ~ t(3) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−10 +−5 +0 +5 +10 +ε ~ t(3) +t +β1(t) +True Beta +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +Figure 3: Average of ˆβ(t)’s in Scenario III with n = 300 based on 100 replicates under Case 1 (top) and Case 2 (bottom), respectively. +Left/middle/right: β0(t)/β1(t)/β2(t). +4. Data analysis +We analyze the Tecator data which is available from http://lib.stat.cmu.edu/datasets/tecator. In this dataset, there +are 215 finely chopped pure meat samples (datasets C, M, and T). For each sample, measurements are available on a +spectrometric curve of spectra of absorbances measured at 100 channels with wavelength range 850-1050nm, as well +as moisture, fat, and protein. The latter three are measured in percent and determined by analytic chemistry. In this +analysis, we study how fat can be modeled as a function of the spectrometric curve X(t) with t being the wavelength +and the two scalar covariates moisture z1 and protein z2. As developed above, we also incorporate the interactions +between the spectrometric curve and scalar covariates in modeling. By introducing local sparsity, we can potentially +distinguish “useful” regions of spectra that are informative for modeling fat from the “noisy” ones. Prior to analysis, +the range of wavelength t is mapped to [0, 1]. There are 31 equally spaced knots. Following the official guidance of +this dataset, we use 129 samples (dataset C) as training for estimation, 43 samples (dataset M) for tuning parameter +selection, and 43 samples (dataset T) for performance evaluation. +We first conduct exploratory regression analysis and present the findings in Appendix IV. Skewed residuals are +observed, which justifies quantile regression. In addition, there is no obvious lack-of-fit under quantile regression. +The estimation results for the functional effects are shown in Figure 4, where we consider τ = 0.3, 0.5, and 0.7. It is +observed that the effects are locally sparse. In addition, the “main effect, interaction” hierarchy is satisfied. For the +scalar effects, the estimates are: (ˆµ, ˆγ1, ˆγ2) = (5.785, −0.068, −0.073) for τ = 0.3, (5.899, −0.074, −0.057) for τ = 0.5, +and (5.930, −0.081, −0.038) for τ = 0.7. The differences across different quantile values partly justify the need for +quantile-based estimation. We recognize that a single split may not be sufficiently informative. As such, we conduct +100 random splittings of the original data, and the sizes of the three sets (under each splitting) are the same as above. +In Table 2, we present the mean (standard deviation) for each scalar estimate. The 100 sets of estimated functional +effects are available from the authors. In addition, we also present the results of prediction error. Overall, taking the +local sparsity, interpretability pertained to the variable selection hierarchy, and prediction performance into account, +the analysis with the proposed approach and τ = 0.7 is recommended as the final one. +10 + +Table 1: Scenario I under Case 1: mean (sd) based on 100 replicates. For the quantile-based methods, τ = 0.5. +n +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +ISE0(×102) +300 +β0 +2.786(3.002) +0.338(1.219) +0.135(0.941) +3.511(3.380) +0.815(2.484) +0.266(1.035) +β1 +2.524(1.514) +0.251(0.638) +0.279(0.861) +2.873(1.759) +0.407(0.926) +0.428(1.046) +β2 +2.341(1.485) +0.245(0.792) +0.291(0.746) +2.668(1.707) +0.617(1.154) +0.568(1.085) +500 +β0 +1.514(1.102) +0.127(0.316) +0.002(0.004) +2.135(1.675) +0.269(0.845) +0.050(0.121) +β1 +1.533(1.050) +0.132(0.352) +0.043(0.228) +1.969(1.500) +0.254(0.668) +0.143(0.496) +β2 +1.652(1.037) +0.126(0.356) +0.049(0.220) +1.766(1.281) +0.153(0.471) +0.078(0.308) +ISE1(×102) +300 +β0 +7.742(4.976) +7.995(5.764) +7.542(5.389) +9.606(6.020) +9.969(6.746) +9.688(6.056) +β1 +4.034(2.838) +3.943(2.829) +3.852(2.840) +4.189(3.039) +4.425(3.664) +4.248(3.296) +β2 +3.764(2.482) +4.212(4.063) +4.387(3.739) +4.580(3.134) +5.689(5.218) +6.092(6.107) +500 +β0 +5.347(2.526) +5.008(2.955) +5.412(2.948) +6.620(3.213) +7.254(4.033) +7.697(3.869) +β1 +2.776(1.827) +2.583(1.925) +3.045(2.118) +3.304(2.091) +3.587(2.848) +3.996(2.681) +β2 +2.471(1.618) +2.363(2.010) +2.636(2.153) +3.016(2.118) +3.502(2.922) +3.910(2.907) +RMSEγ +300 +0.054(0.044) +0.052(0.039) +0.051(0.040) +0.068(0.048) +0.068(0.053) +0.065(0.051) +500 +0.044(0.029) +0.042(0.029) +0.042(0.029) +0.051(0.037) +0.050(0.036) +0.047(0.033) +fTPRR +300 +β0 +1.000(0.000) +0.997(0.009) +0.999(0.004) +1.000(0.000) +0.997(0.012) +0.999(0.004) +β1 +1.000(0.000) +0.998(0.006) +0.998(0.007) +1.000(0.000) +0.997(0.010) +0.997(0.009) +β2 +1.000(0.000) +0.996(0.020) +0.997(0.009) +1.000(0.000) +0.998(0.008) +0.994(0.018) +500 +β0 +1.000(0.000) +0.999(0.003) +0.999(0.003) +1.000(0.000) +0.999(0.004) +0.999(0.002) +β1 +1.000(0.000) +0.999(0.005) +0.998(0.005) +1.000(0.000) +0.998(0.011) +0.999(0.004) +β2 +1.000(0.000) +0.999(0.006) +0.998(0.007) +1.000(0.000) +0.998(0.009) +0.999(0.004) +fTNR +300 +β0 +0.000(0.001) +0.756(0.205) +0.850(0.103) +0.000(0.001) +0.708(0.231) +0.793(0.156) +β1 +0.000(0.001) +0.882(0.148) +0.913(0.115) +0.001(0.001) +0.854(0.177) +0.895(0.120) +β2 +0.001(0.001) +0.894(0.160) +0.922(0.089) +0.000(0.001) +0.847(0.175) +0.883(0.155) +500 +β0 +0.001(0.002) +0.750(0.174) +0.879(0.044) +0.001(0.001) +0.691(0.198) +0.763(0.152) +β1 +0.001(0.001) +0.885(0.151) +0.961(0.054) +0.001(0.001) +0.878(0.116) +0.920(0.078) +β2 +0.001(0.001) +0.893(0.113) +0.958(0.044) +0.001(0.001) +0.894(0.100) +0.920(0.119) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0.2 +0.3 +t +β0(t) +τ = 0.3 +τ = 0.5 +τ = 0.7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +t +β1(t) +τ = 0.3 +τ = 0.5 +τ = 0.7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +t +β2(t) +τ = 0.3 +τ = 0.5 +τ = 0.7 +Figure 4: Estimated functional effects using the proposed approach. +5. Discussion +In this article, we have considered a more sophisticated functional data analysis model. The most significant ad- +vancement comes from the interaction analysis. A new estimation and variable selection method has been developed, +and its theoretical and numerical properties have been carefully investigated. The proposed model can be poten- +11 + +Table 2: Average estimates and prediction error for the intercept and scalar covariate effects based on 100 random partitions. +Method +τ +Estimate +Prediction Error +ˆµ +ˆγ1 +ˆγ2 +Alt.1 +5.103(0.416) +-0.070(0.011) +-0.029(0.031) +0.039(0.006) +Alt.2 +5.215(0.570) +-0.084(0.015) +0.015(0.057) +0.037(0.006) +Alt.3 +5.246(0.613) +-0.084(0.015) +0.015(0.059) +0.037(0.005) +Alt.4 +0.3 +5.375(0.424) +-0.073(0.013) +-0.035(0.040) +0.038(0.008) +0.5 +5.431(0.442) +-0.071(0.009) +-0.046(0.022) +0.034(0.007) +0.7 +5.486(0.465) +-0.072(0.009) +-0.047(0.020) +0.025(0.004) +Alt.5 +0.3 +5.720(0.493) +-0.076(0.010) +-0.047(0.025) +0.038(0.009) +0.5 +5.829(0.548) +-0.076(0.008) +-0.053(0.016) +0.034(0.007) +0.7 +5.809(0.562) +-0.078(0.009) +-0.046(0.019) +0.025(0.005) +Proposed +0.3 +5.707(0.386) +-0.074(0.007) +-0.050(0.023) +0.037(0.009) +0.5 +5.886(0.396) +-0.076(0.006) +-0.057(0.012) +0.033(0.007) +0.7 +5.908(0.386) +-0.077(0.007) +-0.053(0.014) +0.025(0.005) +tially extended to include more complicated interactions (for example, between functional effects) and have higher +dimensions. It will also be of interest to examine more practical applications. +Acknowledgements +We thank the associate editor and reviewers for careful review and insightful comments. This study has been +partly supported by the National Natural Science Foundation of China [11971404], National Bureau of Statistics of +China [2022LZ34], Fundamental Research Funds for the Central Universities, Research Funds of Renmin University +of China [21XNH152], and NIH [CA204120]. +12 + +References +[1] G. Aneiros, S. Novo, P. Vieu, Variable selection in functional regression models: A review, Journal of Multivariate Analysis (2021) 104871. +[2] H. Cardot, F. Ferraty, P. Sarda, Spline estimators for the functional linear model, Statistica Sinica (2003) 571–591. +[3] R. Fan, Y. Wang, J. L. Mills, A. F. Wilson, J. E. Bailey-Wilson, M. Xiong, Functional linear models for association analysis of quantitative +traits, Genetic epidemiology 37 (7) (2013) 726–742. +[4] F. Yao, H.-G. M¨uller, J.-L. 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Wang, Partially linear additive quantile regression in ultra-high dimension, The Annals of Statistics 44 (1) (2016) 288–317. +[24] C. J. Stone, Additive regression and other nonparametric models, The annals of Statistics 13 (2) (1985) 689–705. +Appendix +I. Lemma 1 and remarks +Lemma 1. Consider Mn + 1 equally spaced knots 0 = t0 < t1 < · · · < tMn = T in the domain [0, T]. For the smooth +functional main effect and interactions, we have: +q +� +k=1 +1 +T +� T +0 +pλ1(|βk(t)|)dt + 1 +T +� T +0 +pλ2(∥β(t)∥2)dt += lim +Mn→∞ +1 +Mn +q +� +k=1 +Mn +� +l=1 +pλ1 +� +Mn +1 +2 T − 1 +2 ∥βk[l]∥ +� ++ lim +Mn→∞ +1 +Mn +Mn +� +l=1 +pλ2 +� +Mn +1 +2 T − 1 +2 ∥β[l]∥ +� +, +where ∥βk[l]∥ = ( +� tl +tl−1 β2 +k(t)dt)1/2 and ∥β[l]∥ = (�q +k=0 +� tl +tl−1 β2 +k(t)dt)1/2. +This lemma can be derived from Theorem 1 of [12]. It shows that the penalty evaluated over the whole domain +is asymptotically equivalent to the sum over a large number of subregions. This nicely matches the spline basis +expansion framework. For each subregion, we note that the penalty still has a sparse group form. As such, the “main +effect, interaction” hierarchy is expected to hold for each subregion (and so the whole domain). +13 + +II: Proof of Theorem 1 +Let C be a generic positive constant which may take different values under different circumstances. Denote: +g∗(Xi(t), zi) = +� T +0 +Xi(t)β∗ +0(t)dt + +q +� +k=1 +zik +� T +0 +Xi(t)β∗ +k(t)dt. +Recall that supt∈[0,T] +���β∗ +k(t) − B⊤(t)b∗ +k +��� = O(M−r +n ). With the boundedness Condition 3, we have g∗(Xi(t), zi) = ψ⊤ +i b∗ + +O(M−r +n ). Denote the empirical version of the projection of z·k onto the spline approximation of the functional covariate +space as h·k = Ψ ˆ̟k, where z·k is the kth column of Z and ˆ̟k is the minimizer of: +min +̟k∈Rqn +n +� +i=1 +fi(0)(zik − ψ⊤ +i ̟k)2. +The solution to the above problem is ˆ̟k = (Ψ⊤BnΨ)−1Ψ⊤Bnz·k. Let H be the n × q matrix with the kth column being +h·k. We define the projection matrix P = Ψ(Ψ⊤BnΨ)−1Ψ⊤Bn ∈ Rn×n, and it is obvious that H = PZ. Thus we have +ˇZ = (ˇz1, · · · , ˇzn)⊤ = (In − P)Z. +Define ˜zi = n− 1 +2 ˇzi ∈ Rq, Ψ2 +B = Ψ⊤BnΨ ∈ Rqn×qn, and ˜ψi = Ψ−1 +B ψi ∈ Rqn. Following [23], we reparameterize the +quantile loss function as: +ρτ +� +yi − ψ⊤ +i b − z⊤ +i γ +� += ρτ +� +ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +i θ2 − uni +� +, +where θ1 = √n(γ − γ∗) ∈ Rq, θ2 = ΨB(b − b∗) + Ψ−1 +B Ψ⊤BnZ(γ − γ∗) ∈ Rqn and uni = ψ⊤ +i b∗ − g∗(Xi(t), zi). Let +θ = (θ⊤ +1 , θ⊤ +2 )⊤. The objective function under the reparameterization is: +˜Q(θ) =1 +n +n +� +i=1 +ρτ(ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +i θ2 − uni) + +q +� +k=1 +Mn +� +l=1 +pλ1(∥bk∥Wl) ++ +Mn +� +l=1 +pλ2(∥b∥Wl) + η +q +� +k=0 +b⊤ +k Vbk. +Define: +Di(θ) =ρτ(ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +i θ2 − uni) − ρτ(ǫi − uni) + (˜z⊤ +i θ1 + ˜ψ⊤ +i θ2)Dτ(ǫi) +− E[ρτ(ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +i θ2 − uni) − ρτ(ǫi − uni)], +where Dτ(ǫi) = τ − I(ǫi < 0). We first state the following lemmas. +Lemma 2. Let dn = qn + q. Under Conditions 1-4, for any positive constant L, we have: +sup +∥θ∥2≤L √dn +1 +dn +������� +n +� +i=1 +Di(θ) +������� = op(1). +Proof. Proof follows that of Lemma B.1 in [23] under Conditions 1-4. +Lemma 3. Let ˜θ1 = √n +� ˇZ⊤Bn ˇZ +�−1 ˇZDτ(ǫ), where Dτ(ǫ) = (Dτ(ǫ1), · · · , Dτ(ǫn))⊤. Under Conditions 1-4, we have +∥˜θ1∥2 = Op(1). +Proof. Proof follows from that of Lemma 5 (1) in [23]. +Proof of Theorem 1 (1) Here we show that there exists a local minimizer ˆθ = (ˆθ⊤ +1 , ˆθ⊤ +2 )⊤ of (6) such that ∥ˆθ∥2 = +Op( √Mn) and ∥ˆθ1∥2 = Op(1). +Note that dn = O(Mn). To prove ∥ˆθ∥2 = Op( √Mn), it is sufficient to show that, for any δ > 0, there exists a +sufficiently large positive constant L such that: +P +� +inf +∥θ∥2≤L √dn +˜Q(θ) > ˜Q(0) +� +≥ 1 − δ. +(A.1) +14 + +That is, with probability at least 1 − δ, there exists a local minimizer such that ∥ˆθ∥2 ≤ L √dn. +We first show that, for a sufficiently large positive L, there exists a positive constant C such that: +inf +∥θ∥2=L √dn +1 +n +n +� +i=1 +� +ρτ(ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +i θ2 − uni) − ρτ(ǫi − uni) +� +> CL2dn/n +(A.2) +with probability tending to one. From Lemma 2, we have: +sup +∥θ∥2≤L √dn +1 +n +������� +n +� +i=1 +Di(θ) +������� += +sup +∥θ∥2≤L √dn +1 +n +������� +n +� +i=1 +ρτ(ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +i θ2 − uni) − +n +� +i=1 +ρτ(ǫi − uni) +− +n +� +i=1 +E +� +ρτ(ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +i θ2 − uni) − ρτ(ǫi − uni) +� ++ +n +� +i=1 +(˜z⊤ +i θ1 + ˜ψ⊤ +i θ2)Dτ(ǫi) +������� = op(dn/n). +Denote Fn1 = n−1 �n +i=1 E[ρτ(ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +i θ2 − uni) − ρτ(ǫi − uni)] and Fn2 = n−1 �n +i=1(˜z⊤ +i θ1 + ˜ψ⊤ +i θ2)Dτ(ǫi). Following +similar arguments as in the proof of Lemma 4 in [23], we can show that for a sufficiently large positive L, Fn1 has +asymptotically a lower bound of CL2dn/n and Fn2 = Op(d1/2 +n /n). Therefore, (A.2) is proved. +Let D denote the domain [0, T]. For given λ1, λ2 and Mn, and for each β∗ +k(t), we divide D into three parts: the first +part D[1] +k += {t ∈ D : |β∗ +k(t)| ≥ Cξ max(λ1, λ2)} for some constant C > 1, the second part D[2] +k += {t ∈ D : β∗ +k(t) = 0}, +and the third part D[3] +k += {t ∈ D : 0 < |β∗ +k(t)| < Cξ max(λ1, λ2)}. Since max(λ1, λ2) → 0 as n → ∞, D[3] +k +shrinks to the +empty set ∅ as n → ∞. +Next, we consider the penalty terms. With ∥θ∥2 = O( √dn) and the definition of θ, we have ∥γ − γ0∥2 = O( √dn/n). +In addition, +∥ΨB(b − b∗)∥2 +2 ≤ 2∥θ2∥2 +2 + 2∥Ψ−1 +B Ψ⊤BnZ(γ − γ0)∥2 +2 = O(dn). +(A.3) +The last equality holds because ∥Ψ−1 +B Ψ⊤BnZ(γ − γ0)∥2 +2 = O(n∥γ − γ0∥2 +2) by Conditions 1 and 3. Then we have +∥bk − b∗ +k∥2 = O(dnn−1/2). Notice that β∗ +k(t) = B⊤(t)b∗ +k + O(M−r +n ). For a subregion Il ⊂ D[1] +k , k = 0, · · · , q, with +Mn = O(dn), Condition 4, and n− +r +2r+1 / min(λ1, λ2) = o(1), we have: +∥b∗ +k∥Wl = +� +Mn +T +� tl +tl−1 +β∗2 +k (t)dt + O(M−r +n ) ≥ Cξ max(λ1, λ2). +Applying some inequality techniques, we can derive ∥bk∥Wl ≥ Cξ max(λ1, λ2). With the properties of MCP and +C > 1, we have pλ1(∥bk∥Wl) = pλ1(∥b∗ +k∥Wl) and pλ2(∥b∥Wl) = pλ2(∥b∗∥Wl) for l satisfying Il ⊂ D[1] +k . For a subregion +Il ⊂ D[2] +k ∩ (∪k′�kD[1] +k′ ), by the choice of b∗, we have ∥b∗ +k∥Wl = 0 and ∥b∗∥Wl ≥ Cξ max(λ1, λ2), and thus pλ1(∥bk∥Wl) ≥ +pλ1(∥b∗ +k∥Wl) = 0 and pλ2(∥b∥Wl) = pλ2(∥b∗∥Wl). For a subregion Il ⊂ D[2] +k ∩ (∪k′�kD[1] +k′ )c, we have ∥b∗∥Wl = 0, and thus +pλ1(∥bk∥Wl) ≥ pλ1(∥b∗ +k∥Wl) = 0 and pλ2(∥b∥Wl) ≥ pλ2(∥b∗∥Wl) = 0. Summarizing the above three cases, we have: +q +� +k=1 +Mn +� +l=1 +pλ1 +�∥bk∥Wl +� ≥ +q +� +k=1 +Mn +� +l=1 +pλ1 +� +∥b∗ +k∥Wl +� +(A.4) +and +Mn +� +l=1 +pλ1 +�∥b∥Wl +� ≥ +Mn +� +l=1 +pλ1 +�∥b∗∥Wl +� . +(A.5) +15 + +Also, by the Cauchy-Schwarz inequality and η = o(n−1/2), we have: +q +� +k=0 +ηb⊤ +k Vbk − +q +� +k=0 +ηb∗⊤ +k Vb∗ +k += +q +� +k=0 +η +� +(bk − b∗ +k)⊤V(bk − b∗ +k) + 2(bk − b∗ +k)⊤Vb∗ +k +� +≤ +O(ηdnn−1 + ηn−1/2) = o(n−1), +(A.6) +where the inequality follows from the fact that ∥bk − b∗ +k∥2 = O(dnn−1/2), λmax(V) = O(d−1 +n ) and supj |V· jb∗ +k| ≤ Cd−1 +n , +where Vj· is the jth row of V for j = 1, · · · , Mn + d. +Combining (A.2), (A.4), (A.5) and (A.6), for ∥θ∥2 = L √dn and a sufficiently large L, we prove (A.1). Therefore, +there exists a local minimizer ˆθ such that ∥ˆθ∥2 = Op( √dn). Similar to (A.3), it follows that ∥ΨB(ˆb − b∗)∥2 = Op( √dn), +and thus ∥ˆb − b∗∥2 = Op(dnn−1/2). Then we have: +� T +0 +(ˆβk(t) − β∗ +k(t))2dt +≤ +2 +� T +0 +(ˆβk(t) − B⊤(t)b∗ +k)2dt + 2 +� T +0 +(B⊤(t)b∗ +k − β∗ +k(t))2dt += +O(d−1 +n ∥ˆb − b∗∥2 +2) + Op(M−2r +n +) = Op(n− 2r +2r+1 ), +where the first inequality follows from the triangle inequality, and the last equality is due to dn = O(Mn) and Condition +4. +Next, we examine the convergencerate of ˆθ1. To verify ∥ˆθ1∥2 = Op(1), it is sufficient to show that ∥ˆθ1−˜θ1∥2 = op(1) +under Conditions 1-4. Define: +˜Qi(θ1, ˜θ1, θ2) = ρτ(ǫi − ˜z⊤ +i θ1 − ˜ψ⊤ +2 θ2 − uni) − ρτ(ǫi − ˜z⊤ +i ˜θ1 − ˜ψ⊤ +2 θ2 − uni). +We first show that for any positive constants M and C, +P + +inf +∥θ1−˜θ1∥2≥M +∥θ2∥2≤C √dn +n +� +i=1 +˜Qi(θ1, ˜θ1, θ2) > 0 + +→ 1. +(A.7) +Following the proof of Lemma 6 in [23], we have: +sup +∥θ1−˜θ1∥2≤M +∥θ2∥2≤C √dn +������� +n +� +i=1 +˜Qi(θ1, ˜θ1, θ2) − 1 +2(θ1 − ˜θ1)⊤ +�1 +n +ˇZ⊤Bn ˇZ +� +(θ1 − ˜θ1)(1 + op(1)) +������� = op(1). +By Conditions 1 and 3, for any ∥θ1 − ˜θ1∥2 > M, +1 +2(θ1 − ˜θ1)⊤ +�1 +n +ˇZ⊤Bn ˇZ +� +(θ1 − ˜θ1) > CM, +for some positive constant C, and thus (A.7) holds. Combining (A.1), (A.7) and Lemma 3, we have that there exists a +local minimizer ˆθ1 of (6) such that ∥ˆθ1∥2 = Op(1), and thus ∥ˆγ − γ∗∥2 = Op( √1/n). +Proof of Theorem 1 (2) +We need to show that ˆβk(t) = 0 for all t ∈ D[2] +k +with probability tending to one. Denote ˆb[l] +k = (ˆbk,l, · · · , ˆbk,l+d)⊤. +We need to prove that the local minimizer (ˆb⊤, ˆγ⊤)⊤ satisfies ∥ˆb[l] +k ∥2 = 0 for all l such that Il ⊂ D[2] +k +with probability +tending to one for k = 0, · · · , q. By the way of contradiction, assume that ∥ˆb[l⋆] +k ∥2 � 0 for some l⋆ with Il⋆ ⊂ D[2] +k . +Let ˜bk be the same as ˆbk except that ∥˜b[l⋆] +k ∥2 = 0. Note that ∥˜b[l] +k ∥2 = 0 is equivalent to ∥˜bk∥Wl = 0 for l = 1, · · · , Mn. +Since ∥b∗[l⋆] +k +∥2 = 0 and ∥b∗ +k − ˆbk∥2 = Op(Mn/ √n), we have ∥ˆb[l⋆] +k ∥2 = O(Mn/ √n), and thus ∥ˆbk∥Wl⋆ = O( √Mn/n) by +16 + +λmax(Wl⋆) = O(M−1 +n ). Below we prove that: +1 +n +n +� +i=1 +ρτ(yi − ψ⊤ +i ˆb − z⊤ +i ˆγ) + +q +� +k=1 +Mn +� +l=1 +pλ1(∥ˆbk∥Wl) + +Mn +� +l=1 +pλ2(∥ˆb∥Wl) + η +q +� +k=0 +ˆb⊤ +k V ˆbk +> 1 +n +n +� +i=1 +ρτ(yi − ψ⊤ +i ˜b − z⊤ +i ˆγ) + +q +� +k=1 +Mn +� +l=1 +pλ1(∥˜bk∥Wl) + +Mn +� +l=1 +pλ2(∥˜b∥Wl) + η +q +� +k=0 +˜b⊤ +k V ˜bk, +(A.8) +with probability tending to one, and this leads to a contradiction. Therefore, we conclude that ∥ˆb[l] +k ∥2 = 0 for all +l ⊂ D[2] +k +in probability. Furthermore, by the definition of ˆβk(t), we have ˆβk(t) = 0 for all t ∈ D[2] +k +with probability +tending to one. +By the convexity of the quantile loss function, we have +1 +n +n +� +i=1 +(ρτ(yi − ψ⊤ +i ˆb − z⊤ +i ˆγ) − ρτ(yi − ψ⊤ +i ˜b − z⊤ +i ˆγ)) +≥ +−1 +n +n +� +i=1 +(τ − 1(yi ≤ ψ⊤ +i ˜b + z⊤ +i ˆγ))ψ[kl⋆]⊤ +i +ˆb[l⋆] +k +(A.9) += +−1 +n +n +� +i=1 +(τ − 1(ǫi ≤ 0)) ψ[kl⋆]⊤ +i +ˆb[l⋆] +k +−1 +n +n +� +i=1 +(1(ǫi ≤ 0) − 1(ǫi ≤ ψ⊤ +i (˜b − b∗) + z⊤ +i (ˆγ − γ∗) + uni))ψ[kl⋆]⊤ +i +ˆb[l⋆] +k , +where ψ[kl⋆] +i += (ψi,k(Mn+d)+l⋆, · · · , ψi,k(Mn+d)+l⋆+d)⊤. For the first term on the right hand side of the last equation of (A.9), +by Conditions 1 and 3, we have: +1 +n +n +� +i=1 +(τ − 1(ǫi ≤ 0)) ψ[kl⋆]⊤ +i +ˆb[l⋆] +k += Op(n−1/2∥ˆbk∥Wl⋆). +For the second term, since supi |ψ⊤ +i (˜b − b∗) + z⊤ +i (ˆγ − γ∗) + uni| = Op( √Mn/n), we have: +E + + +1 +n +n +� +i=1 +� +1(ǫi ≤ 0) − 1(ǫi ≤ ψ⊤ +i (˜b − b∗) + z⊤ +i (ˆγ − γ∗) + uni) +� +ψ[kl⋆]⊤ +i +ˆb[l⋆] +k + +2 +≤ +E + + +1 +n +n +� +i=1 +����1(ǫi ≤ C +� +Mn/n) − 1(ǫi ≤ −C +� +Mn/n) +���� × +����ψ[kl⋆]⊤ +i +ˆb[l⋆] +k +���� + +2 += +E + +1 +n2 +n +� +i=1 +1(−C +� +Mn/n ≤ ǫi ≤ C +� +Mn/n) × +����ψ[kl⋆]⊤ +i +ˆb[l⋆] +k +���� +2 ++ +� +i�i′ +1 +n2 E +� +1(−C +� +Mn/n ≤ ǫi ≤ C +� +Mn/n)1(−C +� +Mn/n ≤ ǫi′ ≤ C +� +Mn/n) +× +����ψ[kl⋆]⊤ +i +ψ[kl⋆] +i′ +���� × ∥ˆb[l⋆] +k ∥2 +2 +� +≤ +Cn−2 � +nM1/2 +n n−1/2 + n2Mnn−1� +∥ˆbk∥2 +Wl⋆ = O(Mnn−1)∥ˆbk∥2 +Wl⋆. +Therefore, the second term is bounded by Op( √Mn/n∥ˆbk∥Wl⋆), which dominates the first term. Also, since λmax(V) = +17 + +O(M−1 +n ), we have: +η +q +� +k=0 +ˆb⊤ +k V ˆbk − η +q +� +k=0 +˜b⊤ +k V ˜bk += +η +q +� +k=0 +[(ˆbk − ˜bk)⊤V(ˆbk − ˜bk) + 2(ˆbk − ˜bk)⊤V ˜bk] += +η +q +� +k=0 +ˆb[l⋆]⊤ +k +V[l⋆] ˆb[l⋆] +k +≤ CηM−1 +n ∥ˆb[l⋆] +k ∥2 +2 += +Op(η +� +Mn/n∥ˆbk∥Wl⋆), +(A.10) +where V[l⋆] is the submatrix of V with entries vi j, i, j = l⋆, · · · , l⋆ + d. Since η = op(n−1/2), together with the above +discussions on (A.9) and (A.10), it follows that: +1 +n +n +� +i=1 +(ρτ(yi − ψ⊤ +i ˆb − z⊤ +i ˆγ) − ρτ(yi − ψ⊤ +i ˜b − z⊤ +i ˆγ)) + η +q +� +k=0 +ˆb⊤ +k V ˆbk − η +q +� +k=0 +˜b⊤ +k V ˜bk += Op( +� +Mn/n∥ˆbk∥Wl⋆). +(A.11) +Next, we examine the functional sparse group penalty. For Il⋆ ⊂ D[2] +0 , we have ∥ˆb0∥Wl⋆ ≥ ∥˜b0∥Wl⋆ = 0 and +∥ˆbk∥Wl⋆ = ∥˜bk∥Wl⋆ = 0 for all k = 1, · · · , q by the design of the sparse group penalty. And thus +q +� +k=1 +Mn +� +l=1 +pλ1(∥ˆbk∥Wl) = +q +� +k=1 +Mn +� +l=1 +pλ1(∥˜bk∥Wl). +(A.12) +Also, for l⋆ such that Il⋆ ⊂ D[2] +0 , we have: +Mn +� +l=1 +pλ2(∥ˆb∥Wl) − +Mn +� +l=1 +pλ2(∥˜b∥Wl) = pλ2(∥ˆb∥Wl⋆) ≥ λ2 +2 ∥ˆb∥Wl⋆ ≥ λ2 +2 ∥ˆbk∥Wl⋆, +(A.13) +by ∥ˆb∥Wl⋆ = O( √Mn/n), M−r +n / min(λ1, λ2) = o(1), and Condition 4. For Il⋆ ⊂ D[2] +k +and k � 0, we have ∥ˆbk∥Wl⋆ ≥ +∥˜bk∥Wl⋆ = 0 and ∥ˆbk′∥Wl⋆ = ∥˜bk′∥Wl⋆ for k′ � k. Since ∥ˆbk∥Wl⋆ = O( √Mn/n) and M−r +n / min(λ1, λ2) = o(1), we have: +q +� +k=1 +Mn +� +l=1 +(pλ1(∥ˆbk∥Wl) − pλ1(∥˜bk∥Wl)) = pλ1(∥ˆbk∥Wl⋆) ≥ λ1 +2 ∥ˆbk∥Wl⋆. +(A.14) +For Il⋆ ⊂ D[2] +k +and k � 0, when ∥˜b∥Wl⋆ ≥ λ2ξ, we have ∥ˆb∥Wl⋆ ≥ λ2ξ and +Mn +� +l=1 +(pλ2(∥ˆb∥Wl) − pλ2(∥˜b∥Wl)) = pλ2(∥ˆb∥Wl⋆) − pλ2(∥˜b∥Wl⋆) = 0. +(A.15) +For Il⋆ ⊂ D[2] +k +and k � 0, when ∥˜b∥Wl⋆ < λ2ξ, we have +Mn +� +l=1 +(pλ2(∥ˆb∥Wl) − pλ2(∥˜b∥Wl)) = λ2 +� min(λ2ξ,∥ˆb∥Wl⋆ ) +∥˜b∥Wl⋆ +� +1 − +t +λ2ξ +� ++ +dt ≥ 0. +(A.16) +Combining (A.14) - (A.16), we obtain that for Il⋆ ⊂ D[2] +k , k = 0, · · · , q, +q +� +k=1 +Mn +� +l=1 +(pλ1(∥ˆbk∥Wl) − pλ1(∥˜bk∥Wl)) + +Mn +� +l=1 +(pλ2(∥ˆb∥Wl) − pλ2(∥˜b∥Wl)) ≥ 1 +2 min(λ1, λ2)∥ˆbk∥Wl⋆. +18 + +Combining the above result with (A.11), M−r +n / min(λ1, λ2) = o(1), and Condition 4, we prove (A.8) with probability +tending to one. This completes the proof. +19 + +III. Additional numerical results +Table 3: Scenario I under Case 2: mean (sd) based on 100 replicates. For the quantile-based methods, τ = 0.5. +n +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +ISE0(×102) +300 +β0 +10.493(8.442) +2.012(4.538) +2.012(4.538) +8.146(5.853) +1.433(4.109) +1.433(4.109) +β1 +12.101(9.790) +5.474(9.373) +3.730(6.966) +8.487(7.166) +1.683(4.468) +1.429(4.155) +β2 +10.785(8.806) +4.577(8.575) +3.505(7.119) +6.868(5.097) +1.400(3.650) +1.053(2.812) +500 +β0 +7.896(5.606) +1.723(4.768) +1.012(4.486) +5.330(3.898) +0.813(2.203) +0.741(2.338) +β1 +6.918(5.365) +2.780(6.136) +1.545(5.678) +4.238(2.501) +0.224(0.744) +0.184(0.702) +β2 +8.477(17.375) +6.062(36.792) +5.032(36.771) +4.794(3.148) +0.329(1.289) +0.538(1.900) +ISE1(×102) +300 +β0 +36.417(24.229) +37.843(26.623) +37.843(26.623) +25.041(16.974) +28.583(20.912) +28.583(20.912) +β1 +18.563(14.815) +18.940(15.699) +19.167(15.704) +11.394(10.309) +12.484(11.503) +14.196(13.308) +β2 +18.751(21.966) +18.976(20.397) +19.377(20.527) +11.266(13.571) +12.363(13.028) +15.932(20.021) +500 +β0 +21.969(19.410) +24.553(27.016) +25.836(26.973) +14.997(8.513) +17.765(10.305) +18.207(11.352) +β1 +9.430(7.378) +11.483(9.900) +12.761(9.607) +7.438(6.724) +8.232(6.798) +8.674(6.646) +β2 +11.958(21.889) +14.100(30.410) +14.360(30.392) +7.330(6.497) +7.672(7.130) +9.201(7.945) +RMSEγ +300 +0.135(0.105) +0.126(0.104) +0.130(0.102) +0.100(0.080) +0.091(0.071) +0.089(0.073) +500 +0.114(0.091) +0.112(0.087) +0.112(0.088) +0.083(0.060) +0.083(0.060) +0.087(0.062) +fTPR +300 +β0 +1.000(0.001) +0.989(0.026) +0.988(0.027) +1.000(0.000) +0.981(0.034) +0.988(0.029) +β1 +1.000(0.001) +0.976(0.048) +0.990(0.036) +1.000(0.001) +0.972(0.055) +0.989(0.038) +β2 +1.000(0.004) +0.981(0.053) +0.995(0.019) +1.000(0.001) +0.964(0.109) +0.990(0.043) +500 +β0 +1.000(0.000) +0.990(0.029) +0.996(0.010) +1.000(0.000) +0.992(0.018) +0.991(0.020) +β1 +1.000(0.000) +0.994(0.027) +0.991(0.026) +1.000(0.000) +0.993(0.028) +0.986(0.031) +β2 +1.000(0.000) +0.992(0.030) +0.996(0.013) +1.000(0.000) +0.993(0.026) +0.993(0.026) +fTNR +300 +β0 +0.002(0.005) +0.490(0.326) +0.661(0.301) +0.002(0.003) +0.601(0.321) +0.728(0.314) +β1 +0.003(0.004) +0.674(0.307) +0.768(0.268) +0.003(0.003) +0.830(0.233) +0.871(0.234) +β2 +0.003(0.006) +0.715(0.300) +0.802(0.272) +0.003(0.002) +0.840(0.225) +0.891(0.195) +500 +β0 +0.004(0.008) +0.589(0.309) +0.706(0.212) +0.003(0.004) +0.709(0.252) +0.754(0.270) +β1 +0.003(0.005) +0.746(0.299) +0.861(0.179) +0.004(0.007) +0.905(0.119) +0.925(0.130) +β2 +0.004(0.005) +0.753(0.303) +0.853(0.194) +0.003(0.003) +0.912(0.064) +0.912(0.159) +20 + +Table 4: Scenario I under Case 3: mean (sd) based on 100 replicates. For the quantile-based methods, τ = 0.3. +n +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +ISE0(×102) +300 +β0 +2.868(1.821) +0.249(0.988) +0.195(0.940) +0.472(0.311) +0.014(0.015) +0.007(0.008) +β1 +5.730(3.193) +2.993(5.608) +2.583(4.649) +1.602(1.101) +0.067(0.252) +0.031(0.136) +β2 +3.774(3.799) +1.328(4.369) +1.634(4.465) +0.789(0.965) +0.013(0.082) +0.066(0.476) +500 +β0 +2.397(1.708) +0.031(0.177) +0.029(0.186) +0.245(0.127) +0.013(0.012) +0.008(0.024) +β1 +3.791(2.452) +1.115(2.226) +1.031(1.968) +0.832(0.695) +0.008(0.020) +0.002(0.010) +β2 +2.469(1.873) +0.332(1.145) +0.371(1.290) +0.345(0.259) +0.003(0.005) +0.001(0.004) +ISE1(×102) +300 +β0 +10.983(7.331) +13.227(10.367) +12.700(9.883) +2.060(1.673) +1.182(1.048) +1.132(0.973) +β1 +19.786(23.473) +22.760(25.114) +22.748(27.854) +8.944(11.601) +8.294(11.829) +8.141(11.990) +β2 +4.117(3.009) +4.802(5.488) +5.202(5.639) +0.979(0.668) +0.455(0.371) +0.502(0.453) +500 +β0 +8.707(4.691) +9.069(6.688) +9.654(6.500) +1.045(0.593) +0.570(0.384) +0.679(0.440) +β1 +12.106(14.426) +15.345(16.454) +15.402(16.512) +5.110(5.191) +4.649(4.600) +4.803(4.928) +β2 +3.021(1.586) +3.097(2.661) +3.726(3.473) +0.472(0.315) +0.274(0.229) +0.344(0.307) +RMSEγ +300 +0.221(0.087) +0.221(0.083) +0.219(0.084) +0.028(0.022) +0.021(0.017) +0.022(0.017) +500 +0.209(0.072) +0.210(0.073) +0.210(0.073) +0.019(0.013) +0.015(0.010) +0.016(0.012) +fTPR +300 +β0 +1.000(0.000) +0.991(0.018) +0.992(0.019) +1.000(0.000) +1.000(0.000) +1.000(0.000) +β1 +1.000(0.000) +0.982(0.048) +0.976(0.076) +1.000(0.000) +0.997(0.017) +0.998(0.014) +β2 +1.000(0.000) +0.996(0.013) +0.991(0.022) +1.000(0.000) +1.000(0.000) +1.000(0.000) +500 +β0 +1.000(0.000) +0.995(0.011) +0.993(0.017) +1.000(0.000) +1.000(0.000) +1.000(0.000) +β1 +1.000(0.000) +0.979(0.057) +0.972(0.059) +1.000(0.000) +1.000(0.001) +1.000(0.001) +β2 +1.000(0.000) +0.997(0.011) +0.991(0.024) +1.000(0.000) +1.000(0.000) +1.000(0.000) +fTNR +300 +β0 +0.001(0.002) +0.825(0.204) +0.829(0.182) +0.001(0.003) +0.776(0.043) +0.793(0.035) +β1 +0.000(0.001) +0.718(0.262) +0.770(0.201) +0.001(0.001) +0.909(0.080) +0.937(0.049) +β2 +0.001(0.001) +0.890(0.174) +0.896(0.138) +0.001(0.001) +0.933(0.030) +0.933(0.057) +500 +β0 +0.001(0.001) +0.863(0.080) +0.886(0.114) +0.002(0.003) +0.785(0.035) +0.809(0.043) +β1 +0.000(0.001) +0.809(0.173) +0.854(0.168) +0.001(0.001) +0.943(0.030) +0.955(0.020) +β2 +0.000(0.001) +0.930(0.085) +0.938(0.094) +0.002(0.002) +0.939(0.014) +0.948(0.012) +21 + +Table 5: Scenario I under Case 3: mean (sd) based on 100 replicates. For the quantile-based methods, τ = 0.5. +n +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +ISE0(×102) +300 +β0 +2.697(1.892) +0.079(0.384) +0.045(0.243) +0.485(0.301) +0.018(0.030) +0.007(0.006) +β1 +4.981(3.161) +2.547(5.349) +2.135(5.060) +1.628(1.227) +0.053(0.175) +0.065(0.245) +β2 +3.301(2.561) +0.888(2.575) +1.119(2.698) +0.794(0.705) +0.014(0.053) +0.006(0.044) +500 +β0 +2.038(1.510) +0.113(0.483) +0.013(0.105) +0.263(0.153) +0.010(0.009) +0.005(0.005) +β1 +3.587(2.548) +1.189(1.921) +0.808(1.555) +0.936(0.605) +0.020(0.083) +0.020(0.092) +β2 +2.181(1.581) +0.320(1.199) +0.313(1.237) +0.376(0.320) +0.003(0.005) +0.001(0.002) +ISE1(×102) +300 +β0 +9.413(6.606) +10.574(8.484) +10.629(8.146) +2.130(1.724) +1.086(0.929) +1.060(0.937) +β1 +15.693(18.630) +18.776(21.860) +17.804(21.592) +6.430(8.397) +5.291(7.408) +5.119(7.025) +β2 +4.031(2.731) +4.373(3.918) +4.883(4.300) +0.902(0.638) +0.433(0.357) +0.445(0.375) +500 +β0 +7.600(4.344) +7.386(5.456) +8.241(5.269) +1.167(0.643) +0.591(0.423) +0.633(0.433) +β1 +11.086(11.832) +12.879(12.976) +13.214(12.947) +4.871(5.566) +3.898(4.967) +4.022(4.662) +β2 +2.667(1.868) +2.783(2.685) +3.349(3.062) +0.513(0.380) +0.237(0.178) +0.272(0.196) +RMSEγ +300 +0.066(0.048) +0.065(0.047) +0.064(0.046) +0.029(0.022) +0.037(0.030) +0.020(0.017) +500 +0.053(0.034) +0.053(0.037) +0.054(0.038) +0.020(0.015) +0.014(0.010) +0.015(0.011) +fTPR +300 +β0 +1.000(0.000) +0.996(0.011) +0.995(0.012) +1.000(0.000) +1.000(0.000) +1.000(0.000) +β1 +1.000(0.000) +0.988(0.038) +0.985(0.033) +1.000(0.000) +0.999(0.011) +1.000(0.004) +β2 +1.000(0.000) +0.996(0.011) +0.992(0.019) +1.000(0.000) +1.000(0.001) +1.000(0.000) +500 +β0 +1.000(0.000) +0.997(0.009) +0.995(0.010) +1.000(0.000) +1.000(0.000) +1.000(0.000) +β1 +1.000(0.000) +0.990(0.032) +0.983(0.036) +1.000(0.000) +1.000(0.001) +0.999(0.004) +β2 +1.000(0.000) +0.998(0.006) +0.994(0.013) +1.000(0.000) +1.000(0.000) +1.000(0.000) +fTNR +300 +β0 +0.000(0.001) +0.834(0.104) +0.860(0.132) +0.001(0.002) +0.772(0.049) +0.796(0.030) +β1 +0.000(0.001) +0.735(0.236) +0.802(0.187) +0.001(0.001) +0.909(0.083) +0.926(0.075) +β2 +0.001(0.001) +0.890(0.146) +0.901(0.126) +0.001(0.001) +0.928(0.041) +0.944(0.023) +500 +β0 +0.001(0.002) +0.814(0.155) +0.897(0.067) +0.002(0.002) +0.787(0.033) +0.813(0.029) +β1 +0.000(0.001) +0.762(0.239) +0.878(0.144) +0.001(0.001) +0.937(0.048) +0.947(0.036) +β2 +0.000(0.001) +0.906(0.137) +0.948(0.079) +0.002(0.002) +0.940(0.013) +0.950(0.009) +22 + +Table 6: Scenario I under Case 3: mean (sd) based on 100 replicates. For the quantile-based methods, τ = 0.7. +n +Alt.1 +Alt.2 +Alt.3 +Alt.4 +Alt.5 +Proposed +ISE0(×102) +300 +β0 +2.974(2.044) +0.173(0.544) +0.071(0.374) +0.449(0.315) +0.015(0.015) +0.010(0.011) +β1 +5.555(4.347) +2.825(6.555) +2.698(6.350) +1.609(1.076) +0.102(0.370) +0.100(0.398) +β2 +3.464(2.341) +0.506(1.962) +0.994(2.512) +0.794(0.744) +0.032(0.176) +0.046(0.265) +500 +β0 +2.255(1.649) +0.087(0.289) +0.116(0.555) +0.244(0.126) +0.011(0.011) +0.005(0.005) +β1 +4.016(3.093) +1.590(3.059) +1.404(2.473) +0.845(0.652) +0.013(0.094) +0.007(0.064) +β2 +2.466(2.001) +0.448(1.402) +0.591(1.596) +0.362(0.262) +0.003(0.004) +0.001(0.001) +ISE1(×102) +300 +β0 +10.082(7.045) +12.193(9.736) +11.593(8.968) +1.946(1.971) +1.238(1.487) +1.221(1.371) +β1 +17.222(21.815) +21.428(26.327) +20.306(24.611) +6.689(6.976) +6.246(6.744) +6.558(7.101) +β2 +4.586(2.880) +5.329(3.963) +5.427(4.056) +0.830(0.648) +0.459(0.457) +0.460(0.400) +500 +β0 +8.083(4.343) +8.494(5.980) +8.496(5.556) +1.015(0.566) +0.621(0.450) +0.667(0.455) +β1 +12.375(13.481) +14.495(14.698) +14.452(15.006) +5.086(4.879) +4.556(4.963) +4.979(5.090) +β2 +3.081(2.565) +3.393(3.144) +3.630(3.384) +0.460(0.256) +0.294(0.292) +0.315(0.262) +RMSEγ +300 +0.190(0.087) +0.192(0.085) +0.192(0.086) +0.030(0.022) +0.021(0.016) +0.021(0.017) +500 +0.220(0.064) +0.221(0.066) +0.221(0.066) +0.019(0.014) +0.015(0.011) +0.016(0.012) +fTPR +300 +β0 +1.000(0.000) +0.990(0.021) +0.992(0.015) +1.000(0.000) +1.000(0.000) +1.000(0.000) +β1 +1.000(0.000) +0.974(0.107) +0.985(0.031) +1.000(0.000) +1.000(0.002) +1.000(0.004) +β2 +1.000(0.000) +0.990(0.022) +0.989(0.022) +1.000(0.000) +1.000(0.000) +1.000(0.000) +500 +β0 +1.000(0.000) +0.995(0.012) +0.996(0.009) +1.000(0.000) +1.000(0.000) +1.000(0.000) +β1 +1.000(0.000) +0.993(0.022) +0.990(0.022) +1.000(0.000) +1.000(0.000) +1.000(0.001) +β2 +1.000(0.000) +0.998(0.009) +0.994(0.016) +1.000(0.000) +1.000(0.000) +1.000(0.000) +fTNR +300 +β0 +0.001(0.002) +0.817(0.201) +0.854(0.145) +0.001(0.002) +0.770(0.041) +0.788(0.034) +β1 +0.000(0.001) +0.737(0.292) +0.793(0.190) +0.001(0.001) +0.908(0.083) +0.925(0.080) +β2 +0.001(0.002) +0.897(0.167) +0.884(0.157) +0.001(0.002) +0.924(0.043) +0.933(0.055) +500 +β0 +0.001(0.001) +0.820(0.151) +0.837(0.172) +0.002(0.004) +0.786(0.037) +0.812(0.030) +β1 +0.001(0.001) +0.766(0.263) +0.828(0.201) +0.001(0.001) +0.943(0.024) +0.953(0.028) +β2 +0.001(0.002) +0.892(0.149) +0.917(0.116) +0.002(0.002) +0.942(0.013) +0.951(0.009) +23 + +IV. Additional data analysis results +We conduct exploratory regression analysis with the proposed penalty. For the lack-of-fit, we consider the mean- +based (Alt.3) and the proposed quantile-based. With the proposed approach, we consider τ = 0.3, 0.5, 0.7. In the left +panel of Figure 5, we plot the estimated densities. It is observed that the residuals are left-skewed, which suggests the +sensibility of quantile-based analysis. Different quantiles lead to different results, which has been commonly observed +in the literature. In addition, the mean estimation is closer to the proposed estimation with τ = 0.3, compared to the +other two quantile values. +−0.4 +−0.2 +0.0 +0.2 +0 +2 +4 +6 +8 +10 +12 +Density +τ = 0.3 +τ = 0.5 +τ = 0.7 +OLS +−1 +0 +1 +2 +−1 +0 +1 +2 +Sample +Estimated +Figure 5: Left: estimated densities of residuals from Alt.3 and the proposed method with τ = 0.3, 0.5, 0.7. Right: Lack-of-fit diagnostic QQ plot. +We also conduct model diagnostics using a QQ plot to intuitively assess model fitting. Specifically, we first +randomly generate ˘τ from the uniform distribution on [0,1]. We then fit data using the proposed method with quantile +˘τ and obtain estimator (ˆb(˘τ), ˆγ(˘τ)). Next, we generate the response from the model ˘y = ψ⊤ ˆb(˘τ) + z⊤ ˆγ(˘τ), where (ψ, z) +are randomly selected from the original data. We repeat this process 100 times and obtain a sample of 100 simulated +fat values. The right panel of Figure 5 gives the QQ plot for the simulated and observed fat contents. Most points are +very close to the 45-degree line, which suggests satisfactory model fitting. +24 + diff --git a/qdE2T4oBgHgl3EQfKwYZ/content/tmp_files/load_file.txt b/qdE2T4oBgHgl3EQfKwYZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34a1babc25cede88d97425d6c934aa051b3bd8d1 --- /dev/null +++ b/qdE2T4oBgHgl3EQfKwYZ/content/tmp_files/load_file.txt @@ -0,0 +1,2778 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf,len=2777 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='03705v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='ME] 9 Jan 2023 Locally sparse quantile estimation for a partially functional interaction model Weijuan Lianga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Qingzhao Zhangb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Shuangge Mac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='∗ aSchool of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Renmin University of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' China bDepartment of Statistics and Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' School of Economics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The Wang Yanan Institute for Studies in Economics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' and Fujian Key Lab of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Xiamen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Xiamen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' China cDepartment of Biostatistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Yale School of Public Health,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Connecticut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' USA Abstract Functional data analysis has been extensively conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In this study, we consider a partially functional model, under which some covariates are scalars and have linear effects, while some other variables are functional and have unspecified nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Significantly advancing from the existing literature, we consider a model with inter- actions between the functional and scalar covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' To accommodate long-tailed error distributions which are not uncommon in data analysis, we adopt the quantile technique for estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' To achieve more interpretable estimation, and to accommodate many practical settings, we assume that the functional covariate effects are locally sparse (that is, there exist subregions on which the effects are exactly zero), which naturally leads to a variable/model selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We propose respecting the “main effect, interaction” hierarchy, which postulates that if a subregion has a nonzero effect in an interaction term, then its effect has to be nonzero in the corresponding main functional effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For estimation, identification of local sparsity, and respect of the hierarchy, we propose a penalization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' An effective computational algorithm is developed, and the consistency properties are rigorously established under mild regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Simulation shows the practical effectiveness of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The analysis of the Tecator data further demonstrates its practical applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Overall, this study can deliver a novel and practically useful model and a statistically and numerically satisfactory estimation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Keywords: Partially functional model, interaction analysis, locally sparse estimation, robust estimation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Introduction Functional data analysis has become routine in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' A popular regression setting has a scalar response and functional covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In practice, we may directly observe the functional covariates or their realizations at discrete observational (usually time or space) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In the latter case, estimation of the functional covariates may be first needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For this regression setting, there have been extensive methodological, computational, and theoretical devel- opments as well as data analyses [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In particular, both mean and robust estimations have been developed [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As a natural extension of the aforementioned model, in a partially functional model, there are two types of co- variates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The first type of covariates is functional, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, there are also scalar covariates with linear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Such a model shares some similar spirit with the partially linear regression [8] but may be more complicated in multiple aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As a “natural next step”, we further consider the model with interactions between the functional and scalar covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Interaction is a “basic” concept in data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' However, most of the existing inter- action analyses are limited to parametric covariate effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In the literature, there are a handful of studies that examine interactions in the partially linear models [9, 10], and statistical and computational analysis of such interactions has been shown to be highly nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' To the best of our knowledge, there has been no interaction analysis with partially functional models that consist of two distinct types of covariate effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' ∗Corresponding author Email addresses: weijuanliang@yeah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='net (Weijuan Liang), qzzhang@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='cn (Qingzhao Zhang), shuangge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='ma@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='edu (Shuangge Ma) Preprint submitted to Elsevier January 11, 2023 For the estimation of functional models, both mean and quantile regression methods have been developed, accom- modating “regular” and long-tailed error distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In this article, we consider data with long-tailed errors and quantile estimation, which can be technically more challenging than mean estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In the existing (both quantile and mean regression) studies, it is commonly assumed that the functional covariate effects are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Without addi- tional assumptions/constraints, the estimates are nonzero everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In the past few years, there has been a strong advocacy on locally sparse estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Under such an estimation, there exist continual subregions, on which the estimates are exactly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In terms of both concept and statistical techniques, this has a strong tie with the sparse esti- mation for parametric covariate effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It has been argued that sparse estimation in general can be more interpretable and more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Sparse estimation is “naturally equivalent to” variable/model selection, for which regularization especially penalization techniques have been extensively developed in the past decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Examples of penalized sparse estimation for functionals include [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' If there are no interactions in the model, conceptually, some of the existing penalized sparse methods for function- als can be adapted to the partially functional models, although we note that there has been very limited research in this aspect [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' When interactions are present, however, these methods may lead to a violation of the “main effect, interaction” variable selection hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This hierarchy has been strongly stressed in the recent parametric interaction analysis studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Under this hierarchy, if an interaction effect is identified, then one or both of the corresponding main effects have to be identified, corresponding to the weak and strong hierarchy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It has been argued that in interaction analysis, this hierarchy is statistically sensible and necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the specific model we are interested in, this hierarchy means that, for any specific subregion, if a functional effect is nonzero in an interaction term, then the corresponding main functional effect must be nonzero in this subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This brings additional constraints and complexity to estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' To the best of our knowledge, there is no existing estimation technique that can respect this hierarchy in estimation for our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This study may complement and advance the existing literature in multiple important ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' First, a novel model is developed, which can accommodate not only two distinct types of covariate effects but more importantly their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Such extensions are natural and strongly motivated by practical data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This model includes multiple existing models as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Second, we consider quantile estimation, which is also motivated by many practical data settings and can be more challenging than mean estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It is noted that the proposed model and penalized estimation can also be coupled with mean squares loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Third, locally sparse estimation is conducted, which can lead to more interpretable and more reliable results than those without sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Fourth, as a major advancement, we develop an estimation approach that respects the “main effect, interaction” hierarchy, making this study more aligned with parametric interaction analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Last but not least, this study delivers a useful tool for data considered in Section 4 and those alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Overall, with the significant statistical developments and strong application potential, this study is warranted beyond the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In Section 2, we first describe the data setting, proposed model, and estimation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' An effective computational algorithm is developed, and statistical properties are then rigorously established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Practical performance of the proposed approach is examined using simulation (Section 3) and data analysis (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The article concludes with brief discussions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Additional theoretical developments and numerical results are presented in the Appendix and Supplemental Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Data and model settings Consider a random sample of size n: {Xi(t), zi, yi}n i=1, where Xi(t) is a functional covariate, zi = (zi1, · · · , ziq)⊤ is a q-dimensional vector of scalar covariates, and yi is a scalar response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The proposed model, estimation approach, and statistical and computational properties can be easily extended to data with multiple functional covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Assume that Xi(t), i = 1, · · · , n are independent realizations of an unknown smooth and square-integrable function X(t) on the domain [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Without loss of generality, assume that the functional covariate, scalar covariates, and scalar response have been centered to mean zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Consider the partially functional interaction model: yi = � T 0 Xi(t)β∗ 0(t)dt + q � k=1 zik � T 0 Xi(t)β∗ k(t)dt + q � k=1 zikγ∗ k + ǫi, (1) 2 where β∗ k(t)’s for k = 0, 1, · · · , q are smooth and square-integrable coefficient functions, γ∗ k’s are scalar coefficients of zi, and the error terms ǫi’s are independent of (Xi(t), zi) and satisfy Pr(ǫi ≤ 0|Xi(t), zi) = τ for τ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Note that this assumption accommodates long-tailed error distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As described above, we consider the setting with local sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Take β∗ 0(t) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We say that β∗ 0(t) is locally sparse if there exists a subregion I ⊂ [0, T], and β∗ 0(t) = 0 for all t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Accordingly, X(t) has no contribution to the response for t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Here, we note that there can be more than one region with zero effects, and the region location information is not known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, the proposed approach is flexible enough to also accommodate the case with functional covariate effects being nonzero everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Local sparsity can assist in distinguishing regions with and without effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' And it is easy to see the natural connection with variable selection for parametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The proposed model is more complicated than some existing alternatives as local sparsity may apply to both the main effect and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' With the connection between local sparsity and variable selection, we naturally encounter the “main effect, interaction” variable selection hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In our analysis, we are not interested in the sparsity in γ∗ k’s (although we note that extending to accommodate potential sparsity in γ∗ k’s is relatively easy with the parametric nature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As such, the hierarchy boils down to the relationship between the subregions of β∗ k’s (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=', q) with zero/nonzero effects and those of β∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' More specifically, we say that the hierarchy is satisfied if and only if for any subregion I ⊂ [0, T], if β∗ 0(t) = 0 for all t ∈ I, then β∗ k(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We note that a more rigorous definition should rule out any measure zero set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Estimation For estimating the unknown parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' we propose minimizing the penalized objective function: Q(β(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' γ) =1 n n � i=1 ρτ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edyi − � T 0 Xi(t)β0(t)dt − q � k=1 zik � T 0 Xi(t)βk(t)dt − q � k=1 zikγk \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 + q � k=1 κ T � T 0 pλ1(|βk(t)|)dt + κ T � T 0 pλ2(∥β(t)∥2)dt + η q � k=0 � T 0 β′′2 k (t)dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (2) where ρτ(u) = u(τ−I(u < 0)) is the quantile loss function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' ∥β(t)∥2 = (�q k=0 β2 k(t))1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' pλj(·)’s are penalty functions with tuning parameters λ j’s (for j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' κ is a modifier and will be discussed below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' η is a tuning parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' and β′′ k (t) is the second-order derivative of βk(t) with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Various penalty functions can be adopted here, and pλ1(·) and pλ2(·) do not need to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In our theoretical and numerical developments, we adopt MCP [17] for both pλ1(·) and pλ2(·), where pλj(t) = λ j � |t| 0 � 1 − x λjξ � + dt, λ j ≥ 0, and ξ > 0 is a regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It is expected that, with SCAD and some other penalties, properties will be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In (2), the first term is a “standard” lack-of-fit based on the quantile technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Under the smoothness assumption, the last penalty on derivative has been routinely adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Here we note that a stronger smoothness assumption and correspondingly a higher order derivative can also be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The most significant and innovative advancement is the first and second penalty terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In “ordinary” locally sparse estimation, penalties similar to � T 0 pλ1(|βk(t)|)dt have been adopted [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In our estimation, new challenges are brought by the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Motivated by the sparse group penaliza- tion for parametric models [18], we treat (β0, β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=', βq) as a “group”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For a subregion, κ T � T 0 pλ2(∥β(t)∥2)dt determines whether this group of functionals has overall zero effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' If not, then �q k=1 κ T � T 0 pλ1(|βk(t)|)dt determines which of the q interaction effects are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Note that here no penalty is applied to β0(t), ensuring that the corresponding estimate is nonzero, and hence the hierarchy is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Directly optimizing (2) is challenging with the infinite dimension of the unknown functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Here we adopt a popular B-spline expansion-based technique, which can be preferred with its compact support property, computational efficiency, and satisfactory performance with capturing local sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote HdMn as the linear space spanned by a set of order d + 1 B-spline basis functions B1(t), · · · , BMn+d(t), each with Mn + 1 equally spaced knots 0 = t0 < t1 < · · · < tMn = T in the domain [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In (2), second-order derivatives are taken, corresponding to d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We refer to [19] for the construction of B-spline basis functions and related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote B(t) = (B1(t), · · · , BMn+d(t))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Then we parameterize coefficient functions βk(t) = B(t)⊤bk for k = 0, · · · , q, where bk = (bk,1, · · · , bk,Mn+d)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let Z = (z1, · · · , zn)⊤, X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=', xn)⊤ be the n × (Mn + d) matrix with the (i, j)th entry being xi j = � T 0 Xi(t)B j(t)dt, and 3 U = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=', un)⊤ be the n × q(Mn + d) matrix with ui = zi ⊗ xi, where ⊗ is the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Further denote Ψ = (X, U), which is n × qn with qn = (q + 1) × (Mn + d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The first term of (2) can be rewritten as: 1 n n � i=1 ρτ(yi − ψ⊤ i b − z⊤ i γ), (3) where b = (b⊤ 0 , · · · , b⊤ q )⊤ and γ = (γ1, · · · , γq)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In Lemma 1 (Appendix), we examine approximating the sparse group penalty under this basis expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In particular, setting the modifier κ = Mn, we have: q � k=1 Mn T � T 0 pλ1(|βk(t)|)dt + Mn T � T 0 pλ2(∥β(t)∥2)dt ≈ q � k=1 Mn � l=1 pλ1 �∥bk∥Wl � + Mn � l=1 pλ2 �∥b∥Wl � , (4) where Wl is the (Mn + d) × (Mn + d) matrix with the (i, j)th entry wli j = Mn T � tl tl−1 Bi(t)B j(t)dt if l ≤ i, j ≤ l + d, and wli j = 0 otherwise, ∥bk∥Wl = (b⊤ k Wlbk)1/2, and ∥b∥Wl = (�q k=0 b⊤ k Wlbk)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let V be the (Mn + d) × (Mn + d) matrix with the (i, j)th entry vi j = � T 0 d2Bi(t) dt2 d2B j(t) dt2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Then, q � k=0 � T 0 β′′2 k (t)dt = q � k=0 b⊤ k Vbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (5) With (3), (4) and (5), we propose estimating (b∗, γ∗) by minimizing the following objective function: Q(b, γ) =1 n n � i=1 ρτ � yi − ψ⊤ i b − z⊤ i γ � + q � k=1 Mn � l=1 pλ1 �∥bk∥Wl � + Mn � l=1 pλ2 �∥b∥Wl � + η q � k=0 b⊤ k Vbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (6) Denote (ˆb, ˆγ) as the minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Then the estimate of β∗ k(t) is ˆβk(t) = B⊤(t)ˆbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Computation To accommodate the non-differentiable quantile loss function, we resort to the majorize-minimization (MM) tech- nique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, we adopt the local quadratic approximation (LQA) technique for the sparse group penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The proposed algorithm is iterative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' At the (m + 1)th iteration, with estimate b(m) k from the mth iteration, we have: pλ1 �∥bk∥Wl � ≈ pλ1(∥b(m) k ∥Wl) + 1 2 p′ λ1(∥b(m) k ∥Wl) ∥b(m) k ∥Wl (∥bk∥2 Wl − ∥b(m) k ∥2 Wl) = 1 2 p′ λ1(∥b(m) k ∥Wl) ∥b(m) k ∥Wl ∥bk∥2 Wl + G0(b(m) k ), where p′ λ1(t) = λ1(1 − |t|/(λ1ξ))+ is the first-order derivative of pλ1(t), and G0(b(m) k ) is a function of b(m) k and does not depend on bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As such,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' we can obtain the LQA approximation of the sparse group penalty as: q � k=1 Mn � l=1 pλ1 �∥bk∥Wl � + Mn � l=1 pλ2 �∥b∥Wl � ≈ b⊤ ˘W(m)b + G1(b(m)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (7) 4 where ˘W(m) = diag( ˘W(m) 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' ˘W(m) q ) is the qn × qn block diagonal matrix with: ˘W(m) 0 = 1 2 Mn � l=1 p′ λ2 � ∥b(m)∥Wl � ∥b(m)∥Wl Wl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' and ˘W(m) k = 1 2 Mn � l=1 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed p′ λ1(∥b(m) k ∥Wl) ∥b(m) k ∥Wl + p′ λ2 � ∥b(m)∥Wl � ∥b(m)∥Wl \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8Wl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' for k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' and G1(b(m)) is free of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let Φ be the n × dn matrix with the ith row being φi = (x⊤ i , u⊤ i , z⊤ i )⊤ ∈ Rdn and dn = qn + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote the coefficient vector as ω = (b⊤, γ⊤)⊤ ∈ Rdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let ˜V = diag(V, · · · , V, 0q) and ˜W(m) = diag( ˘W(m), 0q) be dn × dn block-diagonal matrices, where 0q is the q × q matrix with all entries being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' With the MM algorithm, at the (m + 1)th iteration, given the residual value r(m) = y − Φω(m), the quantile loss is majorized at r(m) = (r(m) 1 , · · · , r(m) n )⊤ by the quadratic function: ξ(r|r(m)) = 1 n n � i=1 1 4 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed r2 i ̺ + |r(m) i | + (4τ − 2)ri + c \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 , where r = (r1, · · · , rn)⊤, ̺ is a small perturbation, and c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The overall objective function at the (m + 1)th iteration is: ˜Q(ω|ω(m)) = ξ(r|r(m)) + ω⊤ ˜W(m) ˜ω + ηω⊤ ˜Vω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The first-order derivative of ˜Q(ω|ω(m)) with respect to ω is: ˜Q′(ω|ω(m)) = 1 2n n � i=1 φi � 1 − 2τ − ri/(̺ + |r(m) i |) � + 2 ˜W(m)ω + 2η ˜Vω = 1 2nΦ⊤v̺(ω|ω(m)) + 2 ˜W(m)ω + 2η ˜Vω, where v̺(ω|ω(m)) = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed1 − 2τ − r1 ̺ + |r(m) 1 | , · · · , 1 − 2τ − rn ̺ + |r(m) n | \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⊤ is a length-n column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The second-order derivative of ˜Q(ω|ω(m)) with respect to ω is: ˜Q′′(ω|ω(m)) = 1 2n n � i=1 φiφ⊤ i ̺ + |r(m) i | + 2 ˜W(m) + 2η ˜V = 1 2nΦ⊤R(m)Φ + 2 ˜W(m) + 2η ˜V, where R(m) = diag \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 ̺ + |r(m) 1 | , · · · , 1 ̺ + |r(m) n | \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 is an n × n diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Then the Gauss-Newton step direction is: ∆(m) ̺ (ω|ω(m)) = − � ˜Q′′(ω|ω(m)) �−1 ˜Q′(ω|ω(m)) = − � Φ⊤R(m)Φ + 4n ˜W(m) + 4nη ˜V �−1 � Φ⊤v̺(ω|ω(m)) + 4n ˜W(m)ω + 4nη ˜Vω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (8) Overall, the proposed computational algorithm proceeds as follows: Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Initialize ˆω as ˆω(0) = (Φ⊤Φ + nη ˜V)−1Φ⊤y and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Given ˆω(m), compute ˜W(m), R(m), and v̺( ˆω(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Update: ˆω(m+1) = ˆω(m) + ∆(m) ̺ ( ˆω(m)), and m = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 5 Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Repeat 2 until convergence, which is concluded if the norm of the difference between the estimates from two consecutive iterations is smaller than a prespecified cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The final estimate of ω is obtained by further setting the elements of ˆω with absolute values smaller than a prespecified threshold to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This algorithm is built on the MM and LQA techniques, both of which have been well examined in published literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Convergence of the algorithm can be established following the literature and is achieved in all of our numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' With the LQA, finite iterations cannot lead to sparse estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Following published studies, a cutoff (whose value is not crucial) is imposed in Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In our numerical study, we use 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As in the literature, the value of Mn is also not crucial since the smoothness of estimation is controlled by the roughness penalty, as opposed to the number of knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In our numerical study, we use cubic B-splines with 71 equally spaced knots to estimate β(t)’s, following [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Note that other knot placement strategies can be considered such as some data-driven methods putting knots at certain quantiles of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Following [20, 21], we set λ2 = � q + 1λ1 and ξ = 6 and perform a grid search for the optimal (η, λ1) based on prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' More details are provided in the numerical studies below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The R code implementing the proposed algorithm is publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='com/weijuanliang12138/SHLoS- R-Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Theoretical properties Let fi(·) and Fi(·) be the probability density function and distribution function of ǫi given (Xi(t), zi), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote Bn = diag{f1(0), · · · , fn(0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We assume the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For i = 1, · · · , n, in a neighborhood of zero, fi is continuous and satisfies 0 < c ≤ fi ≤ C < ∞, where c and C are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, the first-order derivative f ′ i has a uniform upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For k = 0, · · · , q, β∗ k(t) belongs to the H¨older space Cα,ν([0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Specifically, |β∗(α) k (x1) − β∗(α) k (x2)| ≤ C1|x1 − x2|ν for a constant C1, positive integer α, and ν ∈ (0, 1], and for all 0 ≤ x1, x2 ≤ 1, where β∗(α) k (·) is the αth-order derivative of β∗ k(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let r = a + ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Assume r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' There exist positive constants C2 and C3, such that ( � T 0 X(t)2dt)1/2 ≤ C2 < ∞ and |zk| ≤ C3 < ∞ for k = 1, · · · , q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, there exist positive constants C4, C5, C6, and C7, such that C4M−1 n ≤ λmin(n−1ΨΨ⊤) ≤ λmax(n−1ΨΨ⊤) ≤ C5M−1 n , C6 ≤ λmin(n−1 ˇZ ˇZ⊤) ≤ λmax(n−1 ˇZ ˇZ⊤) ≤ C7, where ˇZ = (In − Ψ(Ψ⊤BnΨ)−1Ψ⊤Bn)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Mn = O(n 1 2r+1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Condition 1 is common in the quantile regression literature and weaker than those assumed with mean estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Condition 2 ensures that there exists b∗ k ∈ RMn+d such that supt∈[0,T] |β∗ k(t) − B(t)⊤b∗ k| = O(M−r n ) for k = 0, · · · , q [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Condition 3 is on the covariates and design matrices, which is analogous to those in [13] and [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Condition 4 is also common in the spline literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote the null region of β∗ k(t) as Nk = {t ∈ [0, T] : β∗ k(t) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The asymptotic properties of the proposed estimator can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Under Conditions 1-4, if n− r 2r+1 / min(λ1, λ2) = o(1), max(λ1, λ2) = o(1) and η = o(n−1/2), then there exists a local minimizer (ˆb, ˆγ) of (6), such that for all k = 0, · · · , q with ˆβk(t) = B⊤(t)ˆbk, (1) � T 0 (ˆβk(t) − β∗ k(t))2dt = Op(n−2r/(2r+1)) and ∥ˆγ − γ∗∥2 = Op(n−1/2), (2) ˆβk(t) = 0 for all t ∈ Nk with probability tending to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Proof is provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This theorem establishes the estimation and selection consistency properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It is observed that the convergence rate of ˆβk(t) is n−r/(2r+1), which is optimal [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The convergence rate of ˆγ is free of Mn – the optimal root-n rate is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The selection consistency holds by result (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' With the design of the penalty, the “main effect, interaction” hierarchy is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Simulation Data is generated from the following model: yi = � 1 0 Xi(t)β∗ 0(t)dt + 2 � k=1 zik � 1 0 Xi(t)β∗ k(t)dt + 2 � k=1 zikγ∗ k + ǫi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (9) We consider three different scenarios of coefficient functions corresponding to various levels of sparsity and number of null regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' All of these functions in each scenario satisfy the “main effect, interaction” hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Scenario I: 60% regions of β∗ 10(t) have contribution to the response, and there is a null region in β∗ 10(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The functional main effect is: β∗ 10(t) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 2(1 − t) sin(2π(t + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2)) 0 ≤ t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 < t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7, 2t sin(2π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 ≤ t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the functional interaction effects, we consider: (1) β∗ 11(t) = β∗ 10(t) for t ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3], and β∗ 11(t) = 0 otherwise, (2) β∗ 12(t) = β∗ 10(t) for t ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7, 1], and β∗ 12(t) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' These functions are demonstrated in Figure 1 by black solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Scenario II: 30% regions of the main effect are nonnull regions, and there are four null regions on the entire domain of β∗ 20(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The functional main effect and interactions are defined as: β∗ 20(t) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 5 sin(10π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, −3 sin(10π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 sin(10π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8, 0 otherwise, β∗ 21(t) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 2(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='25)2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='052 − 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, 5 sin(10π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6, 0 otherwise, and β∗ 22(t) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 sin(10π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6, 4(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='75)2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='052 − 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8, 0 otherwise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' These functions are presented in Figure 2 by black solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Scenario III: 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5% regions of β∗ 30(t) have nonzero effects on the response, and there are eight null regions on the entire domain of the main effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The functional main effect and interactions are defined as: β∗ 30(t) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 4(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1375)2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='01252 − 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='125 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='15, 7 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='175)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='175 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2, −6 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='325)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='325 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='35, 8 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='625, −10 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='725, 5 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='825, −7 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='875)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='875 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='9, 0 otherwise, β∗ 31(t) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 10 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='125)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='125 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='15, 6 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='325)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='325 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='35, 8(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7125)2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='01252 − 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='725, 9 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='875)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='875 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='9, 0 otherwise, 7 and β∗ 32(t) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 5 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='175)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='175 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2, 10(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6125)2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='01252 − 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='625, 7 sin(40π(t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 < t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='825, 0 otherwise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' These functions are demonstrated in Figure 3 by black solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −2 −1 0 1 2 ε ~ N(0, σ2) t β0(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −2 −1 0 1 2 ε ~ N(0, σ2) t β1(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −2 −1 0 1 2 ε ~ N(0, σ2) t β2(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −2 −1 0 1 2 ε ~ t(3) t β0(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −2 −1 0 1 2 ε ~ t(3) t β1(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −2 −1 0 1 2 ε ~ t(3) t β2(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed Figure 1: Average of ˆβ(t)’s in Scenario I with n = 300 based on 100 replicates under Case 1 (top) and Case 2 (bottom), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Left/middle/right: β0(t)/β1(t)/β2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The scalar covariates z·k, k = 1, 2 (where z·k is the k-th column of Z) are generated independently from the standard normal distribution, and the corresponding coefficient vector is γ∗ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The functional covariate Xi(t) is generated as Xi(t) = � ai jB j(t), where ai j’s are generated from a normal distribution with mean zero and standard deviation 5, and each B j(t) is a B-spline basis function with order 5 and 71 equally spaced knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Consider three distributions for ǫi: Case 1: (homoscedasticity) ǫi follows a normal distribution N(0, σ2), and σ is chosen so that the signal-to-noise ratio equals 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Case 2: (homoscedasticity) ǫi follows a t(3) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Case 3: (heteroscedasticity) ǫi = ( 3 2|zi1 � 1 0 Xi(t)β∗ 1(t)dt|)˜ǫi, where ˜ǫi ∼ N(0, 1) − QN(τ), and QN(τ) denotes the τth quantile of a standard norm distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Note that here the model is misspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For comparison, we consider the following alternatives: (a) Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 adopts the mean squares lack-of-fit and a smooth- ness penalty (which is the last term of the proposed approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As such, it can control smoothness as in many published studies but cannot conduct selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (b) Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 adopts the mean squares lack-of-fit and the “functional MCP penalty + smoothness penalty”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It computes conditional mean of the response and does not have a mechanism to respect the “main effect, interaction” hierarchy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (c) Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 adopts the mean squares lack-of-fit and the same penalty as the pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As such, the only difference lies in the measured conditional quantity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (d) Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 adopts the same quantile-based loss as the proposed approach and the penalty in Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (e) Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 adopts the same quantile-based loss as the proposed approach and the penalty in Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the quantile-based approaches, we set τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 for homoscedasticity errors 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −4 0 2 4 6 8 ε ~ N(0, σ2) t β0(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −4 0 2 4 6 8 ε ~ N(0, σ2) t β1(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −4 0 2 4 6 8 ε ~ N(0, σ2) t β1(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −4 0 2 4 6 8 ε ~ t(3) t β0(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 −4 0 2 4 6 8 ε ~ t(3) t β1(t) True Beta Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed Figure 2: Average of ˆβ(t)’s in Scenario II with n = 300 based on 100 replicates under Case 1 (top) and Case 2 (bottom), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Left/middle/right: β0(t)/β1(t)/β2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (Cases 1 and 2) and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 for heteroscedasticity errors (Case 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We consider sample size n = 300, 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For each simulation replicate, we generate an independent dataset under the same setting with sample size 500 and select the optimal tunings corresponding to the best prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Summary statistics are computed based on 100 independent replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Performance is evaluated using the following criteria: (a) Average integrated squared errors on null region (ISE0): ISE0k = 1 l0k � Nk(ˆβk(t) − β∗ k(t))2dt, where l0k is the length of null region Nk of β∗ k(t), k = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (b) Average integrated squared errors on nonnull region (ISE1): ISE1k = 1 l1k � Nc k (ˆβk(t) − β∗ k(t))2dt, where l1k is the length of nonnull region Nc k of β∗ k(t), k = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (c) Root mean squared errors of γ∗ (RMSEγ): RMSEγ = ∥ˆγ − γ∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (d) Average proportion of nonnull regions that are correctly identified (fTPR), which is the functional counterpart of true positive rate in parametric variable selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (e) Average proportion of null regions that are correctly identified (fTNR), which is the functional counterpart of true negative rate in parametric variable selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The results for Scenario I under Case 1 are provided in Table 1, and those for Scenario I under Cases 2 and 3 are provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, the results for all cases under Scenarios II and III are provided in the supplemental materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Figures 1-3 present the estimated β(t)’s for Scenario I-III under Cases 1 and 2 with n = 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Overall, the findings are highly “as expected”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In particular, when the errors are normally distributed, the mean-based methods can be advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' However, with Cases 2 and 3, the superiority of the quantile-based methods is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, it is observed that introducing local sparsity can improve estimation, and that respecting the hierarchy can further improve selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As a representative example, consider Scenario 1 under Case 2 (Table 3, Appendix) and n = 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The ISE0(×102) for β2 are 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='785, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='577, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='505, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='868, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='400, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='053 for the five alternative and proposed approaches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The corresponding fTNR values are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='003 (Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='715 (Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='802 (Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='003 (Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='840 (Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='891 (proposed), and the fTPR values are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Furthermore, it is observed that, as the proportion of signal regions increases, it gets easier to identify sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed Figure 3: Average of ˆβ(t)’s in Scenario III with n = 300 based on 100 replicates under Case 1 (top) and Case 2 (bottom), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Left/middle/right: β0(t)/β1(t)/β2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Data analysis We analyze the Tecator data which is available from http://lib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='edu/datasets/tecator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In this dataset, there are 215 finely chopped pure meat samples (datasets C, M, and T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For each sample, measurements are available on a spectrometric curve of spectra of absorbances measured at 100 channels with wavelength range 850-1050nm, as well as moisture, fat, and protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The latter three are measured in percent and determined by analytic chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In this analysis, we study how fat can be modeled as a function of the spectrometric curve X(t) with t being the wavelength and the two scalar covariates moisture z1 and protein z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As developed above, we also incorporate the interactions between the spectrometric curve and scalar covariates in modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' By introducing local sparsity, we can potentially distinguish “useful” regions of spectra that are informative for modeling fat from the “noisy” ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Prior to analysis, the range of wavelength t is mapped to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' There are 31 equally spaced knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Following the official guidance of this dataset, we use 129 samples (dataset C) as training for estimation, 43 samples (dataset M) for tuning parameter selection, and 43 samples (dataset T) for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We first conduct exploratory regression analysis and present the findings in Appendix IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Skewed residuals are observed, which justifies quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, there is no obvious lack-of-fit under quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The estimation results for the functional effects are shown in Figure 4, where we consider τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It is observed that the effects are locally sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, the “main effect, interaction” hierarchy is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the scalar effects, the estimates are: (ˆµ, ˆγ1, ˆγ2) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='785, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='068, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='073) for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='899, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='074, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='057) for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='930, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='081, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='038) for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The differences across different quantile values partly justify the need for quantile-based estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We recognize that a single split may not be sufficiently informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As such, we conduct 100 random splittings of the original data, and the sizes of the three sets (under each splitting) are the same as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In Table 2, we present the mean (standard deviation) for each scalar estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The 100 sets of estimated functional effects are available from the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, we also present the results of prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Overall, taking the local sparsity, interpretability pertained to the variable selection hierarchy, and prediction performance into account, the analysis with the proposed approach and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 is recommended as the final one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 10 Table 1: Scenario I under Case 1: mean (sd) based on 100 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the quantile-based methods, τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 Figure 4: Estimated functional effects using the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Discussion In this article, we have considered a more sophisticated functional data analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The most significant ad- vancement comes from the interaction analysis.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='014) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='025(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='005) tially extended to include more complicated interactions (for example, between functional effects) and have higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It will also be of interest to examine more practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Acknowledgements We thank the associate editor and reviewers for careful review and insightful comments.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Wang, Partially linear additive quantile regression in ultra-high dimension, The Annals of Statistics 44 (1) (2016) 288–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Stone, Additive regression and other nonparametric models, The annals of Statistics 13 (2) (1985) 689–705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Lemma 1 and remarks Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Consider Mn + 1 equally spaced knots 0 = t0 < t1 < · · · < tMn = T in the domain [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the smooth functional main effect and interactions, we have: q � k=1 1 T � T 0 pλ1(|βk(t)|)dt + 1 T � T 0 pλ2(∥β(t)∥2)dt = lim Mn→∞ 1 Mn q � k=1 Mn � l=1 pλ1 � Mn 1 2 T − 1 2 ∥βk[l]∥ � + lim Mn→∞ 1 Mn Mn � l=1 pλ2 � Mn 1 2 T − 1 2 ∥β[l]∥ � , where ∥βk[l]∥ = ( � tl tl−1 β2 k(t)dt)1/2 and ∥β[l]∥ = (�q k=0 � tl tl−1 β2 k(t)dt)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This lemma can be derived from Theorem 1 of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It shows that the penalty evaluated over the whole domain is asymptotically equivalent to the sum over a large number of subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This nicely matches the spline basis expansion framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For each subregion, we note that the penalty still has a sparse group form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' As such, the “main effect, interaction” hierarchy is expected to hold for each subregion (and so the whole domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 13 II: Proof of Theorem 1 Let C be a generic positive constant which may take different values under different circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote: g∗(Xi(t), zi) = � T 0 Xi(t)β∗ 0(t)dt + q � k=1 zik � T 0 Xi(t)β∗ k(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Recall that supt∈[0,T] ���β∗ k(t) − B⊤(t)b∗ k ��� = O(M−r n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' With the boundedness Condition 3, we have g∗(Xi(t), zi) = ψ⊤ i b∗ + O(M−r n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote the empirical version of the projection of z·k onto the spline approximation of the functional covariate space as h·k = Ψ ˆ̟k, where z·k is the kth column of Z and ˆ̟k is the minimizer of: min ̟k∈Rqn n � i=1 fi(0)(zik − ψ⊤ i ̟k)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The solution to the above problem is ˆ̟k = (Ψ⊤BnΨ)−1Ψ⊤Bnz·k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let H be the n × q matrix with the kth column being h·k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We define the projection matrix P = Ψ(Ψ⊤BnΨ)−1Ψ⊤Bn ∈ Rn×n, and it is obvious that H = PZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Thus we have ˇZ = (ˇz1, · · · , ˇzn)⊤ = (In − P)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Define ˜zi = n− 1 2 ˇzi ∈ Rq, Ψ2 B = Ψ⊤BnΨ ∈ Rqn×qn, and ˜ψi = Ψ−1 B ψi ∈ Rqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Following [23], we reparameterize the quantile loss function as: ρτ � yi − ψ⊤ i b − z⊤ i γ � = ρτ � ǫi − ˜z⊤ i θ1 − ˜ψ⊤ i θ2 − uni � , where θ1 = √n(γ − γ∗) ∈ Rq, θ2 = ΨB(b − b∗) + Ψ−1 B Ψ⊤BnZ(γ − γ∗) ∈ Rqn and uni = ψ⊤ i b∗ − g∗(Xi(t), zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let θ = (θ⊤ 1 , θ⊤ 2 )⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The objective function under the reparameterization is: ˜Q(θ) =1 n n � i=1 ρτ(ǫi − ˜z⊤ i θ1 − ˜ψ⊤ i θ2 − uni) + q � k=1 Mn � l=1 pλ1(∥bk∥Wl) + Mn � l=1 pλ2(∥b∥Wl) + η q � k=0 b⊤ k Vbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Define: Di(θ) =ρτ(ǫi − ˜z⊤ i θ1 − ˜ψ⊤ i θ2 − uni) − ρτ(ǫi − uni) + (˜z⊤ i θ1 + ˜ψ⊤ i θ2)Dτ(ǫi) − E[ρτ(ǫi − ˜z⊤ i θ1 − ˜ψ⊤ i θ2 − uni) − ρτ(ǫi − uni)], where Dτ(ǫi) = τ − I(ǫi < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We first state the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let dn = qn + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Under Conditions 1-4, for any positive constant L, we have: sup ∥θ∥2≤L √dn 1 dn ������� n � i=1 Di(θ) ������� = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Proof follows that of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 in [23] under Conditions 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let ˜θ1 = √n � ˇZ⊤Bn ˇZ �−1 ˇZDτ(ǫ), where Dτ(ǫ) = (Dτ(ǫ1), · · · , Dτ(ǫn))⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Under Conditions 1-4, we have ∥˜θ1∥2 = Op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Proof follows from that of Lemma 5 (1) in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Proof of Theorem 1 (1) Here we show that there exists a local minimizer ˆθ = (ˆθ⊤ 1 , ˆθ⊤ 2 )⊤ of (6) such that ∥ˆθ∥2 = Op( √Mn) and ∥ˆθ1∥2 = Op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Note that dn = O(Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' To prove ∥ˆθ∥2 = Op( √Mn), it is sufficient to show that, for any δ > 0, there exists a sufficiently large positive constant L such that: P � inf ∥θ∥2≤L √dn ˜Q(θ) > ˜Q(0) � ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1) 14 That is, with probability at least 1 − δ, there exists a local minimizer such that ∥ˆθ∥2 ≤ L √dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We first show that, for a sufficiently large positive L, there exists a positive constant C such that: inf ∥θ∥2=L √dn 1 n n � i=1 � ρτ(ǫi − ˜z⊤ i θ1 − ˜ψ⊤ i θ2 − uni) − ρτ(ǫi − uni) � > CL2dn/n (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2) with probability tending to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' From Lemma 2, we have: sup ∥θ∥2≤L √dn 1 n ������� n � i=1 Di(θ) ������� = sup ∥θ∥2≤L √dn 1 n ������� n � i=1 ρτ(ǫi − ˜z⊤ i θ1 − ˜ψ⊤ i θ2 − uni) − n � i=1 ρτ(ǫi − uni) − n � i=1 E � ρτ(ǫi − ˜z⊤ i θ1 − ˜ψ⊤ i θ2 − uni) − ρτ(ǫi − uni) � + n � i=1 (˜z⊤ i θ1 + ˜ψ⊤ i θ2)Dτ(ǫi) ������� = op(dn/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote Fn1 = n−1 �n i=1 E[ρτ(ǫi − ˜z⊤ i θ1 − ˜ψ⊤ i θ2 − uni) − ρτ(ǫi − uni)] and Fn2 = n−1 �n i=1(˜z⊤ i θ1 + ˜ψ⊤ i θ2)Dτ(ǫi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Following similar arguments as in the proof of Lemma 4 in [23], we can show that for a sufficiently large positive L, Fn1 has asymptotically a lower bound of CL2dn/n and Fn2 = Op(d1/2 n /n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Therefore, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let D denote the domain [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For given λ1, λ2 and Mn, and for each β∗ k(t), we divide D into three parts: the first part D[1] k = {t ∈ D : |β∗ k(t)| ≥ Cξ max(λ1, λ2)} for some constant C > 1, the second part D[2] k = {t ∈ D : β∗ k(t) = 0}, and the third part D[3] k = {t ∈ D : 0 < |β∗ k(t)| < Cξ max(λ1, λ2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Since max(λ1, λ2) → 0 as n → ∞, D[3] k shrinks to the empty set ∅ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Next, we consider the penalty terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' With ∥θ∥2 = O( √dn) and the definition of θ, we have ∥γ − γ0∥2 = O( √dn/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, ∥ΨB(b − b∗)∥2 2 ≤ 2∥θ2∥2 2 + 2∥Ψ−1 B Ψ⊤BnZ(γ − γ0)∥2 2 = O(dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3) The last equality holds because ∥Ψ−1 B Ψ⊤BnZ(γ − γ0)∥2 2 = O(n∥γ − γ0∥2 2) by Conditions 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Then we have ∥bk − b∗ k∥2 = O(dnn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Notice that β∗ k(t) = B⊤(t)b∗ k + O(M−r n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For a subregion Il ⊂ D[1] k , k = 0, · · · , q, with Mn = O(dn), Condition 4, and n− r 2r+1 / min(λ1, λ2) = o(1), we have: ∥b∗ k∥Wl = � Mn T � tl tl−1 β∗2 k (t)dt + O(M−r n ) ≥ Cξ max(λ1, λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Applying some inequality techniques, we can derive ∥bk∥Wl ≥ Cξ max(λ1, λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' With the properties of MCP and C > 1, we have pλ1(∥bk∥Wl) = pλ1(∥b∗ k∥Wl) and pλ2(∥b∥Wl) = pλ2(∥b∗∥Wl) for l satisfying Il ⊂ D[1] k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For a subregion Il ⊂ D[2] k ∩ (∪k′�kD[1] k′ ), by the choice of b∗, we have ∥b∗ k∥Wl = 0 and ∥b∗∥Wl ≥ Cξ max(λ1, λ2), and thus pλ1(∥bk∥Wl) ≥ pλ1(∥b∗ k∥Wl) = 0 and pλ2(∥b∥Wl) = pλ2(∥b∗∥Wl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For a subregion Il ⊂ D[2] k ∩ (∪k′�kD[1] k′ )c, we have ∥b∗∥Wl = 0, and thus pλ1(∥bk∥Wl) ≥ pλ1(∥b∗ k∥Wl) = 0 and pλ2(∥b∥Wl) ≥ pλ2(∥b∗∥Wl) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Summarizing the above three cases, we have: q � k=1 Mn � l=1 pλ1 �∥bk∥Wl � ≥ q � k=1 Mn � l=1 pλ1 � ∥b∗ k∥Wl � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4) and Mn � l=1 pλ1 �∥b∥Wl � ≥ Mn � l=1 pλ1 �∥b∗∥Wl � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5) 15 Also, by the Cauchy-Schwarz inequality and η = o(n−1/2), we have: q � k=0 ηb⊤ k Vbk − q � k=0 ηb∗⊤ k Vb∗ k = q � k=0 η � (bk − b∗ k)⊤V(bk − b∗ k) + 2(bk − b∗ k)⊤Vb∗ k � ≤ O(ηdnn−1 + ηn−1/2) = o(n−1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6) where the inequality follows from the fact that ∥bk − b∗ k∥2 = O(dnn−1/2), λmax(V) = O(d−1 n ) and supj |V· jb∗ k| ≤ Cd−1 n , where Vj· is the jth row of V for j = 1, · · · , Mn + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='6), for ∥θ∥2 = L √dn and a sufficiently large L, we prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Therefore, there exists a local minimizer ˆθ such that ∥ˆθ∥2 = Op( √dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Similar to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3), it follows that ∥ΨB(ˆb − b∗)∥2 = Op( √dn), and thus ∥ˆb − b∗∥2 = Op(dnn−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Then we have: � T 0 (ˆβk(t) − β∗ k(t))2dt ≤ 2 � T 0 (ˆβk(t) − B⊤(t)b∗ k)2dt + 2 � T 0 (B⊤(t)b∗ k − β∗ k(t))2dt = O(d−1 n ∥ˆb − b∗∥2 2) + Op(M−2r n ) = Op(n− 2r 2r+1 ), where the first inequality follows from the triangle inequality, and the last equality is due to dn = O(Mn) and Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Next, we examine the convergencerate of ˆθ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' To verify ∥ˆθ1∥2 = Op(1), it is sufficient to show that ∥ˆθ1−˜θ1∥2 = op(1) under Conditions 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Define: ˜Qi(θ1, ˜θ1, θ2) = ρτ(ǫi − ˜z⊤ i θ1 − ˜ψ⊤ 2 θ2 − uni) − ρτ(ǫi − ˜z⊤ i ˜θ1 − ˜ψ⊤ 2 θ2 − uni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We first show that for any positive constants M and C, P \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed inf ∥θ1−˜θ1∥2≥M ∥θ2∥2≤C √dn n � i=1 ˜Qi(θ1, ˜θ1, θ2) > 0 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7) Following the proof of Lemma 6 in [23], we have: sup ∥θ1−˜θ1∥2≤M ∥θ2∥2≤C √dn ������� n � i=1 ˜Qi(θ1, ˜θ1, θ2) − 1 2(θ1 − ˜θ1)⊤ �1 n ˇZ⊤Bn ˇZ � (θ1 − ˜θ1)(1 + op(1)) ������� = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' By Conditions 1 and 3, for any ∥θ1 − ˜θ1∥2 > M, 1 2(θ1 − ˜θ1)⊤ �1 n ˇZ⊤Bn ˇZ � (θ1 − ˜θ1) > CM, for some positive constant C, and thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7) and Lemma 3, we have that there exists a local minimizer ˆθ1 of (6) such that ∥ˆθ1∥2 = Op(1), and thus ∥ˆγ − γ∗∥2 = Op( √1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Proof of Theorem 1 (2) We need to show that ˆβk(t) = 0 for all t ∈ D[2] k with probability tending to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Denote ˆb[l] k = (ˆbk,l, · · · , ˆbk,l+d)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We need to prove that the local minimizer (ˆb⊤, ˆγ⊤)⊤ satisfies ∥ˆb[l] k ∥2 = 0 for all l such that Il ⊂ D[2] k with probability tending to one for k = 0, · · · , q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' By the way of contradiction, assume that ∥ˆb[l⋆] k ∥2 � 0 for some l⋆ with Il⋆ ⊂ D[2] k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Let ˜bk be the same as ˆbk except that ∥˜b[l⋆] k ∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Note that ∥˜b[l] k ∥2 = 0 is equivalent to ∥˜bk∥Wl = 0 for l = 1, · · · , Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Since ∥b∗[l⋆] k ∥2 = 0 and ∥b∗ k − ˆbk∥2 = Op(Mn/ √n), we have ∥ˆb[l⋆] k ∥2 = O(Mn/ √n), and thus ∥ˆbk∥Wl⋆ = O( √Mn/n) by 16 λmax(Wl⋆) = O(M−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Below we prove that: 1 n n � i=1 ρτ(yi − ψ⊤ i ˆb − z⊤ i ˆγ) + q � k=1 Mn � l=1 pλ1(∥ˆbk∥Wl) + Mn � l=1 pλ2(∥ˆb∥Wl) + η q � k=0 ˆb⊤ k V ˆbk > 1 n n � i=1 ρτ(yi − ψ⊤ i ˜b − z⊤ i ˆγ) + q � k=1 Mn � l=1 pλ1(∥˜bk∥Wl) + Mn � l=1 pλ2(∥˜b∥Wl) + η q � k=0 ˜b⊤ k V ˜bk, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8) with probability tending to one, and this leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Therefore, we conclude that ∥ˆb[l] k ∥2 = 0 for all l ⊂ D[2] k in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Furthermore, by the definition of ˆβk(t), we have ˆβk(t) = 0 for all t ∈ D[2] k with probability tending to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' By the convexity of the quantile loss function, we have 1 n n � i=1 (ρτ(yi − ψ⊤ i ˆb − z⊤ i ˆγ) − ρτ(yi − ψ⊤ i ˜b − z⊤ i ˆγ)) ≥ −1 n n � i=1 (τ − 1(yi ≤ ψ⊤ i ˜b + z⊤ i ˆγ))ψ[kl⋆]⊤ i ˆb[l⋆] k (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='9) = −1 n n � i=1 (τ − 1(ǫi ≤ 0)) ψ[kl⋆]⊤ i ˆb[l⋆] k −1 n n � i=1 (1(ǫi ≤ 0) − 1(ǫi ≤ ψ⊤ i (˜b − b∗) + z⊤ i (ˆγ − γ∗) + uni))ψ[kl⋆]⊤ i ˆb[l⋆] k , where ψ[kl⋆] i = (ψi,k(Mn+d)+l⋆, · · · , ψi,k(Mn+d)+l⋆+d)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the first term on the right hand side of the last equation of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='9), by Conditions 1 and 3, we have: 1 n n � i=1 (τ − 1(ǫi ≤ 0)) ψ[kl⋆]⊤ i ˆb[l⋆] k = Op(n−1/2∥ˆbk∥Wl⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' since supi |ψ⊤ i (˜b − b∗) + z⊤ i (ˆγ − γ∗) + uni| = Op( √Mn/n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' we have: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1(ǫi ≤ 0) − 1(ǫi ≤ ψ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i (˜b − b∗) + z⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i (ˆγ − γ∗) + uni) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='ψ[kl⋆]⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='ˆb[l⋆] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='����1(ǫi ≤ C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Mn/n) − 1(ǫi ≤ −C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Mn/n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='���� × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='����ψ[kl⋆]⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='ˆb[l⋆] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1(−C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Mn/n ≤ ǫi ≤ C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Mn/n) × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='����ψ[kl⋆]⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='ˆb[l⋆] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i�i′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='n2 E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1(−C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Mn/n ≤ ǫi ≤ C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Mn/n)1(−C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Mn/n ≤ ǫi′ ≤ C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Mn/n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='����ψ[kl⋆]⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='ψ[kl⋆] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='i′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='���� × ∥ˆb[l⋆] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='k ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Cn−2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='nM1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='n n−1/2 + n2Mnn−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='∥ˆbk∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Wl⋆ = O(Mnn−1)∥ˆbk∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='Wl⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Therefore, the second term is bounded by Op( √Mn/n∥ˆbk∥Wl⋆), which dominates the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Also, since λmax(V) = 17 O(M−1 n ), we have: η q � k=0 ˆb⊤ k V ˆbk − η q � k=0 ˜b⊤ k V ˜bk = η q � k=0 [(ˆbk − ˜bk)⊤V(ˆbk − ˜bk) + 2(ˆbk − ˜bk)⊤V ˜bk] = η q � k=0 ˆb[l⋆]⊤ k V[l⋆] ˆb[l⋆] k ≤ CηM−1 n ∥ˆb[l⋆] k ∥2 2 = Op(η � Mn/n∥ˆbk∥Wl⋆), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='10) where V[l⋆] is the submatrix of V with entries vi j, i, j = l⋆, · · · , l⋆ + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Since η = op(n−1/2), together with the above discussions on (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='9) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='10), it follows that: 1 n n � i=1 (ρτ(yi − ψ⊤ i ˆb − z⊤ i ˆγ) − ρτ(yi − ψ⊤ i ˜b − z⊤ i ˆγ)) + η q � k=0 ˆb⊤ k V ˆbk − η q � k=0 ˜b⊤ k V ˜bk = Op( � Mn/n∥ˆbk∥Wl⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='11) Next, we examine the functional sparse group penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For Il⋆ ⊂ D[2] 0 , we have ∥ˆb0∥Wl⋆ ≥ ∥˜b0∥Wl⋆ = 0 and ∥ˆbk∥Wl⋆ = ∥˜bk∥Wl⋆ = 0 for all k = 1, · · · , q by the design of the sparse group penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' And thus q � k=1 Mn � l=1 pλ1(∥ˆbk∥Wl) = q � k=1 Mn � l=1 pλ1(∥˜bk∥Wl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='12) Also, for l⋆ such that Il⋆ ⊂ D[2] 0 , we have: Mn � l=1 pλ2(∥ˆb∥Wl) − Mn � l=1 pλ2(∥˜b∥Wl) = pλ2(∥ˆb∥Wl⋆) ≥ λ2 2 ∥ˆb∥Wl⋆ ≥ λ2 2 ∥ˆbk∥Wl⋆, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='13) by ∥ˆb∥Wl⋆ = O( √Mn/n), M−r n / min(λ1, λ2) = o(1), and Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For Il⋆ ⊂ D[2] k and k � 0, we have ∥ˆbk∥Wl⋆ ≥ ∥˜bk∥Wl⋆ = 0 and ∥ˆbk′∥Wl⋆ = ∥˜bk′∥Wl⋆ for k′ � k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Since ∥ˆbk∥Wl⋆ = O( √Mn/n) and M−r n / min(λ1, λ2) = o(1), we have: q � k=1 Mn � l=1 (pλ1(∥ˆbk∥Wl) − pλ1(∥˜bk∥Wl)) = pλ1(∥ˆbk∥Wl⋆) ≥ λ1 2 ∥ˆbk∥Wl⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='14) For Il⋆ ⊂ D[2] k and k � 0, when ∥˜b∥Wl⋆ ≥ λ2ξ, we have ∥ˆb∥Wl⋆ ≥ λ2ξ and Mn � l=1 (pλ2(∥ˆb∥Wl) − pλ2(∥˜b∥Wl)) = pλ2(∥ˆb∥Wl⋆) − pλ2(∥˜b∥Wl⋆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='15) For Il⋆ ⊂ D[2] k and k � 0, when ∥˜b∥Wl⋆ < λ2ξ, we have Mn � l=1 (pλ2(∥ˆb∥Wl) − pλ2(∥˜b∥Wl)) = λ2 � min(λ2ξ,∥ˆb∥Wl⋆ ) ∥˜b∥Wl⋆ � 1 − t λ2ξ � + dt ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='16) Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='14) - (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='16), we obtain that for Il⋆ ⊂ D[2] k , k = 0, · · · , q, q � k=1 Mn � l=1 (pλ1(∥ˆbk∥Wl) − pλ1(∥˜bk∥Wl)) + Mn � l=1 (pλ2(∥ˆb∥Wl) − pλ2(∥˜b∥Wl)) ≥ 1 2 min(λ1, λ2)∥ˆbk∥Wl⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 18 Combining the above result with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='11), M−r n / min(λ1, λ2) = o(1), and Condition 4, we prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='8) with probability tending to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 19 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Additional numerical results Table 3: Scenario I under Case 2: mean (sd) based on 100 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the quantile-based methods, τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' n Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='1 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 Proposed ISE0(×102) 300 β0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='493(8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='442) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='012(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='538) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='012(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='538) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='146(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='853) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='433(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='109) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='433(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='109) β1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='101(9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='790) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} 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+page_content='951(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='009) 23 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Additional data analysis results We conduct exploratory regression analysis with the proposed penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' For the lack-of-fit, we consider the mean- based (Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3) and the proposed quantile-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' With the proposed approach, we consider τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In the left panel of Figure 5, we plot the estimated densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' It is observed that the residuals are left-skewed, which suggests the sensibility of quantile-based analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Different quantiles lead to different results, which has been commonly observed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' In addition, the mean estimation is closer to the proposed estimation with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, compared to the other two quantile values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='2 0 2 4 6 8 10 12 Density τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5 τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7 OLS −1 0 1 2 −1 0 1 2 Sample Estimated Figure 5: Left: estimated densities of residuals from Alt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3 and the proposed method with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Right: Lack-of-fit diagnostic QQ plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We also conduct model diagnostics using a QQ plot to intuitively assess model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Specifically, we first randomly generate ˘τ from the uniform distribution on [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We then fit data using the proposed method with quantile ˘τ and obtain estimator (ˆb(˘τ), ˆγ(˘τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Next, we generate the response from the model ˘y = ψ⊤ ˆb(˘τ) + z⊤ ˆγ(˘τ), where (ψ, z) are randomly selected from the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' We repeat this process 100 times and obtain a sample of 100 simulated fat values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' The right panel of Figure 5 gives the QQ plot for the simulated and observed fat contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' Most points are very close to the 45-degree line, which suggests satisfactory model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE2T4oBgHgl3EQfKwYZ/content/2301.03705v1.pdf'} diff --git a/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf b/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..39666644920ce60d1511c218cadb53159f42c7e0 --- /dev/null +++ b/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c87e63d2636f8bfc9befd4af8db0770b1707087640234ce83e1f17d5ae5d46aa +size 35570304 diff --git a/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf 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Postal 14-740, 07000 M´exico D.F., M´exico +(b) Instituto de F´ısica Corpuscular, CSIC-Universitat de Val`encia, 46980 Paterna, Spain +(c) Departament de F´ısica Te`orica, Universitat de Val`encia, 46100 Burjassot, Spain +diego.portillo@cinvestav.mx, pablo.escribano@ific.uv.es, avelino.vicente@ific.uv.es +Abstract +The Scotogenic model is a popular scenario that induces radiative Majorana +neutrino masses and includes a weakly-interacting dark matter candidate. We +classify all possible ultraviolet extensions of the Scotogenic model in which (i) the +dark Z2 parity emerges at low energies after the spontaneous breaking of a global +U(1)L lepton number symmetry, and (ii) the low-energy effective theory contains +a naturally small lepton number breaking parameter, suppressed by the mass of a +heavy mediator integrated out at tree-level. We find 24 such models and discuss +two of them in detail to illustrate our setup. We also discuss some general aspects +of the phenomenology of the models in our classification, exploring possible lepton +flavor violating signals, collider signatures and implications for dark matter. The +phenomenological prospects of these scenarios are very rich due to the presence +of additional scalar states, including a massless Goldstone boson. +1 +Introduction +The Scotogenic model [1] is a popular extension of the Standard Model (SM) that addresses +two of the currently most important open questions in physics: the origin of neutrino masses +and the nature of the dark matter (DM) of the Universe. Its popularity stems from its +simplicity. The model extends the SM particle content with three singlet fermions, N1,2,3, +and a scalar doublet, η, all odd under a new Z2 symmetry under which the SM fields are +even. These ingredients suffice to induce Majorana neutrino masses at the 1-loop level and +provide a viable DM candidate, namely the lightest Z2-odd state. +1 +arXiv:2301.05249v1 [hep-ph] 12 Jan 2023 + +Radiative neutrino mass models [2–5] provide a natural suppression for neutrino masses +with loop factors. This is one of the main motivations in favor of this class of models [6]. +In addition, further suppression is introduced in some models by assuming an approximate +lepton number symmetry, broken in a small amount by the presence of a Lagrangian term +with a suppressed coefficient. This is the case of the Scotogenic model, that requires a small +λ5 ≪ 1 quartic parameter to obtain the correct size for neutrino masses with sizable Yukawa +couplings. While this is technically valid, and natural in the sense of ’t Hooft [7], it also calls +for an extension that explains the smallness of the λ5 parameter, possibly relating it to the +breaking of lepton number. +In this work we consider ultraviolet (UV) extensions of the Scotogenic model that provide +a natural explanation for the smallness of the λ5 parameter and in which the Z2 parity of +the model emerges at low energies from a spontaneously broken global U(1) lepton number +symmetry. This endeavor was initiated in [8], where a specific UV model with these proper- +ties was proposed. Here we go beyond specific realizations and classify all possible models +with these features in which a low-energy Scotogenic model is obtained after integrating +out a heavy field at tree-level. Besides one or several massive scalars, the particle spectrum +of the theory will contain a massless Goldstone boson, the majoron [9–12], induced by the +spontaneous breaking of lepton number. These new states are not present in the original +Scotogenic model and lead to novel phenomenological predictions that allow one to probe +our setup. +The rest of the manuscript is organized as follows. First, we set our notation and con- +ventions in Sec. 2, where the Scotogenic model is introduced. A general classification of all +possible UV extensions of the Scotogenic model satisfying the requirements explained above +is given in Sec. 3. Two selected example models will be presented in detail in Secs. 4 and 5. +Some general aspects of the phenomenology of this class of models are discussed in Sec. 6. +Finally, we summarize our results and conclude in Sec. 7. Additional information can be +found in Appendix A, where we discuss scenarios with an accidental Z2 symmetry. +2 +The Scotogenic model +Before we discuss specific UV realizations of our setup, let us introduce our conventions for +the Scotogenic model. The particle content of the Scotogenic model [1] includes, besides the +usual SM fields, three generations of right-handed fermions N, transforming as (1, 0) under +(SU(2)L, U(1)Y), and one scalar η, transforming as (2, 1/2). We also impose the conservation +of an ad-hoc Z2 symmetry, under which η and N are odd while the rest of the fields in the +model are even. The lepton and scalar particle content of the model is shown in Table 1. 1 +The model contains two scalar doublets, the usual Higgs doublet H and the new doublet +η, only distinguished by their Z2 charges. They can be decomposed in terms of their SU(2)L +components as +H = +�H+ +H0 +� +, +η = +�η+ +η0 +� +. +(1) +Once specified the particle content and symmetries of the model we can write down the +1We follow the conventions for the Scotogenic model used in [13]. +2 + +Field +Generations +SU(3)c +SU(2)L +U(1)Y +Z2 +ℓL +3 +1 +2 +-1/2 ++ +eR +3 +1 +1 +-1 ++ +N +3 +1 +1 +0 +− +H +1 +1 +2 +1/2 ++ +η +1 +1 +2 +1/2 +− +Table 1: +Lepton and scalar particle content and representations under the gauge and +discrete symmetries in the Scotogenic model. ℓL and eR are the SM left- and right-handed +leptons, respectively, and H is the SM Higgs doublet. +Lagrangian. The Lagrangian of the model contains the terms +LY = y N �η† ℓL + 1 +2MN N +cN + h.c. , +(2) +where y is a general complex 3 × 3 matrix and MN is a symmetric 3 × 3 mass matrix. The +scalar potential of the model is given by +VUV = m2 +HH†H + m2 +ηη†η + λ1 +2 (H†H)2 + λ2 +2 (η†η)2 ++ λ3(H†H)(η†η) + λ4(H†η)(η†H) + +�λ5 +2 (H†η)2 + h.c. +� +. +(3) +Here m2 +H and m2 +η are parameters with dimensions of mass2. We assume that the minimization +of the scalar potential leads to a vacuum defined by +⟨H0⟩ = vH +√ +2 , +⟨η0⟩ = 0 . +(4) +This vacuum configuration breaks the electroweak symmetry in the usual way but preserves +the Z2 symmetry of the model. As a consequence of this, the lightest Z2-odd state (either +N1 or η0) is completely stable and can play the role of the DM of the Universe. Furthermore, +neutrinos acquire non-zero Majorana masses at the 1-loop level, as shown in Fig. 1. The +resulting 3 × 3 neutrino mass matrix is given by +(mν)αβ = λ5 v2 +H +32π2 +� +n +ynα ynβ +MNn +� +M 2 +Nn +m2 +0 − M 2 +Nn ++ +M 4 +Nn +� +m2 +0 − M 2 +Nn +�2 log M 2 +Nn +m2 +0 +� +, +(5) +where m2 +0 = m2 +η + (λ3 + λ4) v2 +H/2 and MNn are the diagonal elements of the MN matrix. +One can easily estimate that in order to obtain neutrino masses of the order of 0.1 eV with +Scotogenic states in the TeV scale and Yukawas of order 1, λ5 must be of order ∼ 10−10. The +smallness of this parameter is protected by lepton number, and thus is technically natural [7]. +However, it is not explained in the context of the Scotogenic model. +3 + +νL +νL +H0 +H0 +η +η +N +N +Figure 1: Neutrino mass generation in the Scotogenic model. This Feynman diagram shows +the relevant gauge eigenstates involved in the 1-loop contribution to neutrino masses. +3 +Ultraviolet extensions of the Scotogenic model +3.1 +General considerations +The Scotogenic model has two features that call for a refinement, namely, the origin of the +Z2 symmetry and λ5 ≪ 1. Although these features do not pose any theoretical problem, +they can be regarded as ad-hoc ingredients in an otherwise very natural framework. We are +thus interested in an UV extension of the Scotogenic model that provides an explanation +for them. More specifically, we want to classify all possible UV scenarios that lead to the +Scotogenic model at low energies after integrating out a heavy scalar field S, with mS ≫ vH, +and satisfy the following two requirements: +(A) The Scotogenic Z2 is obtained as a remnant after the spontaneous breaking of a U(1)L +lepton number symmetry by the VEV of one or several singlet scalar fields σ: +U(1)L +⟨σ⟩ +−−−−−→ Z2 +(B) The (H†η)2 operator is forbidden in the UV theory due to U(1)L conservation, but an +operator of the form (H†η)2σn, with n ≥ 1, is generated after integrating out S. After +the singlets get VEVs and U(1)L is spontaneously broken, this will induce an effective +λ5 coupling, which will be naturally suppressed by the large mS energy scale. +In this work we will concentrate on global U(1)L lepton number symmetries, tree-level com- +pletions of the λ5 operator and UV models with one or two σ singlets. Gauged versions of +the lepton number symmetry, higher-order completions and models with additional singlets +are left for future work. +The models we are looking for induce neutrino masses `a la Scotogenic, with variations of +the neutrino mass diagram in Fig. 1. This diagram has an internal scalar line (with η0) and +an internal fermion line (with N). The analogous diagrams in the UV extended models will +include the heavy scalar S in the loop and one or several external legs with σ singlets (or +4 + +Topology +Diagram +Required operators +I +η +H† +S +η +H† +σA +σB +(σAH†S ˜H),(σB˜η†S†η) +II +H† +H† +S +η +η +σA +σB +(σAH†Sη),(σBH†S†η) +III +H† +H† +S +η +η +σA +σB +(σAσBH†S),(H†ηS†η) +IV +H† +H† +S +η +η +σA +σB +(H†SH†η),(σAσBS†η) +Table 2: (H†η)2σAσB operator in the UV theory. +σ insertions, for short). After these considerations, there are two classes of models that can +be already discarded: +• Models without σ insertions in the scalar line. These models can be discarded because +the (H†η)2 operator would be allowed in the UV theory. +This would preclude an +explanation of λ5 ≪ 1. In addition, η would acquire a VEV. +• Models without σ insertions in the fermion line. The U(1)L charge of the N singlet +fermions must necessarily vanish if the σN +cN operator is absent and their Majorana +masses are explicitly introduced in the Lagrangian. However, in this case N will be +even under the Z2 symmetry obtained after spontaneous U(1)L breaking. This scenario +does not correspond to the Scotogenic model. Nevertheless, an additional accidental +Z2 symmetry may appear, as explained in Appendix A. +We are thus left with neutrino mass topologies with σ insertions in both internal lines. +The scalar line leads to an operator (H†η)2σn after the heavy S is integrated out. In fact, +due to the renormalizability of the UV theory, n can be at most 2. Therefore, the resulting +operator will be of the form +Oλ5 = (H†η)2σAσB . +(6) +This generic expression includes cases with only one σ insertion (for instance, σB = ∅) and +cases in which both σ insertions in the scalar line correspond to the same field (σA = σB). +5 + +All possible topologies are shown in Table 2. Finally, the fermion line simply corresponds +to a σ − N − N Yukawa interaction. In the following we will always assume the presence of +the operator σN +cN (for models with one σ field) or σ1N +cN (for models with two σ fields), +and we will not draw it. The coefficient of this operator will be denoted by κ. Therefore, +once the singlet scalar gets a VEV, ⟨σ(1)⟩ = +vσ(1) +√ +2 , the Majorana mass matrix for the singlet +fermions N is generated, 2 +MN = +√ +2 κ vσ(1) . +(7) +3.2 +Model classification +In the following we will refer to a specific model using the notation ξ(A, B), where ξ = +{I, II, III, IV} denotes the topology for the (H†η)2σAσB operator, as listed in Table 2, and +A and B denote the singlets involved in the vertices where σA,B are coupled. Since we only +consider UV theories with at most two different singlets, A and B can only take the values +∅, 1, 2, 1∗, where ∅ indicates that no σ enters the corresponding vertex and σ1∗ ≡ σ∗ +1. It is +important to mention that we do not consider scenarios with A, B = 2∗ because they lead +to a redefinition of the charge qσ2 → −qσ2. 3 Therefore, in principle each topology has 16 +different variations depending on the way the σA,B singlets are coupled. However, we can +reduce this number by taking into account the following arguments: +• A ̸= B is required to forbid the term (H†η σA)2 in the effective Lagrangian. If this +specific combination is allowed, then the term (H†η σA) is too. This trilinear interaction +induces a non-zero VEV for η after both H and σA acquire their VEVs, hence breaking +the Scotogenic Z2 symmetry. +• A ̸= B∗ is also required. Otherwise, (H†η)2σAσ∗ +A is allowed by the U(1)L symmetry +and then the operator (H†η)2 is present in the UV theory. +• In all ξ(1, ∅) and ξ(∅, 1) models the effective operator leading to the λ5 coupling is +Oλ5 = (H†η)2σ. This implies the relation 2qη + qσ = 0. In addition, the Yukawa +coupling σN +cN implies 2qN + qσ = 0. Hence, the charges for η and N must satisfy +qη = qN and then the N ˜η†ℓL Yukawa term is forbidden by U(1)L. Similarly, in all +ξ(1∗, ∅) and ξ(∅, 1∗) models one finds qη = −qN and then qN = 1 +2 in order to allow the +term N ˜η†ℓL. +With these considerations, there are only 8 possibilities left in each of the four topologies. +However, there are duplicities. Models based on topologies III and IV are symmetric with +respect to the exchange σA ↔ σB (i.e. ξ(A, B) = ξ(B, A) with ξ = III, IV). Similarly, +II(A, B) ∼ II(B, A) by redefining qS → −qS. This further reduces the number of funda- +mentally different UV models. In total, we find 24 (20 + 4, because in II-models S can be +2In models with two σ fields such that qσ1 = qσ2 or qσ1 = −qσ2, an additional Yukawa term σ2N +cN or +σ∗ +2N +cN would be present. Here qσ1 and qσ2 denote the U(1)L charges of σ1 and σ2, respectively. This would +lead to MN = +√ +2 (κ1 vσ1 + κ2 vσ2) without affecting our discussion. We note, however, that in such models +both σ singlets are esentially copies of the same field. +3In the following, we will denote the U(1)L charge of the field X as qX. Furthermore, qℓL = qeR = 1 and +qH = 0, as usual. +6 + +Topology +A +B +qN +qη +qσ1 +qσ2 +qS +(SU(2)L, U(1)Y)S +1 +I +1∗ +∅ +1 +2 +− 1 +2 +−1 +- +−1 +(3, 1) +2 +I +∅ +1∗ +1 +2 +− 1 +2 +−1 +- +0 +(3, 1) +3 +I +2 +∅ +qN +qN − 1 +−2qN +2 − 2qN +2qN − 2 +(3, 1) +4 +I +∅ +2 +qN +qN − 1 +−2qN +2 − 2qN +0 +(3, 1) +5 +I +1 +2 +qN +qN − 1 +−2qN +2 +2qN +(3, 1) +6 +I +2 +1 +qN +qN − 1 +−2qN +2 +−2 +(3, 1) +7 +I +1∗ +2 +qN +qN − 1 +−2qN +2 − 4qN +−2qN +(3, 1) +8 +I +2 +1∗ +qN +qN − 1 +−2qN +2 − 4qN +4qN − 2 +(3, 1) +9-10 +II +1∗ +∅ +1 +2 +− 1 +2 +−1 +- +− 1 +2 +(3, 0) or (1, 0) +11-12 +II +2 +∅ +qN +qN − 1 +−2qN +2 − 2qN +qN − 1 +(3, 0) or (1, 0) +13-14 +II +1 +2 +qN +qN − 1 +−2qN +2 +1 + qN +(3, 0) or (1, 0) +15-16 +II +1∗ +2 +qN +qN − 1 +−2qN +2 − 4qN +1 − 3qN +(3, 0) or (1, 0) +17 +III +1∗ +∅ +1 +2 +− 1 +2 +−1 +- +−1 +(2, 1/2) +18 +III +2 +∅ +qN +qN − 1 +−2qN +2 − 2qN +2qN − 2 +(2, 1/2) +19 +III +1 +2 +qN +qN − 1 +−2qN +2 +2qN − 2 +(2, 1/2) +20 +III +1∗ +2 +qN +qN − 1 +−2qN +2 − 4qN +2qN − 2 +(2, 1/2) +21 +IV +1∗ +∅ +1 +2 +− 1 +2 +−1 +- +1 +2 +(2, 1/2) +22 +IV +2 +∅ +qN +qN − 1 +−2qN +2 − 2qN +1 − qN +(2, 1/2) +23 +IV +1 +2 +qN +qN − 1 +−2qN +2 +1 − qN +(2, 1/2) +24 +IV +1∗ +2 +qN +qN − 1 +−2qN +2 − 4qN +1 − qN +(2, 1/2) +Table 3: UV extended models satisfying conditions (A) and (B). For each model we show +the U(1)L charges of N, η, σ1, σ2 and S, as well as the (SU(2)L, U(1)Y) representation of +S. Models that become any of the models in this list after renaming the fields or redefining +their U(1)L charges are not included, as explained in the text. +7 + +an SU(2)L singlet or triplet) different UV theories. They are listed in Table 3, where the +U(1)L charges of N, η, σA,B and S, as well as the (SU(2)L, U(1)Y) representation of S in +each model, are shown. Some comments are in order: +(i) The (SU(2)L, U(1)Y) representation of the heavy scalar S depends on the topology. In +I-models S transforms as (3, 1), in II-models we have two possibilities, (3, 0) or (1, 0), +while in III- and IV-models S transforms as (2, 1/2). +(ii) In all the models in Table 3, the global U(1)L symmetry may be spontaneously broken +to a Z2 parity, under which N and η are odd. +In all the ξ(1∗, ∅) models and in +I(∅, 1∗), the conservation of U(1)L restricts the lepton number charges of N, η, σA,B +and S, which must take precise values, and this automatically implies a remnant Z2 +that corresponds to the usual Scotogenic parity. The model studied in Ref. [8], which +corresponds to model I(1∗, ∅) in our notation, is a good example of this. In the rest +of the models, the conservation of U(1)L leaves one of the charges to be chosen freely. +We decided to choose qN. In this case, these are the restrictions to recover the dark +Z2 parity from U(1)L breaking: +• qN cannot be an integer. +• If qN = α +β, with α, β ∈ Z, then α and β have to be odd and even, respectively. +• GCD(α, β) = 1, where GCD stands for Greatest Common Divisor. Therefore, α +and β must be coprime. +The first restriction comes from the requirement of N and η being both odd under the +remnant Scotogenic Z2. The relation qη = qN −1 implies that if qN is even, then qη must +be odd, and vice versa. Then, N and η will transform differently under the remnant Z2 +symmetry. As an example of this consider the model I(1, 2) with qN = 2. In this case, +the solution for the rest of the U(1)L charges in the model is qη = 1, qσ1 = −4, qσ2 = +2 and qS = 4. +The global lepton number symmetry gets spontaneously broken as +U(1)L → Z2, but with N and η charged under Z2 as + and −, respectively, and this +does not reproduce the Scotogenic model. +Similarly, if qN = +α +β, after normalizing +all U(1)L charges so that they become integer numbers (multiplying by β) we obtain +˜qη = β − α and ˜qN = α. Hence, for η and N to be odd under Z2, α and β must be odd +and even, respectively. Finally, the third restriction is required to guarantee that n = 2 +after U(1)L breaks to the discrete symmetry Zn. As an example we take model I(1,2), +where n ≡ GCD(˜qσ1, ˜qσ2, ˜qS) = GCD(−2α, 2β, 2α) = 2GCD(α, β) = 2. We checked for +all the working models that GCD(˜qσ1, ˜qσ2, ˜qS) or GCD(˜qσ1, ˜qσ2), depending on whether +S acquires a VEV or not, always reduces to GCD(α, β) = 1. Also, we want qN = α +β to +be irreducible. +(iii) In all models, and for all possible values of qN in agreement with the restrictions +listed in the previous item, η never acquires an induced VEV. This is crucial for the +consistency of the Scotogenic model. +(iv) It is clear that in all models of the form ξ(A, ∅) or ξ(∅, B), a trilinear coupling µ +participates in the generation of the λ5 coupling, induced after the breaking of U(1)L. +8 + +νL +νL +H0 +σ +η0 +η0 +N +N +y +y +κ +H0 +σ +S +β +µ +Figure 2: Neutrino mass generation in an extended Scotogenic model with one σ field. This +Feynman diagram shows the relevant gauge eigenstates involved in the 1-loop contribution +to neutrino masses. In our notation, this is a IV(1∗, ∅) model. +This is perfectly consistent, but requires the assumption µ ≪ mS to justify λ5 ≪ 1. +This poses a theoretical issue, since µ is a parameter of the UV theory. In contrast, +in models of the form ξ(A, B) with A, B ̸= ∅, the λ5 coupling will only depend on the +σA,B VEVs, induced at low energies and naturally small compared to mS. +(v) Finally, we note that in I-models the U(1)L charges of the particles N, η and σA,B +remain the same after the non-trivial change A ↔ B. For instance, this is the case in +models I(1, 2) and I(2, 1). +This concludes our classification of all possible UV extensions of the Scotogenic model +satisfying our requirements (A) and (B). We will now illustrate it with two specific example +models. An additional example can be found in [8]. +4 +An UV extended Scotogenic model with one σ field +Our first example model is an UV extension of the Scotogenic model with one σ field. +Another example of this class of models can be found in [8]. +4.1 +Ultraviolet theory +We consider an extension of the Scotogenic model with two new particles: the SU(2)L doublet +S and the singlet σ, both scalars. The Z2 Scotogenic parity is replaced by a global U(1)L +lepton number symmetry. Table 4 shows the scalar and leptonic fields of the model and their +representations under the gauge and global symmetries. +We want to explain the smallness of the Scotogenic’s λ5 coupling. Our strategy will be +to forbid it in our original Lagrangian and make it arise effectively at low energies once the +9 + +Field +Generations +SU(3)c +SU(2)L +U(1)Y +U(1)L +ℓL +3 +1 +2 +-1/2 +1 +eR +3 +1 +1 +-1 +1 +N +3 +1 +1 +0 +qN +H +1 +1 +2 +1/2 +0 +η +1 +1 +2 +1/2 +qη +σ +1 +1 +1 +0 +qσ +S +1 +1 +2 +1/2 +qS +Table 4: Lepton and scalar particle content and representations under the gauge and global +symmetries in an UV extension of the Scotogenic model with one σ field. +scalar σ acquires a VEV and we integrate out S. We also impose that, after symmetry +breaking, the effective λ5 coupling would induce neutrino masses as shown in Fig. 2. In our +notation, this is a IV(1∗, ∅) model. This requires the presence of the operators +N�η†ℓL +, +σN +cN +, +H†SH†η +, +σ∗S†η , +(8) +which in turn imply the following set of equations for the U(1)L charges of the model: +−qN + qη + 1 = 0 , +(9) +qσ + 2 qN = 0 , +(10) +qS + qη = 0 , +(11) +−qσ − qS + qη = 0 . +(12) +This system of linear equations has a unique solution: +qN = 1 +2 , +(13) +qη = −1 +2 , +(14) +qσ = −1 , +(15) +qS = 1 +2 . +(16) +With this solution, the operators +N +cN +, +N �H†ℓL +, +� +H†η +�2 +(17) +are automatically forbidden due to U(1)L conservation. One should note that if we chose +the operator σS†η instead of σ∗S†η, no solution for the resulting system of equations would +exist. Indeed, if one replaces −qσ by qσ in Eq. (12), the combination of the resulting equation +with Eqs. (10) and (11) leads to qN = qη, which is incompatible with Eq. (9). This illustrates +why ξ(1, ∅) models are not compatible with our requirements. +10 + +Having fixed the quantum numbers of all the particles in the model, we proceed to write +its Lagrangian. The new Yukawa interactions are given by +LY = y N �η† ℓL + κ σN +cN + h.c. , +(18) +where y is a general complex 3 × 3 matrix and κ is a complex symmetric 3 × 3 matrix. The +scalar potential of the model can be written as +VUV = m2 +HH†H + m2 +SS†S + m2 +σσ∗σ + m2 +ηη†η + λ1 +2 (H†H)2 + λ2 +2 (η†η)2 ++ λS +2 (S†S)2 + λσ +2 (σ∗σ)2 + λ3(H†H)(η†η) + λS +3 (H†H)(S†S) ++ λσ +3(H†H)(σ†σ) + ληS +3 (η†η)(S†S) + λησ +3 (η†η)(σ∗σ) ++ λσS +3 (σ∗σ)(S†S) + λ4(H†η)(η†H) + λHS +4 (H†S)(S†H) ++ ληS +4 (S†η)(η†S) + +� +β(H†SH†η) + µ(σ∗S†η) + h.c. +� +. +(19) +Here µ is a trilinear parameter with dimensions of mass while m2 +H, m2 +η and m2 +σ have dimen- +sions of mass2. Other Lagrangian terms are allowed by the gauge symmetries of the model +but forbidden by U(1)L. +4.2 +Effective theory +We will now assume that mS is much larger than any other energy scale in the theory. At +energies well below mS, all physical processes can be properly described by an effective field +theory in which the heavy field S has been integrated out. We now present this effective +theory, obtained after integrating out S at tree-level. The effective potential at low energies +can be written as +VIR = m2 +HH†H + m2 +ηη†η + m2 +σσ∗σ + λ1 +2 (H†H)2 + λ2 +2 (η†η)2 + λσ +2 (σ∗σ)2 ++ λ3(H†H)(η†η) + λσ +3(H†H)(σ∗σ) + +� +λησ +3 − |µ|2 +m2 +S +� +(σ∗σ)(η†η) ++ +� +λ4 − |β|2(H†H) +m2 +S +� +(H†η)(η†H) − +� βµ +m2 +S +σ∗(H†η)2 + h.c. +� ++ O +� 1 +m4 +S +� +. +(20) +Assuming that CP is conserved in the scalar sector, the neutral fields H0 and σ can be +decomposed as +H0 = 1 +√ +2(vH + φ + iA) , +σ = 1 +√ +2(vσ + ρ + iJ) , +(21) +with +vH +√ +2 and +vσ +√ +2 the VEVs of H0 and σ, respectively. +These VEVs are determined by +minimizing the scalar potential in Eq. (20). The resulting tadpole equations are given by +dVIR +dH0 +���� +⟨H0,σ⟩={ vH +√ +2 , vσ +√ +2 } += vH +√ +2 +� +m2 +H + λ1v2 +H +2 ++ λσ +3v2 +σ +2 +� +, +(22) +dVIR +dσ +���� +⟨H0,σ⟩={ vH +√ +2 , vσ +√ +2 } += vσ +√ +2 +� +mσ2 + λσ +3v2 +H +2 ++ λσv2 +σ +2 +� +, +(23) +11 + +where we have only written the non-trivial equations and these are evaluated at the VEVs +of each scalar field. As we see from Eq. (20), once σ acquires a VEV, the operator (H†η)2 +is generated, with an effective λ5 coupling that is naturally suppressed by the mass of the +heavy field S, +λ5 +2 = − βµvσ +√ +2m2 +S +≪ 1 . +(24) +This follows from the assumption µ ≪ mS. As explained in Sec. 3, this is perfectly valid. +However, it poses a theoretical problem since µ is parameter of the UV theory. A model +without this issue will be discussed below in Sec. 5. We now proceed to the computation of +the scalar spectrum of the model. In the bases {φ, ρ} for the CP-even states and {A, J} for +the CP-odd ones, the squared mass matrices read +M2 +R = +� m2 +H + 1 +2 (3λ1v2 +H + λσ +3v2 +σ) +λσ +3vHvσ +λσ +3vHvσ +m2 +σ + 1 +2 (λσ +3v2 +H + 3λσv2 +σ) +� +, +(25) +and +M2 +I = +� +m2 +H + +λ1v2 +H +2 ++ λσ +3 v2 +σ +2 +0 +0 +m2 +σ + +λ2 +σv2 +H +2 ++ λσv2 +σ +2 +� +, +(26) +respectively. The above expressions can be reduced using Eqs. (22) and (23). When this is +done, the resulting M2 +I becomes identically zero. This implies the existence of two massless +Goldstone bosons. One of them (A) corresponds to the state that is eaten up by the Z +boson and becomes its longitudinal component, while the other (J) is associated to the +spontaneous breaking of the global U(1)L symmetry, the so-called majoron. On the other +hand, the reduction of M2 +R with Eqs. (22) and (23) leads to +M2 +R = +� +λ1v2 +H +λσ +3vHvσ +λσ +3vHvσ +λσv2 +σ +� +. +(27) +This matrix can be brought to diagonal form as V T +R M2 +RVR = � +M2 +R = diag(m2 +h, m2 +Φ), where +VR is a unitary matrix that can be parametrized as +VR = +� cos θ +− sin θ +sin θ +cos θ +� +. +(28) +The mixing angle θ is given by +tan(2θ) = +2(M2 +R)12 +(M2 +R)11 − (M2 +R)22 += +2rλσ +3 +r2λ1 − λσ +≈ −2rλσ +3 +λσ ++ O(r2) , +(29) +with r ≡ vH/vσ. For vσ ∼ TeV, r ≪ 1 and simple approximate expressions can be obtained. +The lightest of the two mass eigenstates is the well-known Higgs-like state h, with mass +mh ≈ 125 GeV, discovered at the LHC. In addition, the model contains the heavy scalar Φ, +with a mass of the order of vσ. We focus now on the Z2-odd scalars η+ and η0. The neutral +η0 field can be decomposed as +η0 = 1 +√ +2(ηR + iηI) . +(30) +12 + +Their masses are given by +mη+ = m2 +η + v2 +H +2 λeff +3 , +(31) +m2 +ηR = m2 +η + v2 +H +2 +� +λeff +3 + λeff +4 − +√ +2 βµvσ +m2 +S +� +, +(32) +m2 +ηI = m2 +η + v2 +H +2 +� +λeff +3 + λeff +4 + +√ +2 βµvσ +m2 +S +� +, +(33) +where we have defined +λeff +3 ≡ λ3 + λησ +3 +v2 +σ +v2 +H +− µ2 +v2 +σ +v2 +Hm2 +S +, +(34) +λeff +4 ≡ λ4 − β2v2 +H +2m2 +S +. +(35) +The mass square difference between ηR and ηI is given by +m2 +ηR − m2 +ηI = − +√ +2 βµvσ +m2 +S +v2 +H = λ5v2 +H , +(36) +exactly as in the usual Scotogenic model. Finally, the spontaneous breaking of U(1)L by +the VEV of σ induces a Majorana mass term for the N singlets, with MN = +√ +2 κ vσ. This +leads to Majorana neutrino masses at 1-loop, as shown in Fig. 2. The 3 × 3 neutrino mass +matrix is given by usual Scotogenic formula in Eq. (5), where λ5 is the effective coupling in +Eq. (24). Due to the additional scalar states, including a massless majoron with couplings to +charged leptons, the phenomenology of this model is richer than that of the usual Scotogenic +scenario. This will be discussed in Sec. 6. +5 +An UV extended Scotogenic model with two σ fields +We consider now an UV extension of the Scotogenic model with two σ fields. +5.1 +Ultraviolet theory +We enlarge the Scotogenic particle content with three new particles: the scalar SU(2)L +singlets S, σ1 and σ2. +Again, instead of the usual Z2 Scotogenic parity, a global U(1)L +lepton number symmetry is introduced. Table 5 shows the scalar and leptonic fields of the +model and their representations under the gauge and global symmetries. +We consider the 1-loop generation of neutrino masses by the diagram in Fig. 3. In our +notation, this is a II(1, 2) model. For this mechanism to take place, the operators +N�η†ℓL +, +σ1N +cN +, +σ1H†Sη +, +σ2H†S∗η +(37) +13 + +Field +Generations +SU(3)c +SU(2)L +U(1)Y +U(1)L +ℓL +3 +1 +2 +-1/2 +1 +eR +3 +1 +1 +-1 +1 +N +3 +1 +1 +0 +qN +H +1 +1 +2 +1/2 +0 +η +1 +1 +2 +1/2 +qη +σ1 +1 +1 +1 +0 +qσ1 +σ2 +1 +1 +1 +0 +qσ2 +S +1 +1 +1 +0 +qS +Table 5: Lepton and scalar particle content and representations under the gauge and global +symmetries in an UV extension of the Scotogenic model with two σ fields. +νL +νL +H0 +H0 +η0 +η0 +N +N +y +y +κ +σ1 +σ1 +σ2 +S +β1 +β2 +Figure 3: Neutrino mass generation in an extended Scotogenic model with two σ fields. This +Feynman diagram shows the relevant gauge eigenstates involved in the 1-loop contribution +to neutrino masses. In our notation, this is a II(1, 2) model. +must be allowed by the symmetries of the model. This restricts the U(1)L charges of the +fields in the model. In particular, one can write the following set of equations for them: +−qN + qη + 1 = 0 , +(38) +qσ1 + 2 qN = 0 , +(39) +qσ1 + qS + qη = 0 , +(40) +qσ2 − qS + qη = 0 . +(41) +14 + +They can be solved in terms of qN to obtain +qη = qN − 1 , +(42) +qσ1 = −2 qN , +(43) +qσ2 = 2 , +(44) +qS = qN + 1 . +(45) +In addition, we want the operators +N +cN +, +N �H†ℓL +, +� +H†η +�2 +(46) +to be forbidden. In order to forbid the first operator, a Majorana mass term for N, we +just require qN ̸= 0. The second operator would lead to νL-N Dirac mass terms and we +can forbid it by requiring qN ̸= 1. Then, Eq. (42) implies qη ̸= 0 too. Finally, with these +considerations, we choose +qN = 1 +2 , +(47) +which implies +qη = −1 +2 +, +qS = 3 +2 +, +qσ1 = −1 +, +qσ2 = 2 . +(48) +Some comments are in order. +First, the diagram in Fig. 3 has two different σ singlets +attached to the scalar internal line, σ1 and σ2. In principle, one may wonder why we did not +consider the same σ singlet in both vertices as starting point for constructing our model. +That would imply qS = 0 and reduce the number of couplings in the model. However, such +construction would lead to an effective operator (H†η)2σ2 after integrating out the S field. +If this operator is allowed by all symmetries of the model, so is the trilinear (H†η) σ. We will +eventually assume that the σ singlets acquire non-zero VEVs, breaking the original U(1)L. +In the presence of the trilinear (H†η) σ, this would induce a tadpole for η, hence breaking +the Z2 parity of the Scotogenic model. This forces us to discard this possibility and consider +different σ1 and σ2 attached to the internal scalar line. It also illustrates why models with +σA = σB are not compatible with our requirements. Furthermore, one may consider a third +σ3 singlet field coupled to the internal fermion line. While this is possible, we preferred to +choose a charge assignment that allows us to identify σ3 ≡ σ1 and reduce the number of fields +in the model. Finally, once σ1 and σ2 acquire non-zero VEVs, the original U(1)L symmetry +will get broken to one of its Zn subgroups. Here n is the GCD of |qσ1| and |qσ2| after being +normalized to become integer numbers, hence n = 2 and the remnant symmetry is Z2. +Once we know the quantum numbers of all the particles in the model, we can write its +Lagrangian. The new Yukawa interactions are given by +LY = y N �η† ℓL + κ σ1N +cN + h.c. , +(49) +where y is a general complex 3 × 3 matrix and κ is a complex symmetric 3 × 3 matrix. The +15 + +scalar potential of the model is given by +VUV = m2 +HH†H + m2 +SS∗S + m2 +σiσ∗ +i σi + m2 +ηη†η + λ1 +2 (H†H)2 + λ2 +2 (η†η)2 ++ λS +2 (S∗S)2 + λσi +2 (σ∗ +i σi)2 + λ3(H†H)(η†η) + λS +3 (H†H)(S∗S) ++ λσi +3 (H†H)(σ∗ +i σi) + ληS +3 (η†η)(S∗S) + λησi +3 (η†η)(σ∗ +i σi) ++ λσσ +3 (σ∗ +1σ1)(σ∗ +2σ2) + λσiS +3 +(σ∗ +i σi)(S∗S) + λ4(H†η)(η†H) ++ +� +β1(σ1H†Sη) + β2(σ2H†S†η) + µ +√ +2(σ2σ1σ1) + λ0(SSσ1σ∗ +2) + h.c. +� +, +(50) +where we sum over i = 1, 2. Here µ is a trilinear parameter with dimensions of mass while +m2 +H, m2 +η and m2 +σi have dimensions of mass2. Other Lagrangian terms are allowed by the +gauge symmetries of the model but forbidden by U(1)L. +5.2 +Effective theory +In the following we will assume that mS is much larger than any other energy scale in the +model and integrate out the heavy scalar S. If we do this at tree-level, the effective scalar +potential at low energies can be written as +VIR = m2 +H(H†H) + m2 +η(η†η) + m2 +σi(σ∗ +i σi) + λ1 +2 (H†H)2 + λ2 +2 (η†η)2 + λσi +2 (σ∗ +i σi)2 ++ λ3(H†H)(η†η) + λσi +3 (H†H)(σ∗ +i σi) + λησi +3 (η†η)(σ∗ +i σi) + λσσ +3 (σ∗ +1σ1)(σ∗ +2σ2) ++ +� +λ4 − |βi|2 +m2 +S +(σ∗ +i σi) +� +(H†η)(η†H) ++ +� µ +√ +2(σ2σ1σ1) − β1β2 +m2 +S +σ1σ2(H†η)2 + h.c. +� ++ O +� 1 +m4 +S +� +. +(51) +Now, we decompose the neutral fields H0 and σ1,2 as +H0 = 1 +√ +2(vH + φ + i A) , +σi = 1 +√ +2(vσi + ρi + i Ji) , +(52) +where we defined vH +√ +2 and +vσi +√ +2 as the VEVs of the corresponding fields. After this, we can +compute the tadpole equation resulting from the effective potential in Eq. (51), evaluated +at the VEVs of each scalar field. The non-trivial tadpole equations are +dVIR +dH0 +���� +⟨H0,σi⟩={ vH +√ +2 , +vσi +√ +2 } += vH +√ +2 +� +m2 +H + λ1 +v2 +H +2 + λσ1 +3 +v2 +σ1 +2 + λσ2 +3 +v2 +σ2 +2 +� += 0, +(53) +dVIR +dσ1 +���� +⟨H0,σi⟩={ vH +√ +2 , +vσi +√ +2 } += vσ1 +√ +2 +� +m2 +σ1 + µ vσ2 + λσ1 +3 +v2 +H +2 + λσ1 +v2 +σ1 +2 + λσσ +3 +v2 +σ2 +2 +� += 0, +(54) +dVIR +dσ2 +���� +⟨H0,σi⟩={ vH +√ +2 , +vσi +√ +2 } += vσ2 +√ +2 +� +m2 +σ2 + µ v2 +σ1 +2vσ2 ++ λσ2 +3 +v2 +H +2 + λσ2 +v2 +σ2 +2 + λσσ +3 +v2 +σ1 +2 +� += 0. +(55) +16 + +As already explained, as a result of σi acquiring a VEV, lepton number gets spontaneously +broken, leaving a discrete Z2 symmetry, under which all the particles in the model are even +except for N and η, which are odd. Another important consequence of the spontaneous +breaking of lepton number is the generation of the (H†η)2 operator, with a naturally sup- +pressed λ5 coupling due to the 1/m2 +S factor. One finds +λ5 +2 = −vσ1vσ2β1β2 +2m2 +S +≪ 1 , +(56) +where βi are dimensionless parameters of the UV theory and vσi ≪ mS by construction. +This expression clearly corresponds to a II(1, 2) model, following the classification of Sec. 3. +We now consider the scalar spectrum of the model. We will assume that CP is conserved +in the scalar sector, just for the sake of simplicity. In this case, the spectrum contains three +CP-even and three CP-odd gauge eigenstates. In the bases {φ, ρ1, ρ2} and {A, J1, J2}, their +mass matrices are given by +M2 +R = +� +� +� +λ1v2 +H +λσ1 +3 vHvσ1 +λσ2 +3 vHvσ2 +λσ1 +3 vHvσ1 +λσ1v2 +σ1 +vσ1(µ + λσσ +3 vσ2) +λσ2 +3 vHvσ2 +vσ1(µ + λσσ +3 vσ2) +λ2v2 +σ2 − +µv2 +σ1 +2vσ2 +� +� +� +(57) +and +M2 +I = +� +� +� +0 +0 +0 +0 +−2µvσ2 +−µvσ1 +0 +−µvσ1 +− +µv2 +σ1 +2vσ2 +� +� +� , +(58) +respectively. The tadpole equations (53)-(55) were used in the derivation of Eqs. (57) and +(58). The CP-even and CP-odd physical mass eigenstates can be written as linear combi- +nations of {φ, ρ1, ρ2} and {A, J1, J2}, respectively, obtained after the diagonalization of the +matrices M2 +R and M2 +I. Out of the three CP-even mass eigenstates, one can be identified +with the Higgs boson, with mass mh ≃ 125 GeV, discovered at the LHC. In addition, two +massive CP-even scalar fields exist. In what concerns the CP-odd mass eigenstates, their +mass matrix in Eq. (58) can be readily diagonalized as V T +I M2 +I VI = � +M2 +I, where +VI = +� +� +1 +0 +0 +0 +cos θ +− sin θ +0 +sin θ +cos θ +� +� +(59) +is a unitary matrix and � +M2 +I is a diagonal matrix. One obtains +� +M2 +I = +� +� +� +0 +0 +0 +0 +0 +0 +0 +0 +− +µ(v2 +σ1+4v2 +σ2) +2vσ2 +� +� +� , +(60) +thus leading to two massless pseudoscalar bosons. The first one is the Goldstone boson that +becomes the longitudinal component of the Z boson (A), while the second one (a linear +17 + +combination of fields J1 and J2) is associated to the spontaneous breaking of U(1)L and is +the so-called majoron, denoted as J. The J1 − J2 mixing angle is given by +tan(2θ) = +2 (M2 +I)23 +(M2 +I)22 − (M2 +I)33 += +4vσ1vσ2 +4v2 +σ2 − v2 +σ1 +. +(61) +We finally turn our attention to the Z2-odd scalars and decompose the neutral field η0 as +η0 = 1 +√ +2(ηR + i ηI) . +(62) +The mass of the charged η+ and the neutral ηR,I fields are given by +m2 +η+ = m2 +η + v2 +H +2 λeff +3 , +(63) +m2 +ηR = m2 +η + v2 +H +2 +� +λeff +3 + λeff +4 − β1β2vσ1vσ2 +m2 +S +� +, +(64) +m2 +ηI = m2 +η + v2 +H +2 +� +λeff +3 + λeff +4 + β1β2vσ1vσ2 +m2 +S +� +, +(65) +where we defined +λeff +3 ≡ λ3 + λησ1 +3 +v2 +σ1 +v2 +H ++ λησ2 +3 +v2 +σ2 +v2 +H +(66) +λeff +4 ≡ λ4 − β2 +1v2 +σ1 +2m2 +S +− β2 +2v2 +σ2 +2m2 +S +. +(67) +As in the Scotogenic model, the mass difference between ηR and ηI is proportional to the λ5 +coupling: +m2 +ηR − m2 +ηI = −vσ1vσ2β1β2 +m2 +S +v2 +H = λ5v2 +H . +(68) +Finally, the breaking of U(1)L also induces a Majorana mass term for the N singlets, with +MN = +√ +2 κ vσ1. This leads to Majorana neutrino masses at 1-loop, as shown in Fig. 3. +The resulting neutrino mass matrix is given by Eq. (5), with the effective λ5 of Eq. (56). +Furthermore, contrary to the minimal Scotogenic model, this UV extension induces a 1- +loop interaction between the majoron and a pair of charged leptons. +This enriches the +phenomenology of the model, as we discuss in the next Section. +6 +Phenomenology +All UV scenarios discussed in our classification of Sec. 3 and illustrated with the two examples +of Secs. 4 and 5 share some common features. They are characterized at low energies by +a Scotogenic model extended with a massless pseudoscalar, the majoron J, and one or +several massive scalars and pseudoscalars. While some phenomenological implications may +be specific to particular models, there are also some general expectations that we may +highlight. +18 + +Coupling +Upper limit +References +Im See +2.1 × 10−13 +[16] +Im Sµµ +2.1 × 10−9 +[17] +|Seµ| +5.3 × 10−11 +[15] +|Seτ| +5.9 × 10−7 +[15] +|Sµτ| +7.6 × 10−7 +[15] +Table 6: Current limits on the majoron couplings to charged leptons. The limit on Im See +is at 90% C.L. [16]. The limit on Im Sµµ has been obtained by performing a simulation of +the supernova SN1987A [17]. An alternative and more stringent limit Im Sµµ < 2.1 × 10−10 +can be derived with more aggressive assumptions in the simulation. +6.1 +Majoron coupling to charged leptons +The presence of a massless majoron dramatically affects the phenomenology of this class +of models. In fact, models including a majoron are strongly constrained by a variety of +experimental limits, such as those originated by the majoron coupling to a pair of charged +leptons. The relevance of these limits depends on the flavor structure of the couplings [14], +which necessarily depends on the specific model. Stringent constraints exist for both flavor- +conserving and flavor-violating couplings. Let us write the majoron interaction with charged +leptons as [15], +LℓℓJ = J ¯ℓβ +� +Sβα +L PL + Sβα +R PR +� +ℓα + h.c. . +(69) +Here ℓα,β are the charged leptons with flavors α and β, while PL,R are the usual chiral projec- +tors. We consider all flavor combinations for the SL,R couplings: βα = {ee, µµ, ττ, eµ, eτ, µτ}. +Due to the pseudoscalar nature of majorons, the diagonal Sββ = Sββ +L + Sββ∗ +R +couplings are +purely imaginary. They receive strong constraints from astrophysical observations, due to +the cooling effects induced by the majoron in dense astrophysical media. Flavor off-diagonal +couplings are constrained by the null searches of lepton flavor violation in processes involv- +ing charged leptons. In particular, searches for ℓα → ℓβ J can be used to set bounds on the +combinations +|Sβα| = +����Sβα +L +��� +2 ++ +���Sβα +R +��� +2�1/2 +. +(70) +A compilation of the current limits on the majoron couplings to charged leptons can be +found in Table 6. +While in some scenarios the majoron couplings to charged leptons appear at tree-level [18, +19], in many cases the leading order contribution is induced at the 1-loop level. For instance, +this is the case of the popular type-I seesaw with spontaneous lepton number violation [9, +20,21]. Similarly, in the Scotogenic scenarios discussed in this paper, the majoron coupling +to charged leptons is also induced at 1-loop [8,22] by the Feynman diagram in Fig. 4. Here +19 + +J +ℓ−α +ℓ+ +β +η− +N +N +y +y +gJNN +Figure 4: 1-loop generation of the majoron coupling to a pair of charged leptons in the +Scotogenic scenarios discussed in this work. +gJNN is the J − N − N coupling, which depends on the specific model. It is given by +gJNN = +� +i κ +√ +2 +in models with one σ singlet +i κ +√ +2 cos θ +in models with two σ singlets +, +(71) +where the mixing angle θ is defined in Eq. (59). The prefactor cos θ in models with two σ +singlets is due to the fact that only σ1 has a coupling to N +cN. No other contributions to +the majoron coupling to charged leptons exist at 1-loop. One may wonder about a Feynman +diagram with two scalar lines in the loop, induced by a J η+η− coupling. However, this +contribution vanishes exactly. The reason is the pseudoscalar nature of the majoron. The +J ¯ℓαℓα vertex must be proportional to γ5, but the Lorentz structure of this contribution does +not generate such pseudoscalar coupling. 4 Also, diagrams with gauge bosons vanish due +to the pure singlet nature of N. Therefore, one can find the SL,R couplings introduced in +Eq. (69) by direct computation of the diagram in Fig. 4. The result can be written as [8,22] +Sβα +L = −mℓβ +8π2 +� +y†gJNN Γ y +� +βα , +(72) +Sβα +R = mℓα +8π2 +� +y†gJNN Γ y +� +βα , +(73) +for the non-diagonal couplings and +Sββ = −mℓβ +8π2 +� +y†gJNN Γ y +� +ββ , +(74) +for the diagonal ones. Here mℓβ = {me, mµ, mτ} and we have defined +Γmn = +MNn +� +M 2 +Nn − m2 +η+ +�2 +� +M 2 +Nn − m2 +η+ + m2 +η+ log +m2 +η+ +M 2 +Nn +� +δmn . +(75) +4We also note that the J η+η− coupling is absent in many models, since Lagrangian terms like σ|η|2 or +σ2|η|2 are forbidden by lepton number. Only in models with two σ fields one may have a term of the form +σ1σ2|η|2 (when qσ1 = −qσ2) leading to a J η+η− interaction vertex after symmetry breaking. However, as +explained in the text, even when this term is present, the associated 1-loop contribution to the majoron +coupling to a pair of charged leptons vanishes exactly due to the pseudoscalar nature of the majoron. +20 + +Figure 5: Contours of BR (µ → eJ) in the (MN, mη+) plane. The colored regions correspond +to the regions allowed by the current experimental bound on the branching ratio. On the +left, gJNN has been fixed to 10−1 (blue) and to 10−2 (pink), while rη = 1 has been used. +On the right, the coupling gJNN was not fixed and three different values of the rη ratio have +been considered, 0.1 (pink), 1 (blue) and 2 (green). +We can now study how the bounds on these couplings restrict the parameter space of the +models considered in our classification. In particular, in the following we focus on the 2-body +decay µ → eJ, for which +BR (µ → eJ) = +mµ +32 π Γµ +� +|Seµ +L |2 + |Seµ +R |2� +, +(76) +where Γµ ≈ 3 × 10−19 GeV is the total decay width of the muon. We used a Casas-Ibarra +parametrization [23] properly adapted to the Scotogenic model [24–26] and the best-fit values +obtained in the global fit [27] to neutrino oscillation data in order to express the Yukawa +matrix y in terms of experimentally measured quantities. We assumed that the three singlet +fermions are degenerate, that is MN1 = MN2 = MN3 = MN and we fixed λ5 = 5 × 10−8. +Notice that lower values of this parameter would imply larger values of the Yukawas, thus +further restricting the parameter space of the model. It also proves convenient to define +rη = m0 +mη+ . +(77) +Our results are shown in Fig. 5. On the left-hand side we fixed the coupling gJNN to 10−1 +(blue), and to 10−2 (pink), and we considered rη = 1 in both scenarios. The colored regions +correspond to regions allowed by the experimental bound on the µ → eJ decay, which implies +BR (µ → eJ) < 10−5 [18]. As expected, the larger the J −N −N coupling is, the smaller the +allowed region of the parameter space becomes. We also find that light Scotogenic states can +be made compatible with the µ → eJ bound. This can be easily understood by inspecting +21 + +the non-trivial relation between the masses mη+ and MN and the Yukawa couplings y. Under +the assumptions mentioned above one finds +SL,R ∝ gJNNΓii +� +y†y +� +12 , +(78) +where Γii is any of the diagonal entries of Γ, given by +Γii ∝ MN +M 2 +N − m2 +η+ + m2 +η+ log +m2 +η+ +M2 +N +� +M 2 +N − m2 +η+ +�2 +. +(79) +Eq. (5) implies that the Yukawa product +� +y†y +� +12 is proportional to +� +y†y +� +12 ∝ +1 +MN +(M 2 +N − m2 +0)2 +M 2 +N − m2 +0 + M 2 +N log m2 +0 +M2 +N +. +(80) +Therefore, in the limit rη = 1 one finds +SL,R ∝ gJNN +M 2 +N − m2 +η+ + m2 +η+ log +� +m2 +η+ +M2 +N +� +M 2 +N − m2 +η+ + M 2 +N log +� +m2 +η+ +M2 +N +� . +(81) +For a fixed gJNN value two possibilities arise: (i) if we fix mη+, the Γii +� +y†y +� +12 combination +decreases if MN increases, and (ii) if we fix MN, the Γii +� +y†y +� +12 combination increases if +mη+ increases. Essentially, the involved couplings strongly depend on mη+ and MN and this +dependence may lead to an apparent non-decoupling behavior that explains the results for +the µ → eJ branching ratio observed in Fig. 5. Finally, the right-hand side of this figure +provides complementary information. Here we considered gJNN = i κ +√ +2 = i MN +2 vσ and fixed +vσ = 5 TeV. Since the gJNN coupling grows with MN, for each mη+ there is a maximum +value of MN for which BR (µ → eJ) < 10−5. This can be clearly seen in our results. +6.2 +Collider signatures +Since the spontaneous breaking of U(1)L requires the introduction of additional scalar mul- +tiplets, all models in our classification have extended scalar sectors containing several states +besides the ones in the Scotogenic model. This can be used to probe them at colliders. +One of the CP-even scalars, presumably the lightest, is to be identified with the 125 +GeV state discovered at the LHC. The production cross-section and decay rates of this +state, denoted generally as h, must agree with the values measured by the ATLAS and CMS +collaborations. Since these are very close to those predicted for a pure SM Higgs, h ≈ Re(H0) +is generally required. In particular, mixings with the σ states are strongly constrained, since +they would affect its decay rates in a twofold way. First, the σ states do not couple to the +SM gauge bosons or to quarks. Thus, any mixing would induce a universal reduction of the +h partial decay widths into these states. And second, h can have additional decay models. It +22 + +can decay invisibly to a pair of singlet fermions (h → N1N1) or to pair of majorons (h → JJ). +The former can only take place if mN1 ≤ mh/2. In contrast, since the majoron is massless, +the latter is always kinematically available. We can write the interaction Lagrangian of h +with a pair of majorons as LhJJ = 1 +2 ghJJ h J2, where ghJJ is a dimensionful coupling that +depends on the specific model. This interaction induces the invisible decay h → JJ, with +the decay width given by +Γ(h → JJ) = +g2 +hJJ +32 π mh +. +(82) +If we assume a total Higgs decay width in agreement with the SM expectation, Γh ≈ ΓSM +h += +4.1 MeV [28], the bound on the invisible Higgs branching ratio BR(h → JJ) < 0.19 at +95% C.L. [29], implies ghJJ < 3.1 GeV. This translates into constraints on the parameters +of the scalar potential of the model, which are encoded in ghJJ. For instance, in the model +discussed in [8], this implies that the coefficient of the (H†H)(σ∗σ) operator must be ≲ 10−2. +We note, however, that stronger constraints can be derived by combining invisible and visible +channels, as recently pointed out in [30]. +Finally, all models in our classification also contain additional heavy states. They can +also be searched for at colliders. Their production cross-sections and decay models strongly +depend on the specific realization of our setup and, more specifically, on their gauge com- +position. If they have sizable doublet components, they can in principle be produced at +high rates at the LHC via Drell–Yan processes. In contrast, heavy scalars with a dominant +component in the singlet direction have very suppressed production cross-sections at the +LHC. Due to the constraints discussed above, which imply suppressed mixing between the +SM Higgs doublet and the σ states, this is the most likely scenario in all models discussed +in our classification. +6.3 +Dark matter +In all UV models studied in this paper, a remnant Z2 symmetry is obtained as a result of +the spontaneous breaking of lepton number. This is the Scotogenic Z2 parity, under which +only the usual Scotogenic states N and η are charged. The conservation of Z2 implies that +the lightest of them is completely stable and, in principle, a valid DM candidate. Both +options have been widely studied in the literature. In the case of a scalar candidate, the DM +phenomenology resembles that of the Inert Doublet model [31–35], with the DM production +in the early Universe set by gauge interactions. In contrast, the case of a fermion candidate +typically requires large Yukawa couplings. This leads to tension with bounds from lepton +flavor violation [24], although the observed DM relic density can be achieved [36–40]. +The low energy theories resulting from our UV extended models do not correspond exactly +to the original Scotogenic model. +As explained above and illustrated in Secs. 4 and 5, +additional scalar states are present: the massless majoron and one of several massive scalars. +These new degrees of freedom couple to the Z2-odd states and may affect the resulting DM +phenomenology, which may have some differences with respect to the one in the original +Scotogenic scenario. This has recently been studied in [41,42] for the case of fermion DM. +The main conclusion from these works is that the new scalar states open up new regions in +parameter space in which the DM relic density can match the observed value. In particular, +23 + +annihilations become very efficient when the mass of the DM candidate, mN1, is about half of +the mass of a new scalar state. This implies that one can find the correct DM abundance for +any value of mN1 without resorting to coannihilations, in contrast to the original Scotogenic +model. These models are also expected to have a rich phenomenology at direct and indirect +detection experiments [42]. +7 +Summary and discussion +The Scotogenic model is a very popular scenario for neutrino masses and dark matter. In +this work we have considered extensions of this scenario that naturally explain the smallness +of the quartic λ5 coupling and the origin of the Scotogenic Z2 parity. This is achieved in +UV extensions including a conserved global lepton number symmetry, spontaneously broken +by the VEVs of one or several scalar singlets, and a new heavy state that suppresses all +lepton number violating effects at low energies. We explored all possible models with these +assumptions and found 24 variations. +They are all characterized at low energies by the +presence of a massless Goldstone boson, the majoron, as well as other massive scalars besides +the usual Scotogenic states. Two specific example models are discussed in detail in order +to illustrate the basic ingredients of our setup. In these two models, as well as in all the +variants in our classification, a rich phenomenology is expected, with potential signatures in +collider and lepton flavor violating searches, and implications for dark matter. +Out of the 24 models revealed by our analysis, only one had been previously studied +in the literature, namely [8]. +This illustrates the vast model space beyond the original +Scotogenic model yet to be explored. In fact, there are many variations of the fundamental +setup that keep all the positive features and include additional ingredients. While many +of these modified Scotogenic scenarios may contain unnecessary or redundant ingredients, +other may offer novel ways to address open questions in current particle physics [43]. This +is the main motivation behind the classification presented in this work. +There are several ways in which our analysis can be extended. First of all, we have +considered UV theories that realize the λ5 coupling at tree-level. +In this case, the only +source of suppression is given by the large energy scale mS, assumed to lie well above the +electroweak scale. +Alternatively, the λ5 coupling can also be realized at loop order, as +recently explored in [44]. This possibility leads to many novel extensions of the Scotogenic +setup with, at least potentially, new phenomenological expectations. Another way in which +our analysis can be extended is by considering a local lepton number symmetry. In this case, +the massless majoron that was characteristic in our setup would be replaced by a heavy Z′ +boson, with a dramatic impact on the low-energy phenomenology. However, we note that this +direction requires non-trivial extensions of the fermion particle content in order to cancel out +the usual triangle gauge anomalies. Therefore, a general classification of all possible gauge +models becomes more cumbersome, although interesting too. Finally, variations with non- +universal lepton charges for the N fermions or featuring alternative numbers of generations +for the Scotogenic states can be explored as well. +24 + +Acknowledgements +Work supported by the Spanish grants PID2020-113775GB-I00 (AEI/10.13039/501100011033) +and CIPROM/2021/054 (Generalitat Valenciana). The work of PE is supported by the FPI +grant PRE2018-084599. +AV acknowledges financial support from MINECO through the +Ram´on y Cajal contract RYC2018-025795-I. DPS would like to thank the AHEP group for +the hospitality during his visit. The work of DPS was supported by Ciencia de Frontera +CONACYT project No. 428218 and the program “BECAS CONACYT NACIONALES”. +A +Accidental Z2 symmetries +The dark Z2 parity of the Scotogenic model can also be an accidental symmetry generated +after the σ singlet (or singlets) acquires a VEV. In these scenarios, the symmetry breaking +path is also U(1)L → Z2, but with ℓL, eR and η as the only particles charged under the +discrete symmetry. +In this case, the Yukawa term ¯N ˜η†ℓL and the Majorana mass N +cN +are allowed by all symmetries, while ¯N ˜H†ℓL is forbidden. Furthermore, given that η is the +only Z2-odd scalar, it will always appear in pairs in the effective scalar potential. Therefore, +although the Z2 Scotogenic parity does not emerge as a remnant symmetry after the breaking +of U(1)L, it appears accidentally as a consequence of it. These UV models are not included +in the classification presented in Sec. 3 since they violate requirement (A). However, they +also lead to the Scotogenic model at low energies. +Let us illustrate this possibility with a specific example. 5 Consider the particle content +and charge assignment in Table 7. The new Yukawa interactions in the model are given by +LY = y N �η† ℓL + MN N +cN + h.c. , +(83) +while the scalar potential of the model is written as +VUV = m2 +HH†H + m2 +SS∗S + m2 +σσ∗σ + m2 +ηη†η + λ1 +2 (H†H)2 + λ2 +2 (η†η)2 ++ λS +2 (S∗S)2 + λσ +2 (σ∗σ)2 + λ3(H†H)(η†η) + λS +3 (H†H)(S∗S) ++ λσ +3(H†H)(σ∗σ) + ληS +3 (η†η)(S∗S) + λησ +3 (η†η)(σ∗σ) ++ λσS +3 (σ∗σ)(S∗S) + λ4(H†η)(η†H) + +� +β(σH†ηS) + µ1 H†ηS∗ + µ2 σ S2 + h.c. +� +. +(84) +It is easy to check that other Lagrangian terms are forbidden by U(1)L. This global symmetry +gets spontaneously broken once the electroweak singlet σ acquires a non-zero VEV, leaving +a remnant Z2 under which η, S, ℓL and eR are odd, while the rest of the fields are even. We +can call this symmetry Zrem +2 +. Since qN = 0, N is even under Zrem +2 +, and thus this symmetry +cannot be identified with the Scotogenic dark parity. Nevertheless, the Lagrangian of the +Scotogenic model is still obtained after decoupling the heavy scalar S. This is due to the +fact that a new accidental Z2 parity appears. The only fields charged under this parity are +η and N, while all the other fields in the effective theory are even, therefore, this accidental +symmetry, that we can denote as Zacc +2 , is precisely the Scotogenic Z2. +5This model corresponds to the II′ (1, ∅) model shown below in Table 8. +25 + +Field +Generations +SU(3)c +SU(2)L +U(1)Y +U(1)L +ℓL +3 +1 +2 +-1/2 +1 +eR +3 +1 +1 +-1 +1 +N +3 +1 +1 +0 +0 +H +1 +1 +2 +1/2 +0 +η +1 +1 +2 +1/2 +-1 +σ +1 +1 +1 +0 +2 +S +1 +1 +1 +0 +-1 +Table 7: Lepton and scalar particle content and representations under the gauge and global +symmetries in an UV extension of the Scotogenic model with accidental Z2 symmetry. +Let us now generalize the idea studied in this Appendix. We consider again the set of +models in which (H†η)2 is generated by the topologies shown in Table 2 with the addition of +at most two different singlets σ1,2. There are two possibilities to construct models in which +the Zacc +2 +symmetry is obtained: +(i) Models with qN ̸= 0. In this case we consider the models shown in Table 2 but +impose that N is even under the remnant Zrem +2 +parity while ℓL, eR and η are odd. The +Majorana masses of the N fermions are induced by the κ σ1N +cN Yukawa term. +(ii) Models with qN = 0. This case is excluded from the classification in Sec. 3, which +focuses on qN ̸= 0, and must be discussed independently. In these models the Majorana +mass term MN N +cN is present in the UV theory. +We now proceed to discuss these two cases independently. The first one can be regarded +as a revision of our discussion in Sec. 3, imposing now different conditions on the resulting +models. In fact, the models studied in Sec. 3 could also lead to U(1)L → Zrem +2 +, leaving the +Scotogenic Z2 parity as an accidental symmetry. This will be the case when these conditions +on qN are satisfied: +• qN = 2 z, where z can be any integer number except zero. +• qN = α +β, with α, β ∈ Z and α and β even and odd, respectively. Also, GCD(α, β) = 1 +has to be satisfied. +Notice, however, that models with fixed charges, that is, the ones with only σ1, always have +the Scotogenic symmetry as the remnant symmetry and do not enter this discussion. +Considering now the second possibility, only 11 different models exist and they are listed +in Table 8. Let us denote them as ξ′(A, B), where ξ = {I, II, III, IV} and the prime is used +to distinguish these models from the ones studied in Sec. 3. Each of the 11 models needs +to satisfy any of the following conditions on qσ1 in order to generate the Z2 parity as an +accidental symmetry: +26 + +Topology +A +B +qN +qη +qσ1 +qσ2 +qS +(SU(2)L, U(1)Y)S +1 +I′ +1 +∅ +0 +−1 +2 +- +−2 +(3, 1) +2 +I′ +∅ +1 +0 +−1 +2 +- +0 +(3, 1) +3 +I′ +1 +2 +0 +−1 +qσ1 +2 − qσ1 +−qσ1 +(3, 1) +4-5 +II′ +1 +∅ +0 +−1 +2 +- +−1 +(3, 0) or (1, 0) +6-7 +II′ +1 +2 +0 +−1 +qσ1 +2 − qσ1 +1 − qσ1 +(3, 0) or (1, 0) +8 +III′ +1 +∅ +0 +−1 +2 +- +−2 +(2, 1/2) +9 +III′ +1 +2 +0 +−1 +qσ1 +2 − qσ1 +−2 +(2, 1/2) +10 +IV′ +1 +∅ +0 +−1 +2 +- +1 +(2, 1/2) +11 +IV′ +1 +2 +0 +−1 +qσ1 +2 − qσ1 +1 +(2, 1/2) +Table 8: UV extended models for which the term N +cN is allowed and the Scotogenic Z2 is +an accidental symmetry. For each model we show the U(1)L charges of N, η, σ1, σ2 and S, +as well as the (SU(2)L, U(1)Y) representation of S. 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Toma, “Phenomenological and cosmological implications of a scotogenic three-loop +neutrino mass model,” arXiv:2212.06852 [hep-ph]. +30 + diff --git a/rtE4T4oBgHgl3EQfwQ2l/content/tmp_files/load_file.txt b/rtE4T4oBgHgl3EQfwQ2l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..26b8952bb093b8ac780aa28f7f0f967345884c18 --- /dev/null +++ b/rtE4T4oBgHgl3EQfwQ2l/content/tmp_files/load_file.txt @@ -0,0 +1,902 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf,len=901 +page_content='IFIC/23-01 Ultraviolet extensions of the Scotogenic model Diego Portillo-S´ancheza, Pablo Escribanob, Avelino Vicenteb,c (a) Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Nacional, Apdo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Postal 14-740, 07000 M´exico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=', M´exico (b) Instituto de F´ısica Corpuscular, CSIC-Universitat de Val`encia, 46980 Paterna, Spain (c) Departament de F´ısica Te`orica, Universitat de Val`encia, 46100 Burjassot, Spain diego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='portillo@cinvestav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='mx, pablo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='escribano@ific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='es, avelino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='vicente@ific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='es Abstract The Scotogenic model is a popular scenario that induces radiative Majorana neutrino masses and includes a weakly-interacting dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We classify all possible ultraviolet extensions of the Scotogenic model in which (i) the dark Z2 parity emerges at low energies after the spontaneous breaking of a global U(1)L lepton number symmetry, and (ii) the low-energy effective theory contains a naturally small lepton number breaking parameter, suppressed by the mass of a heavy mediator integrated out at tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We find 24 such models and discuss two of them in detail to illustrate our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We also discuss some general aspects of the phenomenology of the models in our classification, exploring possible lepton flavor violating signals, collider signatures and implications for dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The phenomenological prospects of these scenarios are very rich due to the presence of additional scalar states, including a massless Goldstone boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1 Introduction The Scotogenic model [1] is a popular extension of the Standard Model (SM) that addresses two of the currently most important open questions in physics: the origin of neutrino masses and the nature of the dark matter (DM) of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Its popularity stems from its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The model extends the SM particle content with three singlet fermions, N1,2,3, and a scalar doublet, η, all odd under a new Z2 symmetry under which the SM fields are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' These ingredients suffice to induce Majorana neutrino masses at the 1-loop level and provide a viable DM candidate, namely the lightest Z2-odd state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='05249v1 [hep-ph] 12 Jan 2023 Radiative neutrino mass models [2–5] provide a natural suppression for neutrino masses with loop factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This is one of the main motivations in favor of this class of models [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In addition, further suppression is introduced in some models by assuming an approximate lepton number symmetry, broken in a small amount by the presence of a Lagrangian term with a suppressed coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This is the case of the Scotogenic model, that requires a small λ5 ≪ 1 quartic parameter to obtain the correct size for neutrino masses with sizable Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' While this is technically valid, and natural in the sense of ’t Hooft [7], it also calls for an extension that explains the smallness of the λ5 parameter, possibly relating it to the breaking of lepton number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this work we consider ultraviolet (UV) extensions of the Scotogenic model that provide a natural explanation for the smallness of the λ5 parameter and in which the Z2 parity of the model emerges at low energies from a spontaneously broken global U(1) lepton number symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This endeavor was initiated in [8], where a specific UV model with these proper- ties was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Here we go beyond specific realizations and classify all possible models with these features in which a low-energy Scotogenic model is obtained after integrating out a heavy field at tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Besides one or several massive scalars, the particle spectrum of the theory will contain a massless Goldstone boson, the majoron [9–12], induced by the spontaneous breaking of lepton number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' These new states are not present in the original Scotogenic model and lead to novel phenomenological predictions that allow one to probe our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The rest of the manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' First, we set our notation and con- ventions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 2, where the Scotogenic model is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' A general classification of all possible UV extensions of the Scotogenic model satisfying the requirements explained above is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Two selected example models will be presented in detail in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Some general aspects of the phenomenology of this class of models are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, we summarize our results and conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Additional information can be found in Appendix A, where we discuss scenarios with an accidental Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 2 The Scotogenic model Before we discuss specific UV realizations of our setup, let us introduce our conventions for the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The particle content of the Scotogenic model [1] includes, besides the usual SM fields, three generations of right-handed fermions N, transforming as (1, 0) under (SU(2)L, U(1)Y), and one scalar η, transforming as (2, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We also impose the conservation of an ad-hoc Z2 symmetry, under which η and N are odd while the rest of the fields in the model are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The lepton and scalar particle content of the model is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1 The model contains two scalar doublets, the usual Higgs doublet H and the new doublet η, only distinguished by their Z2 charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' They can be decomposed in terms of their SU(2)L components as H = �H+ H0 � , η = �η+ η0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (1) Once specified the particle content and symmetries of the model we can write down the 1We follow the conventions for the Scotogenic model used in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 2 Field Generations SU(3)c SU(2)L U(1)Y Z2 ℓL 3 1 2 1/2 + eR 3 1 1 1 + N 3 1 1 0 − H 1 1 2 1/2 + η 1 1 2 1/2 − Table 1: Lepton and scalar particle content and representations under the gauge and discrete symmetries in the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' ℓL and eR are the SM left- and right-handed leptons, respectively, and H is the SM Higgs doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The Lagrangian of the model contains the terms LY = y N �η† ℓL + 1 2MN N cN + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' , (2) where y is a general complex 3 × 3 matrix and MN is a symmetric 3 × 3 mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The scalar potential of the model is given by VUV = m2 HH†H + m2 ηη†η + λ1 2 (H†H)2 + λ2 2 (η†η)2 + λ3(H†H)(η†η) + λ4(H†η)(η†H) + �λ5 2 (H†η)2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (3) Here m2 H and m2 η are parameters with dimensions of mass2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We assume that the minimization of the scalar potential leads to a vacuum defined by ⟨H0⟩ = vH √ 2 , ⟨η0⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (4) This vacuum configuration breaks the electroweak symmetry in the usual way but preserves the Z2 symmetry of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' As a consequence of this, the lightest Z2-odd state (either N1 or η0) is completely stable and can play the role of the DM of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Furthermore, neutrinos acquire non-zero Majorana masses at the 1-loop level, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The resulting 3 × 3 neutrino mass matrix is given by (mν)αβ = λ5 v2 H 32π2 � n ynα ynβ MNn � M 2 Nn m2 0 − M 2 Nn + M 4 Nn � m2 0 − M 2 Nn �2 log M 2 Nn m2 0 � , (5) where m2 0 = m2 η + (λ3 + λ4) v2 H/2 and MNn are the diagonal elements of the MN matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' One can easily estimate that in order to obtain neutrino masses of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 eV with Scotogenic states in the TeV scale and Yukawas of order 1, λ5 must be of order ∼ 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The smallness of this parameter is protected by lepton number, and thus is technically natural [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, it is not explained in the context of the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3 νL νL H0 H0 η η N N Figure 1: Neutrino mass generation in the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This Feynman diagram shows the relevant gauge eigenstates involved in the 1-loop contribution to neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3 Ultraviolet extensions of the Scotogenic model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 General considerations The Scotogenic model has two features that call for a refinement, namely, the origin of the Z2 symmetry and λ5 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Although these features do not pose any theoretical problem, they can be regarded as ad-hoc ingredients in an otherwise very natural framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We are thus interested in an UV extension of the Scotogenic model that provides an explanation for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' More specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' we want to classify all possible UV scenarios that lead to the Scotogenic model at low energies after integrating out a heavy scalar field S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' with mS ≫ vH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' and satisfy the following two requirements: (A) The Scotogenic Z2 is obtained as a remnant after the spontaneous breaking of a U(1)L lepton number symmetry by the VEV of one or several singlet scalar fields σ: U(1)L ⟨σ⟩ −−−−−→ Z2 (B) The (H†η)2 operator is forbidden in the UV theory due to U(1)L conservation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' but an operator of the form (H†η)2σn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' with n ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' is generated after integrating out S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' After the singlets get VEVs and U(1)L is spontaneously broken, this will induce an effective λ5 coupling, which will be naturally suppressed by the large mS energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this work we will concentrate on global U(1)L lepton number symmetries, tree-level com- pletions of the λ5 operator and UV models with one or two σ singlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Gauged versions of the lepton number symmetry, higher-order completions and models with additional singlets are left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The models we are looking for induce neutrino masses `a la Scotogenic, with variations of the neutrino mass diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This diagram has an internal scalar line (with η0) and an internal fermion line (with N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The analogous diagrams in the UV extended models will include the heavy scalar S in the loop and one or several external legs with σ singlets (or 4 Topology Diagram Required operators I η H† S η H† σA σB (σAH†S ˜H),(σB˜η†S†η) II H† H† S η η σA σB (σAH†Sη),(σBH†S†η) III H† H† S η η σA σB (σAσBH†S),(H†ηS†η) IV H† H† S η η σA σB (H†SH†η),(σAσBS†η) Table 2: (H†η)2σAσB operator in the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' σ insertions, for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' After these considerations, there are two classes of models that can be already discarded: Models without σ insertions in the scalar line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' These models can be discarded because the (H†η)2 operator would be allowed in the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This would preclude an explanation of λ5 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In addition, η would acquire a VEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Models without σ insertions in the fermion line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The U(1)L charge of the N singlet fermions must necessarily vanish if the σN cN operator is absent and their Majorana masses are explicitly introduced in the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, in this case N will be even under the Z2 symmetry obtained after spontaneous U(1)L breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This scenario does not correspond to the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Nevertheless, an additional accidental Z2 symmetry may appear, as explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We are thus left with neutrino mass topologies with σ insertions in both internal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The scalar line leads to an operator (H†η)2σn after the heavy S is integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In fact, due to the renormalizability of the UV theory, n can be at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Therefore, the resulting operator will be of the form Oλ5 = (H†η)2σAσB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (6) This generic expression includes cases with only one σ insertion (for instance, σB = ∅) and cases in which both σ insertions in the scalar line correspond to the same field (σA = σB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5 All possible topologies are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, the fermion line simply corresponds to a σ − N − N Yukawa interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In the following we will always assume the presence of the operator σN cN (for models with one σ field) or σ1N cN (for models with two σ fields), and we will not draw it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The coefficient of this operator will be denoted by κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Therefore, once the singlet scalar gets a VEV, ⟨σ(1)⟩ = vσ(1) √ 2 , the Majorana mass matrix for the singlet fermions N is generated, 2 MN = √ 2 κ vσ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='2 Model classification In the following we will refer to a specific model using the notation ξ(A, B), where ξ = {I, II, III, IV} denotes the topology for the (H†η)2σAσB operator, as listed in Table 2, and A and B denote the singlets involved in the vertices where σA,B are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Since we only consider UV theories with at most two different singlets, A and B can only take the values ∅, 1, 2, 1∗, where ∅ indicates that no σ enters the corresponding vertex and σ1∗ ≡ σ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' It is important to mention that we do not consider scenarios with A, B = 2∗ because they lead to a redefinition of the charge qσ2 → −qσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3 Therefore, in principle each topology has 16 different variations depending on the way the σA,B singlets are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, we can reduce this number by taking into account the following arguments: A ̸= B is required to forbid the term (H†η σA)2 in the effective Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' If this specific combination is allowed, then the term (H†η σA) is too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This trilinear interaction induces a non-zero VEV for η after both H and σA acquire their VEVs, hence breaking the Scotogenic Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' A ̸= B∗ is also required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Otherwise, (H†η)2σAσ∗ A is allowed by the U(1)L symmetry and then the operator (H†η)2 is present in the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In all ξ(1, ∅) and ξ(∅, 1) models the effective operator leading to the λ5 coupling is Oλ5 = (H†η)2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This implies the relation 2qη + qσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In addition, the Yukawa coupling σN cN implies 2qN + qσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Hence, the charges for η and N must satisfy qη = qN and then the N ˜η†ℓL Yukawa term is forbidden by U(1)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Similarly, in all ξ(1∗, ∅) and ξ(∅, 1∗) models one finds qη = −qN and then qN = 1 2 in order to allow the term N ˜η†ℓL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' With these considerations, there are only 8 possibilities left in each of the four topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, there are duplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Models based on topologies III and IV are symmetric with respect to the exchange σA ↔ σB (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' ξ(A, B) = ξ(B, A) with ξ = III, IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Similarly, II(A, B) ∼ II(B, A) by redefining qS → −qS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This further reduces the number of funda- mentally different UV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In total, we find 24 (20 + 4, because in II-models S can be 2In models with two σ fields such that qσ1 = qσ2 or qσ1 = −qσ2, an additional Yukawa term σ2N cN or σ∗ 2N cN would be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Here qσ1 and qσ2 denote the U(1)L charges of σ1 and σ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This would lead to MN = √ 2 (κ1 vσ1 + κ2 vσ2) without affecting our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We note, however, that in such models both σ singlets are esentially copies of the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3In the following, we will denote the U(1)L charge of the field X as qX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Furthermore, qℓL = qeR = 1 and qH = 0, as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 6 Topology A B qN qη qσ1 qσ2 qS (SU(2)L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' U(1)Y)S 1 I 1∗ ∅ 1 2 − 1 2 −1 −1 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 2 I ∅ 1∗ 1 2 − 1 2 −1 0 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 3 I 2 ∅ qN qN − 1 −2qN 2 − 2qN 2qN − 2 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 4 I ∅ 2 qN qN − 1 −2qN 2 − 2qN 0 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 5 I 1 2 qN qN − 1 −2qN 2 2qN (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 6 I 2 1 qN qN − 1 −2qN 2 −2 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 7 I 1∗ 2 qN qN − 1 −2qN 2 − 4qN −2qN (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 8 I 2 1∗ qN qN − 1 −2qN 2 − 4qN 4qN − 2 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 9-10 II 1∗ ∅ 1 2 − 1 2 −1 − 1 2 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) or (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) 11-12 II 2 ∅ qN qN − 1 −2qN 2 − 2qN qN − 1 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) or (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) 13-14 II 1 2 qN qN − 1 −2qN 2 1 + qN (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) or (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) 15-16 II 1∗ 2 qN qN − 1 −2qN 2 − 4qN 1 − 3qN (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) or (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) 17 III 1∗ ∅ 1 2 − 1 2 −1 −1 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 18 III 2 ∅ qN qN − 1 −2qN 2 − 2qN 2qN − 2 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 19 III 1 2 qN qN − 1 −2qN 2 2qN − 2 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 20 III 1∗ 2 qN qN − 1 −2qN 2 − 4qN 2qN − 2 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 21 IV 1∗ ∅ 1 2 − 1 2 −1 1 2 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 22 IV 2 ∅ qN qN − 1 −2qN 2 − 2qN 1 − qN (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 23 IV 1 2 qN qN − 1 −2qN 2 1 − qN (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 24 IV 1∗ 2 qN qN − 1 −2qN 2 − 4qN 1 − qN (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) Table 3: UV extended models satisfying conditions (A) and (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' For each model we show the U(1)L charges of N, η, σ1, σ2 and S, as well as the (SU(2)L, U(1)Y) representation of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Models that become any of the models in this list after renaming the fields or redefining their U(1)L charges are not included, as explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 7 an SU(2)L singlet or triplet) different UV theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' They are listed in Table 3, where the U(1)L charges of N, η, σA,B and S, as well as the (SU(2)L, U(1)Y) representation of S in each model, are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Some comments are in order: (i) The (SU(2)L, U(1)Y) representation of the heavy scalar S depends on the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In I-models S transforms as (3, 1), in II-models we have two possibilities, (3, 0) or (1, 0), while in III- and IV-models S transforms as (2, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (ii) In all the models in Table 3, the global U(1)L symmetry may be spontaneously broken to a Z2 parity, under which N and η are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In all the ξ(1∗, ∅) models and in I(∅, 1∗), the conservation of U(1)L restricts the lepton number charges of N, η, σA,B and S, which must take precise values, and this automatically implies a remnant Z2 that corresponds to the usual Scotogenic parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The model studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' [8], which corresponds to model I(1∗, ∅) in our notation, is a good example of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In the rest of the models, the conservation of U(1)L leaves one of the charges to be chosen freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We decided to choose qN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this case, these are the restrictions to recover the dark Z2 parity from U(1)L breaking: qN cannot be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' If qN = α β, with α, β ∈ Z, then α and β have to be odd and even, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' GCD(α, β) = 1, where GCD stands for Greatest Common Divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Therefore, α and β must be coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The first restriction comes from the requirement of N and η being both odd under the remnant Scotogenic Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The relation qη = qN −1 implies that if qN is even, then qη must be odd, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Then, N and η will transform differently under the remnant Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' As an example of this consider the model I(1, 2) with qN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this case, the solution for the rest of the U(1)L charges in the model is qη = 1, qσ1 = −4, qσ2 = 2 and qS = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The global lepton number symmetry gets spontaneously broken as U(1)L → Z2, but with N and η charged under Z2 as + and −, respectively, and this does not reproduce the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Similarly, if qN = α β, after normalizing all U(1)L charges so that they become integer numbers (multiplying by β) we obtain ˜qη = β − α and ˜qN = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Hence, for η and N to be odd under Z2, α and β must be odd and even, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, the third restriction is required to guarantee that n = 2 after U(1)L breaks to the discrete symmetry Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' As an example we take model I(1,2), where n ≡ GCD(˜qσ1, ˜qσ2, ˜qS) = GCD(−2α, 2β, 2α) = 2GCD(α, β) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We checked for all the working models that GCD(˜qσ1, ˜qσ2, ˜qS) or GCD(˜qσ1, ˜qσ2), depending on whether S acquires a VEV or not, always reduces to GCD(α, β) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Also, we want qN = α β to be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (iii) In all models, and for all possible values of qN in agreement with the restrictions listed in the previous item, η never acquires an induced VEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This is crucial for the consistency of the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (iv) It is clear that in all models of the form ξ(A, ∅) or ξ(∅, B), a trilinear coupling µ participates in the generation of the λ5 coupling, induced after the breaking of U(1)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 8 νL νL H0 σ η0 η0 N N y y κ H0 σ S β µ Figure 2: Neutrino mass generation in an extended Scotogenic model with one σ field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This Feynman diagram shows the relevant gauge eigenstates involved in the 1-loop contribution to neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In our notation, this is a IV(1∗, ∅) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This is perfectly consistent, but requires the assumption µ ≪ mS to justify λ5 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This poses a theoretical issue, since µ is a parameter of the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In contrast, in models of the form ξ(A, B) with A, B ̸= ∅, the λ5 coupling will only depend on the σA,B VEVs, induced at low energies and naturally small compared to mS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (v) Finally, we note that in I-models the U(1)L charges of the particles N, η and σA,B remain the same after the non-trivial change A ↔ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' For instance, this is the case in models I(1, 2) and I(2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This concludes our classification of all possible UV extensions of the Scotogenic model satisfying our requirements (A) and (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We will now illustrate it with two specific example models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' An additional example can be found in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4 An UV extended Scotogenic model with one σ field Our first example model is an UV extension of the Scotogenic model with one σ field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Another example of this class of models can be found in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 Ultraviolet theory We consider an extension of the Scotogenic model with two new particles: the SU(2)L doublet S and the singlet σ, both scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The Z2 Scotogenic parity is replaced by a global U(1)L lepton number symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Table 4 shows the scalar and leptonic fields of the model and their representations under the gauge and global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We want to explain the smallness of the Scotogenic’s λ5 coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Our strategy will be to forbid it in our original Lagrangian and make it arise effectively at low energies once the 9 Field Generations SU(3)c SU(2)L U(1)Y U(1)L ℓL 3 1 2 1/2 1 eR 3 1 1 1 1 N 3 1 1 0 qN H 1 1 2 1/2 0 η 1 1 2 1/2 qη σ 1 1 1 0 qσ S 1 1 2 1/2 qS Table 4: Lepton and scalar particle content and representations under the gauge and global symmetries in an UV extension of the Scotogenic model with one σ field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' scalar σ acquires a VEV and we integrate out S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We also impose that, after symmetry breaking, the effective λ5 coupling would induce neutrino masses as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In our notation, this is a IV(1∗, ∅) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This requires the presence of the operators N�η†ℓL , σN cN , H†SH†η , σ∗S†η , (8) which in turn imply the following set of equations for the U(1)L charges of the model: −qN + qη + 1 = 0 , (9) qσ + 2 qN = 0 , (10) qS + qη = 0 , (11) −qσ − qS + qη = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (12) This system of linear equations has a unique solution: qN = 1 2 , (13) qη = −1 2 , (14) qσ = −1 , (15) qS = 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (16) With this solution, the operators N cN , N �H†ℓL , � H†η �2 (17) are automatically forbidden due to U(1)L conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' One should note that if we chose the operator σS†η instead of σ∗S†η, no solution for the resulting system of equations would exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Indeed, if one replaces −qσ by qσ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (12), the combination of the resulting equation with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (10) and (11) leads to qN = qη, which is incompatible with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This illustrates why ξ(1, ∅) models are not compatible with our requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 10 Having fixed the quantum numbers of all the particles in the model, we proceed to write its Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The new Yukawa interactions are given by LY = y N �η† ℓL + κ σN cN + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' , (18) where y is a general complex 3 × 3 matrix and κ is a complex symmetric 3 × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The scalar potential of the model can be written as VUV = m2 HH†H + m2 SS†S + m2 σσ∗σ + m2 ηη†η + λ1 2 (H†H)2 + λ2 2 (η†η)2 + λS 2 (S†S)2 + λσ 2 (σ∗σ)2 + λ3(H†H)(η†η) + λS 3 (H†H)(S†S) + λσ 3(H†H)(σ†σ) + ληS 3 (η†η)(S†S) + λησ 3 (η†η)(σ∗σ) + λσS 3 (σ∗σ)(S†S) + λ4(H†η)(η†H) + λHS 4 (H†S)(S†H) + ληS 4 (S†η)(η†S) + � β(H†SH†η) + µ(σ∗S†η) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (19) Here µ is a trilinear parameter with dimensions of mass while m2 H, m2 η and m2 σ have dimen- sions of mass2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Other Lagrangian terms are allowed by the gauge symmetries of the model but forbidden by U(1)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='2 Effective theory We will now assume that mS is much larger than any other energy scale in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' At energies well below mS, all physical processes can be properly described by an effective field theory in which the heavy field S has been integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We now present this effective theory, obtained after integrating out S at tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The effective potential at low energies can be written as VIR = m2 HH†H + m2 ηη†η + m2 σσ∗σ + λ1 2 (H†H)2 + λ2 2 (η†η)2 + λσ 2 (σ∗σ)2 + λ3(H†H)(η†η) + λσ 3(H†H)(σ∗σ) + � λησ 3 − |µ|2 m2 S � (σ∗σ)(η†η) + � λ4 − |β|2(H†H) m2 S � (H†η)(η†H) − � βµ m2 S σ∗(H†η)2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' � + O � 1 m4 S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (20) Assuming that CP is conserved in the scalar sector, the neutral fields H0 and σ can be decomposed as H0 = 1 √ 2(vH + φ + iA) , σ = 1 √ 2(vσ + ρ + iJ) , (21) with vH √ 2 and vσ √ 2 the VEVs of H0 and σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' These VEVs are determined by minimizing the scalar potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The resulting tadpole equations are given by dVIR dH0 ���� ⟨H0,σ⟩={ vH √ 2 , vσ √ 2 } = vH √ 2 � m2 H + λ1v2 H 2 + λσ 3v2 σ 2 � , (22) dVIR dσ ���� ⟨H0,σ⟩={ vH √ 2 , vσ √ 2 } = vσ √ 2 � mσ2 + λσ 3v2 H 2 + λσv2 σ 2 � , (23) 11 where we have only written the non-trivial equations and these are evaluated at the VEVs of each scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' As we see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (20), once σ acquires a VEV, the operator (H†η)2 is generated, with an effective λ5 coupling that is naturally suppressed by the mass of the heavy field S, λ5 2 = − βµvσ √ 2m2 S ≪ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (24) This follows from the assumption µ ≪ mS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3, this is perfectly valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, it poses a theoretical problem since µ is parameter of the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' A model without this issue will be discussed below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We now proceed to the computation of the scalar spectrum of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In the bases {φ, ρ} for the CP-even states and {A, J} for the CP-odd ones, the squared mass matrices read M2 R = � m2 H + 1 2 (3λ1v2 H + λσ 3v2 σ) λσ 3vHvσ λσ 3vHvσ m2 σ + 1 2 (λσ 3v2 H + 3λσv2 σ) � , (25) and M2 I = � m2 H + λ1v2 H 2 + λσ 3 v2 σ 2 0 0 m2 σ + λ2 σv2 H 2 + λσv2 σ 2 � , (26) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The above expressions can be reduced using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (22) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' When this is done, the resulting M2 I becomes identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This implies the existence of two massless Goldstone bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' One of them (A) corresponds to the state that is eaten up by the Z boson and becomes its longitudinal component, while the other (J) is associated to the spontaneous breaking of the global U(1)L symmetry, the so-called majoron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' On the other hand, the reduction of M2 R with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (22) and (23) leads to M2 R = � λ1v2 H λσ 3vHvσ λσ 3vHvσ λσv2 σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (27) This matrix can be brought to diagonal form as V T R M2 RVR = � M2 R = diag(m2 h, m2 Φ), where VR is a unitary matrix that can be parametrized as VR = � cos θ − sin θ sin θ cos θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (28) The mixing angle θ is given by tan(2θ) = 2(M2 R)12 (M2 R)11 − (M2 R)22 = 2rλσ 3 r2λ1 − λσ ≈ −2rλσ 3 λσ + O(r2) , (29) with r ≡ vH/vσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' For vσ ∼ TeV, r ≪ 1 and simple approximate expressions can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The lightest of the two mass eigenstates is the well-known Higgs-like state h, with mass mh ≈ 125 GeV, discovered at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In addition, the model contains the heavy scalar Φ, with a mass of the order of vσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We focus now on the Z2-odd scalars η+ and η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The neutral η0 field can be decomposed as η0 = 1 √ 2(ηR + iηI) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (30) 12 Their masses are given by mη+ = m2 η + v2 H 2 λeff 3 , (31) m2 ηR = m2 η + v2 H 2 � λeff 3 + λeff 4 − √ 2 βµvσ m2 S � , (32) m2 ηI = m2 η + v2 H 2 � λeff 3 + λeff 4 + √ 2 βµvσ m2 S � , (33) where we have defined λeff 3 ≡ λ3 + λησ 3 v2 σ v2 H − µ2 v2 σ v2 Hm2 S , (34) λeff 4 ≡ λ4 − β2v2 H 2m2 S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (35) The mass square difference between ηR and ηI is given by m2 ηR − m2 ηI = − √ 2 βµvσ m2 S v2 H = λ5v2 H , (36) exactly as in the usual Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, the spontaneous breaking of U(1)L by the VEV of σ induces a Majorana mass term for the N singlets, with MN = √ 2 κ vσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This leads to Majorana neutrino masses at 1-loop, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The 3 × 3 neutrino mass matrix is given by usual Scotogenic formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (5), where λ5 is the effective coupling in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Due to the additional scalar states, including a massless majoron with couplings to charged leptons, the phenomenology of this model is richer than that of the usual Scotogenic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5 An UV extended Scotogenic model with two σ fields We consider now an UV extension of the Scotogenic model with two σ fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 Ultraviolet theory We enlarge the Scotogenic particle content with three new particles: the scalar SU(2)L singlets S, σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Again, instead of the usual Z2 Scotogenic parity, a global U(1)L lepton number symmetry is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Table 5 shows the scalar and leptonic fields of the model and their representations under the gauge and global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We consider the 1-loop generation of neutrino masses by the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In our notation, this is a II(1, 2) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' For this mechanism to take place, the operators N�η†ℓL , σ1N cN , σ1H†Sη , σ2H†S∗η (37) 13 Field Generations SU(3)c SU(2)L U(1)Y U(1)L ℓL 3 1 2 1/2 1 eR 3 1 1 1 1 N 3 1 1 0 qN H 1 1 2 1/2 0 η 1 1 2 1/2 qη σ1 1 1 1 0 qσ1 σ2 1 1 1 0 qσ2 S 1 1 1 0 qS Table 5: Lepton and scalar particle content and representations under the gauge and global symmetries in an UV extension of the Scotogenic model with two σ fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' νL νL H0 H0 η0 η0 N N y y κ σ1 σ1 σ2 S β1 β2 Figure 3: Neutrino mass generation in an extended Scotogenic model with two σ fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This Feynman diagram shows the relevant gauge eigenstates involved in the 1-loop contribution to neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In our notation, this is a II(1, 2) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' must be allowed by the symmetries of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This restricts the U(1)L charges of the fields in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In particular, one can write the following set of equations for them: −qN + qη + 1 = 0 , (38) qσ1 + 2 qN = 0 , (39) qσ1 + qS + qη = 0 , (40) qσ2 − qS + qη = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (41) 14 They can be solved in terms of qN to obtain qη = qN − 1 , (42) qσ1 = −2 qN , (43) qσ2 = 2 , (44) qS = qN + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (45) In addition, we want the operators N cN , N �H†ℓL , � H†η �2 (46) to be forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In order to forbid the first operator, a Majorana mass term for N, we just require qN ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The second operator would lead to νL-N Dirac mass terms and we can forbid it by requiring qN ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (42) implies qη ̸= 0 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, with these considerations, we choose qN = 1 2 , (47) which implies qη = −1 2 , qS = 3 2 , qσ1 = −1 , qσ2 = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (48) Some comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' First, the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3 has two different σ singlets attached to the scalar internal line, σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In principle, one may wonder why we did not consider the same σ singlet in both vertices as starting point for constructing our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' That would imply qS = 0 and reduce the number of couplings in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, such construction would lead to an effective operator (H†η)2σ2 after integrating out the S field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' If this operator is allowed by all symmetries of the model, so is the trilinear (H†η) σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We will eventually assume that the σ singlets acquire non-zero VEVs, breaking the original U(1)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In the presence of the trilinear (H†η) σ, this would induce a tadpole for η, hence breaking the Z2 parity of the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This forces us to discard this possibility and consider different σ1 and σ2 attached to the internal scalar line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' It also illustrates why models with σA = σB are not compatible with our requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Furthermore, one may consider a third σ3 singlet field coupled to the internal fermion line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' While this is possible, we preferred to choose a charge assignment that allows us to identify σ3 ≡ σ1 and reduce the number of fields in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, once σ1 and σ2 acquire non-zero VEVs, the original U(1)L symmetry will get broken to one of its Zn subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Here n is the GCD of |qσ1| and |qσ2| after being normalized to become integer numbers, hence n = 2 and the remnant symmetry is Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Once we know the quantum numbers of all the particles in the model, we can write its Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The new Yukawa interactions are given by LY = y N �η† ℓL + κ σ1N cN + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' , (49) where y is a general complex 3 × 3 matrix and κ is a complex symmetric 3 × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The 15 scalar potential of the model is given by VUV = m2 HH†H + m2 SS∗S + m2 σiσ∗ i σi + m2 ηη†η + λ1 2 (H†H)2 + λ2 2 (η†η)2 + λS 2 (S∗S)2 + λσi 2 (σ∗ i σi)2 + λ3(H†H)(η†η) + λS 3 (H†H)(S∗S) + λσi 3 (H†H)(σ∗ i σi) + ληS 3 (η†η)(S∗S) + λησi 3 (η†η)(σ∗ i σi) + λσσ 3 (σ∗ 1σ1)(σ∗ 2σ2) + λσiS 3 (σ∗ i σi)(S∗S) + λ4(H†η)(η†H) + � β1(σ1H†Sη) + β2(σ2H†S†η) + µ √ 2(σ2σ1σ1) + λ0(SSσ1σ∗ 2) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' � , (50) where we sum over i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Here µ is a trilinear parameter with dimensions of mass while m2 H, m2 η and m2 σi have dimensions of mass2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Other Lagrangian terms are allowed by the gauge symmetries of the model but forbidden by U(1)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='2 Effective theory In the following we will assume that mS is much larger than any other energy scale in the model and integrate out the heavy scalar S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' If we do this at tree-level, the effective scalar potential at low energies can be written as VIR = m2 H(H†H) + m2 η(η†η) + m2 σi(σ∗ i σi) + λ1 2 (H†H)2 + λ2 2 (η†η)2 + λσi 2 (σ∗ i σi)2 + λ3(H†H)(η†η) + λσi 3 (H†H)(σ∗ i σi) + λησi 3 (η†η)(σ∗ i σi) + λσσ 3 (σ∗ 1σ1)(σ∗ 2σ2) + � λ4 − |βi|2 m2 S (σ∗ i σi) � (H†η)(η†H) + � µ √ 2(σ2σ1σ1) − β1β2 m2 S σ1σ2(H†η)2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' � + O � 1 m4 S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (51) Now, we decompose the neutral fields H0 and σ1,2 as H0 = 1 √ 2(vH + φ + i A) , σi = 1 √ 2(vσi + ρi + i Ji) , (52) where we defined vH √ 2 and vσi √ 2 as the VEVs of the corresponding fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' After this, we can compute the tadpole equation resulting from the effective potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (51), evaluated at the VEVs of each scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The non-trivial tadpole equations are dVIR dH0 ���� ⟨H0,σi⟩={ vH √ 2 , vσi √ 2 } = vH √ 2 � m2 H + λ1 v2 H 2 + λσ1 3 v2 σ1 2 + λσ2 3 v2 σ2 2 � = 0, (53) dVIR dσ1 ���� ⟨H0,σi⟩={ vH √ 2 , vσi √ 2 } = vσ1 √ 2 � m2 σ1 + µ vσ2 + λσ1 3 v2 H 2 + λσ1 v2 σ1 2 + λσσ 3 v2 σ2 2 � = 0, (54) dVIR dσ2 ���� ⟨H0,σi⟩={ vH √ 2 , vσi √ 2 } = vσ2 √ 2 � m2 σ2 + µ v2 σ1 2vσ2 + λσ2 3 v2 H 2 + λσ2 v2 σ2 2 + λσσ 3 v2 σ1 2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (55) 16 As already explained, as a result of σi acquiring a VEV, lepton number gets spontaneously broken, leaving a discrete Z2 symmetry, under which all the particles in the model are even except for N and η, which are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Another important consequence of the spontaneous breaking of lepton number is the generation of the (H†η)2 operator, with a naturally sup- pressed λ5 coupling due to the 1/m2 S factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' One finds λ5 2 = −vσ1vσ2β1β2 2m2 S ≪ 1 , (56) where βi are dimensionless parameters of the UV theory and vσi ≪ mS by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This expression clearly corresponds to a II(1, 2) model, following the classification of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We now consider the scalar spectrum of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We will assume that CP is conserved in the scalar sector, just for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this case, the spectrum contains three CP-even and three CP-odd gauge eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In the bases {φ, ρ1, ρ2} and {A, J1, J2}, their mass matrices are given by M2 R = � � � λ1v2 H λσ1 3 vHvσ1 λσ2 3 vHvσ2 λσ1 3 vHvσ1 λσ1v2 σ1 vσ1(µ + λσσ 3 vσ2) λσ2 3 vHvσ2 vσ1(µ + λσσ 3 vσ2) λ2v2 σ2 − µv2 σ1 2vσ2 � � � (57) and M2 I = � � � 0 0 0 0 −2µvσ2 −µvσ1 0 −µvσ1 − µv2 σ1 2vσ2 � � � , (58) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The tadpole equations (53)-(55) were used in the derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (57) and (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The CP-even and CP-odd physical mass eigenstates can be written as linear combi- nations of {φ, ρ1, ρ2} and {A, J1, J2}, respectively, obtained after the diagonalization of the matrices M2 R and M2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Out of the three CP-even mass eigenstates, one can be identified with the Higgs boson, with mass mh ≃ 125 GeV, discovered at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In addition, two massive CP-even scalar fields exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In what concerns the CP-odd mass eigenstates, their mass matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (58) can be readily diagonalized as V T I M2 I VI = � M2 I, where VI = � � 1 0 0 0 cos θ − sin θ 0 sin θ cos θ � � (59) is a unitary matrix and � M2 I is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' One obtains � M2 I = � � � 0 0 0 0 0 0 0 0 − µ(v2 σ1+4v2 σ2) 2vσ2 � � � , (60) thus leading to two massless pseudoscalar bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The first one is the Goldstone boson that becomes the longitudinal component of the Z boson (A), while the second one (a linear 17 combination of fields J1 and J2) is associated to the spontaneous breaking of U(1)L and is the so-called majoron, denoted as J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The J1 − J2 mixing angle is given by tan(2θ) = 2 (M2 I)23 (M2 I)22 − (M2 I)33 = 4vσ1vσ2 4v2 σ2 − v2 σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (61) We finally turn our attention to the Z2-odd scalars and decompose the neutral field η0 as η0 = 1 √ 2(ηR + i ηI) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (62) The mass of the charged η+ and the neutral ηR,I fields are given by m2 η+ = m2 η + v2 H 2 λeff 3 , (63) m2 ηR = m2 η + v2 H 2 � λeff 3 + λeff 4 − β1β2vσ1vσ2 m2 S � , (64) m2 ηI = m2 η + v2 H 2 � λeff 3 + λeff 4 + β1β2vσ1vσ2 m2 S � , (65) where we defined λeff 3 ≡ λ3 + λησ1 3 v2 σ1 v2 H + λησ2 3 v2 σ2 v2 H (66) λeff 4 ≡ λ4 − β2 1v2 σ1 2m2 S − β2 2v2 σ2 2m2 S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (67) As in the Scotogenic model, the mass difference between ηR and ηI is proportional to the λ5 coupling: m2 ηR − m2 ηI = −vσ1vσ2β1β2 m2 S v2 H = λ5v2 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (68) Finally, the breaking of U(1)L also induces a Majorana mass term for the N singlets, with MN = √ 2 κ vσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This leads to Majorana neutrino masses at 1-loop, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The resulting neutrino mass matrix is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (5), with the effective λ5 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Furthermore, contrary to the minimal Scotogenic model, this UV extension induces a 1- loop interaction between the majoron and a pair of charged leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This enriches the phenomenology of the model, as we discuss in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 6 Phenomenology All UV scenarios discussed in our classification of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3 and illustrated with the two examples of Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4 and 5 share some common features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' They are characterized at low energies by a Scotogenic model extended with a massless pseudoscalar, the majoron J, and one or several massive scalars and pseudoscalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' While some phenomenological implications may be specific to particular models, there are also some general expectations that we may highlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 18 Coupling Upper limit References Im See 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 × 10−13 [16] Im Sµµ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 × 10−9 [17] |Seµ| 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='3 × 10−11 [15] |Seτ| 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='9 × 10−7 [15] |Sµτ| 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='6 × 10−7 [15] Table 6: Current limits on the majoron couplings to charged leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The limit on Im See is at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The limit on Im Sµµ has been obtained by performing a simulation of the supernova SN1987A [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' An alternative and more stringent limit Im Sµµ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 × 10−10 can be derived with more aggressive assumptions in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 Majoron coupling to charged leptons The presence of a massless majoron dramatically affects the phenomenology of this class of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In fact, models including a majoron are strongly constrained by a variety of experimental limits, such as those originated by the majoron coupling to a pair of charged leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The relevance of these limits depends on the flavor structure of the couplings [14], which necessarily depends on the specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Stringent constraints exist for both flavor- conserving and flavor-violating couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Let us write the majoron interaction with charged leptons as [15], LℓℓJ = J ¯ℓβ � Sβα L PL + Sβα R PR � ℓα + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (69) Here ℓα,β are the charged leptons with flavors α and β, while PL,R are the usual chiral projec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We consider all flavor combinations for the SL,R couplings: βα = {ee, µµ, ττ, eµ, eτ, µτ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Due to the pseudoscalar nature of majorons, the diagonal Sββ = Sββ L + Sββ∗ R couplings are purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' They receive strong constraints from astrophysical observations, due to the cooling effects induced by the majoron in dense astrophysical media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Flavor off-diagonal couplings are constrained by the null searches of lepton flavor violation in processes involv- ing charged leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In particular, searches for ℓα → ℓβ J can be used to set bounds on the combinations |Sβα| = ����Sβα L ��� 2 + ���Sβα R ��� 2�1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (70) A compilation of the current limits on the majoron couplings to charged leptons can be found in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' While in some scenarios the majoron couplings to charged leptons appear at tree-level [18, 19], in many cases the leading order contribution is induced at the 1-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' For instance, this is the case of the popular type-I seesaw with spontaneous lepton number violation [9, 20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Similarly, in the Scotogenic scenarios discussed in this paper, the majoron coupling to charged leptons is also induced at 1-loop [8,22] by the Feynman diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Here 19 J ℓ−α ℓ+ β η− N N y y gJNN Figure 4: 1-loop generation of the majoron coupling to a pair of charged leptons in the Scotogenic scenarios discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' gJNN is the J − N − N coupling, which depends on the specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' It is given by gJNN = � i κ √ 2 in models with one σ singlet i κ √ 2 cos θ in models with two σ singlets , (71) where the mixing angle θ is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The prefactor cos θ in models with two σ singlets is due to the fact that only σ1 has a coupling to N cN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' No other contributions to the majoron coupling to charged leptons exist at 1-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' One may wonder about a Feynman diagram with two scalar lines in the loop, induced by a J η+η− coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, this contribution vanishes exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The reason is the pseudoscalar nature of the majoron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The J ¯ℓαℓα vertex must be proportional to γ5, but the Lorentz structure of this contribution does not generate such pseudoscalar coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4 Also, diagrams with gauge bosons vanish due to the pure singlet nature of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Therefore, one can find the SL,R couplings introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (69) by direct computation of the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The result can be written as [8,22] Sβα L = −mℓβ 8π2 � y†gJNN Γ y � βα , (72) Sβα R = mℓα 8π2 � y†gJNN Γ y � βα , (73) for the non-diagonal couplings and Sββ = −mℓβ 8π2 � y†gJNN Γ y � ββ , (74) for the diagonal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Here mℓβ = {me, mµ, mτ} and we have defined Γmn = MNn � M 2 Nn − m2 η+ �2 � M 2 Nn − m2 η+ + m2 η+ log m2 η+ M 2 Nn � δmn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (75) 4We also note that the J η+η− coupling is absent in many models, since Lagrangian terms like σ|η|2 or σ2|η|2 are forbidden by lepton number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Only in models with two σ fields one may have a term of the form σ1σ2|η|2 (when qσ1 = −qσ2) leading to a J η+η− interaction vertex after symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, as explained in the text, even when this term is present, the associated 1-loop contribution to the majoron coupling to a pair of charged leptons vanishes exactly due to the pseudoscalar nature of the majoron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 20 Figure 5: Contours of BR (µ → eJ) in the (MN, mη+) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The colored regions correspond to the regions allowed by the current experimental bound on the branching ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' On the left, gJNN has been fixed to 10−1 (blue) and to 10−2 (pink), while rη = 1 has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' On the right, the coupling gJNN was not fixed and three different values of the rη ratio have been considered, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 (pink), 1 (blue) and 2 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We can now study how the bounds on these couplings restrict the parameter space of the models considered in our classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In particular, in the following we focus on the 2-body decay µ → eJ, for which BR (µ → eJ) = mµ 32 π Γµ � |Seµ L |2 + |Seµ R |2� , (76) where Γµ ≈ 3 × 10−19 GeV is the total decay width of the muon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We used a Casas-Ibarra parametrization [23] properly adapted to the Scotogenic model [24–26] and the best-fit values obtained in the global fit [27] to neutrino oscillation data in order to express the Yukawa matrix y in terms of experimentally measured quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We assumed that the three singlet fermions are degenerate, that is MN1 = MN2 = MN3 = MN and we fixed λ5 = 5 × 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Notice that lower values of this parameter would imply larger values of the Yukawas, thus further restricting the parameter space of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' It also proves convenient to define rη = m0 mη+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (77) Our results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' On the left-hand side we fixed the coupling gJNN to 10−1 (blue), and to 10−2 (pink), and we considered rη = 1 in both scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The colored regions correspond to regions allowed by the experimental bound on the µ → eJ decay, which implies BR (µ → eJ) < 10−5 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' As expected, the larger the J −N −N coupling is, the smaller the allowed region of the parameter space becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We also find that light Scotogenic states can be made compatible with the µ → eJ bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This can be easily understood by inspecting 21 the non-trivial relation between the masses mη+ and MN and the Yukawa couplings y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Under the assumptions mentioned above one finds SL,R ∝ gJNNΓii � y†y � 12 , (78) where Γii is any of the diagonal entries of Γ, given by Γii ∝ MN M 2 N − m2 η+ + m2 η+ log m2 η+ M2 N � M 2 N − m2 η+ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (79) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (5) implies that the Yukawa product � y†y � 12 is proportional to � y†y � 12 ∝ 1 MN (M 2 N − m2 0)2 M 2 N − m2 0 + M 2 N log m2 0 M2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (80) Therefore, in the limit rη = 1 one finds SL,R ∝ gJNN M 2 N − m2 η+ + m2 η+ log � m2 η+ M2 N � M 2 N − m2 η+ + M 2 N log � m2 η+ M2 N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (81) For a fixed gJNN value two possibilities arise: (i) if we fix mη+, the Γii � y†y � 12 combination decreases if MN increases, and (ii) if we fix MN, the Γii � y†y � 12 combination increases if mη+ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Essentially, the involved couplings strongly depend on mη+ and MN and this dependence may lead to an apparent non-decoupling behavior that explains the results for the µ → eJ branching ratio observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, the right-hand side of this figure provides complementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Here we considered gJNN = i κ √ 2 = i MN 2 vσ and fixed vσ = 5 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Since the gJNN coupling grows with MN, for each mη+ there is a maximum value of MN for which BR (µ → eJ) < 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This can be clearly seen in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='2 Collider signatures Since the spontaneous breaking of U(1)L requires the introduction of additional scalar mul- tiplets, all models in our classification have extended scalar sectors containing several states besides the ones in the Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This can be used to probe them at colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' One of the CP-even scalars, presumably the lightest, is to be identified with the 125 GeV state discovered at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The production cross-section and decay rates of this state, denoted generally as h, must agree with the values measured by the ATLAS and CMS collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Since these are very close to those predicted for a pure SM Higgs, h ≈ Re(H0) is generally required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In particular, mixings with the σ states are strongly constrained, since they would affect its decay rates in a twofold way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' First, the σ states do not couple to the SM gauge bosons or to quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Thus, any mixing would induce a universal reduction of the h partial decay widths into these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' And second, h can have additional decay models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' It 22 can decay invisibly to a pair of singlet fermions (h → N1N1) or to pair of majorons (h → JJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The former can only take place if mN1 ≤ mh/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In contrast, since the majoron is massless, the latter is always kinematically available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We can write the interaction Lagrangian of h with a pair of majorons as LhJJ = 1 2 ghJJ h J2, where ghJJ is a dimensionful coupling that depends on the specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This interaction induces the invisible decay h → JJ, with the decay width given by Γ(h → JJ) = g2 hJJ 32 π mh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (82) If we assume a total Higgs decay width in agreement with the SM expectation, Γh ≈ ΓSM h = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 MeV [28], the bound on the invisible Higgs branching ratio BR(h → JJ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='19 at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' [29], implies ghJJ < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This translates into constraints on the parameters of the scalar potential of the model, which are encoded in ghJJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' For instance, in the model discussed in [8], this implies that the coefficient of the (H†H)(σ∗σ) operator must be ≲ 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We note, however, that stronger constraints can be derived by combining invisible and visible channels, as recently pointed out in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, all models in our classification also contain additional heavy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' They can also be searched for at colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Their production cross-sections and decay models strongly depend on the specific realization of our setup and, more specifically, on their gauge com- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' If they have sizable doublet components, they can in principle be produced at high rates at the LHC via Drell–Yan processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In contrast, heavy scalars with a dominant component in the singlet direction have very suppressed production cross-sections at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Due to the constraints discussed above, which imply suppressed mixing between the SM Higgs doublet and the σ states, this is the most likely scenario in all models discussed in our classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='3 Dark matter In all UV models studied in this paper, a remnant Z2 symmetry is obtained as a result of the spontaneous breaking of lepton number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This is the Scotogenic Z2 parity, under which only the usual Scotogenic states N and η are charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The conservation of Z2 implies that the lightest of them is completely stable and, in principle, a valid DM candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Both options have been widely studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In the case of a scalar candidate, the DM phenomenology resembles that of the Inert Doublet model [31–35], with the DM production in the early Universe set by gauge interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In contrast, the case of a fermion candidate typically requires large Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This leads to tension with bounds from lepton flavor violation [24], although the observed DM relic density can be achieved [36–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The low energy theories resulting from our UV extended models do not correspond exactly to the original Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' As explained above and illustrated in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 4 and 5, additional scalar states are present: the massless majoron and one of several massive scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' These new degrees of freedom couple to the Z2-odd states and may affect the resulting DM phenomenology, which may have some differences with respect to the one in the original Scotogenic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This has recently been studied in [41,42] for the case of fermion DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The main conclusion from these works is that the new scalar states open up new regions in parameter space in which the DM relic density can match the observed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In particular, 23 annihilations become very efficient when the mass of the DM candidate, mN1, is about half of the mass of a new scalar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This implies that one can find the correct DM abundance for any value of mN1 without resorting to coannihilations, in contrast to the original Scotogenic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' These models are also expected to have a rich phenomenology at direct and indirect detection experiments [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 7 Summary and discussion The Scotogenic model is a very popular scenario for neutrino masses and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this work we have considered extensions of this scenario that naturally explain the smallness of the quartic λ5 coupling and the origin of the Scotogenic Z2 parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This is achieved in UV extensions including a conserved global lepton number symmetry, spontaneously broken by the VEVs of one or several scalar singlets, and a new heavy state that suppresses all lepton number violating effects at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We explored all possible models with these assumptions and found 24 variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' They are all characterized at low energies by the presence of a massless Goldstone boson, the majoron, as well as other massive scalars besides the usual Scotogenic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Two specific example models are discussed in detail in order to illustrate the basic ingredients of our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In these two models, as well as in all the variants in our classification, a rich phenomenology is expected, with potential signatures in collider and lepton flavor violating searches, and implications for dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Out of the 24 models revealed by our analysis, only one had been previously studied in the literature, namely [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This illustrates the vast model space beyond the original Scotogenic model yet to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In fact, there are many variations of the fundamental setup that keep all the positive features and include additional ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' While many of these modified Scotogenic scenarios may contain unnecessary or redundant ingredients, other may offer novel ways to address open questions in current particle physics [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This is the main motivation behind the classification presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' There are several ways in which our analysis can be extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' First of all, we have considered UV theories that realize the λ5 coupling at tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this case, the only source of suppression is given by the large energy scale mS, assumed to lie well above the electroweak scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Alternatively, the λ5 coupling can also be realized at loop order, as recently explored in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This possibility leads to many novel extensions of the Scotogenic setup with, at least potentially, new phenomenological expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Another way in which our analysis can be extended is by considering a local lepton number symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this case, the massless majoron that was characteristic in our setup would be replaced by a heavy Z′ boson, with a dramatic impact on the low-energy phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, we note that this direction requires non-trivial extensions of the fermion particle content in order to cancel out the usual triangle gauge anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Therefore, a general classification of all possible gauge models becomes more cumbersome, although interesting too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Finally, variations with non- universal lepton charges for the N fermions or featuring alternative numbers of generations for the Scotogenic states can be explored as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 24 Acknowledgements Work supported by the Spanish grants PID2020-113775GB-I00 (AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='13039/501100011033) and CIPROM/2021/054 (Generalitat Valenciana).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The work of PE is supported by the FPI grant PRE2018-084599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' AV acknowledges financial support from MINECO through the Ram´on y Cajal contract RYC2018-025795-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' DPS would like to thank the AHEP group for the hospitality during his visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The work of DPS was supported by Ciencia de Frontera CONACYT project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 428218 and the program “BECAS CONACYT NACIONALES”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' A Accidental Z2 symmetries The dark Z2 parity of the Scotogenic model can also be an accidental symmetry generated after the σ singlet (or singlets) acquires a VEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In these scenarios, the symmetry breaking path is also U(1)L → Z2, but with ℓL, eR and η as the only particles charged under the discrete symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this case, the Yukawa term ¯N ˜η†ℓL and the Majorana mass N cN are allowed by all symmetries, while ¯N ˜H†ℓL is forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Furthermore, given that η is the only Z2-odd scalar, it will always appear in pairs in the effective scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Therefore, although the Z2 Scotogenic parity does not emerge as a remnant symmetry after the breaking of U(1)L, it appears accidentally as a consequence of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' These UV models are not included in the classification presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3 since they violate requirement (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' However, they also lead to the Scotogenic model at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Let us illustrate this possibility with a specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5 Consider the particle content and charge assignment in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The new Yukawa interactions in the model are given by LY = y N �η† ℓL + MN N cN + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' , (83) while the scalar potential of the model is written as VUV = m2 HH†H + m2 SS∗S + m2 σσ∗σ + m2 ηη†η + λ1 2 (H†H)2 + λ2 2 (η†η)2 + λS 2 (S∗S)2 + λσ 2 (σ∗σ)2 + λ3(H†H)(η†η) + λS 3 (H†H)(S∗S) + λσ 3(H†H)(σ∗σ) + ληS 3 (η†η)(S∗S) + λησ 3 (η†η)(σ∗σ) + λσS 3 (σ∗σ)(S∗S) + λ4(H†η)(η†H) + � β(σH†ηS) + µ1 H†ηS∗ + µ2 σ S2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (84) It is easy to check that other Lagrangian terms are forbidden by U(1)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This global symmetry gets spontaneously broken once the electroweak singlet σ acquires a non-zero VEV, leaving a remnant Z2 under which η, S, ℓL and eR are odd, while the rest of the fields are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We can call this symmetry Zrem 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Since qN = 0, N is even under Zrem 2 , and thus this symmetry cannot be identified with the Scotogenic dark parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Nevertheless, the Lagrangian of the Scotogenic model is still obtained after decoupling the heavy scalar S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This is due to the fact that a new accidental Z2 parity appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The only fields charged under this parity are η and N, while all the other fields in the effective theory are even, therefore, this accidental symmetry, that we can denote as Zacc 2 , is precisely the Scotogenic Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 5This model corresponds to the II′ (1, ∅) model shown below in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 25 Field Generations SU(3)c SU(2)L U(1)Y U(1)L ℓL 3 1 2 1/2 1 eR 3 1 1 1 1 N 3 1 1 0 0 H 1 1 2 1/2 0 η 1 1 2 1/2 1 σ 1 1 1 0 2 S 1 1 1 0 1 Table 7: Lepton and scalar particle content and representations under the gauge and global symmetries in an UV extension of the Scotogenic model with accidental Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Let us now generalize the idea studied in this Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We consider again the set of models in which (H†η)2 is generated by the topologies shown in Table 2 with the addition of at most two different singlets σ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' There are two possibilities to construct models in which the Zacc 2 symmetry is obtained: (i) Models with qN ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In this case we consider the models shown in Table 2 but impose that N is even under the remnant Zrem 2 parity while ℓL, eR and η are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The Majorana masses of the N fermions are induced by the κ σ1N cN Yukawa term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' (ii) Models with qN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This case is excluded from the classification in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3, which focuses on qN ̸= 0, and must be discussed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In these models the Majorana mass term MN N cN is present in the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We now proceed to discuss these two cases independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' The first one can be regarded as a revision of our discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3, imposing now different conditions on the resulting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' In fact, the models studied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3 could also lead to U(1)L → Zrem 2 , leaving the Scotogenic Z2 parity as an accidental symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' This will be the case when these conditions on qN are satisfied: qN = 2 z, where z can be any integer number except zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' qN = α β, with α, β ∈ Z and α and β even and odd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Also, GCD(α, β) = 1 has to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Notice, however, that models with fixed charges, that is, the ones with only σ1, always have the Scotogenic symmetry as the remnant symmetry and do not enter this discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Considering now the second possibility, only 11 different models exist and they are listed in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Let us denote them as ξ′(A, B), where ξ = {I, II, III, IV} and the prime is used to distinguish these models from the ones studied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Each of the 11 models needs to satisfy any of the following conditions on qσ1 in order to generate the Z2 parity as an accidental symmetry: 26 Topology A B qN qη qσ1 qσ2 qS (SU(2)L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' U(1)Y)S 1 I′ 1 ∅ 0 −1 2 −2 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 2 I′ ∅ 1 0 −1 2 0 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 3 I′ 1 2 0 −1 qσ1 2 − qσ1 −qσ1 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1) 4-5 II′ 1 ∅ 0 −1 2 −1 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) or (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) 6-7 II′ 1 2 0 −1 qσ1 2 − qσ1 1 − qσ1 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) or (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 0) 8 III′ 1 ∅ 0 −1 2 −2 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 9 III′ 1 2 0 −1 qσ1 2 − qσ1 −2 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 10 IV′ 1 ∅ 0 −1 2 1 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) 11 IV′ 1 2 0 −1 qσ1 2 − qσ1 1 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 1/2) Table 8: UV extended models for which the term N cN is allowed and the Scotogenic Z2 is an accidental symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' For each model we show the U(1)L charges of N, η, σ1, σ2 and S, as well as the (SU(2)L, U(1)Y) representation of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Models that become any of the models in this list after renaming the fields or redefining their U(1)L charges are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' qσ1 = 2 z, where z can be any integer number, including zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' 6 qσ1 = α β, with α, β ∈ Z and α and β even and odd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Also, GCD(α, β) = 1 has to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' We finally point out that in none of the above scenarios η gets an induced VEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Ma, “Verifiable radiative seesaw mechanism of neutrino mass and dark matter,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE4T4oBgHgl3EQfwQ2l/content/2301.05249v1.pdf'} 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With the recent advances in deep learning (DL), an increasing number of APR +techniques have been proposed to leverage neural networks to learn bug-fixing patterns from massive open- +source code repositories. Such learning-based techniques usually treat APR as a neural machine translation +(NMT) task, where buggy code snippets (i.e., source language) are translated into fixed code snippets (i.e., +target language) automatically. Benefiting from the powerful capability of DL to learn hidden relationships +from previous bug-fixing datasets, learning-based APR techniques have achieved remarkable performance. +In this paper, we provide a systematic survey to summarize the current state-of-the-art research in the +learning-based APR community. We illustrate the general workflow of learning-based APR techniques and +detail the crucial components, including fault localization, patch generation, patch ranking, patch validation, +and patch correctness phases. We then discuss the widely-adopted datasets and evaluation metrics and outline +existing empirical studies. We discuss several critical aspects of learning-based APR techniques, such as +repair domains, industrial deployment, and the open science issue. We highlight several practical guidelines +on applying DL techniques for future APR studies, such as exploring explainable patch generation and +utilizing code features. Overall, our paper can help researchers gain a comprehensive understanding about the +achievements of the existing learning-based APR techniques and promote the practical application of these +techniques. Our artifacts are publicly available at https://github.com/QuanjunZhang/AwesomeLearningAPR. +CCS Concepts: • Software and its engineering → Software testing and debugging; Software testing and +debugging. +Additional Key Words and Phrases: Automatic Program Repair, Deep Learning, Neural Machine Translation, +AI and Software Engineering +ACM Reference Format: +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen. 2023. A Survey of Learning- +based Automated Program Repair . ACM Trans. Softw. Eng. Methodol. 0, 0, Article 1 ( 2023), 51 pages. https: +//doi.org/10.1145/nnnnnnn.nnnnnnn +∗Chunrong Fang is the corresponding author. +Authors’ addresses: Quanjun Zhang, quanjun.zhang@smail.nju.edu.cn, State Key Laboratory for Novel Software Technology, +Nanjing University, Nanjing, Jiangsu, China, 210093; Chunrong Fang, fangchunrong@nju.edu.cn, State Key Laboratory for +Novel Software Technology, Nanjing University, Nanjing, Jiangsu, China, 210093; Yuxiang Ma, 502022320009@smail.nju. +edu.cn, State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, Jiangsu, China, 210093; Weisong +Sun, weisongsun@smail.nju.edu.cn, State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, +Jiangsu, China, 210093; Zhenyu Chen, zychen@nju.edu.cn, State Key Laboratory for Novel Software Technology, Nanjing +University, Nanjing, Jiangsu, China, 210093. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee +provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and +the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. +Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires +prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2022 Association for Computing Machinery. +1049-331X/2023/0-ART1 $15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. +arXiv:2301.03270v1 [cs.SE] 9 Jan 2023 + +1:2 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +1 +INTRODUCTION +Modern software systems continuously evolve with inevitable bugs due to deprecating of old +features, adding of new functionalities, and refactoring of system architecture [155]. These inevitable +bugs have been widely recognized as notoriously costly and destructive, such as costing billions +of dollars annually across the world [17, 160]. The recorded quantity of bugs is increased at a +tremendous speed due to the increasing scale and complexity of software systems [42]. Manual +debugging can be an extremely time-consuming and error-prone task in the software development +and maintenance process. For example, previous reports show that software debugging accounts for +over 50% of the cost in software development [18]. Considering the promising future in relieving +manual debugging efforts, automated program repair (APR), which aims to automatically fix +software bugs without human intervention, has been a very active area in academia and industry. +As a promising research area, APR has been extensively investigated in the literature and has +made substantial progress on the number of correctly-fixed bugs [111]. A living APR review +reports [112] that a growing number of papers get published each year with various exquisitely +implemented APR tools being released. Over the past decade, researchers have proposed a variety of +APR techniques to generate patches [88] [11] [156], including heuristic-based, constraint-based and +pattern-based. Among these traditional techniques, pattern-based APR employs pre-defined repair +patterns to transform buggy code snippets into correct ones and has been widely recognized as +state-of-the-art [87, 164, 165]. However, existing pattern-based techniques mainly rely on manually- +designed repair templates, which require massive effort and professional knowledge to craft in +practice. Besides, these templates are usually designed for specific types of bugs (e.g., null pointer +exception) and thus are challenging to apply to unseen bugs, limiting the repair effectiveness. +Recently, inspired by the advance of deep learning (DL), a variety of learning-based APR tech- +niques have been proposed to learn the bug-fixing patterns automatically from large corpora of +source code [147]. Compared with traditional APR techniques, learning-based techniques can be +applied to a wider range of scenarios (e.g., multi-languages) with parallel bug-fixing pairs. These +learning-based techniques handle the program repair problem as a neural machine translation +(NMT) task, which translates a code sequence from a source language (i.e., buggy code snippets) +into a target language (i.e., correct code snippets). Existing NMT repair models are typically built +on the top of the encoder-decoder architecture [150]. The encoder extracts the hidden status of +buggy code snippets with the necessary context, and the decoder takes the encoder’s hidden status +and generates the correct code snippets [56, 79, 91]. Thanks to the powerful ability of DL to learn +hidden and intricate relationships from massive code corpora, learning-based APR techniques have +achieved remarkable performance in the last couple of years. +The impressive progress of learning-based APR has shown the substantial benefits of exploiting +DL for APR and further revealed its promising future in follow-up research. However, a mass +of existing studies from different organizations (e.g., academia and industry) and communities +(e.g., software engineering and artificial intelligence) make it difficult for interested researchers to +understand state-of-the-arts and improve upon them. Besides, compared with traditional techniques, +learning-based techniques heavily rely on the quality of code corpora and model architectures, +posing several challenges (e.g., code representation and patch ranking) in developing mature NMT +repair models. For example, most learning-based techniques adopt different training datasets, and +there exist various strategies to process the code snippets (e.g., the code context, abstraction, and +tokenization). Besides, researchers design different code representations (e.g., sequence, tree, and +graph) to extract code features, which require corresponding encoder-decoder architectures (e.g., +RNN, LSTM, and transformer) to learn the transformation patterns. Furthermore, execution-based +(e.g., plausible and correctness patches) and match-based (e.g., accuracy and BLUE) metrics are +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:3 +adopted in different studies. Such multitudinous design choices hinder developers from conducting +follow-up research on the learning-based APR direction. +In this paper, we summarize existing work and provide a retrospection of the learning-based APR +field after years of development. Community researchers can have a thorough understanding of the +advantages and limitations of the existing learning-based APR techniques. We illustrate the typical +workflow of learning-based APR and discuss different detailed techniques that appeared in the +papers we collected. Based on our analysis, we point out the current challenges and suggest possible +future directions for learning-based APR research. Overall, our work provides a comprehensive +review of the current progress of the learning-based APR community, enabling researchers to +obtain an overview of this thriving field and make progress toward advanced practices. +Contributions. To sum up, the main contributions of this paper are as follows: +• Survey Methodology. We conduct a detailed analysis of 112 relevant studies that used DL +techniques in terms of publication trends, distribution of publication venues and languages. +• Learning-based APR. We describe the typical framework of leveraging advances in DL tech- +niques to repair software bugs and discuss the key factors, including fault localization, data +pre-processing, patch generation, patch ranking, patch validation and patch correctness. +• Dataset and Metric. We perform a comprehensive analysis of the critical factors that impact +the performance of DL models in APR, including 53 collected datasets and evaluation metrics +in two categories. +• Empirical studies. We detail existing empirical studies performed to better understand the +process of learning-based APR and facilitate future studies. +• Some Discussions. We discuss some other crucial areas (e.g., security vulnerability and syntax +error) where learning-based APR techniques are applied, as well as certain known industrial +deployments. We demonstrate the trend of employing pre-trained models on APR recently. +We list the available learning-based tools and reveal the essential open science problem. +• Outlook and challenges. We pinpoint open research challenges of using DL in APR and provide +several practical guidelines on applying DL for future learning-based APR studies. +Comparison with Existing Surveys. Gazzola et al. [42] present a survey to organize the repair +techniques published up to January 2017. Monperrus et al. [111] present a bibliography of behavioral +and state repair papers. Unlike existing surveys mainly covering traditional techniques, our work +focuses on the learning-based APR, particularly the integration of DL techniques in the repair +phase (e.g., patch generation and correctness), repair domains (e.g., vulnerability and syntax errors) +and challenges. Besides, our survey summarizes the existing studies until Nov 2022. +Paper Organization. The remainder of this paper is organized as follows. Section 2 presents +the research methodology about how we collect relevant papers from several databases following +specific keywords. Section 3 introduces some common concepts encountered in the learning- +based APR field. Section 4 presents the typical workflow of learning-based APR and discusses +the vital components of the workflow in detail. Section 5 extends the discussion on the collection +of datasets and standard evaluation metrics of learning-based APR techniques. Section 6 details +some discussions, including repair applications, industrial deployments, employment of pre-trained +models and the open science problem. Section 7 provides practical guidelines. Section 8 draws the +conclusions. +2 +SURVEY METHODOLOGY +In this section, we present details of our systematic literature review methodology following +Petersen et al. [124]. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:4 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +Google Scholar +ACM Digital Library +IEEE Digital Library +Group1 +repair related keywords +Group2 +DL related keywords +discussion and selection +program repair; bug fix; … +deep; learning; machine; … +Automated Search +filter by year +342 papers +filter by pages +(remove duplications) +283 papers +filter irrelevant papers +87 papers +add missed citations +112 papers +Figure 1. General workflow of the paper collection +Search Process. For this survey, we select papers by mainly searching the Google Scholar repository, +ACM Digital Library, and IEEE Explorer Digital Library at the end of November 2022. Following +existing DL for SE surveys [154, 170], we divide the search keywords used for searching papers into +two groups: (1) a APR-related group containing some commonly used keywords related to program +repair; and (2) a DL-related group containing some keywords related to deep learning or machine +learning. Considering a significant amount of relevant papers from both SE and AI communities, +following Zhang et al. [179], we first try to collect some papers from the community-driven website1 +and the living review of APR by Monperrus [112], and then conclude some frequent words in the +titles of these papers. The search strategy can capture the most relevant studies while achieving +better efficiency than a purely manual search. Finally, we identify a search string including several +DL-related terms frequently appearing in APR papers that make use of DL techniques, listed as +follows. +(“program repair” OR “software repair” OR “automatic repair” OR “code repair” OR “bug repair” +OR “bug fix” OR “code fix” OR “automatic fix” OR “patch generation” OR “fix generation” OR +“code transformation” OR “code edit” OR “fix error”) AND (“neural” OR “machine” OR “deep” OR +“learning” OR “transformer/transformers” OR “model/models” OR “transfer” OR “supervised”) +Study selection. Once the potentially relevant studies based on our search strategy are collected, +we perform a filtering and deduplication phase to exclude papers not aligned with the study goals. +We first attempt to filter out the papers before 2016, considering that Long et al. [91] propose +the first learning-based APR study in 2016. We then filter out any paper less than 7 pages and +the duplicated papers, resulting in 283 papers in total. We then scrutinize the remaining papers +manually to decide whether it is relevant to the learning-based APR field. We obtained 87 papers at +last. To be as much comprehensive as possible, we include the relevant papers that we miss with +our searches but were cited in the papers we selected. We manually analyzed all these cited papers +by scanning the papers and finally collected 112 papers in our survey. The general workflow of +how we collected papers is shown in Figure 1. +1http://program-repair.org/bibliography.html +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:5 +3 +4 +6 +13 +13 +25 +47 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +Number of papers +Year +Figure 2. Collected learning-based APR papers from 2016 to 2022 +44% +20% +18% +13% +5% +Java +C +Python +JavaScript +C++ +Figure 3. Paper distribution on program languages +Trend Observation. Figure 2 shows the collected papers from 2016 to 2022. It is found that the +number of learning-based APR papers has increased rapidly since 2020, indicating that more +researchers are considering DL as a promising solution to fixing software bugs. One reason behind +this phenomenon is that traditional APR techniques have reached a plateau and researchers hope +to find a brand-new way to address the problem. Another non-negligible reason is that DL has +proved its potential in various tasks, including natural language translation, which is similar to +bug fixing to some extent. Figure 3 presents an overview of the programming languages targeted +by learning-based APR techniques in our survey. We can find Java occupies a large proportion, +which is understandable as Java is widely adopted in modern software systems nowadays and the +most targeted language in existing mature datasets (e.g., Defects4J). We also find that the collected +papers cover a wide range of programming languages (i.e., Java, JavaScript, Python, C and C++). +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:6 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +repair strategy +test suite +correct patch +plausible patch +developer +generated patch +overfitting patch +suspicious code +fault localization +deployment +Localization Phase +Repair Phase +buggy program +Verification Phase +Figure 4. Overview of APR +For example, there exist several papers [96, 159] involving multiple programming language repair. +The probable reason may be that learning-based APR techniques usually regard APR as an NMT +problem, independent of programming languages. +3 +BACKGROUND AND CONCEPTS +In this section, we will introduce some background information and common concepts in the +learning-based APR field. +3.1 +Automated Program Repair +The primary objective of APR techniques is to identify and fix software bugs without human +intervention. In the software development and maintenance process, after a designed functionality +is implemented, developers usually write some test suites (e.g., Junit test cases) to check the +functionality. If there exist test suites that make the functionality fail, developers adopt the failing +test suites to analyze the symptoms and the root cause of the bug, and attempt to fix the bug by +making some changes to suspicious code elements. More generally, we can give the following +definition. +Definition 3.1. APR: Given a buggy program 𝑃, the corresponding specification 𝑆 that makes 𝑃 +fail, the transformation operators𝑂 and the allowed maximum edit distance𝜖, APR can be formalized +as a function 𝐴𝑃𝑅(𝑃,𝑆,𝑂,𝜖). 𝑃𝑇 is the set of its all possible program variants by enumerating all +operators 𝑂 on 𝑃. The problem of APR is to find a program variant 𝑃 ′ (𝑃 ′ ∈ 𝑃𝑇) that satisfies 𝑆 and +the changes satisfies 𝜖 (𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑃, 𝑃 ′) ≤ 𝜖). +The specification 𝑆 denotes a relation between inputs and outputs and most APR techniques +usually adopt a test suite as a specification. In other words, APR aims to find a minimal change to 𝑃 +that passes all available test suites. The maximum edit distance 𝜖 limits the range of changes based +on the competent programmer hypothesis [119], which assumes that experienced programmers are +capable of writing almost correct programs and most bugs can be fixed by small changes. If 𝜖 is set +to ∞, 𝐴𝑃𝑅(𝑃,𝑆,𝑂,𝜖) becomes a program synthesizing problem that aims to synthesize a program +to satisfy 𝑆. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:7 +The typical workflow of APR techniques is illustrated in Figure 4, which is usually composed +of three parts: (1) off-the-shelf fault localization techniques are applied to outline the buggy code +snippets [1] [7]; (2) these snippets are modified based on a set of transformation rules or patterns to +generate new various program variants (i.e., candidate patches); (3) the original test suite is adopted +as the oracle to verify all candidate patches. Specifically, a candidate patch passing the original +test suite is called a plausible patch. A plausible patch, which is also semantically equivalent to the +developer patch, denotes a correct patch. +However, such specifications (i.e., test suites) are inherently incomplete as programs have infinite +domains. It is fundamentally challenging to ensure the correctness of the plausible patches (i.e., +overfitting issue) due to the weak test suites in practice. Existing studies have demonstrated that +manually identifying the overfitting patches is time-consuming and may harm the debugging +performance of developers [137, 141]. The overfitting issue is a critical challenge in both traditional +and learning-based APR techniques. We will discuss the issue in Section 4.7. +3.1.1 +Patch Generation Techniques. In the literature, numerous traditional APR techniques have +been proposed to generate patches from different aspects, which can be categorized into three +classes. We list them as follows. (1) Heuristic-based repair techniques. These techniques usually +apply heuristic strategies (e.g., genetic algorithm) to build search space from previous patches +and generate valid patches by exploring the search space [73, 101, 178]. For example, SimFix [56] +builds an abstract search space from existing patches and a concrete search space from similar code +snippets in the buggy project. SimFix then utilizes the intersection of the above two search spaces +to search the final patch by basic heuristics (e.g., syntactic distance). (2) Constraint-based repair +techniques. These techniques usually treat APR as a constraint-solving task and rely on SMT solvers +to return a feasible solution [36, 102, 106]. For example, Nopol [167] relies on an SMT solver to +solve the condition synthesis problem after identifying potential locations of patches by angelic fix +localization and collecting test execution traces of the program. (3) Pattern-based repair techniques. +These techniques usually design certain repair templates by manually analyzing specific software +bugs and generate patches by applying such templates to buggy code snippets [66, 86, 87]. For +example, TBar [87] revisits the effectiveness of pattern-based APR techniques by systematically +summarizing a variety of repair patterns from the literature. +In addition to the above traditional APR techniques, researchers attempt to fix software bugs +enriched by DL techniques due to the large-scale open-source source code repositories. Such +learning-based techniques have demonstrated promising results and are getting growing attention +recently, which is the focus of our work (introduced in Section 3.2). +3.2 +Neural Machine Translation +Sequence-to-sequence (Seq2Seq) is an advanced DL framework widely used in some NLP tasks (e.g., +machine translation [58] and text summarization [113]). A Seq2Seq model usually consists of two +components (i.e., an encoder and a decoder) to learn mappings between two sequences. Inspired by +the success of Seq2Seq models in text generation tasks, program repair can be formulated as an +NMT task. The learning-based APR problem is formally defined as follows: +Definition 3.2. Learning-based APR: Given a buggy code snippet 𝑋𝑖 = [𝑥1, . . . ,𝑥𝑛] with 𝑛 code +tokens and a fixed code snippet 𝑌𝑖 = [𝑦1, . . . ,𝑦𝑚] with𝑚 code tokens, the problem of program repair +is formalized to maximize the conditional probability: 𝑃 (𝑌 | 𝑋) = �𝑚 +𝑖=1 𝑃 (𝑦𝑖 | 𝑦1, . . . ,𝑦𝑖−1;𝑥1, . . . ,𝑥𝑛). +In other words, the objective of an NMT repair model is to learn the mapping between a buggy +code snippet 𝑋 and a fixed code snippet 𝑌. Then the parameters of the model are updated by using +the training dataset, so as to optimize the mapping (i.e., maximizing 𝑃). In the literature, recurrent +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:8 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +Figure 5. Detailed workflow of Learning-based APR +neural network architecture (RNN) is widely used in existing learning-based APR techniques +[24, 48, 146, 147]. Besides, researchers use long short-term memory (LSTM) architecture to capture +the long-distance dependencies among code sequences [20, 107]. Recently, as a variant of the +Seq2Seq model, Transformer [150] has been considered the state-of-the-art NMT repair architecture +due to the self-attention mechanism [25, 26, 40]. +4 +LEARNING-BASED APR +In this section, we will discuss the workflow of learning-based APR tools and introduce some +popular learning-based APR techniques with several examples. +4.1 +Overall Workflow +Figure 5 illustrates the typical framework of existing learning-based APR techniques. The framework +can be generally divided into seven phrases: fault localization, data pre-processing, input encoding, +output decoding, patch ranking, patch validation, and patch correctness assessment. We now discuss +the phrases in detail as follows. +• In the fault localization phase, a given buggy program is taken as the input and a list of +suspicious code elements (e.g., statements or methods) is returned. +• In the data pre-processing phase, a given software buggy code snippet (e.g., buggy state- +ment) is taken as the input and the processed code tokens are returned. According to existing +learning-based APR studies [25, 26], there generally exist three potential ways to pre-process +the buggy code: code context, abstraction, and tokenization. First, code context information +refers to other correlated non-buggy lines within the buggy program. Previous work has +demonstrated that NMT-based repair models reveal diverse code changes to fix bugs under +different contexts [24]. Second, code abstraction renames some special words (e.g., string and +number literals) to a pool of predefined tokens. Code abstraction has been proven to be an +effective method in reducing vocabulary size. Third, code tokenization splits source code into +words or subwords, which are then converted to ids through a look-up table. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +③ Patch Generation Phase +② Data Preprocessing Phase +@ Patch Ranking Phase +? Patch Validation Phase +ei +α1 +e1 +ai +a2 +e2 +e3 +a3 +e3 +@ Patch Correctness PhaseA Survey of Learning-based Automated Program Repair +1:9 +• In the patch generation phase, the processed code tokens are first fed into a word embed- +ding stack to generate representation vectors, which can capture the semantic meaning of +code tokens and their position within a buggy code. Then an encoder stack is implemented +to derive the encoder’s hidden state, which is further passed into a decoder stack. Similar +to the encoder stack, a decoder stack is implemented to take the hidden states provided by +the encoder stack and previously generated tokens as inputs, and returns the probability +distribution of the vocabulary. +• In the patch ranking phase, after the NMT-based repair model is well-trained, a rank +strategy (e.g., beam search) is leveraged to prioritize the candidate patches as prediction +results based on the probability distribution of the vocabulary. +• In the patch validation phase, the generated candidate patches are then verified by the +available program specification, such as functional test suites or static analysis tools. +• In the patch correctness assessment phase, the plausible patches (i.e., passing the exist- +ing specification) are assessed to predict their correctness (i.e., whether the plausible are +overfitting), which are finally manually checked by developers for deployment in the software +pipeline. +4.2 +Fault Localization +Fault localization aims to diagnose buggy program elements (e.g., statements and methods) without +human intervention and has been extensively studied to facilitate the program repair process [162]. +As a crucial start in the learning-based APR pipeline, fault localization provides the repair model +with information about where a software bug is and directly influences the performance of the +repair model. For example, the repair accuracy under normal fault localization is usually lower +than the circumstance under perfect fault localization. +In the literature, fault localization techniques often leverage various static analysis or dynamic +execution information to compute suspiciousness scores (i.e., probability of being faulty) for each +program element. Program elements are then ranked in descending order of their suspiciousness +scores, based on which APR techniques can further be applied. Researchers have proposed a variety +of fault localization techniques, such as spectrum-based [123, 182], mutation-based [77, 120], +slicing-based [10, 99] and learning-based [78, 93] techniques. Among them, spectrum-based fault +localization (SBFL) has been extensively utilized as a general mechanism to localize the statements +that are likely to be faulty in the APR literature. +4.2.1 +Localization Techniques. Similar to traditional APR techniques, some learning-based APR +techniques rely on existing SBFL fault localization approaches to localize the revealed bug. For +example, DLFix [79] adopts Ochiai algorithm to identify a buggy line and extracts all AST nodes +(including intermediate ones) related to that buggy line as a replaced subtree for patch generation. +Recoder [186] also assumes the faulty location of a bug is unknown to APR tools and uses Ochiai +algorithm with GZoltar [130], which is widely used in existing APR tools, such as RewardRepair +[176] and AlphaRepair [165]. Such SBFL techniques exploit runtime information to recognize the +program elements that are likely to be faulty when the buggy program is executed by the available +test suite. The crucial insight is that (1) the program elements executed by more failing test suites +and fewer passing test suites are likely to be faulty; and (2) the program elements executed by more +passing test suites and fewer failing suites are likely to be correct. In particular, SBFL produces a +list of program elements ranked according to their likelihood of being faulty based on the analysis +of the program entities covered by passing and failing tests (e.g., Ochiai and Tarantula [85]). +However, Liu et al. [85] have demonstrated that the fault localization techniques may introduce +a significant bias in the evaluation of APR techniques. The vast majority of learning-based APR +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:10 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +techniques consider repairing software bugs under perfect-based fault localization techniques. +Perfect-based fault localization techniques assume that the genuine localization of the bug is +known. Thus, perfect-based fault localization can provide a fair assessment of APR techniques and +the assessment is independent of the localization techniques. For example, CoCoNut [96] manually +checks the bug-fixing pairs in Defects4J benchmark and extracts the changed statements as inputs +to the repair model. Subsequently, recent learning-based APR techniques adopt the same or similar +processing method to conduct perfect localization, such as CIRCLE [159], CURE [57], SelfAPR [175] +and AlphaRepair [165]. +Besides, there exist some techniques attempting to perform fault localization on their own. For +example, DeepFix [48] proposes an end-to-end approach in which the network reports a ranked list +of potentially erroneous lines with a beam search mechanism. Similarly, Prophet [91] designs a fault +localization algorithm to return a ranked list of program candidate lines to modify by analyzing +dynamic execution traces of the test suite. Szalontai et al. [138] first localize the nonidiomatic +code snippets by LSTM networks and predict the nonidiomatic pattern by a feed-forward neural +network, which is fixed by a high-quality alternative. Recently, Meng et al .[107] build a novel +fault localization technique based on deep semantic features and transferred knowledge, which is +further fed to a fix template prioritization model and a template-based APR technique TBar. +4.2.2 +Localization Granularity. APR techniques consider program elements of different granulari- +ties, thus determining the scope of the fault localization. In other words, APR and fault localization +usually work at the same granularity level. For example, if APR techniques focus on repairing +buggy statements (or methods), the fault localization also works at the level of program statements +(or methods). In the literature, a majority of fault localization techniques adopted in learning-based +APR techniques usually record the line of a buggy code snippet [57, 79, 80, 96, 159, 186]. There +also exists little work considering other granularity. For example, Tufano et al. [147] adopt the +NMT-based repair model to learn the translation from buggy to fixed code at the method-level. +4.3 +Data Pre-processing +Data pre-processing phase aims to analyze and parse the identified buggy code snippets, which are +then passed into neural networks for training and inference. In the data pre-processing phase, a +given software buggy code snippet (e.g., a buggy function) is taken as the input and the processed +code tokens are returned. According to existing learning-based repair studies [25, 26], the data +pre-processing phase generally consists of three parts: code abstraction, code context and code +tokenization. +4.3.1 +Code Context. Code context generally refers to other correct statements around the buggy +lines. In the manual repair scenario, the context of the buggy code plays a significant role in +understanding faulty behaviors and reasoning about potential repairs. Developers usually identify +the buggy lines, and then analyze how they interact with the rest of the method’s execution, and +observe the context (e.g., variables and other methods) in order to come up with the possible repair +and pick several tokens from the context to generate the fixed line [64]. In learning-based APR, the +NMT model mimics this process by extracting the code context and the buggy line into a certain +code representation to preserve the necessary context that allows the model to predict the possible +fixes. +Existing learning-based APR techniques typically consider the surrounding source code relevant +to the buggy statement as context. These techniques typically employ context in various ways, +such as extracting code near the buggy statement within the buggy method, class, and even file. +On the one hand, a broad context contains plenty of essential fix ingredients, while such a large +vocabulary size introduces noise that negatively affects the repair performance of the NMT model +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:11 +due to the tricky long-term dependency problem in NMT models. On the other hand, a narrow +context contains too little information to capture the proper semantics of the buggy statement +and leads to incorrect patches generation due to a lack of necessary vocabulary. There seems to +be a trade-off relationship between vocabulary size and context size. In the literature, our survey +concludes the code context into four granularities: context-free, line-level context, method-level +context, and class-level context. +• Context-free means that NMT models only consider buggy statements without any additional +context information [32, 51, 103]. For example, Mashhadi et al. [103] consider single statement +bugs from the ManySStuBs4J dataset and extract the buggy statement as a source side and +the fixed statement as a target side from bug-fixing commits. Ding et al. [32] provide NMT +models with a single program line that contains a buggy statement. However, previous work +demonstrates that fixing nearly 90% of bugs requires new vocabulary relative to the buggy +code. NMT repair models suffer from capturing enough information from the buggy code +alone. +• Statement-level context means that the buggy code and several statements around it are fed to +MNT repair models. For example, TFix [13] extracts the two neighboring statements of the +buggy code as the code context. Chi et al. [26] extract statement-level code changes by the +“git dif” command and employ data-flow dependencies to capture more critical information +around the context. +• Method-level context means that the method to which the buggy line belongs is fully fed into +the model [96, 147, 159]. It is the most commonly used type of context in literature as it often +contains enough information for repairing the bug, such as the type of variables and the +function of this method. For example, Tufano et al .[146] focus on the method-level context +since (1) the functionality to be fixed is usually implemented in program methods; (2) the +methods provide neural networks with meaningful abundant context information, such as +literals and variables. CoCoNuT [20] extracts the entire method of the buggy code as context, +which is encoded as a separate input. +• Class-level context means that the class to which this buggy code belongs is fed into the NMT +model. It is a relatively broad context, while it can provide the model with rich information. +For example, SequenceR [24] considers the class-level context and conducts abstract buggy +context from the buggy class, which captures the most important context around the buggy +source code and reduces the complexity of the input sequence to 1,000 tokens. Hoppity [31] +takes the whole buggy file as the context with a length limit of 500 tokens nodes in the AST. +4.3.2 +Code Abstraction. Code abstraction aims to limit the number of words the NMT models need +to process by renaming raw words (e.g., function names and string literals) to a set of predefined +tokens. Previous work demonstrates that it is challenging for NMT models to learn bug-fixing +transformation patterns due to the huge vocabulary of source code [147]. In particular, NMT models +usually employ a beam-search decoding strategy to output repair candidates by a probability +distribution over all words. The search space can be extremely large with many possible words in +the source code, resulting in inefficient patch generation. +In our survey, a considerable number of learning-based papers we collect employ the abstracted +source code to tackle this problem. Such abstraction operation means the original source code is +not directly fed into the NMT model. Benefiting from the abstracted code, we can (1) reduce the size +of vocabulary significantly and the frequency of specific tokens; (2) filter out irrelevant information +and improve the efficiency of the NMT model. Generally, the natural elements (e.g., identifiers and +literal) in the source code are renamed, while the core semantic information (e.g., idioms) should +be preserved. For example, Tufano et al. [147] propose the first code abstraction approach in the +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:12 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +int GetMaxCommonDivisor(int m, int n){ +int r; +while (n!=0){ +r=m%n; +m=n; +n=r; +} +return n; +} +raw buggy code +(a) raw buggy code +raw fixed code +int GetMaxCommonDivisor(int m, int n){ +int r; +while (n!=0){ +r=m%n; +m=n; +n=r; +} +return m; +} +(b) raw fixed code +TYPE_1 METHOD_1(TYPE_1 VAR_1, TYPE_1 VAR_2){ +TYPE_1 VAR_3; +while (VAR_2!=NUMBER_1){ +VAR_3=VAR_1%VAR_2; +VAR_1=VAR_2; +VAR_2=VAR_3; +} +return VAR_2; +} +abstracted buggy code +(c) abstracted buggy code +abstracted fixed code +TYPE_1 METHOD_1(TYPE_1 VAR_1, TYPE_1 VAR_2){ +TYPE_1 VAR_3; +while (VAR_2!=NUMBER_1){ +VAR_3=VAR_1%VAR_2; +VAR_1=VAR_2; +VAR_2=VAR_3; +} +return VAR_1; +} +(d) abstracted fixed code +Figure 6. A simple example of code abstraction +learning-based APR field by (1 ) adopting a lexer to tokenize the raw source code as a stream of +tokens based on Another Tool for Language Recognition (ANTLR) [122]; (2) passing the stream of +tokens into a parser to identify the role of each identifier and literals (e.g., whether it represents +a variable, method, or type name); (3) replacing each identifier and literal with a unique ID to +generate the abstracted source code. Besides, they extract the idioms (i.e., tokens that appear many +times) and keep their original textual tokens in the abstraction process because such idioms contain +beneficial semantic information. The typical code abstraction example is presented in Figure 6. +Similarly, CoCoNut [96] and CURE [57] only abstract string and number literals except for the +frequent numbers (e.g., 0 and 1). DLFix [79] adopts a novel abstraction strategy to alpha-rename +the variables, so as to learn the fix between methods with similar scenarios while having different +variable names. DLFix also keeps the type of the variable to avoid accidental clashing names +and maintains a mapping table to recover the actual names. Recoder [186] abstracts infrequent +identifiers with placeholders to make the neural network learns to generate placeholders for these +identifiers. +Although a variety of techniques adopt the code abstraction strategy (such as Tufano et al. [147]) +to limit the vocabulary size and make the transformer concentrate on learning common patterns +from different code changes, we still find some techniques prefer raw source code [159, 186]. For +example, developers may name one function as SetHeightValue to indicate that this function can +set the value of height as they want. If this name is abstracted directly as func_1, critical semantic +information would be missed. Instead of renaming rare identifiers through a custom abstraction +process, SequenceR [24] utilizes the copy mechanism to generate candidate patches with a large +set of tokens. Chen et al. [25] adopt the raw source code as they think abstracted code may hide +valuable information about the variable that can be learned by word embedding. A strategy similar +to Chen et al. [25] is also implemented in other learning-based APR techniques, such as in CODIT +[20], CIRCLE [159] and TFix [13]. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:13 +4.3.3 +Code Tokenization. Code tokenization aims to split source code into a stream of tokens, +which are then converted to ids through a look-up table2. These id numbers are in turn used +by the repair models for further processing and training. A simple tokenization approach can be +conducted by dividing the source code into individual characters. The core concept of this char-level +tokenization is that although the source code has many different words, it has a limited number of +characters. This approach is straightforward and leads to an exceeding small vocabulary. However, +it leads to a relatively long tokenized sequence with the splitting of each world into all characters. +More importantly, it is pretty difficult for repair models to meaningful input representations as +characters alone do not have semantic meaning. Generally, there exist two main granularities +of code tokenizers used in learning-based APR techniques: word-level tokenizers and [40] and +subword-level tokenization [26]. +The word-level tokenization means that a sentence is divided according to its words (e.g., space- +separated), which is widely used in NLP tasks. However, different from natural language, words +(e.g., variable names) in programming languages can be created arbitrarily, leading to a more +irregular vocabulary. Previous work [24] demonstrates that it is difficult for NMT models to handle +a code vocabulary size larger than 560,000 tokens. This kind of granularity often causes the out- +of-vocabulary (OOV) problem for infrequent tokens, and the model could be more efficient for an +excessively large vocabulary size. To address this issue, VRepair employs a word-level tokenization +to tokenize C source code and the copy mechanism to deal with the out-of-vocabulary problem. +For example, CoCoNut [96] designs a code-aware space-separated tokenization algorithm that is +specific to programming languages by (1) separating operators from variables as they might not be +space-separated; (2) considering underscores, camel letters, and numbers as separators as many +words are composed of multiple words without separation (e.g., SetHeightValue); (3) introducing a +new token to mark where the camel case split occurs to regenerate source code from the +list of tokens generated correctly. +The subword-level tokenization splits rare tokens into multiple subwords instead of directly +adding full tokens into the vocabulary. Besides, the frequent words should not be split into smaller +subwords. This kind of granularity can reduce the vocabulary size significantly and is widely used in +the learning-based APR field. Technically, there exist several subword-level tokenization techniques, +such as byte-pair encoding (BPE), byte-level byte-pair encoding (BBPE) [135] and SentencePiece +[68], listed as follows. +(1) BPE tokenizer generally needs to be trained upon a given dataset by (1) leveraging a pre- +tokenizer to splits the dataset into words by space-separated tokenization; (2) creating a +set of unique words and counting the frequency of each word in the dataset; (3) building a +base vocabulary with all symbols that occur in the set of unique words and learning merge +rules to form a new symbol from two symbols of the base vocabulary; (4) repeating the +above process until the vocabulary is reduced to a reasonable size, which is a pre-defined +hyperparameter, before training the tokenizer. For example, VulRepair [40] employs a BPE +algorithm to train a subword tokenizer on eight different programming languages (i.e., Ruby, +JavaScript, Go, Python, Java, PHP, C, C#) [157] and is suitable for tokenizing source code. In +the learning-based APR literature, a majority of repair studies adopt BPE as the tokenization +technique, such as CURE [57], CoCoNut [96], SeqTrans [26]. The results have demonstrated +the effectiveness of BPE in reducing vocabulary size and mitigating the OOV problem by +extracting the most frequent subwords and merging the most frequent byte pair iteratively. +(2) BBPE refines BPE by employing bytes as the base vocabulary, ensuring that every base +character is included with a proper vocabulary size. For example, AlphaRepair [165] builds a +2https://huggingface.co/Salesforce/codet5-base/blob/main/vocab.json +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:14 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +BBPE-based tokenizer to reduce the vocabulary size by breaking uncommon long words into +meaningful subwords. +(3) SentencePiece contains the space in the base vocabulary and utilizes the existing BPE al- +gorithm (e.g., BPE) to create the desired vocabulary by regarding the source code as a raw +input stream. In the literature, before entering source code into the neural network, sev- +eral learning-based APR techniques use SentencePiece to divide words into a sequence of +subtokens, such as SelfAPR[175], RewardRepair [176] and CIRCLE [159]. +4.4 +Patch Generation +In the learning-based APR context, to apply NMT repair models to high-level programming lan- +guages, the code snippets need to be converted to embedding vectors. Then an NMT repair model +is built on top of the encoder-decoder architecture [150] to learn the repair patterns automatically. +Finally, the mapping from buggy code to fixed code is optimized by updating the parameters of the +designed model. Thus, it is crucial to determine (1) how to represent the source code (with which +format) as input for word embedding, referred to as code representation; and (2) how to design the +specific architecture (with which neural network) as encoder-decoder for repair transformation +learning, referred to as model architecture. +In the literature, various strategies have been proposed to represent the source code as the input +for NMT repair models, which can be categorized into three classes: sequence-based, tree-based and +graph-based representation. +4.4.1 +Sequence-based Generation. These techniques divide the textual source code as a sequence of +tokens and treat APR as a token-to-token translation task based on a sequence-to-sequence model. +Code Representation. Considering the buggy lines and the context, there generally exist four +different ways to sequence the textual code tokens. +(1) Raw representation. +Similar to NMT, which translates a sentence from one source language (e.g., English) to +another target language(e.g., Chinese), most sequence-based techniques directly feed the +model with the buggy code snippet [147]. For example, Tufano et al. [147] extract the buggy +method and train an NMT model for method-to-method translation. The size of this code +snippet depends on the choice of the buggy code and code context. However, the raw +representation is unaware of the difference between the buggy code and the code context, as +these two parts are sent into the encoder together. As a result, the transformation rules may +be applied in some correct lines, limiting the repair performance. +(2) Context representation. +The context representation splits the buggy code and the code context, then feeds them +into two encoders separately. Under this circumstance, the model is aware of the difference +between buggy code and the corresponding context. For example, Lutellier et al. [57, 96] +attempt to encode these two parts separately and then merge the encoding vectors. However, +it is challenging to merge the two separated encoding vectors and eliminate the semantic +gaps between the two encoders. +(3) Prompt representation. +The prompt representation refers to a text-in-text-out input format and can effectively +concatenates different input components with some prefixed prompt [127]. The prefixed +prompt is a piece of tokens inserted in the input, so that the original task can be formulated +as a language modeling task. For example, Yuan et al. [159] employs manually designed +prompt template to convert buggy code and corresponding context into a unified fill-in-the- +blank format. In particular, they employ “Buggy line:” and “Context:” to denote the buggy +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:15 +code and code context, and then employ “The fixed code is:” to guide the NMT model to +generate candidate patches according to the previous input. This mechanism has been proven +effective in bridging the gap between pre-trained tasks and the downstream task, facilitating +fine-tuning pre-trained models for APR. +(4) Mask representation. +The mask representation replaces the buggy code with mask tokens and queries NMT models +to fill the masks with the correct code lines. This mechanism views the APR problem as a +cloze task and usually adopts the pre-trained model as the query model in the learning-based +APR. For example, Xia et al. [165] transform the original buggy code into a comment and +generate multiple mask lines with templates. The input is represented by comment buggy +code, context before buggy code, mask lines and context after buggy code. In particular, the +buggy code is masked randomly from one token to the whole line, and researchers expect to +generate every possible patch for different situations within a limited candidate patch size. +Compared with the above three representation strategies, the mask representation can adopt +pre-trained models to predict randomly masked tokens to perform cloze-style APR without +any additional training on the bug-fixing dataset. +Model Architecture. Sequenced-based techniques usually treat the source code as a sequence of +tokens and adopt existing sequence-to-sequence architectures in the NLP field instead of designing +new network architectures. For example, CoCoNut [96] adopts two fully convolutional (FConv) +encoders to represent the buggy lines and the context separately. One common encoder architecture +is long short-term memory (LSTM), and it resolves the long-term dependency problem of the +RNN module by introducing the gate mechanism and ensures that short-term memory is not +neglected. For example, SequenceR [24] is based on an LSTM encoder-decoder architecture with +copy mechanism. As a powerful kind of DL architecture, transformer can model global dependencies +between input and output effectively thanks to the attention mechanism and has been adopted in +existing APR studies, such as Bug-Transformer [171], SeqTrans [26] and VRepair [25]. +Recently, the usage of pre-trained models has gradually attracted the attention of researchers in +the learning-based APR community. Such models are first pre-trained by self-supervised training +on a large-scale unlabeled corpus (e.g., CodeSearchNet), and then transferred to benefit multiple +downstream tasks by fine-tuning on a limited labeled corpus. For example, Mashhadi et al. [103] +employ CodeBERT, a bimodal pre-trained language model for both natural and programming +languages, to fix Java single-line bugs by fine-tuning on the ManySStuBs4J small and large datasets. +CURE [57] applies a pre-trained GPT module to further revise an NMT-based APR architecture (i.e., +CoCoNut). CIRCLE [159] proposes a T5-based program repair framework equipped with continual +learning ability across multiple languages. We will discuss the application of pre-trained models in +Section 6.3. +State-of-the-arts. In the following, we discuss these individual sequence-based patch generation +techniques in more detail. +Tufano et al .[146] design an NMT model to generate the same patches applied by developers +under a narrow context. They reduce the vocabulary size by mapping the method of the code to a +specific ID and feed the model with pairs of methods before and after the patch. The model can +replicate up to 36% of the buggy code. Moreover, it can be applied to refactoring and other code +relating activities. +Current works aim at exploring fixes in a limited search space, which may not contain the correct +patches. Hata et al .[51] follow the recently NMT-based approach and use an encoder-decoder +model Ratchet with multi-layer attention to fix bugs. They perform an empirical study with five +large software projects. Moreover, they collect a fine-grained dataset from these projects and try +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:16 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +to ignore noisy data. They train and evaluate Ratchet on this dataset. Results show that Ratchet +performs at least as well as pattern-based APR tools. Besides, Ratchet’s output was considered +helpful in fixing the bugs on many occasions, even if the fix was not 100% correct. +Lutellier et al. [96] propose CoCoNut, a novel generate&validate technique with a new context- +aware NMT architecture that separately inputs the buggy line and method context. They further +combine CNN (FConv architecture) with the NMT model to improve the accuracy of generated +patches. After collecting a large dataset from four programming languages and training the model +on it, CoCoNut is then evaluated on six benchmarks(also from four programming languages). It +turns out that CoCoNut outperforms previous APR tools and is capable of fixing 300 more bugs +other APR tools fail to. Moreover, CoCoNut proves that FConv architecture can outperform LSTM. +Further, Jiang et al. [57] propose CURE, a novel NMT-based program repair technique to fix Java +bugs. They pre-train a programming language model on a large corpus and combine it with NMT +architecture to learn code syntax and fix patterns. They also apply a code-aware search strategy +and a new subword tokenization technique to improve the accuracy of generated patches. This +model outperforms SequenceR and CoCoNut APR tools on Defects4J and QuixBugs benchmarks +under different beam search sizes. +Lutellier et al. [95] propose ENCORE, a new end-to-end APR technique that leverages the NMT +model to generate bug fixes for Java programs. Evaluating ENCORE on two Java benchmarks +proves that it can fix diverse bugs, and further experiments on Python, C++, and JS benchmarks +proves that it can handle bugs in different programming languages. They also present attention +maps to explain why certain fixes are generated or not by ENCORE. +Chen et al. [24] propose SequenceR, a novel end-to-end approach based on sequence-to-sequence +learning. They combine LSTM encoder-decoder architecture with copy mechanism to address +the problem of a large vocabulary. First, they apply state-of-the-art fault localization techniques +to identify the buggy method and the suspicious buggy lines. Then, they perform a novel buggy +context abstraction process that intelligently organizes the fault localization data into a suitable +representation for the deep learning model. Finally, SequenceR generates multiple patches for the +buggy code. Although their approach can only be applied to single-line buggy code, this model +outperforms the APR tool of Tufano et al. on Defects4J benchmarks. Moreover, they prove that +copy mechanism can improve the accuracy of generated patches. +Ye et al. [176] introduce RewardRepair as a neural program repair approach for fixing bugs in +Java code based on transformer. They apply a novel training strategy and feed the model with +compiling and testing execution information to improve the quality of generated patches. This +model is then evaluated on four benchmarks, Defects4J v1 and v2, Bugs.jar, and QuixBugs. Results +show that this model has lower cross-entropy than previous APR tools. Besides, RewardRepair +outperforms SequenceR, CoCoNut and CURE in terms of top-k accuracy on Defects4J benchmarks. +Previous neural program repair approaches focus on supervised training and lack project-specific +knowledge. Ye et al. [175] propose, SelfAPR, a self-supervised training approach with test execution +diagnostics based on a transformer neural network. SelfAPR consists of two components, training +sample generator and neural network optimization. The first part generates perturbed programs +with a perturbing model and tests it to capture compile errors and execute failures information. +The second part is fed with the previous information and outputs n best patches with beam search. +SelfAPR is capable of repairing ten bugs that are never repaired before by the supervised neural +repair models. Moreover, evaluation results prove the effectiveness of self-supervised training and +its components. +Yao et al. [171] propose a new transformer-based APR technique, Bug-Transformer, to fix buggy +code snippets. It applies a novel token pair encoding (TPE) approach to reduce vocabulary size by +compressing code structure while preserving semantic information. Besides, they apply a novel +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:17 +rename mechanism to preserve semantic features for code abstraction. Bug-Transformer leverages +the transformer architecture and is fine-tuned for learning tasks. It is then evaluated on Java +benchmarks and outperforms other baseline models such as Bug2Fix. +Rahman et al. [128] present a bidirectional LSTM model for code evaluation and repair. They first +train the model as a Seq2Seq model with abundant source code. Then they fine-tune it for error +detection and provide suggestions for code repair. The model is evaluated on Aizu Online Judge +(AOJ) system, and the result shows that this model outperforms previous RNN and LSTM models. +It also proves to be useful for novice programmers and accelerates the code evaluation process. +Huang et al. [55] propose an enhanced transformer-based APR technique by introducing a +general pyramid encoder, which is added in between layers of regular multi-layer encoders. For +the purpose of testing the generality of the pyramid encoder, they combined this encoder with +different attention mechanisms. They conduct experiments on Juliet Test Suite for C/C++ and Java +to evaluate seq2seq models. Results show that seq2seq models can be well applied in providing +suggestions to potential errors and have a decent correct rate in code auto-correction. Besides, their +results on transfer learning point out a way of processing this small dataset using the pre-trained +model as an encoder, which boosts the performance by a large amount. +4.4.2 +Tree-based Generation. Sequence-based techniques usually adopt sequence-to-sequence +models for patch generation. However, these techniques ignore code structure information because +they are designed for NLP, which is significantly different from programming language with strict +syntactic and grammatical rules. The generated patches of these techniques may suffer from syntax +errors that cause compilers to fail. As a result, researchers recently propose various tree-based +generation techniques by considering the syntactic structure of source code. These techniques treat +the APR problem as a tree transformation learning task. +Code Representation. A common solution is to parse the source code into an AST and adopt a +tree-aware model to perform patch generation, i.e., structure-aware representation. For example, +given a bug-fixing method pair 𝑀𝑏 and 𝑀𝑓 representing the buggy and fixed method, DLFix [79] +first extracts a buggy AST for 𝑀𝑏 (i.e.,𝑇𝑏), a fixed AST for 𝑀𝑓 (i.e.,𝑇𝑓 ), a buggy sub-AST (i.e,𝑇 𝑠 +𝑏 ) and +a fixed sub-AST (i.e., 𝑇 𝑠 +𝑓 ) between 𝑇𝑏 and 𝑇𝑓 . DLFix then adopts an existing summarization model +to encoder 𝑇 𝑠 +𝑏 as a single node 𝑆𝑠 +𝑏. Finally, the buggy method 𝑀𝑏 can be represented as a context +tree by replacing 𝑇 𝑠 +𝑏 in 𝑇𝑏 with 𝑆𝑠 +𝑏 and a sub-changed tree 𝑇 𝑠 +𝑏 . The fixed method 𝑀𝑓 is represented +in a similar way. +As tree-based representation contains the structure information, which cannot be directly de- +ployed to sequenced-based neural models. Thus, an additional code representation strategy is +utilized to parse the tree representation as a sequential traverse sequence, i.e., sequential-traverse +representation. For example, Tang et al. [140] parse the source code into AST representation, which +is further translated into a sequence of rules. The sequence of rules can be processed by the vanilla +transformer [150] while capturing the grammar and syntax information. CODIT [20] first identifies +the edited AST nodes (i.e., the inserting, deleting, and updating) and selects the minimal subtree of +each AST. CODIT then collects the edit context by including the nodes that connect the root of the +method to the root of the changed tree. CODIT expands the considered context until the context +exceeds a maximum tree size. Given each bug-fixing method pair, CODIT extracts a buggy AST +and fixed AST, and then converts the ASTs to their tree representation. +Model Architecture. Most NMT-based APR models treat patch generation as a machine translation +from buggy code to a fixed one. However, such models could not capture the information of code +structures and suffer from handling the context of the code. Tree-based encoders consider the +structure features of source code, such as AST. For example, DLFix [79] parses the source code +to AST and adopts a tree-based LSTM to represent the changed and context sub-trees. Besides, +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:18 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +Devlin et al. [30] encode the AST with a sequential bidirectional LSTM by enumerating a depth-first +traversal of the nodes. +State-of-the-arts. In the following, we discuss these individual tree-based patch generation tech- +niques in more detail. +One variant of LSTM is tree-LSTM architecture which leverages tree structure AST or graph +to parse the syntax information of buggy code. For example, Yi et al .[79] propose a tree-based +model with 2 layers. The first layer is used to learn context information, which is implemented by +a tree-based LSTM. Another layer is used to catch code transformation between changes. After +training, filtering and re-ranking (with the help of a CNN layer), the model generates a bunch of +patches. DLFix is designed for single-statement bug fixing, and it shows the potential of a tree-based +model in bug fixing. +Li et al[80]. propose DEAR, a learning-based approach for multi-hunk multi-statement fixes. +They design a fault localization technique based on traditional SBFL and data flow analysis. This +technique can acquire multi-hunks that need to be fixed together. They further design a two-tier +tree-based LSTM with an attention layer for fixing multiple statements in the suitable fixing context. +Moreover, they apply cycle training to learn code transformation and fix need-to-be-fixed-together +bugs detected by the fault localization technique mentioned above. This approach outperforms +many DL APR techniques such as DLFix and CoCoNut. Besides, it can fix multi-statement bugs +which other APR tools fail to fix. +Chakraborty et al .[20] propose a tree-based APR technique CODIT to learn code changes from +the wild and generate patches for software bugs. CODIT transforms the correct (or buggy) code +snippet into the parse tree and generates the deleted (or added) subtree. CODIT then predicts the +structural changes using a tree-based translation model among the subtrees and employs token +names to concrete the structure using a token generation model. The former tree-based model takes +the previous code tree structure and generates a new tree with the maximum likelihood, while the +latter token generation model takes tokens and types of tokens in the code and generates new tokens +with the help of LSTM. The authors conduct a real-world bug-fixing dataset Code-Change-Data +from 48 open-source projects and employ Pull-Request-Data [24] and Defect4J [60]. The results +on these three datasets illustrate CODIT outperforms existing seq2seq models, highlighting the +potential of the tree-based models in APR. +Devlin et al. [30] present a novel model SSC (Share, Specialize, and Compete) to repair semantic +bugs, which means fixing non-syntactic bugs in source code. The input code snippet is encoded +with a neural network on the AST level, and each repair type is associated with its own specialized +neural module, which emits a score for every repair candidate of that type. The authors conduct a +large-scale corpus by mining code snippets from real-world Python projects on GitHub. Results +indicate that it outperforms existing sequence-to-sequence models with an attention mechanism. +Yu et al. [177] present a novel approach to predict code transformation at AST level based on +structural information for Java programs. For structured prediction of source code transforms, +they establish a conditional random field (CRF) for the transform prediction, then define the +feature functions used in CRF, and finally train the CRF model for prediction. They use the learned +model to predict transforms for the new, unseen buggy code snippets. They conduct a large-scale +experimental evaluation on a large dataset of 4,590,679 bug-fixing commits. The results show the +great performance of the proposed technique to generate patches by predicting code structure +transforms. +4.4.3 +Graph-based Generation. These techniques transform source code into graph representations +with contextual information and frame the APR problem in terms of learning a sequence of graph +transformations. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:19 +Code Representation. To capture the neighbor relations between AST nodes, Recoder [186] treats +AST as a directional graph where the nodes denote AST nodes and the edges denote the relationship +between each node and its children and left sibling. Besides, Xu et al. [166] consider the context +structure by data and control dependencies captured by a data dependence graph (i.e., DDG) and a +control dependence graph (i.e., CDG). +Model Architecture. Existing graph-based APR techniques usually design graph neural networks +and their variants to capture graph representation and perform patch generation. For example, +Hoppity [31] adopts a gated graph neural network (GGNN) to treat the AST as a graph, where a +candidate patch is generated by a sequence of predictions, including the position of graph nodes +and corresponding graph edits. Besides, Xu et al. [166] design a graph neural network (GNN) for +obtaining a graph representation by first converting DDG and CDG into two graph representations +and then fusing them. +State-of-the-arts. In the following, we discuss these individual graph-based patch generation +techniques in more detail. +Zhu et al. [186] propose Recoder, a syntax-guided edit decoder that uses a novel provider/decider +architecture. Recoder takes a buggy statement and its context as input and generates edits as output +by (1) embedding the buggy statement and its context by a code reader; (2) embedding the partial +AST of the edits by a AST reader; (3) embedding a path from the root node to a non-terminal node +by a tree path reader; and (4) producing a probability of each choice for expanding the non-terminal +node based on previous embeddings. The authors evaluate Recoder on four widely-adopted Java +benchmarks: Defects4J v1.2 with 395 bugs, Defects4J v2.0 with 420 bugs, QuixBugs with 40 bugs +and IntroClassJava with 297 bugs. The results indicate that Recoder is the first learning-based APR +technique that outperforms existing traditional techniques on these four Java benchmarks. +Xu et al. [166] introduce M3V, a new multi-modal multi-view context embedding approach to +predict repair operators for buggy Java code. They apply a GNN with multi-view graph-based +context structure embedding to capture data and control dependencies. They also present a tree- +LSTM with tree-based context signature embedding for capturing high-level semantics. After M3V +is evaluated on repairing two common types of bugs: null pointer exceptions and index out of +bounds, results show that M3V is effective in predicting repair operators. +Dinella et al. [31] introduce HOPPITY, an end-to-end learning-based tool for detecting and fixing +bugs in JS programs. They apply one step graph edit which is called graph transformation for the +model and feed the model with the graph structure of buggy code. This model is then trained to +detect and fix more complex and diverse bugs which require adding or deleting code. It outperforms +other tools on the same baseline with or without the perfect bug locations. +Nguyen et al. [115] propose GRAPHIX, a medium-scale graph edit model which is pre-trained +with deleted sub-tree reconstruction. This model is trained with both abstract and concrete code +to learn both structural and semantic code patterns, and it suggests that abstraction may be +unnecessary. It is then evaluated on the Java benchmark from Tufano et al. [147] and it turns +out that this model is as competitive as large-scale transformer models and outperforms other +state-of-art APR tools. +Tang et al. [139] propose a novel end-to-end approach Grasp for repairing buggy Java programs. +They represent the buggy method as a graph to retain structural information and apply the Graph- +to-Sequence model to capture information from the graph, overcoming the problem of information +missing. Grasp is then evaluated on the Defects4J benchmark as well as real-world bugs from +open-source projects and it achieves good results. +Yasunaga et al. [172] propose DrRepair to repair C/C++ bugs. They parse the buggy source code +into a joint graph representation with diagnostic feedback that captures the semantic structure. +The graph representation takes all identifiers in the source code and any symbols in the diagnostic +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:20 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +feedback as nodes, and connects the same symbols as edges. They then design a GNN model for +learning the graph representation. Besides, they apply a self-supervised learning paradigm that can +generate extra patches by corrupting unlabeled programs. They also discover that pre-training on +unlabeled programs improves accuracy. The model is evaluated on DeepFix and SPoC datasets and +it outperforms existing state-of-art APR tools. +4.5 +Patch Ranking +The patch generation is a search process for the maximum in the combinatorial space. Given the +max output length l and the size of vocabulary V, the total number of candidate patches that the +decoder can generate reaches 𝑉 𝑙, all of which it is impossible to validate in practice. Developers +may spend a considerable amount of effort to assess the correctness of the generated candidate +patches manually. In such a scenario, only inspecting fewer repair candidates (e.g., Top-1 and Top-5) +that have a high probability of being correct is more practical and reduces the valuable manual +effort. As a result, a patch ranking strategy is crucial to ensure the inference efficiency of the model +and relieve the burden of patch validation. +Beam search is an effective heuristic search algorithm to rank the outputs in previous NMT +applications [157] and is the most common patch ranking strategy in learning-based APR studies, +such as CIRCLE [159], SelfAPR [175], RewardRepair [176] and Recoder [186]. In particular, for every +iteration, the beam search algorithm selects the 𝑘 most probable tokens for the patch (corresponding +to beam size 𝑘) and ranks them according to the likelihood estimation score of the next 𝑑 prediction +steps (corresponding to search depth 𝑑). At last, the top 𝑘 most likely patches are maintained for +validation in the next procedure. Beam search provides a great trade-off between repair accuracy +versus inference cost via its flexible choice of beam size. +However, the vanilla beam search considers only the log probability to generate the next token +while ignoring the code-related information, such as variables. Thus, it may generate high-score +patches with unknown variables, leading to uncompilable candidate patches. In addition to directly +applying the existing beam search strategy, researchers design some novel strategies to filter out +low-probability patches. For example, CURE [57] designs a code-aware beam search strategy to +generate more compilable and correct patches based on valid-identifier check and length control +components. The code-aware strategy first performs static analysis to identify all valid tokens +used for sequence generation and then prompts beam search to generate sequences of a similar +length to the buggy line. DLFix [79] first derives the possible candidate patches by program analysis +filtering and ranks the list of possible patches by a CNN-based binary classification model. The +classifier adopts a Word2Vec model as the encoder stack at the char-level, followed by a CNN stack +as the learning stack (containing a Convolutional layer, pooling, and fully connected layers), and a +softmax function as the classification stack. Then DLFix ranks the given list of patches based on +their possibilities of being a correct patch. Further, DEAR [80] applies a set of filters to verify the +program semantics and ranks the candidate patches in the same manner as DLFix does. +Besides, AlphaRepair [165] designs a patch ranking strategy based on a masked language model. +In particular, given a candidate patch, AlphaRepair calculates its priority score by (1) extracting all +generated tokens; (2) masking out only one of the tokens; (3) querying CodeBERT to obtain the +conditional probability of that token; (4) repeating the same process for all other previous mask +tokens; and (5) computing the joint score which is an average of the individual token probabilities. +4.6 +Patch Validation +Patch validation takes a ranked list of candidate patches generated by NMT models as the input and +returns the plausible patches for deployment, which is a crucial phase in the learning-based APR +pipeline. However, developers may spend a considerable amount of effort to inspect the candidate +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:21 +patches manually. Thus, researchers usually recompile the buggy program with the generated +patch to check if it can pass the available test suite. In such a scenario, hundreds or even thousands +of candidate patches can be filtered automatically (e.g., 1000 candidate patches per bug in CIRCLE +[159]), which may benefit its adoption in practice. +Similar to traditional APR techniques, most learning-based techniques adopt a test-based vali- +dation strategy (i.e., executing available test suites against each candidate patch) to assess patch +correctness [57, 79, 80, 96, 159, 186]. For example, CIRCLE [159] filters out the candidate patches +that do not compile or do not pass available test suites. There generally exist two criteria for the +validation process: (1) the passing test suites that make the buggy program pass should still pass +on the patched program; and (2) the fault-triggering test suites that fail on the buggy program +should pass on the patched program. All candidate patches are checked until a plausible patch (i.e., +a patch passing all test suites) is found. Finally, CIRCLE stops the validation process and reports +the plausible patch for manual investigation. +However, it can be extremely time-consuming to compile a large number of candidate patches +and repeat all test executions to identify plausible patches. For example. CURE [57] generates +10,000 candidate patches per bug and validates the top 5,000 ones considering the overhead time. +Similarly, AlphaRepair [165] returns at most 5,000 candidate patches for each bug. To reduce the +validation cost, some learning-based APR techniques return an acceptable amount of candidate +patches. For example, RewardRepair configures the beam size as 200 and outputs the 200 best +patches per bug. Similarly, SelfAPR adopts a beam search size of 50 and Recoder generates 100 +valid candidate patches for validation. Besides, similar to traditional APR techniques [56, 87], there +exist several learning-based ones limiting maximum time for validation. For example, DEAR [80] +and DLFix [79] set a 5-hour running-time limit for patch generation and validation. +In addition to the above strategies in patch validation, the learning-based APR community +benefits from some optimizations to speed up the dynamic execution. For example, AlphaRepair +[165] adopts the UniAPR [22] tool to validate the candidate patches on-the-fly. +Inspired by the PraPR work, Chen et al. [22] present UniAPR as the first unified on-the-fly patch +validation framework to speed up APR techniques for JVM-based languages at both the source +and byte-code levels. They leverage the JVM HotSwap mechanism and Java Agent technology to +implement this framework。Besides, they apply the JVM resetting technique based on the ASM +byte-code manipulation framework. Since previous work shows that on-the-fly patch validation +can be imprecise, they reset the JVM state right after each patch execution to address such an issue. +The evaluation shows that this work can speed up state-of-the-art representative APR tools. +Bento et al. [12] introduce SeAPR, the first self-boosted patch validation tool. Based on the idea +that patches similar to earlier high-quality/low-quality patches should be promoted/degraded, they +leverage the patch-execution information on its similarity with the executed patches to update each +patch’s priority score. The evaluation shows that SeAPR can substantially speed up the studied APR +techniques and its performance is stable under different formulae for computing patch priority. +Since previous APR techniques often neglect the impact of test selection for each patch, Lou et +al. [92] conduct an extensive study to investigate the impact of Regression Test Selection (RTS) +on APR. They explore three representative RTS techniques for 12 state-of-the-art APR systems +at different levels (i.e., class/method/statement levels) with over 2M patches. Results show that +all studied RTS techniques can substantially improve APR efficiency and should be considered in +future APR work. Besides, method- and statement-level RTS substantially outperform class-level +RTS, and are more recommended for APR. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:22 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +4.7 +Patch Correctness +Patch correctness is an additional phase for developers to further filter out overfitting patches after +patch validation, so as to improve the quality of returned patches. As discussed in Section 4.6, a +majority of existing learning-based APR techniques usually leverage the developer-written test +suites as the program specification to assess the correctness of the generated patches. However, +the test suite is an incomplete specification as it only describes a part of the program’s behavioral +space. As a result, it is fundamentally difficult to achieve high precision for returned patches due +to the incomplete program specification [70]. The plausible patch passing the available test suites +may not generalize to other potential test suites, leading to a long-standing challenge of APR +(i.e., the overfitting issue) [70]. Previous studies [90, 91] have demonstrated that a majority of the +overfitting patches are equivalent to a single modification that deletes the buggy functionality +and does not actually fix the detected bugs. Under the circumstances, it takes enormous time +and effort to manually filter out the overfitting patches, even resulting in a negative debugging +performance [141, 184]. Different from some traditional APR techniques that guide the repair +process to generate patches with a high probability of being correct, DL techniques lead to an +end-to-end repair mechanism and the patches are generated in a black-box manner. The overfitting +issue in learning-based APR is more significant and severe. +In the literature, researchers have proposed a mass of automated patch correctness assessment +(APCA) techniques to identify whether a plausible patch is indeed correct or overfitting [143]. +There are usually two types of traditional APCA techniques based on the employed patch features: +static and dynamic [156]. The former focuses on the transformation patterns or the static syntactic +similarity, while the latter relies on the dynamic execution outcomes by additional test suites +from automated test generation tools (e.g., Evosuite [39]). Recently, inspired by large-scale patch +benchmarks being released, some learning-based APCA techniques have been proposed to predict +patch correctness with the assistance of DL models [142, 143, 174]. In general, such learning-based +APCA techniques extract the code features by code embedding and build a classifier model to +directly perform patch prediction. We view patch correctness as an essential component of the +learning-based APR pipeline and focus on such APCA techniques that employ DL models. +Now, we list the existing learning-based techniques to predict patch correctness automated as +follows. +Lin et al. [82] propose Cache, a novel context-aware code change embedding technique for the +patch correctness task. They leverage context information of unchanged code and parse the AST +nodes to capture the code structure information. They conduct various experiments to evaluate +Cache on diverse patch benchmarks. The results show that Cache achieves significantly better +performance than both previous representation learning techniques and existing APCA techniques. +Ye et al. [174] propose ODS, a learning-based approach to identify overfitting patches based on +static code features and supervised learning. ODS first defines and extracts a set of 202 static code +features from the AST to represent a candidate patch. ODS then adopts the gradient boosting with +the captured code features and patch correctness labels to train a classifier for patch correctness +classification. They conduct on three benchmarks (i.e., Defects4J, Bugs.jar and Bears) and the results +show that ODS achieves an accuracy of 71.9% in detecting overfitting patches from 26 projects, and +outperforms other state-of-the-art techniques. +Considering most existing APCA techniques evaluated on limited datasets, Wang et al. [158] +conduct an extensive empirical study of patch correctness on Java programs. First, they collect +a large-scale real-world dataset for patch correctness, containing 1,988 patches generated by +the recent PraPR [43] APR tool. Then they revisit state-of-the-art APCA techniques on the new +dataset, including static, dynamic, and learning-based ones. Results show that learning-based +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:23 +techniques tend to suffer from the overfitting issue. Besides, the performance of dynamic techniques +significantly drops when encountering patches with more complicated changes. +Tian et al. [145] attempt to formulate the patch correctness assessment problem as a question +answering problem, which can assess the semantic correlation between a bug report (question) +and a patch description (answer). They introduce QUATRAIN, a supervised learning approach that +exploits a deep NLP model to predict patch correctness based on the relatedness of a bug report with +a patch description. QUATRAIN first mines bug reports for bug datasets automatically and generates +patch descriptions by existing commit message generation models. QUATRAIN then leverages an +NLP model to capture the semantic correlation between bug reports and patch descriptions. They +evaluate QUATRAIN on a large dataset of 9135 patches from three Java datasets (i.e., Defects4j, +Bugs.jar, and Bears). The results demonstrate that QUATRAIN achieves comparable or better +performance against other state-of-the-art dynamic and static techniques. Besides, QUATRAIN is +proven practical in learning the relationship between bug reports and code change descriptions for +the patch prediction task. +Different from most existing studies focusing on Java programs, Yan et al. [168] propose Crex to +predict patch correctness in C programs based on execution semantics. They first leverage transfer +learning to extract semantics from micro-traces in buggy C code on the function level. They then +perform semantic similarity computation to denote patch correctness. They evaluate Crex on a set +of 212 patches generated by the CoCoNut APR tool on CodeFlaws programs. The experimental +results indicate that Crex can achieve high precision and recall in predicting patch correctness. +Tian et al. [142] introduce BATS, an unsupervised learning-based approach to predict patch +correctness based on failing test specifications. BATS first constructs a search space of historical +patches with failing test cases. Given a plausible patch, BATS identifies similar failing test cases in +the search space. BATS then calculates the similarity of historical patches and the plausible patch +based on the failing test cases. The plausible patch is predicted as correct if the similarity score is +larger than a predefined threshold; otherwise it is predicted as incorrect. After collecting plausible +patches from 32 APR tools to construct a large dataset, they evaluate the performance of BATS on +Defects4J benchmarks with some standard classification metrics (e.g., recall). BATS outperforms +existing techniques in identifying correct patches and filtering out incorrect patches. +Csuvik et al. [28] present a Doc2Vec model to explore the nature of similarity-based approach for +patch correctness assessment. They feed the model with a token sequence under a simple rule and +the model measure the similarity between plausible patches and original programs. They find that +plain source code embeddings fail to capture nuanced code semantics, thus a more sophisticated +technique is needed to validate patches correctly. +Different from traditional APCA techniques relying on dynamic information or manually-crafted +heuristics, Tian et al. [143] investigate the feasibility of code representation learning to encode +the properties of patch correctness. They consider different representation learning techniques +(i.e., Doc2Vec, BERT, code2vec, and CC2Vec) to get embedding vectors for code changes, including +pre-trained models and the retraining of models. They also investigate the discriminative power +of learned features in a classification training pipeline (i.e., Decision tree, Logistic regression, and +Naive Bayes) for patch correctness. Based on previous work [143], Tian et al. [144] further leverage +representation learning models and supervised learning algorithms to investigate the feasibility of +statically predicting patch correctness. They implement two patch correctness predicting frame- +works, Leopard and Panther (upgraded version of Leopard), to investigate the discriminative power +of the deep learned features by training machine learning classifiers to predict correct patches. +Besides, they run exploratory experiments assessing the possibility of selecting cutoff similarity +scores between learned embeddings of buggy code and patched code snippets for heuristically +filtering out incorrect patches. After evaluating several models on the same dataset, they find that +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:24 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +the performance of these models on learned embedding features is promising when compared +against the state-of-the-art techniques which applies dynamic execution traces. +Ghanbari et al. [44] propose a novel technique Shibboleth for patch correctness assessment via +ranking and classification. It leverages the impact of the patches on both production code and test +suite coverage and relies on a simpler set of assumptions. They collect a curated and annotated data +set of generated and human-written patches, and they evaluate the model on this dataset. Results +show that Shibboleth outperforms state-of-the-art patch classification techniques. +Phung et al. [125] present MIPI, a novel approach for patch correctness assessment. Based on +a discovery that the distance between the method name and correct patches is smaller than that +between the method name and incorrect patches, they decide to extract the intention of developers +by analyzing method name to help distinguish the incorrect patches. Thus, their method does +not require any test cases or noisy source code that are not clearly determined whether they are +faulty or clean. The evaluation shows that MIPI is more precise and less destructive than existing +heuristic-based patch assessment techniques. +5 +EMPIRICAL EVALUATION +In this section, we introduce existing widely-adopted datasets in the learning-based APR field and +discuss common evaluation metrics for evaluating repair performance. +5.1 +Dataset +Different from previous APR techniques conducted in a traditional pipeline (e.g., generating patches +by heuristic strategies), the process of learning-based APR techniques is two-fold (1) a training +process with supervised learning on large labeled datasets (e.g., CoCoNut [96]); and (2) an evaluation +process on limited labeled datasets (e.g., Defects4J [60]). Benefiting from a large amount of research +effort in the learning-based APR community, there are several existing benchmarks to evaluate +NMT techniques for automatically repairing bugs. Now we discuss the widely adopted datasets in +the literature. +Defects4J [60] is the most widely-adopted benchmark in learning-based APR studies, which +contains 395 known and reproducible real-world bugs from six open-source Java projects. To +facilitate reproducible studies, each bug contains a buggy version and a fixed version, as well as +a corresponding test suite that triggers that bug. Defects v2.0 provides 420 additional real-world +bugs from 17 Java projects, which is adopted by some recent studies [165, 186]. QuixBugs [83] is a +multi-lingual parallel bug-fixing dataset in Python and Java used in [159, 165]. QuixBugs contains +40 small classic algorithms with a bug on a single line, along with the test suite. Bugs.jar [132] +contains 1,158 real bugs from 8 large open-source Java projects, each of which has a fault-revealing +test suite. ManyBugs [72] contains 185 real-world bugs from 9 open-source C projects and each +bug has a corresponding developer patch and test suite. IntroClass [72] consists of 998 bugs in six +small student-written programming assignments for C language. Due to a well-defined test suite, +These dataset is effective in evaluating the correctness of generated patches by dynamic program +behavior. +However, NMT-based APR techniques employ neural network techniques to learn the bug-fixing +patterns from the training dataset. The high-quality test suite requires massive manual effort, so +those datasets are usually scarce to train a reliable NMT repair model. To make experiment results +more persuasive, lots of large-scale datasets have been conducted recently. Such datasets contain +bug-fixing code pairs for the model to learn how to transform a buggy code into the expected fixed +code. In particular, researchers usually mine open-source projects from code platforms (e.g., GitHub) +and extract the commits by fixing-related keywords. Then the unqualified commits are filtered out +by pre-defined rules (e.g., non-code changes). For example, Tufano et al. [147] extract the bug-fixing +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:25 +Table 1. Detailed information on collected datasets +ID +Name +Language +#Bugs +Test Suite +Training +Testing +Techniques +1 +Bears +Java +251 +yes +yes +yes +[74][176] +2 +BFP medium +Java +65454 +no +yes +yes +[21] [26] [34] [115] [53] +[139] [140] [147] [171] +3 +BFP small +Java +58350 +no +yes +yes +[21] [26] [34] [115] [53] +[139] [140] [147] [171] +4 +BigFix +Java +1.824 M +no +yes +yes +[79] [80] +5 +Bugs2Fix +Java +92849 +no +yes +yes +[24] [27] +6 +Bugs.jar +Java +1158 +yes +yes +yes +[79] [176] [143] +7 +Code-Change-Data +Java +44372 +no +yes +yes +[20] +8 +CodeXGlue +Java +122 K +no +no +yes +[27] +9 +CodRep +Java +58069 +no +yes +yes +[24] [176] +10 +CPatMiner +Java +44 K +no +yes +yes +[80] +11 +DeepRepair +Java +374 +no +yes +no +[161] +12 +Defects4J +Java +835 +yes +yes +yes +[20] [23] [74] [80] [95] +[175] [98] [139] [142] +[143] [151] +13 +Function-SStuBs4J +Java +21047 +no +yes +yes +[116] +14 +IntroClassJava +Java +998 +yes +yes +yes +[23] [186] +15 +Java-med +Java +7454 +no +yes +no +[62] +16 +ManySStuBs4J large +Java +63923 +no +yes +yes +[103] +17 +ManySStuBs4J small +Java +10231 +no +yes +yes +[103] [143] +18 +MegaDiff +Java +663029 +no +yes +no +[25] +19 +Ponta +Java +624 +no +yes +yes +[26] +20 +Pull-Request-Data +Java +10666 +no +yes +yes +[20] [146] +21 +Ratchet +Java +35 K +no +yes +yes +[51] +22 +Recoder +Java +103585 +no +yes +no +[186] +23 +TRANSFER +Java +408091 +no +yes +no +[107] +24 +Mesbah +Java +4.8 M +no +yes +yes +[108] +25 +AOJ +C +2482 +no +yes +yes +[128] +26 +Big-Vul +C +3745 +no +yes +yes +[25] +27 +Code4Bench +C +25 K +yes +yes +yes +[148] +28 +CodeHunt +C +195 K +yes +yes +yes +[152] +29 +CVEFixes +C +8482 +no +yes +yes +[40][153] +30 +DeepFix +C +6971 +yes +yes +yes +[48] [47] [172] [173] [49] +[59] +31 +ManyBugs +C +185 +yes +yes +yes +[96] [159] [151] +32 +Prophet +C +69 +yes +yes +yes +[91] [95] +33 +Prutor +C +6971 +yes +yes +yes +[110] [169] +34 +BugAID +JS +105133 +no +yes +yes +[95] [96] [151] [159] +35 +BugsJS +JS +453 +yes +yes +yes +[69] +36 +HOPPITY +JS +363 K +no +yes +yes +[31] +37 +KATANA +JS +114 K +no +yes +yes +[136] +38 +REPTORY +JS +407 K +no +yes +yes +[114] +39 +TFix +JS +100 K +no +yes +yes +[13] +40 +ETH Py150 +Python +150 K +no +yes +yes +[52] [131] [149] +41 +GitHub-Python +Python +3 M +no +yes +yes +[173] +42 +Mester +Python +13 K +no +yes +yes +[138] +43 +PyPIBug +Python +2374 +no +yes +yes +[6] [131] +44 +SSB-9M +Python +9 M +no +yes +no +[131] +45 +VUDENC +Python +10 K +no +yes +yes +[185] +46 +Chhatbar +Python +286 +yes +no +yes +[180] +47 +SPoC +C++ +18356 +yes +yes +yes +[172] +48 +QuixBugs +Java Python +40 +yes +yes +yes +[23] [25] [33] [151] [57] +[95] [96] [143] [186] +[159] [165] +49 +DeepDebug +Java Python +523 +no +yes +yes +[33] [34] +50 +MSR20 +C,C++ +188K +no +yes +yes +[185] +51 +CoCoNut +C,Java JS Python +24 M +yes +yes +no +[57] [96] [159] [176] +52 +CodeFlaw +C Python +3902 +yes +yes +yes +[16] [96] [168] +53 +ENCORE +Java Python,JS,C++ +9.2 M +no +yes +no +[95] +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:26 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +commits between March 2011 and October 2017 on GitHub and release two BFP datasets for small +(i.e., 0∼50 tokens) and medium (i.e., 50∼100 tokens) methods, consisting of 58k (58,350) and 65k +(65,455) bug-fixing samples, respectively. Recoder [186] releases a dataset of 103,585 bug-fixing +pairs by crawling Java projects on GitHub between March 2011 and March 2018. Further, CoCoNut +[96] provides five datasets across four languages (i.e., Java, Python, C and JavaScript) by extracting +commits from GitHub projects, resulting in more than twenty million bug-fixing pairs. +Table 1 presents the description of all involved datasets in our survey. The first two columns list +the dataset name and the third column lists the programming languages the dataset covers. The +fourth column lists the number of bugs the dataset contains. The fifth column indicates whether +the dataset has corresponding test suites. The sixth and seventh columns indicate whether the +dataset is used in the training and evaluation process. The last column lists some learning-based +studies employing the dataset. +Among the collected datasets in our survey, we find that training datasets usually contain +buggy-fixing pairs while evaluation datasets may additionally contain test suites to validate the +correctness of generated patches. For example, existing studies [96, 159] generally adopt some +datasets like Defects4J as the evaluation datasets while adopting other datasets like CoCoNut as the +training datasets. Besides, we find some studies [146, 147] adopt the same dataset for training and +evaluation without executing test suites. For example, Tufano [147] split BFP dataset into training +and evaluation parts and evaluate the repair performance by match-based metric. +Table 1 also presents the programming languages of all datasets. It can be found that the collected +datasets mainly involve five languages (i.e., Java, JavaScript, Python, C and C++). Among them, +similar to traditional APR, Java is the most targeted language in the learning-based APR techniques. +Besides, researchers conduct lots of datasets in other languages (e.g., Python), indicating that +learning-based APR techniques begin to consider more languages in practice. For Java, researchers +prefer the traditionally-dominated Defects4J dataset and the recently-released BFP dataset. For +other program languages, researchers have different choices for datasets due in part to the lack of +publicly-accepted datasets. We also find that some recent datasets involve multi-languages, such +as CoCoNut [96] and QuixBugs [83], while the traditional APR techniques mainly focus on Java +language [35]. The possible reasons lie in that (1) traditional techniques are widely conducted on +the same benchmark Defects4J while some additional datasets have been released along with the +application of DL; (2) traditional techniques may rely on language-specific features to generate +patches, which is challenging to apply to other languages (e.g., PraPR adopting JVM bytecode [43]), +while learning-based techniques treat APR as an NMT task similar to NLP, which is independent of +specific programming languages. +5.2 +Metric +Evaluation metrics play a crucial role in the growth of the learning-based APR field as they serve as +the standard to quantitatively define how good an NMT repair model is. In this section, we discuss +the common evaluation metrics in learning-based APR. +5.2.1 +Execution-based Metrics. In general, learning-based APR techniques predict some candidate +patches with high probability as the outputs. The generated patches are evaluated by executing the +available test suites to determine whether to report them to the developers for deployment. We list +the standard metrics as follows. +(1) Compilable Patch. Such a candidate patch makes the patched buggy program compile +successfully. +(2) Plausible Patch. Such a compilable patch fixes the buggy functionality without harming +existing functionality (i.e., passing all available test suites). +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:27 +(3) Correct Patch. Such a plausible patch is semantically or syntactically equivalent to the +developer patch (i.e., generalizing the potential test suite). +5.2.2 +Match-based Metrics. However, it is time-consuming to evaluate generated patches on +dynamic execution for all available test suites. Besides, test suites may not always be available +in large-scale evaluation datasets. More recently, an increasing number of studies evaluate the +performance by code token matching between the generated patch and the ground truth (i.e., +developer-written patches), listed as follows. +(1) Accuracy. Accuracy measures the percentage of candidate patches in which the sequence +predicted by the model equals the ground truth. As learning-based APR techniques usu- +ally employ a beam-search strategy, the beam-search strategy reports the 𝑘 sequences (i.e., +sequence of terms representing the fixed code) with the highest probability. Researchers +consider these 𝑘 final sequences as candidate patches for a given buggy code snippet. Then +Accuracy@K value is defined as follows. +𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦@𝐾 = +�𝑛 +𝑖=1 1{𝑚𝑎𝑡𝑐ℎ(�𝑘 +𝑗=1 𝑐 𝑗 +𝑖 )} +𝑛 +(1) +where 1 denotes whether 𝐶𝑖 contains a predicted repair sequence equal to the ground truth +repair sequence. The sequence accuracy is 1 if any predicted sequence among the 𝑘 outputs +matches the ground truth sequence, and it is 0 otherwise. +(2) BLEU. BLUE (Bilingual Evaluation Understudy) [121] score measures how similar the +predicted candidate patch and the ground truth is. Given a size 𝑛, BLEU splits the candidate +patch and ground truth into n-grams and determines how many n-grams of the candidate +patch appear in the reference patch. The BLEU score ranges between 0 (the sequences are +completely different) and 1 (the sequences are identical). +Compared with execution-based metrics, accuracy and BLUE evaluate the candidate patch by +matching the tokens of the candidate patch and ground truth without dynamic execution. These +two metrics can be employed to evaluate the performance of a mass of candidate patches in a limited +time and thus have been commonly adopted in the learning-based APR community [146, 147, 159]. +However, accuracy and BLUE are initially designed in NLP tasks and may be improper to evaluate +the program repair task due to the differences between natural language and source code, For +example, accuracy refers to the perfect prediction, which ignores that different code snippets may +have the same semantic logic. Besides, BLEU is originally designed for natural language sentences +by token-level matching, neglecting important syntactic and semantic features of codes. To address +the above concerns, recently researchers adopt a variant of BLEU (i.e., CodeBLEU [129]) to evaluate +the performance of learning-based APR techniques [94]. Compared with BLEU, CodeBLEU further +considers the weighted n-gram match, the syntactic AST match, and the semantic data-flow match. +In particular, the n-gram match assigns different weights for different n-grams, the syntactic match +considers the AST information in the evaluation score by matching the sub-trees, and the semantic +match employs a data-flow structure to measure semantic similarity. +5.3 +Empirical Study +Despite an emerging research area, a variety of learning-based APR techniques have been proposed +and continuously achieved promising results in terms of the number of fixed bugs in the litera- +ture [96, 159]. In addition to developing new repair techniques that address technical challenges, +the learning-based APR research field is benefiting from several empirical studies. These empiri- +cal studies systematically explore the impact of different components (e.g., code representation), +providing insights into future learning-based APR work. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:28 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +Tufano et al. [147] conduct a systematic empirical study to investigate the capability of utilizing +NMT models to fix software bugs from open-source bug-fixing commits. They first mine the +bug-fixing commits by message patterns from projects in GitHub repositories and filter out the +low-quality commits by specific rules. They then extract correct and buggy code pairs at the +method-level by GumTree and design a code abstraction strategy to reduce vocabulary size. Finally, +they construct two datasets (i.e., small and medium BFPs) and train NMT models to translate the +buggy method into the correct method. The results demonstrate that NMT models are able to fix a +considerable number of buggy methods in the wild, proving the applicability of NMT for APR. +Ding et al. [32] empirically investigate to what extent program repair is like machine translation. +They reveal that there exist essential differences between seq2seq models and translation models in +terms of task design and architectural design. The translation model is inappropriate for program +repair due to the lack of vocabulary and immediate context. Besides, the translation model usually +keeps up most tokens from the bug code while replacing only a small number, which is not ideal for +program repair. Finally, they implement an edit-based model by adapting the seq2seq models used +for translation to generate edits rather than raw tokens, which leads to promising improvement. +Namavar et al. [114] conduct a systematic study to understand the effect of code representation +on learning-based APR performance. In particular, they implement REPTORY as a tool for controlled +experiments to assess the accuracy of different code representations (e.g., AST variants) and the +functionality of four different embeddings (e.g., GloVe). They conduct 21 experiments with different +models to evaluate their automatic patchability and perceived usefulness as well as accuracy. The +results reveal that mixed code representation with Golve embedding outperforms other settings. +Moreover, they find that bug type affects the accuracy of different code representations. +Recently, Xia et al. [164] present the first extensive evaluation on large programming language +models (PLM) for program repair. They select nine state-of-art pre-trained PLMs with different +types (i.e., infilling and generative models) and parameter sizes (i.e., ranging from 125M to 20B). +They design three different repair settings for PLMs (i.e., complete function generation, correct +code infilling, and single line generation). They then conduct experiments on 5 datasets across +3 different languages to compare different PLMs in the number of bugs fixed, generation speed +and compilation rate . They also compare the performance of PLMs against state-of-the-art APR +techniques and results demonstrate the promising future of directly adopting PLMs for APR. +6 +DISCUSSION +In this section, we will discuss several prevalent applications of learning-based repair and list some +papers for reference. +6.1 +Domain Repair +6.1.1 +Vulnerability Repair. Software vulnerability generally refers to the security flaws in the +concrete implementation of hardware, software, or protocols. Malicious attackers can exploit +unresolved security vulnerabilities to get access to the system without authorization or even +paralyze the system. Such vulnerabilities open a range of threats to cyber security, resulting in +severe economic damage and fatal consequences. For example, the Log4Shell vulnerability (CVE- +2021-44228) from Apache Log4j library3 allows attackers to run arbitrary code on any affected +system4 and is widely recognized as the most severe vulnerability in the last decade (e.g., 93% of +the cloud enterprise environment are vulnerable to Log4Shell5). Nowadays, the number of exposed +3https://logging.apache.org/log4j/2.x/ +4https://www.ftc.gov/policy/advocacy-research/tech-at-ftc/2022/01/ftc-warns-companies-remediate-log4j-security- +vulnerability +5https://www.wiz.io/blog/10-days-later-enterprises\-halfway-through-patching-log4shell +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:29 +security vulnerabilities recorded by the National Vulnerability Database (NVD)6 has been increasing +at a striking speed, affecting millions of software systems annually. +However, it is incredibly time-consuming and labor-intensive for security experts to repair such +security vulnerabilities manually due to the strikingly increasing number of detected vulnerabilities +and the complexity of modern software systems [41, 184]. For example, previous studies report +that the average time for repairing severe vulnerabilities is 256 days7 and the life spans of 50% of +vulnerabilities even exceed 438 days [75]. It is incredibly time-critical to patch reported security +vulnerabilities as a belated vulnerability repair could expose software systems to attack [81, 84], +posing enormous risks to millions of users around the globe and costing billions of dollars in +financial losses [67]. Given the potentially disastrous effect when software vulnerabilities are +exploited, a mass of learning-based studies has recently been conducted on automated software +vulnerability repair [25, 40]. +We list the recent learning-based vulnerability repair studies in details as follows. +Chen et al. [25] propose VRepair, a learning-based approach to repair security vulnerabilities +based on the transformer and transfer learning. VRepair is first trained on a large bug-fixing dataset +and is then transferred to a relatively small vulnerability-fixing dataset. VRepair uses a transformer +neural network model to generate potential patches that are likely to be correct based on the +training data. The results show that VRepair trained on a bug-fixing dataset already fix some +vulnerabilities. Besides, they demonstrate the knowledge learned from the program repair task can +be transferred to the vulnerability repair task. In particular, VRepair with the transfer learning +gains a better repair performance than that only trained on a vulnerability-fixing or bug-fixing +dataset. +Fu et al. [40] propose VulRepair, a T5-based automated vulnerability repair technique based on +subword tokenization and pre-training components. They compare VulRepair with two competitive +baseline approaches, VRepair and CodeBERT on a C benchmark – CVEFixes. Besides, they analyze +the impact of adopted components (i.e., tokenization and pre-training) and conduct an ablation study +to investigate the contribution of each component. The results show that VulRepair outperforms +other state-of-the-art vulnerability repair techniques and it is capable of repairing the Top-10 most +dangerous CWEs. +Chi et al. [26] propose SeqTrans, a learning-based appraoach to provide suggestions for automati- +cally repairing vulnerability. SeqTrans first uses Gumtree to search for differences between different +commits and then traverses the whole AST to label the variables. SeqTrans then traverses up the +leaf nodes, localizes the statement with vulnerability and generates code change pairs, which is fed +into the NMT model. As SeqTrans requires a massive amount of training data, SeqTrans is first +trained on a bug-fixing dataset (i.e., source domain) and fine-tuned on a vulnerability-fixing dataset +(i.e., target domain). SeqTrans is proven to achieve better repair accuracy than existing techniques +(e.g., SequenceR) and performs very well in certain kinds of vulnerabilities (e.g., CWE-287). +Harer et al. [50] apply a GAN-based approach to train a NMT model for learning to automatically +repair the source code containing security vulnerabilities. They apply an NMT model as the +generator and employ two novel generator loss functions instead of the traditional negative +likelihood loss. They also design a discriminator to distinguish the output generated by the NMT +model and oracle output. This approach can be used in the absence of paired bug-fixing datasets, +thus reducing the requirements of datasets. The authors evaluate the proposed approach on SATE +IV dataset and prove the promising results in fixing vulnerabilities. They also demonstrate the +proposed approach can be applicable to other tasks, such as grammatical error correction. +6https://www.nist.gov/ +7https://www.securitymagazine.com/articles/95929-average-time-to-fix-severe-vulnerabilities-is-256-days +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:30 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +Huang et al. [54] propose to apply large pre-trained models for vulnerability repair to overcome +the shortcomings of learning-based APR techniques. They compare the performance of CodeBERT +and GraphCodeBERT on a C/C++ vulnerability dataset with five CWE types. They discover that +GraphCodeBERT with a data flow graph is significantly better than CodeBERT without documenting +code dependencies. They also demonstrate that such pre-trained models outperform learning-based +APR techniques (e.g., CoCoNut [96] and DLFix [79]) and more data-dependent features (e.g., data +flow and control flow) will help to repair more complex vulnerabilities. +Zhou et al. [185] propose a novel approach SFVP for automatically fixing vulnerabilities based on +the attention-mechanism model. SPVF first extracts the security properties from descriptions of the +vulnerabilities (e.g., CWE category). SPVF then designs the pointer generator network to combine +the AST representation and the security properties. The authors evaluate SPVF on two public +C/C++ and Python vulnerability-fixing datasets and results show that it outperforms state-of-the-art +SeqTrans [26]. +Ma et al. [97] introduce a novel tool, VuRLE, to autimatically detect and repair vulnerabilities +in Java programs. In the learning phase, it generates templates by analyzing edits from repair +examples. First, it extracts edit blocks by performing AST diff. Then, it compares each edit block +with the other edit blocks, and produces groups of similar edit blocks. Finally, for each edit group, +VuRLE generates a repair template for each pair of edit blocks that are adjacent to each other. In +the repairing phase, VuRLE detects and repairs vulnerabilities by selecting the most appropriate +template. It applies repair templates in order of their matching score until it detects no redundant +code. Evaluation results on real-world vulnerabilities show that VuRLE outperforms another APR +tool in fixing vulnerabilities. +6.1.2 +Syntax Errors. In the learning-based APR field, semantic errors (i.e., test-triggering errors) +have attracted considerable attention. Such errors usually refer to any case where the actual program +behavior is not expected by developers. Existing learning-based APR techniques usually expect +that the programs under repair are syntactically correct and these techniques are not applicable +for syntax errors. Novice programmers are more likely to make syntax errors (e.g., replacing a +“∗” with an “𝑥”) that make compilers fail. Previous studies have indicated the long-term challenge +from a wide range of syntax mistakes, consuming a lot of time for novices and their instructors. +Recently, the release of high-quality novice error data and the emergence of trustworthy deep +learning models have raised the possibility of designing and training DL models to fix syntax errors +automatically. +Now, we list the recent learning-based APR studies that focus on syntax errors as follows. +Ahmed et al. [3] propose SynShine, a machine learning-based approach to fix syntax errors in +Java programs. They apply a three-stage syntax repair tool: BlockFix for recovering block structure, +LineFix for fixing line errors, and UnkFix for recovering unknown tokens. SynShine leverages +RoBERTa pre-training, uses compiler errors, and generates fixes using multi-label classification. +After being evaluated on a dataset collected from the Blackbox repository, SynShine outperforms +other state-of-art tools on different token ranges. They have also integrated SynShine with the +VSCode IDE for public usage. +Previous works mainly focus on logical errors and assume that the program should be compiled +successfully. Gupta et al. [48]. propose DeepFix to fix multiple errors in a program interactively. +They apply the RNN encoder-decoder to serve as the seq2seq network. To implement the iterative +repair for multiple errors, they decide to repair one bug each time. An oracle is applied after the +decoder to decide whether the program needs further repair after one patch is generated. Although +this approach is only evaluated on C program language written by students in an introductory +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:31 +programming course, the result shows that DeepFix can fix a variety of errors and has potential in +other languages. +Existing work often applies heuristics on generating buggy code to construct buggy-fixed pairs. +Such synthetically-generated data may not improve the model and generate low-quality patches. +Yasunaga et al. [173] propose Break-It-Fix-It (BIFI), a novel APR tool to address this problem. They +first try to train a breaker with real-world buggy-fixed pairs to generate more realistic. They also +leverage correct paired data to train the fixer. BIFI does not simply collect data, it is also capable of +turning raw unlabeled data into usable paired data with the help of a critic. They then evaluate this +approach on both Python and C benchmarks and it outperforms other state-of-the-art APR tools. +Berabi et al .[13] present TFix to deal with text-to-text prediction problems. They fine-tune a +pre-trained T5 model to generate JavaScript fixes on datasets extracted from GitHub by themselves. +By feeding the model with line context and fine-tuning it according to various error types, they +obtain multiple fine-tuned T5 models. The evaluation shows that TFix generates more patches than +SequenceR and CoCoNut. +Mesbah et al. [108] propose DeepDelta to repair the most costly classes of build-time compilation +failures in Java programs. They perform a large-scale study of compilation errors and collect a +large dataset from logs in Google. They further classify different compilation errors and target +repairing these errors following specific patterns learned from the AST diff files in the dataset. +For the two most prevalent and costly classes of Java compilation errors: missing symbols and +mismatched method signatures, evaluation results show that DeepDelta generates over half of the +correct patches. +Santos et al. [134] propose to leverage language models for repairing syntax errors in Java +programs. They compare n-gram with LSTM models trained on a large corpus of Java projects +from Github about localizing bugs and repairing them. Besides, their methodology does not rely on +buggy code from the same domain as the training data. Evaluation results show that this tool can +localize and suggest corrections for syntax errors, and it is especially useful to novice programmers. +Ahmed et al. [2] introduce an indirect-supervision approach to leverage GitHub code to create +massive amounts of "incorrect-fixed" training pairs for model training. They apply a two-stage +approach, with two different neural networks for learning to model block nesting structure and +code fragments. This approach performs better on the large and diverse BlackBox dataset than +previous work. It also performs well for StackOverflow fragment parsing and helps fix errors for +novice programmers. +Gupta et al. [47] propose RLAssist to address the problem of syntactic error repair in student +programs. They leverage reinforcement learning and train the model using Asynchronous Advan- +tage Actor-Critic (A3C)[109]. A3C uses multiple asynchronous parallel actor-learner threads to +update a shared model, stabilizing the learning process by reducing the correlation of an agent’s +experience. After they evaluate RLAssist on the C benchmark from [48], results show that this +model outperforms the APR tool DeepFix without using any labeled data for training and can help +novice programmers. +Bhatia et al. [14] propose a novel approach for repairing programs committed by students. They +first apply an RNN to repair syntax errors and then formalize the problem of syntax corrections +in programs as a token sequence prediction problem. Then they leverage the constrain-based +technique to find minimal repairs for semantic correctness. This approach is then evaluated on a +Python dataset and results demonstrate the effectiveness of their system. +Hajipour et al. [49] propose an efficient method to fix common programming errors by learning +the distribution over potential patches. To encourage the model to generate diverse fixes even with +a limited number of samples, they propose a novel regularizer that aims to increase the distance +between the two closest candidate fixes. They prove that this approach is capable of generating +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:32 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +multiple diverse fixes with different functionalities. After evaluating the approach on real-world +datasets, they show that this approach outperforms DeepFix. +Wu et al. [163] propose a novel deep supervise learning model, Graph-based Grammar Fix (GGF), +to localize and fix syntax errors. They first parse the erroneous code into ASTs. Since the parser +may crash in the parsing process due to syntax errors, they create so-called sub-AST and build the +graph based on it. To tackle the problem of isolated points and some error edges in the generated +graph, they treat the code snippet as a mixture of token sequences and graphs. Thus, GGF utilizes +a mixture of the GRU and the GGNN as the encoder module and a token replacement mechanism +as the decoder module. The evaluation shows that the architecture used in GGF is quite helpful for +the programming language syntax error correction task. +6.1.3 +Programming Assignments. Ahmed et al. [5] introduce TRACER to generate targeted repairs +for novice programmers in C programs. They leverage buggy student programs in Prutor and +conduct experiments on single-line and multi-line bugs. TRACER first localizes the buggy line, then +abstracts the program, and finally converts it into fixed code. Evaluation on the dataset collected +from IIT-K shows that TRACER achieves high accuracy and student-friendliness of the repair. +Wang et al. [153] propose Sarfgen, a high-level data-driven framework to fix student-submitted +programs for introductory programming exercises. They develop novel program embeddings and +the associated distance metric to efficiently and precisely identify similar programs and compute +program alignment. They also conduct an extensive evaluation of Sarfgen on thousands of student +submissions on 17 different programming exercises from Microsoft DEV204-.1x edx course and +the Microsoft CodeHunt platform. Results show that Sarfgen is effective and it improves existing +systems automation, capability, and scalability. +Wang et al. [152] present dynamic program embeddings which learn from runtime execution +traces to predict error patterns that students would make in their online programming submissions. +They define three program embedding models: 1) variable trace model to obtain a sequence of +variables; 2) state trace model to embed each program state as a numerical vector and feed all +program state embeddings as a sequence to another RNN encoder; 3) dependency enforcement +model to combine the advantages of the previous two approaches. They have proved that dynamic +embeddings overcome critical problems with syntax-based program representations and outperform +other syntactic program embeddings. +Zhang et al. [180] propose a novel approach to repair both semantic and syntactic bugs in +Python programs. They apply a large language model trained on code (LLMC). They also leverage +multimodal prompts, iterative querying, test-case-based few-shot selection, and program chunking +to repair bugs in students’ committed programs. Zhang et al. implement it in MMAPR through +Codex as LLMC. After evaluating MMAPR on real student programs and another baseline (BIFI +and Refactory), it outperforms other state-of-art tools. +Ahmed et al. [4] propose Verifix as a tool to provide feedback for students in programming +tasks. They first align a student-submitted program with a reference solution in terms of control +flow. Then the variables of the two programs are automatically aligned. After that, they turn a +verification problem into a MaxSMT problem if the above verification attempt fails. The solution of +MaxSMT problem leads to a minimal repair. Ahmed et al. are the first to espouse verified repair for +general-purpose programming education and their approach produces small-sized verified patches +as feedback which can be used by struggling students with high confidence. +Li et al. [76] propose AssignmentMender to repair student programs by leveraging both correct +and faulty C programs. This is the first approach that can exploit faulty submissions in generating +patches for programming assignments. The evaluation on the Codeforces benchmark shows that +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:33 +AssignmentMender outperforms several other approaches in feedback generation when only a +small number of reference programs are available. +Since previous works fail to parse ASTs for student programs with syntax errors, Bhatia et al. +[15] present a technique to apply RNN for repairing syntax errors in student programs. They first +train the model with syntactically correct programs. Then, they query the trained model with +student submissions with syntax errors and feed the model with the prefix token sequence. Finally, +the model would predict suffix tokens and repair the syntax error. Evaluation on a dataset obtained +from a MOOC course shows that this approach can provide automated feedback on syntax errors +for students. +6.1.4 +Other Domains. Programming Contests. Fan et al. [37] propose to leverage large pre-trained +model to repair buggy programs generated by Codex model. They collect a Java dataset (LMDefects) +from LeetCode contest containing different levels of tasks. They then compare the performance of +Codex-e and traditional APR tools (TBar and Recoder) on this dataset. Results show that existing +APR techniques (TBar and Recoder) do not perform well at fixing bugs in auto-generated programs. +Fan et al. also define three strategies as instructions:1) fix bugs in the program;2) fx line N;3) and +fix statement S to evaluate their approach. They find that Codex-e performs well under proper +instructions. +Program Synthesis. Gupta et al. [46] present SED as a framework incorporating synthesis, execu- +tion, and debugging stages. SED applies a synthesizer that employs greedy decoding to generate +buggy programs for training and the debugger is fed with synthesized bugs as well as execution +results. SED is then evaluated on the Karel benchmark and it outperforms other beam search +techniques. +Nonidiomatic Snippets. Szalontai et al. [138] present a novel algorithm to localize and substitute +non-idiomatic code snippets in Python programs. They apply a feed-forward and two RNNs +to accomplish the task. Once the code snippet is localized, the model classifies the type of the +nonidiomatic pattern and extracts the key variables. Finally, the model substitutes the code snippet +with a cleaner and more performant alternative. This model is evaluated on a Python dataset and it +achieves good results. +6.2 +Industrial Deployment +As a promising field, APR has been extensively studied in academia and even has drawn growing +attention from industry [8]. For example, Marginean et al. [100] present SapFix, the first end-to- +end deployment of industrial APR in Meta. SapFix is implemented to a continuous integration +environment and deployed into six production systems with tens of millions of code lines. Similar +industrial practice can also be found in other companies, such as Fujitsu [133], Bloomberg [63] and +Alibaba [183]. In addition to the above-mentioned traditional deployment, the industry recently +explores the feasibility of deploying learning-based APR tools. For example, GitHub launches a +product Copilot8, which can provide code suggestions (e.g., fixing bugs) for more than a dozen +programming languages. Copilot is deployed in multiple IDEs, such as VS Code, Visual Studio, +Neovim, and JetBrains. Besides, Microsoft recently releases a new tool Jigsaw9 to fix bugs in +machine-written software. +Now, we summarize the existing learning-based APR techniques and industrial deployment from +enterprises. +Bader et al. [8] present Getafix, the first industrially-deployed automated bug-fixing tool for +Java programs. To be fast enough to suggest fixes in time, this model produces a ranked list of +8https://github.com/features/copilot +9https://www.microsoft.com/en-us/research/blog/jigsaw-fixes-bugs-in-machine-written-software/ +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:34 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +fix candidates based entirely on past fixes and on the context in which a fix is applied. Besides, it +leverages the hierarchical clustering technique for discovering repetitive fix patterns. Moreover, +They apply a statistical ranking technique to enable the model to predict human-like fixes among +the top few suggestions. An evaluation with a large dataset containing six types of common bugs +and their experience of deploying Getafix within Facebook show that the approach accurately +predicts human-like fixes for various bugs, reducing the time developers have to spend on fixing +recurring kinds of bugs. +Baudry et al. [9] present R-HERO, a novel software repair robot to automatically repair bugs on +the single platform GitHub/Travis CI. R-HERO contains six main blocks: a) Continuous integration, +b) Fault localization, c) Patch generation, d) Compilation & Test execution, e) Overfitting prevention, +and f) Pull-request creation. It receives and analyzes the events from a continuous integration (CI) +system. R-HERO leverages continual learning to acquire bug-fixing strategies from the platform +mentioned above. It shows that developers and bots can cooperate fruitfully to produce high-quality, +reliable software systems. +Allamanis et al [6] from Microsoft propose BUGLAB to detect and repair software bugs auto- +matically by self-supervised learning. Similar to BIFI [173], BUGLAB employs a detector model +to repair bugs and a selector model to generate buggy code snippets as the training data of the +detector. The authors create a dataset PYPIBUGS of 2374 real-world bugs from the PyPI packages. +The results show that BUGLAB can fix a number of software bugs and detects some previously +unknown bugs in open-source software. +Drain et al. [33] from Microsoft leverage the same DeepDev-py sequence-to-sequence model +which is pre-trained from BART. They train it on the Python commit data and reversed data to +generate bug-patcher and bug-creator respectively, then leverage them to generate neural bugs to +finally train a back-translation model on Python methods with debugging information and stack +traces. They evaluate the proposed technique on QuixBugs benchmarks and their own benchmarks, +and this model outperforms many APR tools on Python. +Drain et al. [34] from Microsoft introduce DeepDebug, a span-masking pre-trained encoder +decoder transformer as a tool to fix Java methods. The model is pre-trained from BART which is +pre-trained in English. They conduct three pre-training experiments to verify the feasibility of +the model and test it on the Java benchmarks from Tufano et al. [147]. DeepDebug outperforms +many state-of-art APR tools, and adding syntax embeddings along with the standard positional +embeddings helps improve the model. +Hellendoorn et al. [52] from Google conduct experiments for two different models architectures +that leverage both local and global information. They propose sandwich models that apply different +message-passing techniques and GREAT models that add extra information to a transformer. Both +architectures achieve high results and outperform state-of-art tools, proving that a hybrid model +with global information and incorporating structural bias helps improve accuracy. +Hu et al. [53] from AWS AI propose NSEdit to generate patches for Java programs. Given only +the buggy code, NSEdit uses the pre-trained CodeBERT as the encoder and CodeGPT as the decoder +to address the sequence-to-sequence NMT problem. Moreover, it uses a pointer network to select +content-based edit locations. They apply beam search to and design a novel technique to fine-tune +the reranker to rank the top-k patches for the buggy code. The results on BFP benchmarks [147] +indicate that NSEdit outperforms state-of-art APR tools and demonstrate the effectiveness of each +component of the model. +Tang et al. [140] from Microsoft introduce a grammar-guided end-to-end approach to generate +patches, which treats APR as the transformation of grammar rules. They apply structure-aware +modules and three different types of strategies for grammar-based inference algorithms. They +also leverage two encoders and enhance the model with a new tree-based self-attention. The +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:35 +experimental results on BFP datasets [147] demonstrate that the proposed technique outperforms +other state-of-art APR techniques. +Wang et al. [151] from Ping An Technology propose CPR, short for causal program repair, as a +tool to utilize data augmentation strategy for input perturbations. This model can generate patches +for Java, Python, JavaScript, and C based on causally related input-output tokens. Besides, it can +offer explanations by transforming code into explainable graphs on various Seq2Seq models in +APR. They conduct experiments on four programming languages and prove that APR models can +be utilized as causal inference tools. +6.3 +Pre-trained Model-based Repair +In this section, we will discuss the existing studies of pre-trained models on the ARP task. +Pre-trained models have significantly improved performance across a wide range of natural +language processing (NLP) and code-related tasks, such as machine translation, defect detection +and code classification [45, 94]. Typically, the models are pre-trained to derive generic vector +representation by self-supervised training on a large-scale unlabeled corpus and then are transferred +to benefit multiple downstream tasks by fine-tuning on a limited labeled corpus [29]. The application +of existing pre-trained models to program repair is usually divided into two categories: universal and +specific pre-trained model-based APR techniques. The former aims to propose universal pre-trained +models for multiple code-related tasks (including program repair), while the latter only focuses on +program repair by designing a novel APR technique based on pre-trained models. +6.3.1 +Universal Pre-trained Model-based APR Techniques. Existing pre-trained models generally +adopt the encoder-decoder transformer architecture, which can be classified into three types: +encoder-only, decoder-only, and encoder-decoder models. Encoder-only models (e.g., CodeBERT +[38]) usually pre-train a bidirectional transformer where tokens can attend to each other. Encoder- +only models are good at understanding tasks (e.g., code search), but their bidirectionality nature +requires an additional decoder for generation tasks. Decoder-only models (e.g., CodeGPT [19]) are +pre-trained using unidirectional language modeling that only allows tokens to attend to the previous +tokens and themselves to predict the next token. Decoder-only models are good at auto-regressive +tasks like code completion, but the unidirectional framework is sub-optimal for understanding tasks. +Encoder-decoder models (e.g., CodeT5 [127]) often make use of denoising pre-training objectives +that corrupt the source input and require the decoder to recover them. Compared to encoder-only +and decoder-only models that favor understanding and auto-regressive tasks, encoder-decoder +models can support generation tasks like code summarization. +Inspired by the success of pre-trained models in NLP, many recent attempts have been adopted +to boost numerous code-related tasks (such as program repair) with pre-trained models (e.g., +CodeBERT) [38, 45]. In the context of APR, an encoder stack takes a sequence of code tokens as +input to map a buggy code 𝑋𝑖 = [𝑥1, . . . ,𝑥𝑛] into a fixed-length intermediate hidden state, while +the decoder stack takes the hidden state vector as an input to generate the output sequence of +tokens 𝑌𝑖 = [𝑦1, . . . ,𝑦𝑛]. Researchers treat the APR problem as a generation task, and consider +encoder-decoder or encoder-only (with an additional decoder) pre-traiend models, which are usually +evaluated by BFP dataset from Tufano et al. [147]. +We summarize existing pre-trained models involving the program repair task as follows. +Feng et al. [38] present a bimodal pre-trained model (i.e., CodeBERT) for natural language and +programming language with a transformer-based architecture. CodeBERT utilizes two pre-training +objectives (i.e., masked language modeling and replaced token detection) to support both code +search and code documentation generation tasks. To support program repair task, Lu et al. [94] +leverage CodeBERT as the encoder, which is connected with a randomly initialized decoder. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:36 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +Wang et al. [157] present a pre-trained encoder-decoder model (i.e., CodeT5) that considers the +code token type information based on T5 architecture. CodeT5 employs a unified framework to +support code understanding (e.g., clone detection) and generation tasks (e.g., program repair) and +allows for multi-task learning. The most crucial feature of CodeT5 is that the code semantics of +identifiers are taken into consideration. Assigned by developers, identifiers often convey rich code +semantics and thus a novel identifier-aware objective is added to the training of CodeT5. +Guo et al. [45] present the first structure-aware pre-trained model (i.e., GraphCodeBERT) that +learns code representation from source code and data flow. Unlike existing models focusing on +syntactic-level information (e.g., AST), GraphCodeBERT takes semantic-level information of code +(e.g., data flow) for pre-training with a transformer-based architecture. The results on BFP datasets +[147] demonstrate the advantage of leveraging code structure information to repair software bugs. +Mastropaolo et al. [104] propose pre-trained text-to-text transfer transformer (T5) to address +four code-related tasks, namely automatic bug fixing, injection of code mutants, generation of +assert statements in test methods, and code summarization. They apply BFP small and BFP medium +datasets to train and evaluate the bug-fixing task, and then compare other state-of-art learning- +based APR tools on the same benchmark. Moreover, they have done single-task fine-tuning and +multi-task fine-tuning to fully evaluate the function of the pre-trained T5 model. Although multi- +task fine-tuning does not improve the result of code-related tasks, single-task fine-tuning does +prove that this model outperforms other tools on the same benchmarks. +Niu et al. [117] propose a seq2seq pre-trained model (i.e., SPT) by three ode-specific tasks (code- +AST prediction, masked sequence to sequence and method name generation) and fine-tune on the +generation tasks (i.e., code summarization, code completion, program repair and code translation) +and classification task (i.e., code search). +6.3.2 +Specific Pre-trained Model-based APR Techniques. In addition to those above-mentioned +typical pre-trained models that involve program repair, researchers have adopted pre-trained +models to design novel APR techniques (e.g., CURE integrates GPT into CoCoNut architecture +[57]). We summarize existing APR studies that employ pre-trained models as follows. +Existing learning-based APR techniques can only generate patches for a single programming +language and most of them are developed offline. Yuan et al. [159] propose CIRCLE, a T5-based +APR technique targeting multiple programming languages with continual learning. CIRCLE first +employs a pre-trained model as a repair skeleton, then designs a prompt template to bridge the gap +between pre-trained tasks and program repair. To further strengthen the continual learning ability, +CIRCLE applies a difficulty-based rehearsal method to achieve lifelong learning without access to +the entire historical data and an elastic regularization to resolve catastrophic forgetting. Finally, to +perform the multi-lingual repair, CIRCLE designs a simple but effective re-repairing mechanism to +eliminate incorrectly generated patches caused by multiple programming languages. +Chakraborty et al. [21] present MODIT, a novel multi-modal NMT-based tool, to automatically +generate fixes for buggy code. They leverage three modalities of information: edit location, edit +code context, and commit messages (natural language guidance from the developer). They conduct +many experiments and the evaluation shows that, through pre-training, MODIT improves the +ability to generate patches. Also, leveraging additional modalities of information could benefit the +source code repairing. +Kolak et al. [65] propose to apply large pre-trained language models to generate patches for +one-line bugs in Java and Python programs. They consider pre-trained models with a wide range of +sizes (e.g., GPT-2 with 160M, 0.4B, and 2.7B parameters and CodeX 12B parameters) for evaluation +and comparison. After evaluating these models on the QuixBugs benchmark, they discover that +larger language models are more promising in guiding patch selection in APR work. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:37 +Richter et al. [131] propose RealiT for localizing and fixing bugs in Python programs. They first +pre-train a transformer model on large numbers of mutant bugs and then fine-tune it with a small +set of real bugs. After evaluating RealiT on the PyPIBug benchmark, they prove that training on +both mutant and real-world bugs can significantly improve the performance of the model and +abundant mutant bugs also improve the model’s ability to localize and fix bugs. +Zhang et al. [181] propose CoditT5, a pre-trained language model for software-related edit +tasks. CoditT5 is pre-trained on both program languages and natural language comments. Zhang +et al. fine-tune it for three down-streaming tasks: comment updating, bug fixing, and automatic +code review. For bug-fixing, they fine-tune it with Java datasets BFP small and BFP medium. The +evaluation shows that CoditT5 outperforms other state-pf-art tools on three down-streaming tasks. +Mashhadi et al. [103] apply CodeBERT, a pre-trained neural network model, for fixing Java +bugs. They fine-tune and evaluate it on ManySStuBs4J datasets and find it is capable of generating +patches. Their approach gets rid of the limitation of token length and vocabulary problems, thus +this model is more efficient and effective. This model can generate patches for different types of +bugs and outperform other state-of-art APR tools in terms of the accuracy of generated patches. +Lajko et al. [69] propose to apply a Generative Pre-trained Transformer (GPT) for generating +patches. Specifically, they apply GPT-2 medium model to repair JS programs. First, the model is +fine-tuned on datasets mined from GitHub. Then, it is evaluated on the same dataset and it achieves +good results if it could generate more candidate patches. Lajko et al. consider using large models +and it can achieve better results. +Further, Prenner et al. [126] propose to apply Codex, a GPT-3 like model trained on a large +corpus, to localize and repair bugs on multi-language benchmarks. They conduct experiments to +evaluate Codex under different prompt conditions and they also compare the model with other +state-of-art APR tools. Results show that despite not being trained for APR, Codex still performs +well on the Quixbugs benchmark. Besides, Codex repairs more bugs in Python than those in Java. +Kang et al. [61] present GLAD, a novel learning-based APR tool targeting fixing if statement +omission faults (i.e., faults in which necessary code is missing). By leveraging generative pre- +trained Language Models (LMs) instead of machine translation models, GLAD does not require the +localization of a buggy line. Moreover, GLAD applies a grammar-based beam search to constrain +the output of the model and efficiently reduces the validation cost by performing dynamic ranking +of candidate patches using a debugger. Evaluation results on Defects4J benchmark show that GLAD +is capable of fixing bugs other APR tools fail to do. +Xia et al. [165] introduce AlphaRepair as a cloze-style APR tool to directly query a pre-trained +model for generating patches. They apply the newly pre-trained CodeBERT as an example under +zero-shot learning settings. They try to mask the buggy line in the source code with different +templates or strategies and feed the whole source code into the model with the buggy line as a +“comment". Then with a large number of patches this model generated, they propose probabilistic +patch ranking to determine top-k plausible patches. After evaluating this technique on both Java +and Python benchmarks, it outperforms other state-of-art APR tools and proves that a pre-trained +model with no fine-tuning is feasible. +6.4 +DL for Traditional APR +These approaches attempt to boost traditional APR techniques by utilizing deep learning or machine +learning. +Long et al. [91] propose Prophet, a patch-generation system for repairing defects. It uses dynamic +analysis on the given test suite to get the program points for the patch to modify. Then, the SPR[89] +is used to generate search space. With a trained probabilistic model, Prophet ranks the candidate +patches, which are validated by executing the test suites. They collect eight projects from Github +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:38 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +and get 777 patches to train their model and test it on a benchmark[71]. The result shows that +Prophet can generate patches correctly with the learned knowledge compared with previous patch +generation systems. +Chen et al. [23] propose a novel search-based technique called LIANA, which is based on a +designed learning-to-rank prioritization mode. It is based on the idea of repeatedly updating a +statistical model online based on the intermediate validation results of an ongoing program repair +process. The model is first trained offline and updated repeatedly after the generating progress +starts. The most up-to-date model is used to generate fixes and prioritize those that are more likely +to include the correct ingredients. +Wang et al. [107] propose TRANSFER, a fault localization and program repair approach with deep +semantic features and transferred knowledge which is obtained by a combination of spectrum-based +and mutation-based localization techniques. They build a fault localization and program repair +dataset respectively and employ existing fix templates designed by TBar. They also design 11 binary +classifications to identify whether one of the 11 bug types they define exists in a statement and +a multi-classification to determine which fix template this statement should apply. The binary +classification, consisting of one embedding layer, one RNN layer, one max pooling layer, and one +dense layer, is fed with spectrum-based, mutation-based, and semantic features and outputs the +probability of containing specific bugs. Although this approach is only tested on Java, it is proven +to outperform many state-of-art approaches. +Li et al. [74] design a novel framework called ARJANMT to leverage both redundancy assumption +and Seq2Seq learning of correct patches to generate fixes for Java methods using NSGA-II algorithm. +This framework combines both ARJA and SequenceR into a unified framework. After evaluating +ARJANMT on two Java benchmarks, results show that it benefits from search-based and NMT-based +techniques and outperforms other state-of-art techniques. +Valueian et al. [148] propose SituRepair for repairing multiple bugs in C programs based on +pre-defined repair patterns. It applies a machine learning model to predict the buggy type and +localization of the buggy code and then repairs them with situational modifications accordingly. +SituRepair is evaluated on a C benchmark Code4Bench and it successfully repairs 3,848 multiple- +fault programs. +6.5 +Open science +Recent years have witnessed an increasing use of DL in traditional SE problems and tasks. In +particular, software bug is a growing quality concern for modern software, and accordingly, APR +has become an actively studied topic in the SE community. According to our survey, various +learning-based APR techniques have been introduced in the last five years (discussed in Section 2). +DL brings a new repair paradigm (i.e., training and repairing) for the APR problem with promising +results. However, due to the nature of DL, learning-based APR techniques face some concerns in +reproducibility, which is quite different from transitional APR techniques. For example, it may +require a large number of machine resources for researchers to reproduce the NMT model’s work. +The cost is even unaffordable for most researchers from academia. Besides, there exists randomness +in the neural network training process, which hinders the reproduction results. +Such challenges posed by DL motivate us to further understand the potential issues with open +science in the learning-based APR area, so as to advance existing techniques by taking advantage +of the general merits of open science. Open science advocates that researchers make their artifacts +(e.g., raw data, dataset, scripts, related models, or any results produced in their work) available +to all levels of researchers [105], so knowledge can be shared without boundaries [118]. While a +mass of DL techniques are proposed to fix software bugs automatically, more support is needed +to investigate the critical open science problem. In particular, we investigate to what extent the +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:39 +Table 2. Results on tool availability +Tool +Language +Hosting Site +Link Accessibility +SA +DA +TA +URL +Wang et al. [152] +C +Github +valid +yes +no +no +https://github.com/keowang/dynamic-program-embedding +Tufano et al. [146] +Java +Google +valid +yes +yes +yes +https://sites.google.com/view/learning-codechanges +Tufano et al. [147] +Java +Google +valid +yes +yes +no +https://sites.google.com/view/learning-fixes +RLAssitst [47] +C +bitbucket +valid +yes +yes +yes +https://bitbucket.org/iiscseal/rlassist +CoCoNut [96] +Java,C,Python,JS +Github +valid +yes +yes +no +https://github.com/lin-tan/CoCoNut-Artifact +DLFix [79] +Java +Github +valid +yes +yes +no +https://github.com/ICSE-2019-AUTOFIX/ICSE-2019-AUTOFIX +Hellendoorn et al. [52] +Python +Github +valid +yes +yes +yes +https://github.com/VHellendoorn/ICLR20-Great +DrRepair [172] +C,C++ +Github +valid +yes +yes +yes +https://github.com/michiyasunaga/DrRepair +Learn2Fix [16] +Python +Github +valid +yes +yes +yes +https://github.com/mboehme/learn2fix +Tian et al. [143] +Java +Github +valid +yes +yes +yes +https://github.com/TruX-DTF/DL4PatchCorrectness +BIFI [173] +Python,C +Github +valid +yes +yes +yes +https://github.com/michiyasunaga/bifi +Recoder [186] +Java +Github +valid +yes +yes +no +https://github.com/pkuzqh/Recoder +SequenceR [24] +Java +Github +valid +yes +yes +yes +https://github.com/kth/SequenceR +TFix [13] +JS +Github +valid +yes +yes +yes +https://github.com/eth-sri/TFix +BugLab [6] +Python +Github +valid +yes +yes +no +https://github.com/microsoft/neurips21-self-supervised-bug-detection-and-repair +Reptory [114] +JS +Github +valid +yes +yes +no +https://github.com/annon-reptory/reptory +RewardRepair [176] +Java +Github +valid +yes +yes +yes +https://github.com/SophieHYe/RewardRepair +CodeBERT [103] +Java +Github +valid +yes +yes +no +https://github.com/EhsanMashhadi/MSR2021-ProgramRepair +R-HERO [9] +Github +valid +no +yes +no +https://github.com/repairnator/open-science-repairnator/tree/master/data/2020-r-hero +Ahmed et al. [2] +Java +zenodo +valid +yes +yes +no +https://doi.org/10.5281/zenodo.3374019 +ODS [174] +Java +Github +valid +yes +yes +yes +https://github.com/SophieHYe/ODSExperiment +CIRCLE [159] +Java,C,JS,Python +Github +valid +no +no +no +https://github.com/2022CIRCLE/CIRCLE +TRANSFER [107] +Java +Github +valid +yes +yes +yes +https://github.com/mxx1219/TRANSFER +DEAR [80] +Java +Github +valid +yes +yes +yes +https://github.com/AutomatedProgramRepair-2021/dear-auto-fix +Cornor et al. [27] +Java +Github +valid +yes +yes +no +https://github.com/WM-SEMERU/hephaestus +BATS [142] +Java +Github +valid +yes +yes +yes +https://github.com/HaoyeTianCoder/BATS +T5 [104] +Java +Github +valid +yes +yes +yes +https://github.com/antonio-mastropaolo/TransferLearning4Code +CompDefect [116] +Java +zenodo +valid +yes +yes +no +https://zenodo.org/record/5353354#.Y4CVdhRByUl +VRepair [25] +C +Github +valid +yes +yes +yes +https://github.com/SteveKommrusch/VRepair +SeqTrans [26] +Java +Github +valid +yes +yes +yes +https://github.com/chijianlei/SeqTrans +VulRepair [40] +C +Github +valid +yes +yes +yes +https://github.com/awsm-research/VulRepair +Crex [168] +C +Github +valid +yes +yes +yes +https://github.com/1993ryan/crex +RealiT [131] +Python +Github +valid +yes +no +yes +https://github.com/cedricrupb/nbfbaselines +GPT-2 [69] +JS +Github +valid +yes +yes +yes +https://github.com/RGAI-USZ/APR22-JS-GPT +CoditT5 [181] +Java +Github +valid +yes +yes +yes +https://github.com/EngineeringSoftware/CoditT5 +SYNSHINE [3] +Java +zenodo +valid +yes +yes +yes +https://zenodo.org/record/4572390#.Y4CY8xRByUk +Verifix [4] +C +Github +valid +yes +yes +yes +https://github.com/zhiyufan/Verifix +Cache [82] +Java +Github +valid +yes +yes +no +https://github.com/Ringbo/Cache +Wang et al. [158] +Java +Github +valid +yes +yes +yes +https://github.com/anonymous0903/patch_correctness +Quatrain [145] +Java +Github +valid +yes +yes +yes +https://github.com/Trustworthy-Software/Quatrain +Shibboleth [44] +Java +Github +valid +yes +yes +yes +https://github.com/ali-ghanbari/shibboleth +Tian et al. [144] +Java +Github +valid +yes +yes +yes +https://github.com/HaoyeTianCoder/Panther +SSC [30] +Python +Github +valid +no +no +no +https://iclr2018anon.github.io/semantic_code_repair/ +Huang et al. [55] +Java,C,C++ +Github +valid +yes +yes +yes +https://github.com/shan-huang-1993/PLC-Pyramid +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:40 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +collected papers make their artifacts publicly available and in what way they provide the relevant +information. +Table 2 shows the tool availability results of the investigated papers. For each paper we collect, +we check whether an accessible link for its tool or data is provided in the main text or footnotes of +the paper. We only present the studies that provide the link of publicly available data or tools due +to limited space, listed in the first column. We then investigate the following five dimensions in +characterizing the availability of each paper: +• Hosting Site. This information indicates which hosting site the available artifact is uploaded +to for public access (e.g., GitHub or Google), if the artifact link is presented in the paper. The +detailed information is listed in the third column. +• Link Accessibility. This information indicates whether the provided link is accessible, such +that we could download the artifacts. The detailed information is listed in the fourth column. +• Source Code Available. This information indicates whether the source code (e.g., training +and evaluation scripts) is available in the artifacts. The detailed information is listed in the +fifth column. +• Dataset Available. This information indicates whether the dataset (e.g., raw data and train- +ing data) is available in the artifacts. The detailed information is listed in the sixth column. +• Trained Model Available. This information indicates whether the trained model (e.g., raw +data and training data) is available in the artifacts. The detailed information is listed in the +seventh column. +We also list the programming languages targeted by the tools in the second column and list the +accessible url links in the last column. After carefully checking the collected papers, we find that +only a few of the papers have made their source code available to the public. For convenient public +access, a majority of papers upload their works to GitHub. The possible reason is that GitHub +is the most popular platform to host open-source code publicly. Meanwhile, we find that several +papers fail to provide the source code, dataset, or already trained model [140, 159]. The possible +reasons may be (1) the artifacts need to be refactored or reorganized for public availableness; (2) +the artifacts are used for further studies; and (3) the artifacts are lost due to some accidents. We +also find while the artifacts are available, some studies yet cannot be reproduced because (1) the +missing of default hyperparameters10; (2) the complexity of environment settings for training11; +and (3) the sufficiency of documentation to reproduce the experiments12. +Overall, most traditional APR tools have provided open-source code and data, which are easy to +reproduce. However, some learning-based APR tools require complex environment settings and +some authors fail to provide high-quality code. Besides, learning-based APR involves abundant +time and expensive equipment to train a model, and thus it is much harder to reproduce a learning- +based APR tool. Therefore, we hope that learning-based APR researchers can provide high-quality +open-source code to construct a unified repair framework for convenient reproduction. +7 +IMPLICATION AND DISCUSSION +Our study reveals the following important practical guidelines for future learning-based APR. +The quality of the training dataset is important. In contrast to traditional APR techniques, +learning-based techniques heavily rely on the quality of the training dataset. A majority of existing +techniques mine bug-fixing pairs from open-source code repositories (e.g., GitHub) and build their +own datasets. However, the training dataset is usually collected by automated tools (e.g., extracting +10https://github.com/lin-tan/CoCoNut-Artifact/issues/11 +11https://github.com/pkuzqh/Recoder/issues/11 +12https://github.com/ICSE-2019-AUTOFIX/ICSE-2019-AUTOFIX/issues/5 +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:41 +commit by fix-related keywords) and then inspected by some filtering rules (e.g., more than five +Java files) [186], which means the quality of the training dataset can be variant. Many training +datasets contain noise (e.g., CoCoNut contains a number of duplicated samples) that may reduce the +performance of the model. Besides, the number of training samples in different techniques varies +greatly (e.g., 3,241,966 in CoCoNut [96] and 2,000 in DLFix [79]). These concerns may introduce bias +when comparing and analyzing learning-based techniques. Thus, a unified standard for training +datasets should be built to reduce the burden on researchers when they evaluate the performance +of different repair models. +More practical evaluation metrics are needed. Recently, when evaluating repair perfor- +mance, an increasing number of learning-based techniques rely on static match-based metrics, +which are derived from NLP. However, such metrics fail to consider that a program’s functionality +can be implemented in various ways, such as different algorithms, data structures, or data flows. +It is unclear whether the match-based metrics can reflect the repair capability of NMT models. +Besides, the relationships between the static match-based and dynamic test execution-based metrics +need to be studied in the future. +Code features need to be studied. Inspired by the advance of machine translation in NLP, +early learning-based APR work treats source code as a sequence of tokens. The follow-up work +has begun to consider complex code features, such as code edit, [186], AST [79] and data flow +graph [115]. However, the most recent technique CIRCLE, treating the APR as a simple machine +translation task on code sequences, still achieves state-of-the-art results. Such observation indicates +that simple features require more attention in future work. Besides, considering the mass of code +representation ways in learning-based APR, it is crucial to conduct a systematic study to explore +the impact of different code representations under various model architectures and benchmarks. +Overfitting issue still exists. Similar to traditional APR techniques, learning-based techniques +usually adopt available test suites to filter incorrect candidate patches. However, the test suite is an +incomplete specification under the program behavioral space. The plausible patches passing the +existing test suite may not satisfy the expected outputs of potential test suites, leading to a long +challenge in APR (i.e., the overfitting issue). Considering the learning-based APR is an end-to-end +repair paradigm (in a black-box manner), which is different from traditional techniques adopting +test suites to guide the repair process, the overfitting issue in learning-based APR is more significant +and severe. Recently, researchers have adopted DL techniques (e.g., code embedding [82, 143]) to +predict the correctness of plausible patches, which is a promising direction to address overfitting +problems. We also recommend designing some advanced NMT repair frameworks to generate +high-quality patches. +Unified repair in urgent. As discussed in Section 4.2, similar to traditional APR techniques, +existing learning-based techniques usually consider fault localization as an additional step in the +repair process and adopt off-the-shelf fault localization tools (e.g., SBFL) to identify suspicious +code element, which is the input of NMT repair models. In the literature, these two tasks (i.e., +fault localization and patch generation) are developing in their own respective fields so far and +little work has explored their potential relationship. Recently, Ni et al. [116] propose CompDefect +to handle defect prediction and repair simultaneously. The powerful capacity of DL to learn the +semantic information of source code for fault localization [78, 93] and program repair [159, 186] +makes it possible to combine the two tasks. +Practical NMT repair model is needed. An increasing number of learning-based techniques +attempt to generate patches by large language models. Although remarkable progress is obtained, +such NMT models contain millions or even billions of parameters. For example, CodeBERT has +125 million parameters and 476 MB model size in total. It is significant to deploy these models in +modern IDEs to assist developers during software development and maintenance. However, these +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +1:42 +Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen +repair models consume huge device resources and run slowly in the development workflow (e.g., +IDEs), limiting their application in practice. In the future, It is promising to reduce the size of these +repair models to deploy in real-world scenarios while maintaining comparable accuracy, such as +model pruning and knowledge distillation. +Model size is not the only option. As discussed before, learning-based APR techniques tend to +employ the growing size of models, achieving better performance. Xia et al. [164] have demonstrated +that larger models usually repair a greater number of bugs, highlighting the promising future of +pre-trained models for APR. However, such large models are difficult to deploy in the development +workflow. Besides, with the release of ever-larger models, there may exist a barrier in the trade-off +between effectiveness and model size. In fact, existing pre-trained models in APR usually treat +source code as natural language, which cannot capture the code features. In the future, investigating +how to bring in code structures (e.g., data flow or control flow) in model training may be a flexible +strategy instead of employing a larger mode size. +Combined with traditional APR techniques. Existing DL techniques are usually adopted as +a patch generator in APR workflow, which takes the buggy code snippets as inputs and returns a +ranked list of candidate patches. Despite remarkable progress, such learning-based APR techniques +are developed separately from traditional techniques. Previous work [186] has demonstrated that +learning-based is complementary to traditional techniques in terms of fixed bugs. Thus, it is flexible +to integrate DL techniques into traditional APR techniques instead of developing a brand-new +end-to-end patch generator. For example, Meng et al. [107] design a multi-classifier to rank the fix +templates for TBar. In the future, researchers can boost existing template-based APR techniques +(e.g., TBar) via mask prediction. +Domain repair techniques are needed. A majority of learning-based techniques focus on +semantic bugs, which are investigated intensively. However, only a small amount of existing tech- +niques consider other types of bugs, such as security vulnerabilities or programming assignments. +The community usually treats fixing these types of bugs as separate tasks. SequenceR [24] has +demonstrated that NMT-based models only trained on a limited bug-fixing corpus can already +fix notable vulnerabilities. These results indicate that bug fixing and vulnerability repair both +aiming to fix errors in the source code have a high degree of similarity and the knowledge learned +from bug fixing can be well transferred to vulnerability repair. Such observation motivates future +researchers to explore their potential relationship and investigate whether these tasks can benefit +each other. Besides, it is promising to migrate existing mature learning-based bug-fixing techniques +to automated vulnerability repair. +Explainable Patch Generation. Traditional APR techniques generate patches along with a +log output, which contains detailed information in the generation process, while learning-based +APR techniques perform an end-to-end patch generation due to the interpretability of DL. Thus, +the developers are unaware of why repair models predict such results, hindering the adoption of +repair models in practice. In the literature, a majority of studies focus on improving repair accuracy, +while minor focus on improving the explainability of such repair models. In the future, advanced +explainable techniques can be considered to make the predictions of NMT repair models more +practical, explainable, and actionable. +8 +CONCLUSION +APR techniques address the long-standing the challenge of fixing software bugs automatically, and +alleviates manual debugging effort significantly, which promotes software testing, validation, and +debugging practices. In the last couple of years, learning-based APR techniques have achieved +promising results, demonstrating the substantial potential of using DL techniques for APR. +ACM Trans. Softw. Eng. Methodol., Vol. 0, No. 0, Article 1. Publication date: 2023. + +A Survey of Learning-based Automated Program Repair +1:43 +In this paper, we provide a comprehensive survey of existing learning-based APR techniques. We +describe the typical learning-based repair framework, involving fault localization, data pre-processing, +patch generation, patch ranking, validation and correctness components. We summarize how ex- +isting learning-based techniques design strategies for these crucial components. We discuss the +metrics, datasets and empirical studies in the learning-based APR community. Finally, we point out +several challenges (such as overfitting issues) and provide possible directions for future study. +9 +ACKNOWLEDGMENTS +This work is supported partially by the National Natural Science Foundation of China (61932012, +62141215). +REFERENCES +[1] Rui Abreu, Peter Zoeteweij, and Arjan JC Van Gemund. 2007. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' China Automated program repair (APR) aims to fix software bugs automatically and plays a crucial role in software development and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' With the recent advances in deep learning (DL), an increasing number of APR techniques have been proposed to leverage neural networks to learn bug-fixing patterns from massive open- source code repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such learning-based techniques usually treat APR as a neural machine translation (NMT) task, where buggy code snippets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', source language) are translated into fixed code snippets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', target language) automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Benefiting from the powerful capability of DL to learn hidden relationships from previous bug-fixing datasets, learning-based APR techniques have achieved remarkable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In this paper, we provide a systematic survey to summarize the current state-of-the-art research in the learning-based APR community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We illustrate the general workflow of learning-based APR techniques and detail the crucial components, including fault localization, patch generation, patch ranking, patch validation, and patch correctness phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We then discuss the widely-adopted datasets and evaluation metrics and outline existing empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We discuss several critical aspects of learning-based APR techniques, such as repair domains, industrial deployment, and the open science issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We highlight several practical guidelines on applying DL techniques for future APR studies, such as exploring explainable patch generation and utilizing code features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Overall, our paper can help researchers gain a comprehensive understanding about the achievements of the existing learning-based APR techniques and promote the practical application of these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Our artifacts are publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/QuanjunZhang/AwesomeLearningAPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CCS Concepts: • Software and its engineering → Software testing and debugging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Software testing and debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Additional Key Words and Phrases: Automatic Program Repair, Deep Learning, Neural Machine Translation, AI and Software Engineering ACM Reference Format: Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning- based Automated Program Repair .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, 0, Article 1 ( 2023), 51 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='nnnnnnn ∗Chunrong Fang is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Authors’ addresses: Quanjun Zhang, quanjun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='zhang@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='cn, State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, Jiangsu, China, 210093;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chunrong Fang, fangchunrong@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='cn, State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, Jiangsu, China, 210093;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Yuxiang Ma, 502022320009@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='cn, State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, Jiangsu, China, 210093;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Weisong Sun, weisongsun@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='nju.' metadata={'source': 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No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='03270v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='SE] 9 Jan 2023 1:2 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen 1 INTRODUCTION Modern software systems continuously evolve with inevitable bugs due to deprecating of old features, adding of new functionalities, and refactoring of system architecture [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These inevitable bugs have been widely recognized as notoriously costly and destructive, such as costing billions of dollars annually across the world [17, 160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The recorded quantity of bugs is increased at a tremendous speed due to the increasing scale and complexity of software systems [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Manual debugging can be an extremely time-consuming and error-prone task in the software development and maintenance process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, previous reports show that software debugging accounts for over 50% of the cost in software development [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Considering the promising future in relieving manual debugging efforts, automated program repair (APR), which aims to automatically fix software bugs without human intervention, has been a very active area in academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As a promising research area, APR has been extensively investigated in the literature and has made substantial progress on the number of correctly-fixed bugs [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A living APR review reports [112] that a growing number of papers get published each year with various exquisitely implemented APR tools being released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Over the past decade, researchers have proposed a variety of APR techniques to generate patches [88] [11] [156], including heuristic-based, constraint-based and pattern-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Among these traditional techniques, pattern-based APR employs pre-defined repair patterns to transform buggy code snippets into correct ones and has been widely recognized as state-of-the-art [87, 164, 165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, existing pattern-based techniques mainly rely on manually- designed repair templates, which require massive effort and professional knowledge to craft in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, these templates are usually designed for specific types of bugs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', null pointer exception) and thus are challenging to apply to unseen bugs, limiting the repair effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, inspired by the advance of deep learning (DL), a variety of learning-based APR tech- niques have been proposed to learn the bug-fixing patterns automatically from large corpora of source code [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Compared with traditional APR techniques, learning-based techniques can be applied to a wider range of scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', multi-languages) with parallel bug-fixing pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These learning-based techniques handle the program repair problem as a neural machine translation (NMT) task, which translates a code sequence from a source language (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', buggy code snippets) into a target language (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', correct code snippets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing NMT repair models are typically built on the top of the encoder-decoder architecture [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The encoder extracts the hidden status of buggy code snippets with the necessary context, and the decoder takes the encoder’s hidden status and generates the correct code snippets [56, 79, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thanks to the powerful ability of DL to learn hidden and intricate relationships from massive code corpora, learning-based APR techniques have achieved remarkable performance in the last couple of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The impressive progress of learning-based APR has shown the substantial benefits of exploiting DL for APR and further revealed its promising future in follow-up research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, a mass of existing studies from different organizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', academia and industry) and communities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', software engineering and artificial intelligence) make it difficult for interested researchers to understand state-of-the-arts and improve upon them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, compared with traditional techniques, learning-based techniques heavily rely on the quality of code corpora and model architectures, posing several challenges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', code representation and patch ranking) in developing mature NMT repair models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, most learning-based techniques adopt different training datasets, and there exist various strategies to process the code snippets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', the code context, abstraction, and tokenization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, researchers design different code representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', sequence, tree, and graph) to extract code features, which require corresponding encoder-decoder architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', RNN, LSTM, and transformer) to learn the transformation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Furthermore, execution-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', plausible and correctness patches) and match-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', accuracy and BLUE) metrics are ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:3 adopted in different studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such multitudinous design choices hinder developers from conducting follow-up research on the learning-based APR direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In this paper, we summarize existing work and provide a retrospection of the learning-based APR field after years of development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Community researchers can have a thorough understanding of the advantages and limitations of the existing learning-based APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We illustrate the typical workflow of learning-based APR and discuss different detailed techniques that appeared in the papers we collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Based on our analysis, we point out the current challenges and suggest possible future directions for learning-based APR research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Overall, our work provides a comprehensive review of the current progress of the learning-based APR community, enabling researchers to obtain an overview of this thriving field and make progress toward advanced practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To sum up, the main contributions of this paper are as follows: Survey Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We conduct a detailed analysis of 112 relevant studies that used DL techniques in terms of publication trends, distribution of publication venues and languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Learning-based APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We describe the typical framework of leveraging advances in DL tech- niques to repair software bugs and discuss the key factors, including fault localization, data pre-processing, patch generation, patch ranking, patch validation and patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Dataset and Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We perform a comprehensive analysis of the critical factors that impact the performance of DL models in APR, including 53 collected datasets and evaluation metrics in two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We detail existing empirical studies performed to better understand the process of learning-based APR and facilitate future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Some Discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We discuss some other crucial areas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', security vulnerability and syntax error) where learning-based APR techniques are applied, as well as certain known industrial deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We demonstrate the trend of employing pre-trained models on APR recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We list the available learning-based tools and reveal the essential open science problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Outlook and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We pinpoint open research challenges of using DL in APR and provide several practical guidelines on applying DL for future learning-based APR studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Comparison with Existing Surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Gazzola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [42] present a survey to organize the repair techniques published up to January 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Monperrus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [111] present a bibliography of behavioral and state repair papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Unlike existing surveys mainly covering traditional techniques, our work focuses on the learning-based APR, particularly the integration of DL techniques in the repair phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', patch generation and correctness), repair domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', vulnerability and syntax errors) and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, our survey summarizes the existing studies until Nov 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Paper Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Section 2 presents the research methodology about how we collect relevant papers from several databases following specific keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Section 3 introduces some common concepts encountered in the learning- based APR field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Section 4 presents the typical workflow of learning-based APR and discusses the vital components of the workflow in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Section 5 extends the discussion on the collection of datasets and standard evaluation metrics of learning-based APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Section 6 details some discussions, including repair applications, industrial deployments, employment of pre-trained models and the open science problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Section 7 provides practical guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Section 8 draws the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 2 SURVEY METHODOLOGY In this section, we present details of our systematic literature review methodology following Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:4 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen Google Scholar ACM Digital Library IEEE Digital Library Group1 repair related keywords Group2 DL related keywords discussion and selection program repair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' bug fix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' … deep;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' machine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' … Automated Search filter by year 342 papers filter by pages (remove duplications) 283 papers filter irrelevant papers 87 papers add missed citations 112 papers Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' General workflow of the paper collection Search Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For this survey, we select papers by mainly searching the Google Scholar repository, ACM Digital Library, and IEEE Explorer Digital Library at the end of November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Following existing DL for SE surveys [154, 170], we divide the search keywords used for searching papers into two groups: (1) a APR-related group containing some commonly used keywords related to program repair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (2) a DL-related group containing some keywords related to deep learning or machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Considering a significant amount of relevant papers from both SE and AI communities, following Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [179], we first try to collect some papers from the community-driven website1 and the living review of APR by Monperrus [112], and then conclude some frequent words in the titles of these papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The search strategy can capture the most relevant studies while achieving better efficiency than a purely manual search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, we identify a search string including several DL-related terms frequently appearing in APR papers that make use of DL techniques, listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (“program repair” OR “software repair” OR “automatic repair” OR “code repair” OR “bug repair” OR “bug fix” OR “code fix” OR “automatic fix” OR “patch generation” OR “fix generation” OR “code transformation” OR “code edit” OR “fix error”) AND (“neural” OR “machine” OR “deep” OR “learning” OR “transformer/transformers” OR “model/models” OR “transfer” OR “supervised”) Study selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Once the potentially relevant studies based on our search strategy are collected, we perform a filtering and deduplication phase to exclude papers not aligned with the study goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We first attempt to filter out the papers before 2016, considering that Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [91] propose the first learning-based APR study in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We then filter out any paper less than 7 pages and the duplicated papers, resulting in 283 papers in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We then scrutinize the remaining papers manually to decide whether it is relevant to the learning-based APR field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We obtained 87 papers at last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To be as much comprehensive as possible, we include the relevant papers that we miss with our searches but were cited in the papers we selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We manually analyzed all these cited papers by scanning the papers and finally collected 112 papers in our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The general workflow of how we collected papers is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1http://program-repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='org/bibliography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='html ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:5 3 4 6 13 13 25 47 0 5 10 15 20 25 30 35 40 45 50 2016 2017 2018 2019 2020 2021 2022 Number of papers Year Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Collected learning-based APR papers from 2016 to 2022 44% 20% 18% 13% 5% Java C Python JavaScript C++ Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Paper distribution on program languages Trend Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Figure 2 shows the collected papers from 2016 to 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is found that the number of learning-based APR papers has increased rapidly since 2020, indicating that more researchers are considering DL as a promising solution to fixing software bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' One reason behind this phenomenon is that traditional APR techniques have reached a plateau and researchers hope to find a brand-new way to address the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Another non-negligible reason is that DL has proved its potential in various tasks, including natural language translation, which is similar to bug fixing to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Figure 3 presents an overview of the programming languages targeted by learning-based APR techniques in our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We can find Java occupies a large proportion, which is understandable as Java is widely adopted in modern software systems nowadays and the most targeted language in existing mature datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Defects4J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We also find that the collected papers cover a wide range of programming languages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Java, JavaScript, Python, C and C++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:6 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen repair strategy test suite correct patch plausible patch developer generated patch overfitting patch suspicious code fault localization deployment Localization Phase Repair Phase buggy program Verification Phase Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Overview of APR For example, there exist several papers [96, 159] involving multiple programming language repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The probable reason may be that learning-based APR techniques usually regard APR as an NMT problem, independent of programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 3 BACKGROUND AND CONCEPTS In this section, we will introduce some background information and common concepts in the learning-based APR field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Automated Program Repair The primary objective of APR techniques is to identify and fix software bugs without human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the software development and maintenance process, after a designed functionality is implemented, developers usually write some test suites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Junit test cases) to check the functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' If there exist test suites that make the functionality fail, developers adopt the failing test suites to analyze the symptoms and the root cause of the bug, and attempt to fix the bug by making some changes to suspicious code elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' More generally, we can give the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' APR: Given a buggy program 𝑃, the corresponding specification 𝑆 that makes 𝑃 fail, the transformation operators𝑂 and the allowed maximum edit distance𝜖, APR can be formalized as a function 𝐴𝑃𝑅(𝑃,𝑆,𝑂,𝜖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 𝑃𝑇 is the set of its all possible program variants by enumerating all operators 𝑂 on 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The problem of APR is to find a program variant 𝑃 ′ (𝑃 ′ ∈ 𝑃𝑇) that satisfies 𝑆 and the changes satisfies 𝜖 (𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑃, 𝑃 ′) ≤ 𝜖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The specification 𝑆 denotes a relation between inputs and outputs and most APR techniques usually adopt a test suite as a specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In other words, APR aims to find a minimal change to 𝑃 that passes all available test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The maximum edit distance 𝜖 limits the range of changes based on the competent programmer hypothesis [119], which assumes that experienced programmers are capable of writing almost correct programs and most bugs can be fixed by small changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' If 𝜖 is set to ∞, 𝐴𝑃𝑅(𝑃,𝑆,𝑂,𝜖) becomes a program synthesizing problem that aims to synthesize a program to satisfy 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:7 The typical workflow of APR techniques is illustrated in Figure 4, which is usually composed of three parts: (1) off-the-shelf fault localization techniques are applied to outline the buggy code snippets [1] [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) these snippets are modified based on a set of transformation rules or patterns to generate new various program variants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', candidate patches);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) the original test suite is adopted as the oracle to verify all candidate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Specifically, a candidate patch passing the original test suite is called a plausible patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A plausible patch, which is also semantically equivalent to the developer patch, denotes a correct patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, such specifications (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', test suites) are inherently incomplete as programs have infinite domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is fundamentally challenging to ensure the correctness of the plausible patches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', overfitting issue) due to the weak test suites in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing studies have demonstrated that manually identifying the overfitting patches is time-consuming and may harm the debugging performance of developers [137, 141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The overfitting issue is a critical challenge in both traditional and learning-based APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We will discuss the issue in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Patch Generation Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, numerous traditional APR techniques have been proposed to generate patches from different aspects, which can be categorized into three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We list them as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (1) Heuristic-based repair techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These techniques usually apply heuristic strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', genetic algorithm) to build search space from previous patches and generate valid patches by exploring the search space [73, 101, 178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, SimFix [56] builds an abstract search space from existing patches and a concrete search space from similar code snippets in the buggy project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SimFix then utilizes the intersection of the above two search spaces to search the final patch by basic heuristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', syntactic distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) Constraint-based repair techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These techniques usually treat APR as a constraint-solving task and rely on SMT solvers to return a feasible solution [36, 102, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Nopol [167] relies on an SMT solver to solve the condition synthesis problem after identifying potential locations of patches by angelic fix localization and collecting test execution traces of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) Pattern-based repair techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These techniques usually design certain repair templates by manually analyzing specific software bugs and generate patches by applying such templates to buggy code snippets [66, 86, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, TBar [87] revisits the effectiveness of pattern-based APR techniques by systematically summarizing a variety of repair patterns from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In addition to the above traditional APR techniques, researchers attempt to fix software bugs enriched by DL techniques due to the large-scale open-source source code repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such learning-based techniques have demonstrated promising results and are getting growing attention recently, which is the focus of our work (introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Neural Machine Translation Sequence-to-sequence (Seq2Seq) is an advanced DL framework widely used in some NLP tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', machine translation [58] and text summarization [113]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Seq2Seq model usually consists of two components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', an encoder and a decoder) to learn mappings between two sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Inspired by the success of Seq2Seq models in text generation tasks, program repair can be formulated as an NMT task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The learning-based APR problem is formally defined as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Learning-based APR: Given a buggy code snippet 𝑋𝑖 = [𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ,𝑥𝑛] with 𝑛 code tokens and a fixed code snippet 𝑌𝑖 = [𝑦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ,𝑦𝑚] with𝑚 code tokens, the problem of program repair is formalized to maximize the conditional probability: 𝑃 (𝑌 | 𝑋) = �𝑚 𝑖=1 𝑃 (𝑦𝑖 | 𝑦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ,𝑦𝑖−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ,𝑥𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In other words, the objective of an NMT repair model is to learn the mapping between a buggy code snippet 𝑋 and a fixed code snippet 𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then the parameters of the model are updated by using the training dataset, so as to optimize the mapping (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', maximizing 𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, recurrent ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:8 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Detailed workflow of Learning-based APR neural network architecture (RNN) is widely used in existing learning-based APR techniques [24, 48, 146, 147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, researchers use long short-term memory (LSTM) architecture to capture the long-distance dependencies among code sequences [20, 107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, as a variant of the Seq2Seq model, Transformer [150] has been considered the state-of-the-art NMT repair architecture due to the self-attention mechanism [25, 26, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4 LEARNING-BASED APR In this section, we will discuss the workflow of learning-based APR tools and introduce some popular learning-based APR techniques with several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Overall Workflow Figure 5 illustrates the typical framework of existing learning-based APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The framework can be generally divided into seven phrases: fault localization, data pre-processing, input encoding, output decoding, patch ranking, patch validation, and patch correctness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We now discuss the phrases in detail as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the fault localization phase, a given buggy program is taken as the input and a list of suspicious code elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', statements or methods) is returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the data pre-processing phase, a given software buggy code snippet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', buggy state- ment) is taken as the input and the processed code tokens are returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' According to existing learning-based APR studies [25, 26], there generally exist three potential ways to pre-process the buggy code: code context, abstraction, and tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' First, code context information refers to other correlated non-buggy lines within the buggy program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Previous work has demonstrated that NMT-based repair models reveal diverse code changes to fix bugs under different contexts [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Second, code abstraction renames some special words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', string and number literals) to a pool of predefined tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Code abstraction has been proven to be an effective method in reducing vocabulary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Third, code tokenization splits source code into words or subwords, which are then converted to ids through a look-up table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ③ Patch Generation Phase ② Data Preprocessing Phase @ Patch Ranking Phase ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Patch Validation Phase ei α1 e1 ai a2 e2 e3 a3 e3 @ Patch Correctness PhaseA Survey of Learning-based Automated Program Repair 1:9 In the patch generation phase, the processed code tokens are first fed into a word embed- ding stack to generate representation vectors, which can capture the semantic meaning of code tokens and their position within a buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then an encoder stack is implemented to derive the encoder’s hidden state, which is further passed into a decoder stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similar to the encoder stack, a decoder stack is implemented to take the hidden states provided by the encoder stack and previously generated tokens as inputs, and returns the probability distribution of the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the patch ranking phase, after the NMT-based repair model is well-trained, a rank strategy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', beam search) is leveraged to prioritize the candidate patches as prediction results based on the probability distribution of the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the patch validation phase, the generated candidate patches are then verified by the available program specification, such as functional test suites or static analysis tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the patch correctness assessment phase, the plausible patches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', passing the exist- ing specification) are assessed to predict their correctness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', whether the plausible are overfitting), which are finally manually checked by developers for deployment in the software pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Fault Localization Fault localization aims to diagnose buggy program elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', statements and methods) without human intervention and has been extensively studied to facilitate the program repair process [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As a crucial start in the learning-based APR pipeline, fault localization provides the repair model with information about where a software bug is and directly influences the performance of the repair model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, the repair accuracy under normal fault localization is usually lower than the circumstance under perfect fault localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, fault localization techniques often leverage various static analysis or dynamic execution information to compute suspiciousness scores (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', probability of being faulty) for each program element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Program elements are then ranked in descending order of their suspiciousness scores, based on which APR techniques can further be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Researchers have proposed a variety of fault localization techniques, such as spectrum-based [123, 182], mutation-based [77, 120], slicing-based [10, 99] and learning-based [78, 93] techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Among them, spectrum-based fault localization (SBFL) has been extensively utilized as a general mechanism to localize the statements that are likely to be faulty in the APR literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Localization Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similar to traditional APR techniques, some learning-based APR techniques rely on existing SBFL fault localization approaches to localize the revealed bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, DLFix [79] adopts Ochiai algorithm to identify a buggy line and extracts all AST nodes (including intermediate ones) related to that buggy line as a replaced subtree for patch generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recoder [186] also assumes the faulty location of a bug is unknown to APR tools and uses Ochiai algorithm with GZoltar [130], which is widely used in existing APR tools, such as RewardRepair [176] and AlphaRepair [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such SBFL techniques exploit runtime information to recognize the program elements that are likely to be faulty when the buggy program is executed by the available test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The crucial insight is that (1) the program elements executed by more failing test suites and fewer passing test suites are likely to be faulty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (2) the program elements executed by more passing test suites and fewer failing suites are likely to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, SBFL produces a list of program elements ranked according to their likelihood of being faulty based on the analysis of the program entities covered by passing and failing tests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Ochiai and Tarantula [85]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [85] have demonstrated that the fault localization techniques may introduce a significant bias in the evaluation of APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The vast majority of learning-based APR ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:10 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen techniques consider repairing software bugs under perfect-based fault localization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Perfect-based fault localization techniques assume that the genuine localization of the bug is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, perfect-based fault localization can provide a fair assessment of APR techniques and the assessment is independent of the localization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, CoCoNut [96] manually checks the bug-fixing pairs in Defects4J benchmark and extracts the changed statements as inputs to the repair model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Subsequently, recent learning-based APR techniques adopt the same or similar processing method to conduct perfect localization, such as CIRCLE [159], CURE [57], SelfAPR [175] and AlphaRepair [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, there exist some techniques attempting to perform fault localization on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, DeepFix [48] proposes an end-to-end approach in which the network reports a ranked list of potentially erroneous lines with a beam search mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similarly, Prophet [91] designs a fault localization algorithm to return a ranked list of program candidate lines to modify by analyzing dynamic execution traces of the test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Szalontai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [138] first localize the nonidiomatic code snippets by LSTM networks and predict the nonidiomatic pattern by a feed-forward neural network, which is fixed by a high-quality alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, Meng et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [107] build a novel fault localization technique based on deep semantic features and transferred knowledge, which is further fed to a fix template prioritization model and a template-based APR technique TBar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Localization Granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' APR techniques consider program elements of different granulari- ties, thus determining the scope of the fault localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In other words, APR and fault localization usually work at the same granularity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, if APR techniques focus on repairing buggy statements (or methods), the fault localization also works at the level of program statements (or methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, a majority of fault localization techniques adopted in learning-based APR techniques usually record the line of a buggy code snippet [57, 79, 80, 96, 159, 186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' There also exists little work considering other granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147] adopt the NMT-based repair model to learn the translation from buggy to fixed code at the method-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3 Data Pre-processing Data pre-processing phase aims to analyze and parse the identified buggy code snippets, which are then passed into neural networks for training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the data pre-processing phase, a given software buggy code snippet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', a buggy function) is taken as the input and the processed code tokens are returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' According to existing learning-based repair studies [25, 26], the data pre-processing phase generally consists of three parts: code abstraction, code context and code tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Code Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Code context generally refers to other correct statements around the buggy lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the manual repair scenario, the context of the buggy code plays a significant role in understanding faulty behaviors and reasoning about potential repairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Developers usually identify the buggy lines, and then analyze how they interact with the rest of the method’s execution, and observe the context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', variables and other methods) in order to come up with the possible repair and pick several tokens from the context to generate the fixed line [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In learning-based APR, the NMT model mimics this process by extracting the code context and the buggy line into a certain code representation to preserve the necessary context that allows the model to predict the possible fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing learning-based APR techniques typically consider the surrounding source code relevant to the buggy statement as context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These techniques typically employ context in various ways, such as extracting code near the buggy statement within the buggy method, class, and even file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' On the one hand, a broad context contains plenty of essential fix ingredients, while such a large vocabulary size introduces noise that negatively affects the repair performance of the NMT model ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:11 due to the tricky long-term dependency problem in NMT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' On the other hand, a narrow context contains too little information to capture the proper semantics of the buggy statement and leads to incorrect patches generation due to a lack of necessary vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' There seems to be a trade-off relationship between vocabulary size and context size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, our survey concludes the code context into four granularities: context-free, line-level context, method-level context, and class-level context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Context-free means that NMT models only consider buggy statements without any additional context information [32, 51, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Mashhadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [103] consider single statement bugs from the ManySStuBs4J dataset and extract the buggy statement as a source side and the fixed statement as a target side from bug-fixing commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [32] provide NMT models with a single program line that contains a buggy statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, previous work demonstrates that fixing nearly 90% of bugs requires new vocabulary relative to the buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' NMT repair models suffer from capturing enough information from the buggy code alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Statement-level context means that the buggy code and several statements around it are fed to MNT repair models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, TFix [13] extracts the two neighboring statements of the buggy code as the code context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [26] extract statement-level code changes by the “git dif” command and employ data-flow dependencies to capture more critical information around the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Method-level context means that the method to which the buggy line belongs is fully fed into the model [96, 147, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is the most commonly used type of context in literature as it often contains enough information for repairing the bug, such as the type of variables and the function of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Tufano et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [146] focus on the method-level context since (1) the functionality to be fixed is usually implemented in program methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) the methods provide neural networks with meaningful abundant context information, such as literals and variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CoCoNuT [20] extracts the entire method of the buggy code as context, which is encoded as a separate input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Class-level context means that the class to which this buggy code belongs is fed into the NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is a relatively broad context, while it can provide the model with rich information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, SequenceR [24] considers the class-level context and conducts abstract buggy context from the buggy class, which captures the most important context around the buggy source code and reduces the complexity of the input sequence to 1,000 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Hoppity [31] takes the whole buggy file as the context with a length limit of 500 tokens nodes in the AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Code Abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Code abstraction aims to limit the number of words the NMT models need to process by renaming raw words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', function names and string literals) to a set of predefined tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Previous work demonstrates that it is challenging for NMT models to learn bug-fixing transformation patterns due to the huge vocabulary of source code [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, NMT models usually employ a beam-search decoding strategy to output repair candidates by a probability distribution over all words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The search space can be extremely large with many possible words in the source code, resulting in inefficient patch generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In our survey, a considerable number of learning-based papers we collect employ the abstracted source code to tackle this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such abstraction operation means the original source code is not directly fed into the NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Benefiting from the abstracted code, we can (1) reduce the size of vocabulary significantly and the frequency of specific tokens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) filter out irrelevant information and improve the efficiency of the NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Generally, the natural elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', identifiers and literal) in the source code are renamed, while the core semantic information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', idioms) should be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147] propose the first code abstraction approach in the ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:12 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen int GetMaxCommonDivisor(int m, int n){ int r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' while (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='=0){ r=m%n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' m=n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' n=r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' } return n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' } raw buggy code (a) raw buggy code raw fixed code int GetMaxCommonDivisor(int m, int n){ int r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' while (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='=0){ r=m%n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' m=n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' n=r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' } return m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' } (b) raw fixed code TYPE_1 METHOD_1(TYPE_1 VAR_1, TYPE_1 VAR_2){ TYPE_1 VAR_3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' while (VAR_2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='=NUMBER_1){ VAR_3=VAR_1%VAR_2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' VAR_1=VAR_2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' VAR_2=VAR_3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' } return VAR_2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' } abstracted buggy code (c) abstracted buggy code abstracted fixed code TYPE_1 METHOD_1(TYPE_1 VAR_1, TYPE_1 VAR_2){ TYPE_1 VAR_3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' while (VAR_2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='=NUMBER_1){ VAR_3=VAR_1%VAR_2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' VAR_1=VAR_2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' VAR_2=VAR_3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' } return VAR_1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' } (d) abstracted fixed code Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A simple example of code abstraction learning-based APR field by (1 ) adopting a lexer to tokenize the raw source code as a stream of tokens based on Another Tool for Language Recognition (ANTLR) [122];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) passing the stream of tokens into a parser to identify the role of each identifier and literals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', whether it represents a variable, method, or type name);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) replacing each identifier and literal with a unique ID to generate the abstracted source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, they extract the idioms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', tokens that appear many times) and keep their original textual tokens in the abstraction process because such idioms contain beneficial semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The typical code abstraction example is presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similarly, CoCoNut [96] and CURE [57] only abstract string and number literals except for the frequent numbers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', 0 and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' DLFix [79] adopts a novel abstraction strategy to alpha-rename the variables, so as to learn the fix between methods with similar scenarios while having different variable names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' DLFix also keeps the type of the variable to avoid accidental clashing names and maintains a mapping table to recover the actual names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recoder [186] abstracts infrequent identifiers with placeholders to make the neural network learns to generate placeholders for these identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Although a variety of techniques adopt the code abstraction strategy (such as Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147]) to limit the vocabulary size and make the transformer concentrate on learning common patterns from different code changes, we still find some techniques prefer raw source code [159, 186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, developers may name one function as SetHeightValue to indicate that this function can set the value of height as they want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' If this name is abstracted directly as func_1, critical semantic information would be missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Instead of renaming rare identifiers through a custom abstraction process, SequenceR [24] utilizes the copy mechanism to generate candidate patches with a large set of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [25] adopt the raw source code as they think abstracted code may hide valuable information about the variable that can be learned by word embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A strategy similar to Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [25] is also implemented in other learning-based APR techniques, such as in CODIT [20], CIRCLE [159] and TFix [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3 Code Tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Code tokenization aims to split source code into a stream of tokens, which are then converted to ids through a look-up table2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These id numbers are in turn used by the repair models for further processing and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A simple tokenization approach can be conducted by dividing the source code into individual characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The core concept of this char-level tokenization is that although the source code has many different words, it has a limited number of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This approach is straightforward and leads to an exceeding small vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, it leads to a relatively long tokenized sequence with the splitting of each world into all characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' More importantly, it is pretty difficult for repair models to meaningful input representations as characters alone do not have semantic meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Generally, there exist two main granularities of code tokenizers used in learning-based APR techniques: word-level tokenizers and [40] and subword-level tokenization [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The word-level tokenization means that a sentence is divided according to its words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', space- separated), which is widely used in NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, different from natural language, words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', variable names) in programming languages can be created arbitrarily, leading to a more irregular vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Previous work [24] demonstrates that it is difficult for NMT models to handle a code vocabulary size larger than 560,000 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This kind of granularity often causes the out- of-vocabulary (OOV) problem for infrequent tokens, and the model could be more efficient for an excessively large vocabulary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To address this issue, VRepair employs a word-level tokenization to tokenize C source code and the copy mechanism to deal with the out-of-vocabulary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, CoCoNut [96] designs a code-aware space-separated tokenization algorithm that is specific to programming languages by (1) separating operators from variables as they might not be space-separated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) considering underscores, camel letters, and numbers as separators as many words are composed of multiple words without separation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', SetHeightValue);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) introducing a new token to mark where the camel case split occurs to regenerate source code from the list of tokens generated correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The subword-level tokenization splits rare tokens into multiple subwords instead of directly adding full tokens into the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, the frequent words should not be split into smaller subwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This kind of granularity can reduce the vocabulary size significantly and is widely used in the learning-based APR field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Technically, there exist several subword-level tokenization techniques, such as byte-pair encoding (BPE), byte-level byte-pair encoding (BBPE) [135] and SentencePiece [68], listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (1) BPE tokenizer generally needs to be trained upon a given dataset by (1) leveraging a pre- tokenizer to splits the dataset into words by space-separated tokenization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) creating a set of unique words and counting the frequency of each word in the dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) building a base vocabulary with all symbols that occur in the set of unique words and learning merge rules to form a new symbol from two symbols of the base vocabulary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (4) repeating the above process until the vocabulary is reduced to a reasonable size, which is a pre-defined hyperparameter, before training the tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, VulRepair [40] employs a BPE algorithm to train a subword tokenizer on eight different programming languages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Ruby, JavaScript, Go, Python, Java, PHP, C, C#) [157] and is suitable for tokenizing source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the learning-based APR literature, a majority of repair studies adopt BPE as the tokenization technique, such as CURE [57], CoCoNut [96], SeqTrans [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results have demonstrated the effectiveness of BPE in reducing vocabulary size and mitigating the OOV problem by extracting the most frequent subwords and merging the most frequent byte pair iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) BBPE refines BPE by employing bytes as the base vocabulary, ensuring that every base character is included with a proper vocabulary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, AlphaRepair [165] builds a 2https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='co/Salesforce/codet5-base/blob/main/vocab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='json ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:14 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen BBPE-based tokenizer to reduce the vocabulary size by breaking uncommon long words into meaningful subwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) SentencePiece contains the space in the base vocabulary and utilizes the existing BPE al- gorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', BPE) to create the desired vocabulary by regarding the source code as a raw input stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, before entering source code into the neural network, sev- eral learning-based APR techniques use SentencePiece to divide words into a sequence of subtokens, such as SelfAPR[175], RewardRepair [176] and CIRCLE [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='4 Patch Generation In the learning-based APR context, to apply NMT repair models to high-level programming lan- guages, the code snippets need to be converted to embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then an NMT repair model is built on top of the encoder-decoder architecture [150] to learn the repair patterns automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, the mapping from buggy code to fixed code is optimized by updating the parameters of the designed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, it is crucial to determine (1) how to represent the source code (with which format) as input for word embedding, referred to as code representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (2) how to design the specific architecture (with which neural network) as encoder-decoder for repair transformation learning, referred to as model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, various strategies have been proposed to represent the source code as the input for NMT repair models, which can be categorized into three classes: sequence-based, tree-based and graph-based representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Sequence-based Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These techniques divide the textual source code as a sequence of tokens and treat APR as a token-to-token translation task based on a sequence-to-sequence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Code Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Considering the buggy lines and the context, there generally exist four different ways to sequence the textual code tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (1) Raw representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similar to NMT, which translates a sentence from one source language (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', English) to another target language(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Chinese), most sequence-based techniques directly feed the model with the buggy code snippet [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147] extract the buggy method and train an NMT model for method-to-method translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The size of this code snippet depends on the choice of the buggy code and code context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, the raw representation is unaware of the difference between the buggy code and the code context, as these two parts are sent into the encoder together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As a result, the transformation rules may be applied in some correct lines, limiting the repair performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) Context representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The context representation splits the buggy code and the code context, then feeds them into two encoders separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Under this circumstance, the model is aware of the difference between buggy code and the corresponding context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Lutellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [57, 96] attempt to encode these two parts separately and then merge the encoding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, it is challenging to merge the two separated encoding vectors and eliminate the semantic gaps between the two encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) Prompt representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The prompt representation refers to a text-in-text-out input format and can effectively concatenates different input components with some prefixed prompt [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The prefixed prompt is a piece of tokens inserted in the input, so that the original task can be formulated as a language modeling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [159] employs manually designed prompt template to convert buggy code and corresponding context into a unified fill-in-the- blank format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, they employ “Buggy line:” and “Context:” to denote the buggy ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:15 code and code context, and then employ “The fixed code is:” to guide the NMT model to generate candidate patches according to the previous input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This mechanism has been proven effective in bridging the gap between pre-trained tasks and the downstream task, facilitating fine-tuning pre-trained models for APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (4) Mask representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The mask representation replaces the buggy code with mask tokens and queries NMT models to fill the masks with the correct code lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This mechanism views the APR problem as a cloze task and usually adopts the pre-trained model as the query model in the learning-based APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [165] transform the original buggy code into a comment and generate multiple mask lines with templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The input is represented by comment buggy code, context before buggy code, mask lines and context after buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, the buggy code is masked randomly from one token to the whole line, and researchers expect to generate every possible patch for different situations within a limited candidate patch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Compared with the above three representation strategies, the mask representation can adopt pre-trained models to predict randomly masked tokens to perform cloze-style APR without any additional training on the bug-fixing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Model Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Sequenced-based techniques usually treat the source code as a sequence of tokens and adopt existing sequence-to-sequence architectures in the NLP field instead of designing new network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, CoCoNut [96] adopts two fully convolutional (FConv) encoders to represent the buggy lines and the context separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' One common encoder architecture is long short-term memory (LSTM), and it resolves the long-term dependency problem of the RNN module by introducing the gate mechanism and ensures that short-term memory is not neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, SequenceR [24] is based on an LSTM encoder-decoder architecture with copy mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As a powerful kind of DL architecture, transformer can model global dependencies between input and output effectively thanks to the attention mechanism and has been adopted in existing APR studies, such as Bug-Transformer [171], SeqTrans [26] and VRepair [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, the usage of pre-trained models has gradually attracted the attention of researchers in the learning-based APR community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such models are first pre-trained by self-supervised training on a large-scale unlabeled corpus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CodeSearchNet), and then transferred to benefit multiple downstream tasks by fine-tuning on a limited labeled corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Mashhadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [103] employ CodeBERT, a bimodal pre-trained language model for both natural and programming languages, to fix Java single-line bugs by fine-tuning on the ManySStuBs4J small and large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CURE [57] applies a pre-trained GPT module to further revise an NMT-based APR architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CoCoNut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CIRCLE [159] proposes a T5-based program repair framework equipped with continual learning ability across multiple languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We will discuss the application of pre-trained models in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' State-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the following, we discuss these individual sequence-based patch generation techniques in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Tufano et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [146] design an NMT model to generate the same patches applied by developers under a narrow context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They reduce the vocabulary size by mapping the method of the code to a specific ID and feed the model with pairs of methods before and after the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The model can replicate up to 36% of the buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, it can be applied to refactoring and other code relating activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Current works aim at exploring fixes in a limited search space, which may not contain the correct patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Hata et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [51] follow the recently NMT-based approach and use an encoder-decoder model Ratchet with multi-layer attention to fix bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They perform an empirical study with five large software projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, they collect a fine-grained dataset from these projects and try ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:16 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen to ignore noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They train and evaluate Ratchet on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that Ratchet performs at least as well as pattern-based APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, Ratchet’s output was considered helpful in fixing the bugs on many occasions, even if the fix was not 100% correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Lutellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [96] propose CoCoNut, a novel generate&validate technique with a new context- aware NMT architecture that separately inputs the buggy line and method context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They further combine CNN (FConv architecture) with the NMT model to improve the accuracy of generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After collecting a large dataset from four programming languages and training the model on it, CoCoNut is then evaluated on six benchmarks(also from four programming languages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It turns out that CoCoNut outperforms previous APR tools and is capable of fixing 300 more bugs other APR tools fail to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, CoCoNut proves that FConv architecture can outperform LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Further, Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [57] propose CURE, a novel NMT-based program repair technique to fix Java bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They pre-train a programming language model on a large corpus and combine it with NMT architecture to learn code syntax and fix patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also apply a code-aware search strategy and a new subword tokenization technique to improve the accuracy of generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This model outperforms SequenceR and CoCoNut APR tools on Defects4J and QuixBugs benchmarks under different beam search sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Lutellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [95] propose ENCORE, a new end-to-end APR technique that leverages the NMT model to generate bug fixes for Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Evaluating ENCORE on two Java benchmarks proves that it can fix diverse bugs, and further experiments on Python, C++, and JS benchmarks proves that it can handle bugs in different programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also present attention maps to explain why certain fixes are generated or not by ENCORE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [24] propose SequenceR, a novel end-to-end approach based on sequence-to-sequence learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They combine LSTM encoder-decoder architecture with copy mechanism to address the problem of a large vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' First, they apply state-of-the-art fault localization techniques to identify the buggy method and the suspicious buggy lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then, they perform a novel buggy context abstraction process that intelligently organizes the fault localization data into a suitable representation for the deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, SequenceR generates multiple patches for the buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Although their approach can only be applied to single-line buggy code, this model outperforms the APR tool of Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' on Defects4J benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, they prove that copy mechanism can improve the accuracy of generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [176] introduce RewardRepair as a neural program repair approach for fixing bugs in Java code based on transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply a novel training strategy and feed the model with compiling and testing execution information to improve the quality of generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This model is then evaluated on four benchmarks, Defects4J v1 and v2, Bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='jar, and QuixBugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that this model has lower cross-entropy than previous APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, RewardRepair outperforms SequenceR, CoCoNut and CURE in terms of top-k accuracy on Defects4J benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Previous neural program repair approaches focus on supervised training and lack project-specific knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [175] propose, SelfAPR, a self-supervised training approach with test execution diagnostics based on a transformer neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SelfAPR consists of two components, training sample generator and neural network optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The first part generates perturbed programs with a perturbing model and tests it to capture compile errors and execute failures information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The second part is fed with the previous information and outputs n best patches with beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SelfAPR is capable of repairing ten bugs that are never repaired before by the supervised neural repair models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, evaluation results prove the effectiveness of self-supervised training and its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [171] propose a new transformer-based APR technique, Bug-Transformer, to fix buggy code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It applies a novel token pair encoding (TPE) approach to reduce vocabulary size by compressing code structure while preserving semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, they apply a novel ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:17 rename mechanism to preserve semantic features for code abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Bug-Transformer leverages the transformer architecture and is fine-tuned for learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is then evaluated on Java benchmarks and outperforms other baseline models such as Bug2Fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Rahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [128] present a bidirectional LSTM model for code evaluation and repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first train the model as a Seq2Seq model with abundant source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then they fine-tune it for error detection and provide suggestions for code repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The model is evaluated on Aizu Online Judge (AOJ) system, and the result shows that this model outperforms previous RNN and LSTM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It also proves to be useful for novice programmers and accelerates the code evaluation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [55] propose an enhanced transformer-based APR technique by introducing a general pyramid encoder, which is added in between layers of regular multi-layer encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For the purpose of testing the generality of the pyramid encoder, they combined this encoder with different attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct experiments on Juliet Test Suite for C/C++ and Java to evaluate seq2seq models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that seq2seq models can be well applied in providing suggestions to potential errors and have a decent correct rate in code auto-correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, their results on transfer learning point out a way of processing this small dataset using the pre-trained model as an encoder, which boosts the performance by a large amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Tree-based Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Sequence-based techniques usually adopt sequence-to-sequence models for patch generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, these techniques ignore code structure information because they are designed for NLP, which is significantly different from programming language with strict syntactic and grammatical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The generated patches of these techniques may suffer from syntax errors that cause compilers to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As a result, researchers recently propose various tree-based generation techniques by considering the syntactic structure of source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These techniques treat the APR problem as a tree transformation learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Code Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A common solution is to parse the source code into an AST and adopt a tree-aware model to perform patch generation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', structure-aware representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, given a bug-fixing method pair 𝑀𝑏 and 𝑀𝑓 representing the buggy and fixed method, DLFix [79] first extracts a buggy AST for 𝑀𝑏 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=',𝑇𝑏), a fixed AST for 𝑀𝑓 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=',𝑇𝑓 ), a buggy sub-AST (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e,𝑇 𝑠 𝑏 ) and a fixed sub-AST (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', 𝑇 𝑠 𝑓 ) between 𝑇𝑏 and 𝑇𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' DLFix then adopts an existing summarization model to encoder 𝑇 𝑠 𝑏 as a single node 𝑆𝑠 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, the buggy method 𝑀𝑏 can be represented as a context tree by replacing 𝑇 𝑠 𝑏 in 𝑇𝑏 with 𝑆𝑠 𝑏 and a sub-changed tree 𝑇 𝑠 𝑏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The fixed method 𝑀𝑓 is represented in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As tree-based representation contains the structure information, which cannot be directly de- ployed to sequenced-based neural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, an additional code representation strategy is utilized to parse the tree representation as a sequential traverse sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', sequential-traverse representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [140] parse the source code into AST representation, which is further translated into a sequence of rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The sequence of rules can be processed by the vanilla transformer [150] while capturing the grammar and syntax information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CODIT [20] first identifies the edited AST nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', the inserting, deleting, and updating) and selects the minimal subtree of each AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CODIT then collects the edit context by including the nodes that connect the root of the method to the root of the changed tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CODIT expands the considered context until the context exceeds a maximum tree size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Given each bug-fixing method pair, CODIT extracts a buggy AST and fixed AST, and then converts the ASTs to their tree representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Model Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Most NMT-based APR models treat patch generation as a machine translation from buggy code to a fixed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, such models could not capture the information of code structures and suffer from handling the context of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Tree-based encoders consider the structure features of source code, such as AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, DLFix [79] parses the source code to AST and adopts a tree-based LSTM to represent the changed and context sub-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:18 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [30] encode the AST with a sequential bidirectional LSTM by enumerating a depth-first traversal of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' State-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the following, we discuss these individual tree-based patch generation tech- niques in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' One variant of LSTM is tree-LSTM architecture which leverages tree structure AST or graph to parse the syntax information of buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Yi et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [79] propose a tree-based model with 2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The first layer is used to learn context information, which is implemented by a tree-based LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Another layer is used to catch code transformation between changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After training, filtering and re-ranking (with the help of a CNN layer), the model generates a bunch of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' DLFix is designed for single-statement bug fixing, and it shows the potential of a tree-based model in bug fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Li et al[80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' propose DEAR, a learning-based approach for multi-hunk multi-statement fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They design a fault localization technique based on traditional SBFL and data flow analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This technique can acquire multi-hunks that need to be fixed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They further design a two-tier tree-based LSTM with an attention layer for fixing multiple statements in the suitable fixing context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, they apply cycle training to learn code transformation and fix need-to-be-fixed-together bugs detected by the fault localization technique mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This approach outperforms many DL APR techniques such as DLFix and CoCoNut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, it can fix multi-statement bugs which other APR tools fail to fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chakraborty et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [20] propose a tree-based APR technique CODIT to learn code changes from the wild and generate patches for software bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CODIT transforms the correct (or buggy) code snippet into the parse tree and generates the deleted (or added) subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CODIT then predicts the structural changes using a tree-based translation model among the subtrees and employs token names to concrete the structure using a token generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The former tree-based model takes the previous code tree structure and generates a new tree with the maximum likelihood, while the latter token generation model takes tokens and types of tokens in the code and generates new tokens with the help of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The authors conduct a real-world bug-fixing dataset Code-Change-Data from 48 open-source projects and employ Pull-Request-Data [24] and Defect4J [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results on these three datasets illustrate CODIT outperforms existing seq2seq models, highlighting the potential of the tree-based models in APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [30] present a novel model SSC (Share, Specialize, and Compete) to repair semantic bugs, which means fixing non-syntactic bugs in source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The input code snippet is encoded with a neural network on the AST level, and each repair type is associated with its own specialized neural module, which emits a score for every repair candidate of that type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The authors conduct a large-scale corpus by mining code snippets from real-world Python projects on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results indicate that it outperforms existing sequence-to-sequence models with an attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [177] present a novel approach to predict code transformation at AST level based on structural information for Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For structured prediction of source code transforms, they establish a conditional random field (CRF) for the transform prediction, then define the feature functions used in CRF, and finally train the CRF model for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They use the learned model to predict transforms for the new, unseen buggy code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct a large-scale experimental evaluation on a large dataset of 4,590,679 bug-fixing commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results show the great performance of the proposed technique to generate patches by predicting code structure transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3 Graph-based Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These techniques transform source code into graph representations with contextual information and frame the APR problem in terms of learning a sequence of graph transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:19 Code Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To capture the neighbor relations between AST nodes, Recoder [186] treats AST as a directional graph where the nodes denote AST nodes and the edges denote the relationship between each node and its children and left sibling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [166] consider the context structure by data and control dependencies captured by a data dependence graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', DDG) and a control dependence graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CDG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Model Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing graph-based APR techniques usually design graph neural networks and their variants to capture graph representation and perform patch generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Hoppity [31] adopts a gated graph neural network (GGNN) to treat the AST as a graph, where a candidate patch is generated by a sequence of predictions, including the position of graph nodes and corresponding graph edits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [166] design a graph neural network (GNN) for obtaining a graph representation by first converting DDG and CDG into two graph representations and then fusing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' State-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the following, we discuss these individual graph-based patch generation techniques in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [186] propose Recoder, a syntax-guided edit decoder that uses a novel provider/decider architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recoder takes a buggy statement and its context as input and generates edits as output by (1) embedding the buggy statement and its context by a code reader;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) embedding the partial AST of the edits by a AST reader;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) embedding a path from the root node to a non-terminal node by a tree path reader;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (4) producing a probability of each choice for expanding the non-terminal node based on previous embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The authors evaluate Recoder on four widely-adopted Java benchmarks: Defects4J v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 with 395 bugs, Defects4J v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='0 with 420 bugs, QuixBugs with 40 bugs and IntroClassJava with 297 bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results indicate that Recoder is the first learning-based APR technique that outperforms existing traditional techniques on these four Java benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [166] introduce M3V, a new multi-modal multi-view context embedding approach to predict repair operators for buggy Java code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply a GNN with multi-view graph-based context structure embedding to capture data and control dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also present a tree- LSTM with tree-based context signature embedding for capturing high-level semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After M3V is evaluated on repairing two common types of bugs: null pointer exceptions and index out of bounds, results show that M3V is effective in predicting repair operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Dinella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [31] introduce HOPPITY, an end-to-end learning-based tool for detecting and fixing bugs in JS programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply one step graph edit which is called graph transformation for the model and feed the model with the graph structure of buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This model is then trained to detect and fix more complex and diverse bugs which require adding or deleting code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It outperforms other tools on the same baseline with or without the perfect bug locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [115] propose GRAPHIX, a medium-scale graph edit model which is pre-trained with deleted sub-tree reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This model is trained with both abstract and concrete code to learn both structural and semantic code patterns, and it suggests that abstraction may be unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is then evaluated on the Java benchmark from Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147] and it turns out that this model is as competitive as large-scale transformer models and outperforms other state-of-art APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [139] propose a novel end-to-end approach Grasp for repairing buggy Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They represent the buggy method as a graph to retain structural information and apply the Graph- to-Sequence model to capture information from the graph, overcoming the problem of information missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Grasp is then evaluated on the Defects4J benchmark as well as real-world bugs from open-source projects and it achieves good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Yasunaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [172] propose DrRepair to repair C/C++ bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They parse the buggy source code into a joint graph representation with diagnostic feedback that captures the semantic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The graph representation takes all identifiers in the source code and any symbols in the diagnostic ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:20 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen feedback as nodes, and connects the same symbols as edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They then design a GNN model for learning the graph representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, they apply a self-supervised learning paradigm that can generate extra patches by corrupting unlabeled programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also discover that pre-training on unlabeled programs improves accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The model is evaluated on DeepFix and SPoC datasets and it outperforms existing state-of-art APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='5 Patch Ranking The patch generation is a search process for the maximum in the combinatorial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Given the max output length l and the size of vocabulary V, the total number of candidate patches that the decoder can generate reaches 𝑉 𝑙, all of which it is impossible to validate in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Developers may spend a considerable amount of effort to assess the correctness of the generated candidate patches manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In such a scenario, only inspecting fewer repair candidates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Top-1 and Top-5) that have a high probability of being correct is more practical and reduces the valuable manual effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As a result, a patch ranking strategy is crucial to ensure the inference efficiency of the model and relieve the burden of patch validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Beam search is an effective heuristic search algorithm to rank the outputs in previous NMT applications [157] and is the most common patch ranking strategy in learning-based APR studies, such as CIRCLE [159], SelfAPR [175], RewardRepair [176] and Recoder [186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, for every iteration, the beam search algorithm selects the 𝑘 most probable tokens for the patch (corresponding to beam size 𝑘) and ranks them according to the likelihood estimation score of the next 𝑑 prediction steps (corresponding to search depth 𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' At last, the top 𝑘 most likely patches are maintained for validation in the next procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Beam search provides a great trade-off between repair accuracy versus inference cost via its flexible choice of beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, the vanilla beam search considers only the log probability to generate the next token while ignoring the code-related information, such as variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, it may generate high-score patches with unknown variables, leading to uncompilable candidate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In addition to directly applying the existing beam search strategy, researchers design some novel strategies to filter out low-probability patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, CURE [57] designs a code-aware beam search strategy to generate more compilable and correct patches based on valid-identifier check and length control components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The code-aware strategy first performs static analysis to identify all valid tokens used for sequence generation and then prompts beam search to generate sequences of a similar length to the buggy line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' DLFix [79] first derives the possible candidate patches by program analysis filtering and ranks the list of possible patches by a CNN-based binary classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The classifier adopts a Word2Vec model as the encoder stack at the char-level, followed by a CNN stack as the learning stack (containing a Convolutional layer, pooling, and fully connected layers), and a softmax function as the classification stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then DLFix ranks the given list of patches based on their possibilities of being a correct patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Further, DEAR [80] applies a set of filters to verify the program semantics and ranks the candidate patches in the same manner as DLFix does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, AlphaRepair [165] designs a patch ranking strategy based on a masked language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, given a candidate patch, AlphaRepair calculates its priority score by (1) extracting all generated tokens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) masking out only one of the tokens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (3) querying CodeBERT to obtain the conditional probability of that token;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (4) repeating the same process for all other previous mask tokens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (5) computing the joint score which is an average of the individual token probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='6 Patch Validation Patch validation takes a ranked list of candidate patches generated by NMT models as the input and returns the plausible patches for deployment, which is a crucial phase in the learning-based APR pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, developers may spend a considerable amount of effort to inspect the candidate ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:21 patches manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, researchers usually recompile the buggy program with the generated patch to check if it can pass the available test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In such a scenario, hundreds or even thousands of candidate patches can be filtered automatically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', 1000 candidate patches per bug in CIRCLE [159]), which may benefit its adoption in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similar to traditional APR techniques, most learning-based techniques adopt a test-based vali- dation strategy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', executing available test suites against each candidate patch) to assess patch correctness [57, 79, 80, 96, 159, 186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, CIRCLE [159] filters out the candidate patches that do not compile or do not pass available test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' There generally exist two criteria for the validation process: (1) the passing test suites that make the buggy program pass should still pass on the patched program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (2) the fault-triggering test suites that fail on the buggy program should pass on the patched program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' All candidate patches are checked until a plausible patch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', a patch passing all test suites) is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, CIRCLE stops the validation process and reports the plausible patch for manual investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, it can be extremely time-consuming to compile a large number of candidate patches and repeat all test executions to identify plausible patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CURE [57] generates 10,000 candidate patches per bug and validates the top 5,000 ones considering the overhead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similarly, AlphaRepair [165] returns at most 5,000 candidate patches for each bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To reduce the validation cost, some learning-based APR techniques return an acceptable amount of candidate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, RewardRepair configures the beam size as 200 and outputs the 200 best patches per bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similarly, SelfAPR adopts a beam search size of 50 and Recoder generates 100 valid candidate patches for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, similar to traditional APR techniques [56, 87], there exist several learning-based ones limiting maximum time for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, DEAR [80] and DLFix [79] set a 5-hour running-time limit for patch generation and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In addition to the above strategies in patch validation, the learning-based APR community benefits from some optimizations to speed up the dynamic execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, AlphaRepair [165] adopts the UniAPR [22] tool to validate the candidate patches on-the-fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Inspired by the PraPR work, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [22] present UniAPR as the first unified on-the-fly patch validation framework to speed up APR techniques for JVM-based languages at both the source and byte-code levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They leverage the JVM HotSwap mechanism and Java Agent technology to implement this framework。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='Besides, they apply the JVM resetting technique based on the ASM byte-code manipulation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Since previous work shows that on-the-fly patch validation can be imprecise, they reset the JVM state right after each patch execution to address such an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The evaluation shows that this work can speed up state-of-the-art representative APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Bento et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [12] introduce SeAPR, the first self-boosted patch validation tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Based on the idea that patches similar to earlier high-quality/low-quality patches should be promoted/degraded, they leverage the patch-execution information on its similarity with the executed patches to update each patch’s priority score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The evaluation shows that SeAPR can substantially speed up the studied APR techniques and its performance is stable under different formulae for computing patch priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Since previous APR techniques often neglect the impact of test selection for each patch, Lou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [92] conduct an extensive study to investigate the impact of Regression Test Selection (RTS) on APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They explore three representative RTS techniques for 12 state-of-the-art APR systems at different levels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', class/method/statement levels) with over 2M patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that all studied RTS techniques can substantially improve APR efficiency and should be considered in future APR work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, method- and statement-level RTS substantially outperform class-level RTS, and are more recommended for APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:22 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='7 Patch Correctness Patch correctness is an additional phase for developers to further filter out overfitting patches after patch validation, so as to improve the quality of returned patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='6, a majority of existing learning-based APR techniques usually leverage the developer-written test suites as the program specification to assess the correctness of the generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, the test suite is an incomplete specification as it only describes a part of the program’s behavioral space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As a result, it is fundamentally difficult to achieve high precision for returned patches due to the incomplete program specification [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The plausible patch passing the available test suites may not generalize to other potential test suites, leading to a long-standing challenge of APR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', the overfitting issue) [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Previous studies [90, 91] have demonstrated that a majority of the overfitting patches are equivalent to a single modification that deletes the buggy functionality and does not actually fix the detected bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Under the circumstances, it takes enormous time and effort to manually filter out the overfitting patches, even resulting in a negative debugging performance [141, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Different from some traditional APR techniques that guide the repair process to generate patches with a high probability of being correct, DL techniques lead to an end-to-end repair mechanism and the patches are generated in a black-box manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The overfitting issue in learning-based APR is more significant and severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, researchers have proposed a mass of automated patch correctness assessment (APCA) techniques to identify whether a plausible patch is indeed correct or overfitting [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' There are usually two types of traditional APCA techniques based on the employed patch features: static and dynamic [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The former focuses on the transformation patterns or the static syntactic similarity, while the latter relies on the dynamic execution outcomes by additional test suites from automated test generation tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Evosuite [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, inspired by large-scale patch benchmarks being released, some learning-based APCA techniques have been proposed to predict patch correctness with the assistance of DL models [142, 143, 174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In general, such learning-based APCA techniques extract the code features by code embedding and build a classifier model to directly perform patch prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We view patch correctness as an essential component of the learning-based APR pipeline and focus on such APCA techniques that employ DL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Now, we list the existing learning-based techniques to predict patch correctness automated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [82] propose Cache, a novel context-aware code change embedding technique for the patch correctness task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They leverage context information of unchanged code and parse the AST nodes to capture the code structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct various experiments to evaluate Cache on diverse patch benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results show that Cache achieves significantly better performance than both previous representation learning techniques and existing APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [174] propose ODS, a learning-based approach to identify overfitting patches based on static code features and supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ODS first defines and extracts a set of 202 static code features from the AST to represent a candidate patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ODS then adopts the gradient boosting with the captured code features and patch correctness labels to train a classifier for patch correctness classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct on three benchmarks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Defects4J, Bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='jar and Bears) and the results show that ODS achieves an accuracy of 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='9% in detecting overfitting patches from 26 projects, and outperforms other state-of-the-art techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Considering most existing APCA techniques evaluated on limited datasets, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [158] conduct an extensive empirical study of patch correctness on Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' First, they collect a large-scale real-world dataset for patch correctness, containing 1,988 patches generated by the recent PraPR [43] APR tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then they revisit state-of-the-art APCA techniques on the new dataset, including static, dynamic, and learning-based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that learning-based ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:23 techniques tend to suffer from the overfitting issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, the performance of dynamic techniques significantly drops when encountering patches with more complicated changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [145] attempt to formulate the patch correctness assessment problem as a question answering problem, which can assess the semantic correlation between a bug report (question) and a patch description (answer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They introduce QUATRAIN, a supervised learning approach that exploits a deep NLP model to predict patch correctness based on the relatedness of a bug report with a patch description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' QUATRAIN first mines bug reports for bug datasets automatically and generates patch descriptions by existing commit message generation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' QUATRAIN then leverages an NLP model to capture the semantic correlation between bug reports and patch descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They evaluate QUATRAIN on a large dataset of 9135 patches from three Java datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Defects4j, Bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='jar, and Bears).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results demonstrate that QUATRAIN achieves comparable or better performance against other state-of-the-art dynamic and static techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, QUATRAIN is proven practical in learning the relationship between bug reports and code change descriptions for the patch prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Different from most existing studies focusing on Java programs, Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [168] propose Crex to predict patch correctness in C programs based on execution semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first leverage transfer learning to extract semantics from micro-traces in buggy C code on the function level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They then perform semantic similarity computation to denote patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They evaluate Crex on a set of 212 patches generated by the CoCoNut APR tool on CodeFlaws programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The experimental results indicate that Crex can achieve high precision and recall in predicting patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [142] introduce BATS, an unsupervised learning-based approach to predict patch correctness based on failing test specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' BATS first constructs a search space of historical patches with failing test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Given a plausible patch, BATS identifies similar failing test cases in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' BATS then calculates the similarity of historical patches and the plausible patch based on the failing test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The plausible patch is predicted as correct if the similarity score is larger than a predefined threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' otherwise it is predicted as incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After collecting plausible patches from 32 APR tools to construct a large dataset, they evaluate the performance of BATS on Defects4J benchmarks with some standard classification metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', recall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' BATS outperforms existing techniques in identifying correct patches and filtering out incorrect patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Csuvik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [28] present a Doc2Vec model to explore the nature of similarity-based approach for patch correctness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They feed the model with a token sequence under a simple rule and the model measure the similarity between plausible patches and original programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They find that plain source code embeddings fail to capture nuanced code semantics, thus a more sophisticated technique is needed to validate patches correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Different from traditional APCA techniques relying on dynamic information or manually-crafted heuristics, Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [143] investigate the feasibility of code representation learning to encode the properties of patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They consider different representation learning techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Doc2Vec, BERT, code2vec, and CC2Vec) to get embedding vectors for code changes, including pre-trained models and the retraining of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also investigate the discriminative power of learned features in a classification training pipeline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Decision tree, Logistic regression, and Naive Bayes) for patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Based on previous work [143], Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [144] further leverage representation learning models and supervised learning algorithms to investigate the feasibility of statically predicting patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They implement two patch correctness predicting frame- works, Leopard and Panther (upgraded version of Leopard), to investigate the discriminative power of the deep learned features by training machine learning classifiers to predict correct patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, they run exploratory experiments assessing the possibility of selecting cutoff similarity scores between learned embeddings of buggy code and patched code snippets for heuristically filtering out incorrect patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After evaluating several models on the same dataset, they find that ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:24 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen the performance of these models on learned embedding features is promising when compared against the state-of-the-art techniques which applies dynamic execution traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ghanbari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [44] propose a novel technique Shibboleth for patch correctness assessment via ranking and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It leverages the impact of the patches on both production code and test suite coverage and relies on a simpler set of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They collect a curated and annotated data set of generated and human-written patches, and they evaluate the model on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that Shibboleth outperforms state-of-the-art patch classification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Phung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [125] present MIPI, a novel approach for patch correctness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Based on a discovery that the distance between the method name and correct patches is smaller than that between the method name and incorrect patches, they decide to extract the intention of developers by analyzing method name to help distinguish the incorrect patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, their method does not require any test cases or noisy source code that are not clearly determined whether they are faulty or clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The evaluation shows that MIPI is more precise and less destructive than existing heuristic-based patch assessment techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 5 EMPIRICAL EVALUATION In this section, we introduce existing widely-adopted datasets in the learning-based APR field and discuss common evaluation metrics for evaluating repair performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Dataset Different from previous APR techniques conducted in a traditional pipeline (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', generating patches by heuristic strategies), the process of learning-based APR techniques is two-fold (1) a training process with supervised learning on large labeled datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CoCoNut [96]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (2) an evaluation process on limited labeled datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Defects4J [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Benefiting from a large amount of research effort in the learning-based APR community, there are several existing benchmarks to evaluate NMT techniques for automatically repairing bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Now we discuss the widely adopted datasets in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Defects4J [60] is the most widely-adopted benchmark in learning-based APR studies, which contains 395 known and reproducible real-world bugs from six open-source Java projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To facilitate reproducible studies, each bug contains a buggy version and a fixed version, as well as a corresponding test suite that triggers that bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Defects v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='0 provides 420 additional real-world bugs from 17 Java projects, which is adopted by some recent studies [165, 186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' QuixBugs [83] is a multi-lingual parallel bug-fixing dataset in Python and Java used in [159, 165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' QuixBugs contains 40 small classic algorithms with a bug on a single line, along with the test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='jar [132] contains 1,158 real bugs from 8 large open-source Java projects, each of which has a fault-revealing test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ManyBugs [72] contains 185 real-world bugs from 9 open-source C projects and each bug has a corresponding developer patch and test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' IntroClass [72] consists of 998 bugs in six small student-written programming assignments for C language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Due to a well-defined test suite, These dataset is effective in evaluating the correctness of generated patches by dynamic program behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, NMT-based APR techniques employ neural network techniques to learn the bug-fixing patterns from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The high-quality test suite requires massive manual effort, so those datasets are usually scarce to train a reliable NMT repair model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To make experiment results more persuasive, lots of large-scale datasets have been conducted recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such datasets contain bug-fixing code pairs for the model to learn how to transform a buggy code into the expected fixed code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, researchers usually mine open-source projects from code platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', GitHub) and extract the commits by fixing-related keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then the unqualified commits are filtered out by pre-defined rules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', non-code changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147] extract the bug-fixing ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='C++ 188K no yes yes [185] 51 CoCoNut C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='Java JS Python 24 M yes yes no [57] [96] [159] [176] 52 CodeFlaw C Python 3902 yes yes yes [16] [96] [168] 53 ENCORE Java Python,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='JS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='C++ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 M no yes no [95] ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:26 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen commits between March 2011 and October 2017 on GitHub and release two BFP datasets for small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', 0∼50 tokens) and medium (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', 50∼100 tokens) methods, consisting of 58k (58,350) and 65k (65,455) bug-fixing samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recoder [186] releases a dataset of 103,585 bug-fixing pairs by crawling Java projects on GitHub between March 2011 and March 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Further, CoCoNut [96] provides five datasets across four languages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Java, Python, C and JavaScript) by extracting commits from GitHub projects, resulting in more than twenty million bug-fixing pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Table 1 presents the description of all involved datasets in our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The first two columns list the dataset name and the third column lists the programming languages the dataset covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The fourth column lists the number of bugs the dataset contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The fifth column indicates whether the dataset has corresponding test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The sixth and seventh columns indicate whether the dataset is used in the training and evaluation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The last column lists some learning-based studies employing the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Among the collected datasets in our survey, we find that training datasets usually contain buggy-fixing pairs while evaluation datasets may additionally contain test suites to validate the correctness of generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, existing studies [96, 159] generally adopt some datasets like Defects4J as the evaluation datasets while adopting other datasets like CoCoNut as the training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, we find some studies [146, 147] adopt the same dataset for training and evaluation without executing test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Tufano [147] split BFP dataset into training and evaluation parts and evaluate the repair performance by match-based metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Table 1 also presents the programming languages of all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It can be found that the collected datasets mainly involve five languages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Java, JavaScript, Python, C and C++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Among them, similar to traditional APR, Java is the most targeted language in the learning-based APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, researchers conduct lots of datasets in other languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Python), indicating that learning-based APR techniques begin to consider more languages in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For Java, researchers prefer the traditionally-dominated Defects4J dataset and the recently-released BFP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For other program languages, researchers have different choices for datasets due in part to the lack of publicly-accepted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We also find that some recent datasets involve multi-languages, such as CoCoNut [96] and QuixBugs [83], while the traditional APR techniques mainly focus on Java language [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The possible reasons lie in that (1) traditional techniques are widely conducted on the same benchmark Defects4J while some additional datasets have been released along with the application of DL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) traditional techniques may rely on language-specific features to generate patches, which is challenging to apply to other languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', PraPR adopting JVM bytecode [43]), while learning-based techniques treat APR as an NMT task similar to NLP, which is independent of specific programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Metric Evaluation metrics play a crucial role in the growth of the learning-based APR field as they serve as the standard to quantitatively define how good an NMT repair model is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In this section, we discuss the common evaluation metrics in learning-based APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Execution-based Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In general, learning-based APR techniques predict some candidate patches with high probability as the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The generated patches are evaluated by executing the available test suites to determine whether to report them to the developers for deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We list the standard metrics as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (1) Compilable Patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such a candidate patch makes the patched buggy program compile successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) Plausible Patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such a compilable patch fixes the buggy functionality without harming existing functionality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', passing all available test suites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:27 (3) Correct Patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such a plausible patch is semantically or syntactically equivalent to the developer patch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', generalizing the potential test suite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Match-based Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, it is time-consuming to evaluate generated patches on dynamic execution for all available test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, test suites may not always be available in large-scale evaluation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' More recently, an increasing number of studies evaluate the performance by code token matching between the generated patch and the ground truth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', developer-written patches), listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (1) Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Accuracy measures the percentage of candidate patches in which the sequence predicted by the model equals the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As learning-based APR techniques usu- ally employ a beam-search strategy, the beam-search strategy reports the 𝑘 sequences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', sequence of terms representing the fixed code) with the highest probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Researchers consider these 𝑘 final sequences as candidate patches for a given buggy code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then Accuracy@K value is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦@𝐾 = �𝑛 𝑖=1 1{𝑚𝑎𝑡𝑐ℎ(�𝑘 𝑗=1 𝑐 𝑗 𝑖 )} 𝑛 (1) where 1 denotes whether 𝐶𝑖 contains a predicted repair sequence equal to the ground truth repair sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The sequence accuracy is 1 if any predicted sequence among the 𝑘 outputs matches the ground truth sequence, and it is 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) BLEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' BLUE (Bilingual Evaluation Understudy) [121] score measures how similar the predicted candidate patch and the ground truth is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Given a size 𝑛, BLEU splits the candidate patch and ground truth into n-grams and determines how many n-grams of the candidate patch appear in the reference patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The BLEU score ranges between 0 (the sequences are completely different) and 1 (the sequences are identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Compared with execution-based metrics, accuracy and BLUE evaluate the candidate patch by matching the tokens of the candidate patch and ground truth without dynamic execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These two metrics can be employed to evaluate the performance of a mass of candidate patches in a limited time and thus have been commonly adopted in the learning-based APR community [146, 147, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, accuracy and BLUE are initially designed in NLP tasks and may be improper to evaluate the program repair task due to the differences between natural language and source code, For example, accuracy refers to the perfect prediction, which ignores that different code snippets may have the same semantic logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, BLEU is originally designed for natural language sentences by token-level matching, neglecting important syntactic and semantic features of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To address the above concerns, recently researchers adopt a variant of BLEU (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CodeBLEU [129]) to evaluate the performance of learning-based APR techniques [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Compared with BLEU, CodeBLEU further considers the weighted n-gram match, the syntactic AST match, and the semantic data-flow match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, the n-gram match assigns different weights for different n-grams, the syntactic match considers the AST information in the evaluation score by matching the sub-trees, and the semantic match employs a data-flow structure to measure semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3 Empirical Study Despite an emerging research area, a variety of learning-based APR techniques have been proposed and continuously achieved promising results in terms of the number of fixed bugs in the litera- ture [96, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In addition to developing new repair techniques that address technical challenges, the learning-based APR research field is benefiting from several empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These empiri- cal studies systematically explore the impact of different components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', code representation), providing insights into future learning-based APR work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:28 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147] conduct a systematic empirical study to investigate the capability of utilizing NMT models to fix software bugs from open-source bug-fixing commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first mine the bug-fixing commits by message patterns from projects in GitHub repositories and filter out the low-quality commits by specific rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They then extract correct and buggy code pairs at the method-level by GumTree and design a code abstraction strategy to reduce vocabulary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, they construct two datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', small and medium BFPs) and train NMT models to translate the buggy method into the correct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results demonstrate that NMT models are able to fix a considerable number of buggy methods in the wild, proving the applicability of NMT for APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [32] empirically investigate to what extent program repair is like machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They reveal that there exist essential differences between seq2seq models and translation models in terms of task design and architectural design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The translation model is inappropriate for program repair due to the lack of vocabulary and immediate context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, the translation model usually keeps up most tokens from the bug code while replacing only a small number, which is not ideal for program repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, they implement an edit-based model by adapting the seq2seq models used for translation to generate edits rather than raw tokens, which leads to promising improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Namavar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [114] conduct a systematic study to understand the effect of code representation on learning-based APR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, they implement REPTORY as a tool for controlled experiments to assess the accuracy of different code representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', AST variants) and the functionality of four different embeddings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', GloVe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct 21 experiments with different models to evaluate their automatic patchability and perceived usefulness as well as accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results reveal that mixed code representation with Golve embedding outperforms other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, they find that bug type affects the accuracy of different code representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [164] present the first extensive evaluation on large programming language models (PLM) for program repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They select nine state-of-art pre-trained PLMs with different types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', infilling and generative models) and parameter sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', ranging from 125M to 20B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They design three different repair settings for PLMs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', complete function generation, correct code infilling, and single line generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They then conduct experiments on 5 datasets across 3 different languages to compare different PLMs in the number of bugs fixed, generation speed and compilation rate .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also compare the performance of PLMs against state-of-the-art APR techniques and results demonstrate the promising future of directly adopting PLMs for APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6 DISCUSSION In this section, we will discuss several prevalent applications of learning-based repair and list some papers for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Domain Repair 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Vulnerability Repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Software vulnerability generally refers to the security flaws in the concrete implementation of hardware, software, or protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Malicious attackers can exploit unresolved security vulnerabilities to get access to the system without authorization or even paralyze the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such vulnerabilities open a range of threats to cyber security, resulting in severe economic damage and fatal consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, the Log4Shell vulnerability (CVE- 2021-44228) from Apache Log4j library3 allows attackers to run arbitrary code on any affected system4 and is widely recognized as the most severe vulnerability in the last decade (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', 93% of the cloud enterprise environment are vulnerable to Log4Shell5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Nowadays, the number of exposed 3https://logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='org/log4j/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='x/ 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='ftc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='gov/policy/advocacy-research/tech-at-ftc/2022/01/ftc-warns-companies-remediate-log4j-security- vulnerability 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='wiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='io/blog/10-days-later-enterprises\\-halfway-through-patching-log4shell ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:29 security vulnerabilities recorded by the National Vulnerability Database (NVD)6 has been increasing at a striking speed, affecting millions of software systems annually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, it is incredibly time-consuming and labor-intensive for security experts to repair such security vulnerabilities manually due to the strikingly increasing number of detected vulnerabilities and the complexity of modern software systems [41, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, previous studies report that the average time for repairing severe vulnerabilities is 256 days7 and the life spans of 50% of vulnerabilities even exceed 438 days [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is incredibly time-critical to patch reported security vulnerabilities as a belated vulnerability repair could expose software systems to attack [81, 84], posing enormous risks to millions of users around the globe and costing billions of dollars in financial losses [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Given the potentially disastrous effect when software vulnerabilities are exploited, a mass of learning-based studies has recently been conducted on automated software vulnerability repair [25, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We list the recent learning-based vulnerability repair studies in details as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [25] propose VRepair, a learning-based approach to repair security vulnerabilities based on the transformer and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' VRepair is first trained on a large bug-fixing dataset and is then transferred to a relatively small vulnerability-fixing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' VRepair uses a transformer neural network model to generate potential patches that are likely to be correct based on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results show that VRepair trained on a bug-fixing dataset already fix some vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, they demonstrate the knowledge learned from the program repair task can be transferred to the vulnerability repair task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, VRepair with the transfer learning gains a better repair performance than that only trained on a vulnerability-fixing or bug-fixing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [40] propose VulRepair, a T5-based automated vulnerability repair technique based on subword tokenization and pre-training components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They compare VulRepair with two competitive baseline approaches, VRepair and CodeBERT on a C benchmark – CVEFixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, they analyze the impact of adopted components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', tokenization and pre-training) and conduct an ablation study to investigate the contribution of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results show that VulRepair outperforms other state-of-the-art vulnerability repair techniques and it is capable of repairing the Top-10 most dangerous CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [26] propose SeqTrans, a learning-based appraoach to provide suggestions for automati- cally repairing vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SeqTrans first uses Gumtree to search for differences between different commits and then traverses the whole AST to label the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SeqTrans then traverses up the leaf nodes, localizes the statement with vulnerability and generates code change pairs, which is fed into the NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As SeqTrans requires a massive amount of training data, SeqTrans is first trained on a bug-fixing dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', source domain) and fine-tuned on a vulnerability-fixing dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', target domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SeqTrans is proven to achieve better repair accuracy than existing techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', SequenceR) and performs very well in certain kinds of vulnerabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CWE-287).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Harer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [50] apply a GAN-based approach to train a NMT model for learning to automatically repair the source code containing security vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply an NMT model as the generator and employ two novel generator loss functions instead of the traditional negative likelihood loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also design a discriminator to distinguish the output generated by the NMT model and oracle output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This approach can be used in the absence of paired bug-fixing datasets, thus reducing the requirements of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The authors evaluate the proposed approach on SATE IV dataset and prove the promising results in fixing vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also demonstrate the proposed approach can be applicable to other tasks, such as grammatical error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='gov/ 7https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='securitymagazine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/articles/95929-average-time-to-fix-severe-vulnerabilities-is-256-days ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:30 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [54] propose to apply large pre-trained models for vulnerability repair to overcome the shortcomings of learning-based APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They compare the performance of CodeBERT and GraphCodeBERT on a C/C++ vulnerability dataset with five CWE types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They discover that GraphCodeBERT with a data flow graph is significantly better than CodeBERT without documenting code dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also demonstrate that such pre-trained models outperform learning-based APR techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CoCoNut [96] and DLFix [79]) and more data-dependent features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', data flow and control flow) will help to repair more complex vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [185] propose a novel approach SFVP for automatically fixing vulnerabilities based on the attention-mechanism model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SPVF first extracts the security properties from descriptions of the vulnerabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CWE category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SPVF then designs the pointer generator network to combine the AST representation and the security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The authors evaluate SPVF on two public C/C++ and Python vulnerability-fixing datasets and results show that it outperforms state-of-the-art SeqTrans [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [97] introduce a novel tool, VuRLE, to autimatically detect and repair vulnerabilities in Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the learning phase, it generates templates by analyzing edits from repair examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' First, it extracts edit blocks by performing AST diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then, it compares each edit block with the other edit blocks, and produces groups of similar edit blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, for each edit group, VuRLE generates a repair template for each pair of edit blocks that are adjacent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the repairing phase, VuRLE detects and repairs vulnerabilities by selecting the most appropriate template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It applies repair templates in order of their matching score until it detects no redundant code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Evaluation results on real-world vulnerabilities show that VuRLE outperforms another APR tool in fixing vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Syntax Errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the learning-based APR field, semantic errors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', test-triggering errors) have attracted considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such errors usually refer to any case where the actual program behavior is not expected by developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing learning-based APR techniques usually expect that the programs under repair are syntactically correct and these techniques are not applicable for syntax errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Novice programmers are more likely to make syntax errors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', replacing a “∗” with an “𝑥”) that make compilers fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Previous studies have indicated the long-term challenge from a wide range of syntax mistakes, consuming a lot of time for novices and their instructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, the release of high-quality novice error data and the emergence of trustworthy deep learning models have raised the possibility of designing and training DL models to fix syntax errors automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Now, we list the recent learning-based APR studies that focus on syntax errors as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [3] propose SynShine, a machine learning-based approach to fix syntax errors in Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply a three-stage syntax repair tool: BlockFix for recovering block structure, LineFix for fixing line errors, and UnkFix for recovering unknown tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SynShine leverages RoBERTa pre-training, uses compiler errors, and generates fixes using multi-label classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After being evaluated on a dataset collected from the Blackbox repository, SynShine outperforms other state-of-art tools on different token ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They have also integrated SynShine with the VSCode IDE for public usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Previous works mainly focus on logical errors and assume that the program should be compiled successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' propose DeepFix to fix multiple errors in a program interactively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply the RNN encoder-decoder to serve as the seq2seq network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To implement the iterative repair for multiple errors, they decide to repair one bug each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' An oracle is applied after the decoder to decide whether the program needs further repair after one patch is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Although this approach is only evaluated on C program language written by students in an introductory ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:31 programming course, the result shows that DeepFix can fix a variety of errors and has potential in other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing work often applies heuristics on generating buggy code to construct buggy-fixed pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such synthetically-generated data may not improve the model and generate low-quality patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Yasunaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [173] propose Break-It-Fix-It (BIFI), a novel APR tool to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first try to train a breaker with real-world buggy-fixed pairs to generate more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also leverage correct paired data to train the fixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' BIFI does not simply collect data, it is also capable of turning raw unlabeled data into usable paired data with the help of a critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They then evaluate this approach on both Python and C benchmarks and it outperforms other state-of-the-art APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Berabi et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [13] present TFix to deal with text-to-text prediction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They fine-tune a pre-trained T5 model to generate JavaScript fixes on datasets extracted from GitHub by themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' By feeding the model with line context and fine-tuning it according to various error types, they obtain multiple fine-tuned T5 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The evaluation shows that TFix generates more patches than SequenceR and CoCoNut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Mesbah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [108] propose DeepDelta to repair the most costly classes of build-time compilation failures in Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They perform a large-scale study of compilation errors and collect a large dataset from logs in Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They further classify different compilation errors and target repairing these errors following specific patterns learned from the AST diff files in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For the two most prevalent and costly classes of Java compilation errors: missing symbols and mismatched method signatures, evaluation results show that DeepDelta generates over half of the correct patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [134] propose to leverage language models for repairing syntax errors in Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They compare n-gram with LSTM models trained on a large corpus of Java projects from Github about localizing bugs and repairing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, their methodology does not rely on buggy code from the same domain as the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Evaluation results show that this tool can localize and suggest corrections for syntax errors, and it is especially useful to novice programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [2] introduce an indirect-supervision approach to leverage GitHub code to create massive amounts of "incorrect-fixed" training pairs for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply a two-stage approach, with two different neural networks for learning to model block nesting structure and code fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This approach performs better on the large and diverse BlackBox dataset than previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It also performs well for StackOverflow fragment parsing and helps fix errors for novice programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [47] propose RLAssist to address the problem of syntactic error repair in student programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They leverage reinforcement learning and train the model using Asynchronous Advan- tage Actor-Critic (A3C)[109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A3C uses multiple asynchronous parallel actor-learner threads to update a shared model, stabilizing the learning process by reducing the correlation of an agent’s experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After they evaluate RLAssist on the C benchmark from [48], results show that this model outperforms the APR tool DeepFix without using any labeled data for training and can help novice programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [14] propose a novel approach for repairing programs committed by students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first apply an RNN to repair syntax errors and then formalize the problem of syntax corrections in programs as a token sequence prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then they leverage the constrain-based technique to find minimal repairs for semantic correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This approach is then evaluated on a Python dataset and results demonstrate the effectiveness of their system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Hajipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [49] propose an efficient method to fix common programming errors by learning the distribution over potential patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To encourage the model to generate diverse fixes even with a limited number of samples, they propose a novel regularizer that aims to increase the distance between the two closest candidate fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They prove that this approach is capable of generating ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:32 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen multiple diverse fixes with different functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After evaluating the approach on real-world datasets, they show that this approach outperforms DeepFix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [163] propose a novel deep supervise learning model, Graph-based Grammar Fix (GGF), to localize and fix syntax errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first parse the erroneous code into ASTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Since the parser may crash in the parsing process due to syntax errors, they create so-called sub-AST and build the graph based on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To tackle the problem of isolated points and some error edges in the generated graph, they treat the code snippet as a mixture of token sequences and graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, GGF utilizes a mixture of the GRU and the GGNN as the encoder module and a token replacement mechanism as the decoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The evaluation shows that the architecture used in GGF is quite helpful for the programming language syntax error correction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3 Programming Assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [5] introduce TRACER to generate targeted repairs for novice programmers in C programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They leverage buggy student programs in Prutor and conduct experiments on single-line and multi-line bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' TRACER first localizes the buggy line, then abstracts the program, and finally converts it into fixed code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Evaluation on the dataset collected from IIT-K shows that TRACER achieves high accuracy and student-friendliness of the repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [153] propose Sarfgen, a high-level data-driven framework to fix student-submitted programs for introductory programming exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They develop novel program embeddings and the associated distance metric to efficiently and precisely identify similar programs and compute program alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also conduct an extensive evaluation of Sarfgen on thousands of student submissions on 17 different programming exercises from Microsoft DEV204-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1x edx course and the Microsoft CodeHunt platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that Sarfgen is effective and it improves existing systems automation, capability, and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [152] present dynamic program embeddings which learn from runtime execution traces to predict error patterns that students would make in their online programming submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They define three program embedding models: 1) variable trace model to obtain a sequence of variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 2) state trace model to embed each program state as a numerical vector and feed all program state embeddings as a sequence to another RNN encoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 3) dependency enforcement model to combine the advantages of the previous two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They have proved that dynamic embeddings overcome critical problems with syntax-based program representations and outperform other syntactic program embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [180] propose a novel approach to repair both semantic and syntactic bugs in Python programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply a large language model trained on code (LLMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also leverage multimodal prompts, iterative querying, test-case-based few-shot selection, and program chunking to repair bugs in students’ committed programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' implement it in MMAPR through Codex as LLMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After evaluating MMAPR on real student programs and another baseline (BIFI and Refactory), it outperforms other state-of-art tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [4] propose Verifix as a tool to provide feedback for students in programming tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first align a student-submitted program with a reference solution in terms of control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then the variables of the two programs are automatically aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After that, they turn a verification problem into a MaxSMT problem if the above verification attempt fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The solution of MaxSMT problem leads to a minimal repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' are the first to espouse verified repair for general-purpose programming education and their approach produces small-sized verified patches as feedback which can be used by struggling students with high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [76] propose AssignmentMender to repair student programs by leveraging both correct and faulty C programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This is the first approach that can exploit faulty submissions in generating patches for programming assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The evaluation on the Codeforces benchmark shows that ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:33 AssignmentMender outperforms several other approaches in feedback generation when only a small number of reference programs are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Since previous works fail to parse ASTs for student programs with syntax errors, Bhatia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [15] present a technique to apply RNN for repairing syntax errors in student programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first train the model with syntactically correct programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then, they query the trained model with student submissions with syntax errors and feed the model with the prefix token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, the model would predict suffix tokens and repair the syntax error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Evaluation on a dataset obtained from a MOOC course shows that this approach can provide automated feedback on syntax errors for students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='4 Other Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Programming Contests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [37] propose to leverage large pre-trained model to repair buggy programs generated by Codex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They collect a Java dataset (LMDefects) from LeetCode contest containing different levels of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They then compare the performance of Codex-e and traditional APR tools (TBar and Recoder) on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that existing APR techniques (TBar and Recoder) do not perform well at fixing bugs in auto-generated programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' also define three strategies as instructions:1) fix bugs in the program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2) fx line N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3) and fix statement S to evaluate their approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They find that Codex-e performs well under proper instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Program Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [46] present SED as a framework incorporating synthesis, execu- tion, and debugging stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SED applies a synthesizer that employs greedy decoding to generate buggy programs for training and the debugger is fed with synthesized bugs as well as execution results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SED is then evaluated on the Karel benchmark and it outperforms other beam search techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Nonidiomatic Snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Szalontai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [138] present a novel algorithm to localize and substitute non-idiomatic code snippets in Python programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply a feed-forward and two RNNs to accomplish the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Once the code snippet is localized, the model classifies the type of the nonidiomatic pattern and extracts the key variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, the model substitutes the code snippet with a cleaner and more performant alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This model is evaluated on a Python dataset and it achieves good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Industrial Deployment As a promising field, APR has been extensively studied in academia and even has drawn growing attention from industry [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Marginean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [100] present SapFix, the first end-to- end deployment of industrial APR in Meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SapFix is implemented to a continuous integration environment and deployed into six production systems with tens of millions of code lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similar industrial practice can also be found in other companies, such as Fujitsu [133], Bloomberg [63] and Alibaba [183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In addition to the above-mentioned traditional deployment, the industry recently explores the feasibility of deploying learning-based APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, GitHub launches a product Copilot8, which can provide code suggestions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', fixing bugs) for more than a dozen programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Copilot is deployed in multiple IDEs, such as VS Code, Visual Studio, Neovim, and JetBrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, Microsoft recently releases a new tool Jigsaw9 to fix bugs in machine-written software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Now, we summarize the existing learning-based APR techniques and industrial deployment from enterprises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Bader et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [8] present Getafix, the first industrially-deployed automated bug-fixing tool for Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To be fast enough to suggest fixes in time, this model produces a ranked list of 8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/features/copilot 9https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/en-us/research/blog/jigsaw-fixes-bugs-in-machine-written-software/ ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:34 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen fix candidates based entirely on past fixes and on the context in which a fix is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, it leverages the hierarchical clustering technique for discovering repetitive fix patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, They apply a statistical ranking technique to enable the model to predict human-like fixes among the top few suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' An evaluation with a large dataset containing six types of common bugs and their experience of deploying Getafix within Facebook show that the approach accurately predicts human-like fixes for various bugs, reducing the time developers have to spend on fixing recurring kinds of bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Baudry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [9] present R-HERO, a novel software repair robot to automatically repair bugs on the single platform GitHub/Travis CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' R-HERO contains six main blocks: a) Continuous integration, b) Fault localization, c) Patch generation, d) Compilation & Test execution, e) Overfitting prevention, and f) Pull-request creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It receives and analyzes the events from a continuous integration (CI) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' R-HERO leverages continual learning to acquire bug-fixing strategies from the platform mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It shows that developers and bots can cooperate fruitfully to produce high-quality, reliable software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Allamanis et al [6] from Microsoft propose BUGLAB to detect and repair software bugs auto- matically by self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similar to BIFI [173], BUGLAB employs a detector model to repair bugs and a selector model to generate buggy code snippets as the training data of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The authors create a dataset PYPIBUGS of 2374 real-world bugs from the PyPI packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results show that BUGLAB can fix a number of software bugs and detects some previously unknown bugs in open-source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Drain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [33] from Microsoft leverage the same DeepDev-py sequence-to-sequence model which is pre-trained from BART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They train it on the Python commit data and reversed data to generate bug-patcher and bug-creator respectively, then leverage them to generate neural bugs to finally train a back-translation model on Python methods with debugging information and stack traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They evaluate the proposed technique on QuixBugs benchmarks and their own benchmarks, and this model outperforms many APR tools on Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Drain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [34] from Microsoft introduce DeepDebug, a span-masking pre-trained encoder decoder transformer as a tool to fix Java methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The model is pre-trained from BART which is pre-trained in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct three pre-training experiments to verify the feasibility of the model and test it on the Java benchmarks from Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' DeepDebug outperforms many state-of-art APR tools, and adding syntax embeddings along with the standard positional embeddings helps improve the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Hellendoorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [52] from Google conduct experiments for two different models architectures that leverage both local and global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They propose sandwich models that apply different message-passing techniques and GREAT models that add extra information to a transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Both architectures achieve high results and outperform state-of-art tools, proving that a hybrid model with global information and incorporating structural bias helps improve accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [53] from AWS AI propose NSEdit to generate patches for Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Given only the buggy code, NSEdit uses the pre-trained CodeBERT as the encoder and CodeGPT as the decoder to address the sequence-to-sequence NMT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, it uses a pointer network to select content-based edit locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply beam search to and design a novel technique to fine-tune the reranker to rank the top-k patches for the buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results on BFP benchmarks [147] indicate that NSEdit outperforms state-of-art APR tools and demonstrate the effectiveness of each component of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [140] from Microsoft introduce a grammar-guided end-to-end approach to generate patches, which treats APR as the transformation of grammar rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply structure-aware modules and three different types of strategies for grammar-based inference algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also leverage two encoders and enhance the model with a new tree-based self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:35 experimental results on BFP datasets [147] demonstrate that the proposed technique outperforms other state-of-art APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [151] from Ping An Technology propose CPR, short for causal program repair, as a tool to utilize data augmentation strategy for input perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This model can generate patches for Java, Python, JavaScript, and C based on causally related input-output tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, it can offer explanations by transforming code into explainable graphs on various Seq2Seq models in APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct experiments on four programming languages and prove that APR models can be utilized as causal inference tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3 Pre-trained Model-based Repair In this section, we will discuss the existing studies of pre-trained models on the ARP task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Pre-trained models have significantly improved performance across a wide range of natural language processing (NLP) and code-related tasks, such as machine translation, defect detection and code classification [45, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Typically, the models are pre-trained to derive generic vector representation by self-supervised training on a large-scale unlabeled corpus and then are transferred to benefit multiple downstream tasks by fine-tuning on a limited labeled corpus [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The application of existing pre-trained models to program repair is usually divided into two categories: universal and specific pre-trained model-based APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The former aims to propose universal pre-trained models for multiple code-related tasks (including program repair), while the latter only focuses on program repair by designing a novel APR technique based on pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='1 Universal Pre-trained Model-based APR Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing pre-trained models generally adopt the encoder-decoder transformer architecture, which can be classified into three types: encoder-only, decoder-only, and encoder-decoder models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Encoder-only models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CodeBERT [38]) usually pre-train a bidirectional transformer where tokens can attend to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Encoder- only models are good at understanding tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', code search), but their bidirectionality nature requires an additional decoder for generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Decoder-only models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CodeGPT [19]) are pre-trained using unidirectional language modeling that only allows tokens to attend to the previous tokens and themselves to predict the next token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Decoder-only models are good at auto-regressive tasks like code completion, but the unidirectional framework is sub-optimal for understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Encoder-decoder models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CodeT5 [127]) often make use of denoising pre-training objectives that corrupt the source input and require the decoder to recover them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Compared to encoder-only and decoder-only models that favor understanding and auto-regressive tasks, encoder-decoder models can support generation tasks like code summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Inspired by the success of pre-trained models in NLP, many recent attempts have been adopted to boost numerous code-related tasks (such as program repair) with pre-trained models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CodeBERT) [38, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the context of APR, an encoder stack takes a sequence of code tokens as input to map a buggy code 𝑋𝑖 = [𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ,𝑥𝑛] into a fixed-length intermediate hidden state, while the decoder stack takes the hidden state vector as an input to generate the output sequence of tokens 𝑌𝑖 = [𝑦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ,𝑦𝑛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Researchers treat the APR problem as a generation task, and consider encoder-decoder or encoder-only (with an additional decoder) pre-traiend models, which are usually evaluated by BFP dataset from Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We summarize existing pre-trained models involving the program repair task as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [38] present a bimodal pre-trained model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CodeBERT) for natural language and programming language with a transformer-based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CodeBERT utilizes two pre-training objectives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', masked language modeling and replaced token detection) to support both code search and code documentation generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To support program repair task, Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [94] leverage CodeBERT as the encoder, which is connected with a randomly initialized decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:36 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [157] present a pre-trained encoder-decoder model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CodeT5) that considers the code token type information based on T5 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CodeT5 employs a unified framework to support code understanding (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', clone detection) and generation tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', program repair) and allows for multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The most crucial feature of CodeT5 is that the code semantics of identifiers are taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Assigned by developers, identifiers often convey rich code semantics and thus a novel identifier-aware objective is added to the training of CodeT5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [45] present the first structure-aware pre-trained model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', GraphCodeBERT) that learns code representation from source code and data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Unlike existing models focusing on syntactic-level information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', AST), GraphCodeBERT takes semantic-level information of code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', data flow) for pre-training with a transformer-based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The results on BFP datasets [147] demonstrate the advantage of leveraging code structure information to repair software bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Mastropaolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [104] propose pre-trained text-to-text transfer transformer (T5) to address four code-related tasks, namely automatic bug fixing, injection of code mutants, generation of assert statements in test methods, and code summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply BFP small and BFP medium datasets to train and evaluate the bug-fixing task, and then compare other state-of-art learning- based APR tools on the same benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, they have done single-task fine-tuning and multi-task fine-tuning to fully evaluate the function of the pre-trained T5 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Although multi- task fine-tuning does not improve the result of code-related tasks, single-task fine-tuning does prove that this model outperforms other tools on the same benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Niu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [117] propose a seq2seq pre-trained model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', SPT) by three ode-specific tasks (code- AST prediction, masked sequence to sequence and method name generation) and fine-tune on the generation tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', code summarization, code completion, program repair and code translation) and classification task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', code search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2 Specific Pre-trained Model-based APR Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In addition to those above-mentioned typical pre-trained models that involve program repair, researchers have adopted pre-trained models to design novel APR techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CURE integrates GPT into CoCoNut architecture [57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We summarize existing APR studies that employ pre-trained models as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing learning-based APR techniques can only generate patches for a single programming language and most of them are developed offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [159] propose CIRCLE, a T5-based APR technique targeting multiple programming languages with continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CIRCLE first employs a pre-trained model as a repair skeleton, then designs a prompt template to bridge the gap between pre-trained tasks and program repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' To further strengthen the continual learning ability, CIRCLE applies a difficulty-based rehearsal method to achieve lifelong learning without access to the entire historical data and an elastic regularization to resolve catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, to perform the multi-lingual repair, CIRCLE designs a simple but effective re-repairing mechanism to eliminate incorrectly generated patches caused by multiple programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [21] present MODIT, a novel multi-modal NMT-based tool, to automatically generate fixes for buggy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They leverage three modalities of information: edit location, edit code context, and commit messages (natural language guidance from the developer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct many experiments and the evaluation shows that, through pre-training, MODIT improves the ability to generate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Also, leveraging additional modalities of information could benefit the source code repairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Kolak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [65] propose to apply large pre-trained language models to generate patches for one-line bugs in Java and Python programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They consider pre-trained models with a wide range of sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', GPT-2 with 160M, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='4B, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='7B parameters and CodeX 12B parameters) for evaluation and comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After evaluating these models on the QuixBugs benchmark, they discover that larger language models are more promising in guiding patch selection in APR work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:37 Richter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [131] propose RealiT for localizing and fixing bugs in Python programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They first pre-train a transformer model on large numbers of mutant bugs and then fine-tune it with a small set of real bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After evaluating RealiT on the PyPIBug benchmark, they prove that training on both mutant and real-world bugs can significantly improve the performance of the model and abundant mutant bugs also improve the model’s ability to localize and fix bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [181] propose CoditT5, a pre-trained language model for software-related edit tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' CoditT5 is pre-trained on both program languages and natural language comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' fine-tune it for three down-streaming tasks: comment updating, bug fixing, and automatic code review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For bug-fixing, they fine-tune it with Java datasets BFP small and BFP medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The evaluation shows that CoditT5 outperforms other state-pf-art tools on three down-streaming tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Mashhadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [103] apply CodeBERT, a pre-trained neural network model, for fixing Java bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They fine-tune and evaluate it on ManySStuBs4J datasets and find it is capable of generating patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Their approach gets rid of the limitation of token length and vocabulary problems, thus this model is more efficient and effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This model can generate patches for different types of bugs and outperform other state-of-art APR tools in terms of the accuracy of generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Lajko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [69] propose to apply a Generative Pre-trained Transformer (GPT) for generating patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Specifically, they apply GPT-2 medium model to repair JS programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' First, the model is fine-tuned on datasets mined from GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then, it is evaluated on the same dataset and it achieves good results if it could generate more candidate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Lajko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' consider using large models and it can achieve better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Further, Prenner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [126] propose to apply Codex, a GPT-3 like model trained on a large corpus, to localize and repair bugs on multi-language benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They conduct experiments to evaluate Codex under different prompt conditions and they also compare the model with other state-of-art APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results show that despite not being trained for APR, Codex still performs well on the Quixbugs benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, Codex repairs more bugs in Python than those in Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [61] present GLAD, a novel learning-based APR tool targeting fixing if statement omission faults (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', faults in which necessary code is missing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' By leveraging generative pre- trained Language Models (LMs) instead of machine translation models, GLAD does not require the localization of a buggy line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Moreover, GLAD applies a grammar-based beam search to constrain the output of the model and efficiently reduces the validation cost by performing dynamic ranking of candidate patches using a debugger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Evaluation results on Defects4J benchmark show that GLAD is capable of fixing bugs other APR tools fail to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [165] introduce AlphaRepair as a cloze-style APR tool to directly query a pre-trained model for generating patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They apply the newly pre-trained CodeBERT as an example under zero-shot learning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They try to mask the buggy line in the source code with different templates or strategies and feed the whole source code into the model with the buggy line as a “comment".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then with a large number of patches this model generated, they propose probabilistic patch ranking to determine top-k plausible patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After evaluating this technique on both Java and Python benchmarks, it outperforms other state-of-art APR tools and proves that a pre-trained model with no fine-tuning is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='4 DL for Traditional APR These approaches attempt to boost traditional APR techniques by utilizing deep learning or machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [91] propose Prophet, a patch-generation system for repairing defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It uses dynamic analysis on the given test suite to get the program points for the patch to modify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Then, the SPR[89] is used to generate search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' With a trained probabilistic model, Prophet ranks the candidate patches, which are validated by executing the test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They collect eight projects from Github ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:38 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen and get 777 patches to train their model and test it on a benchmark[71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The result shows that Prophet can generate patches correctly with the learned knowledge compared with previous patch generation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [23] propose a novel search-based technique called LIANA, which is based on a designed learning-to-rank prioritization mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is based on the idea of repeatedly updating a statistical model online based on the intermediate validation results of an ongoing program repair process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The model is first trained offline and updated repeatedly after the generating progress starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The most up-to-date model is used to generate fixes and prioritize those that are more likely to include the correct ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [107] propose TRANSFER, a fault localization and program repair approach with deep semantic features and transferred knowledge which is obtained by a combination of spectrum-based and mutation-based localization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They build a fault localization and program repair dataset respectively and employ existing fix templates designed by TBar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' They also design 11 binary classifications to identify whether one of the 11 bug types they define exists in a statement and a multi-classification to determine which fix template this statement should apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The binary classification, consisting of one embedding layer, one RNN layer, one max pooling layer, and one dense layer, is fed with spectrum-based, mutation-based, and semantic features and outputs the probability of containing specific bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Although this approach is only tested on Java, it is proven to outperform many state-of-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [74] design a novel framework called ARJANMT to leverage both redundancy assumption and Seq2Seq learning of correct patches to generate fixes for Java methods using NSGA-II algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This framework combines both ARJA and SequenceR into a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After evaluating ARJANMT on two Java benchmarks, results show that it benefits from search-based and NMT-based techniques and outperforms other state-of-art techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Valueian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [148] propose SituRepair for repairing multiple bugs in C programs based on pre-defined repair patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It applies a machine learning model to predict the buggy type and localization of the buggy code and then repairs them with situational modifications accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SituRepair is evaluated on a C benchmark Code4Bench and it successfully repairs 3,848 multiple- fault programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='5 Open science Recent years have witnessed an increasing use of DL in traditional SE problems and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, software bug is a growing quality concern for modern software, and accordingly, APR has become an actively studied topic in the SE community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' According to our survey, various learning-based APR techniques have been introduced in the last five years (discussed in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' DL brings a new repair paradigm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', training and repairing) for the APR problem with promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, due to the nature of DL, learning-based APR techniques face some concerns in reproducibility, which is quite different from transitional APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, it may require a large number of machine resources for researchers to reproduce the NMT model’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The cost is even unaffordable for most researchers from academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, there exists randomness in the neural network training process, which hinders the reproduction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such challenges posed by DL motivate us to further understand the potential issues with open science in the learning-based APR area, so as to advance existing techniques by taking advantage of the general merits of open science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Open science advocates that researchers make their artifacts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', raw data, dataset, scripts, related models, or any results produced in their work) available to all levels of researchers [105], so knowledge can be shared without boundaries [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' While a mass of DL techniques are proposed to fix software bugs automatically, more support is needed to investigate the critical open science problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In particular, we investigate to what extent the ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:39 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Results on tool availability Tool Language Hosting Site Link Accessibility SA DA TA URL Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [152] C Github valid yes no no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/keowang/dynamic-program-embedding Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [146] Java Google valid yes yes yes https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/view/learning-codechanges Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [147] Java Google valid yes yes no https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/view/learning-fixes RLAssitst [47] C bitbucket valid yes yes yes https://bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='org/iiscseal/rlassist CoCoNut [96] Java,C,Python,JS Github valid yes yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/lin-tan/CoCoNut-Artifact DLFix [79] Java Github valid yes yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/ICSE-2019-AUTOFIX/ICSE-2019-AUTOFIX Hellendoorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [52] Python Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/VHellendoorn/ICLR20-Great DrRepair [172] C,C++ Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/michiyasunaga/DrRepair Learn2Fix [16] Python Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/mboehme/learn2fix Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [143] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/TruX-DTF/DL4PatchCorrectness BIFI [173] Python,C Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/michiyasunaga/bifi Recoder [186] Java Github valid yes yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/pkuzqh/Recoder SequenceR [24] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/kth/SequenceR TFix [13] JS Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/eth-sri/TFix BugLab [6] Python Github valid yes yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/microsoft/neurips21-self-supervised-bug-detection-and-repair Reptory [114] JS Github valid yes yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/annon-reptory/reptory RewardRepair [176] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/SophieHYe/RewardRepair CodeBERT [103] Java Github valid yes yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/EhsanMashhadi/MSR2021-ProgramRepair R-HERO [9] Github valid no yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/repairnator/open-science-repairnator/tree/master/data/2020-r-hero Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [2] Java zenodo valid yes yes no https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='3374019 ODS [174] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/SophieHYe/ODSExperiment CIRCLE [159] Java,C,JS,Python Github valid no no no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/2022CIRCLE/CIRCLE TRANSFER [107] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/mxx1219/TRANSFER DEAR [80] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/AutomatedProgramRepair-2021/dear-auto-fix Cornor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [27] Java Github valid yes yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/WM-SEMERU/hephaestus BATS [142] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/HaoyeTianCoder/BATS T5 [104] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/antonio-mastropaolo/TransferLearning4Code CompDefect [116] Java zenodo valid yes yes no https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='org/record/5353354#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='Y4CVdhRByUl VRepair [25] C Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/SteveKommrusch/VRepair SeqTrans [26] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/chijianlei/SeqTrans VulRepair [40] C Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/awsm-research/VulRepair Crex [168] C Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/1993ryan/crex RealiT [131] Python Github valid yes no yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/cedricrupb/nbfbaselines GPT-2 [69] JS Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/RGAI-USZ/APR22-JS-GPT CoditT5 [181] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/EngineeringSoftware/CoditT5 SYNSHINE [3] Java zenodo valid yes yes yes https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='org/record/4572390#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='Y4CY8xRByUk Verifix [4] C Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/zhiyufan/Verifix Cache [82] Java Github valid yes yes no https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/Ringbo/Cache Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [158] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/anonymous0903/patch_correctness Quatrain [145] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/Trustworthy-Software/Quatrain Shibboleth [44] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/ali-ghanbari/shibboleth Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [144] Java Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/HaoyeTianCoder/Panther SSC [30] Python Github valid no no no https://iclr2018anon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='io/semantic_code_repair/ Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [55] Java,C,C++ Github valid yes yes yes https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/shan-huang-1993/PLC-Pyramid ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:40 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen collected papers make their artifacts publicly available and in what way they provide the relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Table 2 shows the tool availability results of the investigated papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For each paper we collect, we check whether an accessible link for its tool or data is provided in the main text or footnotes of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We only present the studies that provide the link of publicly available data or tools due to limited space, listed in the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We then investigate the following five dimensions in characterizing the availability of each paper: Hosting Site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This information indicates which hosting site the available artifact is uploaded to for public access (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', GitHub or Google), if the artifact link is presented in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The detailed information is listed in the third column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Link Accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This information indicates whether the provided link is accessible, such that we could download the artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The detailed information is listed in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Source Code Available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This information indicates whether the source code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', training and evaluation scripts) is available in the artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The detailed information is listed in the fifth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Dataset Available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This information indicates whether the dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', raw data and train- ing data) is available in the artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The detailed information is listed in the sixth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Trained Model Available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' This information indicates whether the trained model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', raw data and training data) is available in the artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The detailed information is listed in the seventh column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We also list the programming languages targeted by the tools in the second column and list the accessible url links in the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' After carefully checking the collected papers, we find that only a few of the papers have made their source code available to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For convenient public access, a majority of papers upload their works to GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The possible reason is that GitHub is the most popular platform to host open-source code publicly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Meanwhile, we find that several papers fail to provide the source code, dataset, or already trained model [140, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The possible reasons may be (1) the artifacts need to be refactored or reorganized for public availableness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) the artifacts are used for further studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (3) the artifacts are lost due to some accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We also find while the artifacts are available, some studies yet cannot be reproduced because (1) the missing of default hyperparameters10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' (2) the complexity of environment settings for training11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' and (3) the sufficiency of documentation to reproduce the experiments12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Overall, most traditional APR tools have provided open-source code and data, which are easy to reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, some learning-based APR tools require complex environment settings and some authors fail to provide high-quality code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, learning-based APR involves abundant time and expensive equipment to train a model, and thus it is much harder to reproduce a learning- based APR tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Therefore, we hope that learning-based APR researchers can provide high-quality open-source code to construct a unified repair framework for convenient reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 7 IMPLICATION AND DISCUSSION Our study reveals the following important practical guidelines for future learning-based APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The quality of the training dataset is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In contrast to traditional APR techniques, learning-based techniques heavily rely on the quality of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A majority of existing techniques mine bug-fixing pairs from open-source code repositories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', GitHub) and build their own datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, the training dataset is usually collected by automated tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', extracting 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/lin-tan/CoCoNut-Artifact/issues/11 11https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/pkuzqh/Recoder/issues/11 12https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='com/ICSE-2019-AUTOFIX/ICSE-2019-AUTOFIX/issues/5 ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:41 commit by fix-related keywords) and then inspected by some filtering rules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', more than five Java files) [186], which means the quality of the training dataset can be variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Many training datasets contain noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', CoCoNut contains a number of duplicated samples) that may reduce the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, the number of training samples in different techniques varies greatly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', 3,241,966 in CoCoNut [96] and 2,000 in DLFix [79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These concerns may introduce bias when comparing and analyzing learning-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, a unified standard for training datasets should be built to reduce the burden on researchers when they evaluate the performance of different repair models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' More practical evaluation metrics are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, when evaluating repair perfor- mance, an increasing number of learning-based techniques rely on static match-based metrics, which are derived from NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, such metrics fail to consider that a program’s functionality can be implemented in various ways, such as different algorithms, data structures, or data flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is unclear whether the match-based metrics can reflect the repair capability of NMT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, the relationships between the static match-based and dynamic test execution-based metrics need to be studied in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Code features need to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Inspired by the advance of machine translation in NLP, early learning-based APR work treats source code as a sequence of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The follow-up work has begun to consider complex code features, such as code edit, [186], AST [79] and data flow graph [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, the most recent technique CIRCLE, treating the APR as a simple machine translation task on code sequences, still achieves state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such observation indicates that simple features require more attention in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, considering the mass of code representation ways in learning-based APR, it is crucial to conduct a systematic study to explore the impact of different code representations under various model architectures and benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Overfitting issue still exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Similar to traditional APR techniques, learning-based techniques usually adopt available test suites to filter incorrect candidate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, the test suite is an incomplete specification under the program behavioral space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The plausible patches passing the existing test suite may not satisfy the expected outputs of potential test suites, leading to a long challenge in APR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', the overfitting issue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Considering the learning-based APR is an end-to-end repair paradigm (in a black-box manner), which is different from traditional techniques adopting test suites to guide the repair process, the overfitting issue in learning-based APR is more significant and severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, researchers have adopted DL techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', code embedding [82, 143]) to predict the correctness of plausible patches, which is a promising direction to address overfitting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We also recommend designing some advanced NMT repair frameworks to generate high-quality patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Unified repair in urgent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='2, similar to traditional APR techniques, existing learning-based techniques usually consider fault localization as an additional step in the repair process and adopt off-the-shelf fault localization tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', SBFL) to identify suspicious code element, which is the input of NMT repair models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, these two tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', fault localization and patch generation) are developing in their own respective fields so far and little work has explored their potential relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Recently, Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [116] propose CompDefect to handle defect prediction and repair simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The powerful capacity of DL to learn the semantic information of source code for fault localization [78, 93] and program repair [159, 186] makes it possible to combine the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Practical NMT repair model is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' An increasing number of learning-based techniques attempt to generate patches by large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Although remarkable progress is obtained, such NMT models contain millions or even billions of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, CodeBERT has 125 million parameters and 476 MB model size in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' It is significant to deploy these models in modern IDEs to assist developers during software development and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, these ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 1:42 Quanjun Zhang, Chunrong Fang, Yuxiang Ma, Weisong Sun, and Zhenyu Chen repair models consume huge device resources and run slowly in the development workflow (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', IDEs), limiting their application in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the future, It is promising to reduce the size of these repair models to deploy in real-world scenarios while maintaining comparable accuracy, such as model pruning and knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Model size is not the only option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' As discussed before, learning-based APR techniques tend to employ the growing size of models, achieving better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [164] have demonstrated that larger models usually repair a greater number of bugs, highlighting the promising future of pre-trained models for APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, such large models are difficult to deploy in the development workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, with the release of ever-larger models, there may exist a barrier in the trade-off between effectiveness and model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In fact, existing pre-trained models in APR usually treat source code as natural language, which cannot capture the code features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the future, investigating how to bring in code structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', data flow or control flow) in model training may be a flexible strategy instead of employing a larger mode size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Combined with traditional APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Existing DL techniques are usually adopted as a patch generator in APR workflow, which takes the buggy code snippets as inputs and returns a ranked list of candidate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Despite remarkable progress, such learning-based APR techniques are developed separately from traditional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Previous work [186] has demonstrated that learning-based is complementary to traditional techniques in terms of fixed bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, it is flexible to integrate DL techniques into traditional APR techniques instead of developing a brand-new end-to-end patch generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' For example, Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' [107] design a multi-classifier to rank the fix templates for TBar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the future, researchers can boost existing template-based APR techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', TBar) via mask prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Domain repair techniques are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A majority of learning-based techniques focus on semantic bugs, which are investigated intensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' However, only a small amount of existing tech- niques consider other types of bugs, such as security vulnerabilities or programming assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' The community usually treats fixing these types of bugs as separate tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' SequenceR [24] has demonstrated that NMT-based models only trained on a limited bug-fixing corpus can already fix notable vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' These results indicate that bug fixing and vulnerability repair both aiming to fix errors in the source code have a high degree of similarity and the knowledge learned from bug fixing can be well transferred to vulnerability repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Such observation motivates future researchers to explore their potential relationship and investigate whether these tasks can benefit each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Besides, it is promising to migrate existing mature learning-based bug-fixing techniques to automated vulnerability repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Explainable Patch Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Traditional APR techniques generate patches along with a log output, which contains detailed information in the generation process, while learning-based APR techniques perform an end-to-end patch generation due to the interpretability of DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Thus, the developers are unaware of why repair models predict such results, hindering the adoption of repair models in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the literature, a majority of studies focus on improving repair accuracy, while minor focus on improving the explainability of such repair models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the future, advanced explainable techniques can be considered to make the predictions of NMT repair models more practical, explainable, and actionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 8 CONCLUSION APR techniques address the long-standing the challenge of fixing software bugs automatically, and alleviates manual debugging effort significantly, which promotes software testing, validation, and debugging practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' In the last couple of years, learning-based APR techniques have achieved promising results, demonstrating the substantial potential of using DL techniques for APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 0, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Publication date: 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' A Survey of Learning-based Automated Program Repair 1:43 In this paper, we provide a comprehensive survey of existing learning-based APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We describe the typical learning-based repair framework, involving fault localization, data pre-processing, patch generation, patch ranking, validation and correctness components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We summarize how ex- isting learning-based techniques design strategies for these crucial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' We discuss the metrics, datasets and empirical studies in the learning-based APR community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' Finally, we point out several challenges (such as overfitting issues) and provide possible directions for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' 9 ACKNOWLEDGMENTS This work is supported partially by the National Natural Science Foundation of China (61932012, 62141215).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE1T4oBgHgl3EQfkATn/content/2301.03270v1.pdf'} +page_content=' REFERENCES [1] Rui Abreu, Peter Zoeteweij, and Arjan JC Van Gemund.' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..826abbf6c4c52924782da5df1bc9c7fe5932479c --- /dev/null +++ b/x9E2T4oBgHgl3EQfMAaP/content/tmp_files/2301.03720v1.pdf.txt @@ -0,0 +1,3012 @@ + +Federated Learning for Energy Constrained IoT devices: A +systematic mapping study +Rachid EL Mokadem, Yann Ben Maissa and Zineb El Akkaoui +{elmokadem.rachid, benmaissa, elakkaoui}@inpt.ac.ma +Telecommunications Systems, Networks and Services Lab, National Institute of Posts and +Telecommunications, Rabat, 10587, Morocco. +Abstract +Federated Machine Learning (Fed ML) is a new distributed machine learning technique applied to +collaboratively train a global model using clients’ local data without transmitting it. Nodes only send +parameter updates (e.g., weight updates in the case of neural networks), which are fused together by the +server to build the global model. By not divulging node data, Fed ML guarantees its confidentiality, a crucial +aspect of network security, which enables it to be used in the context of data-sensitive Internet of Things +(IoT) and mobile applications, such as smart geo-location and the smart grid. However, most IoT devices +are particularly energy constrained, which raises the need to optimize the Fed ML process for efficient +training tasks and optimized power consumption. In this paper, we conduct, to the best of our knowledge, +the first Systematic Mapping Study (SMS) on FedML optimization techniques for energy-constrained IoT +devices. From a total of more than 800 papers, we select 67 that satisfy our criteria and give a structured +overview of the field using a set of carefully chosen research questions. Finally, we attempt to provide an +analysis of the energy-constrained Fed ML state of the art and try to outline some potential +recommendations for the research community. + +Keywords: Federated Machine Learning, Energy Optimization, Internet of Things, Edge and Mobile Computing, On-device +Intelligence +1 Introduction +Context. Machine learning (ML) has become an +important and increasingly used paradigm in +different applications. In the last decade, the IoT +computer systems and their potential applications +(e.g., smart cities, smart grids) have grown +considerably, which would make them benefit +from the capabilities of ML in such a large and +complex context. Furthermore, widespread IoT +adoption in industry and academia (e.g., via rapid +prototyping platforms such as the Raspberry PI™ +and Arduino™) raises expectations for data privacy +preservation and efficient resource utilization in a +wide range of critical applications. Therefore, in +light of ML limitations for distributed systems and +sensitive data, Federated Machine Learning (Fed +ML) was proposed by McMahan et al. in 2016 [46] +to address these constraints. The approach + +delegated model training tasks to client devices, +which collaboratively built a global shared model +that consolidated their respective local data +learning while avoiding any private data from +leaving its original device [32]. Since the seminal +paper, Fed ML has become one of the ”hot topics” +in ML. +Problem. IoT and mobile devices have a major +constraint related to energy sources, and as a +result, the power consumption on these devices +must be optimized for any assigned task. In +particular, a machine learning algorithm is known +to be a highly power-consuming multi-task +process [17]. In a distributed ML setup, nodes must + +continually exchange data with a master node, +which may drive up overhead costs for the system. +FedML attempts to solve this issue by limiting the +exchanged data to the local model’s weights [46], +trained by nodes, instead of voluminous raw data +exchange. At the same time, FedML still requires +improvement to enable resolving further critical +challenges related to IoT and mobile device +characteristics, namely, the limited resources and +energy constraints [54]. As a consequence, several +Fed ML works addressing these aspects have +increasingly been proposed by the scientific +community in the last few years. In light of this +evolving literature, there is a substantial need for a +comprehensive study in order to provide a clear +overview of energy optimization approaches and +propose new research directions for the research +community. + + +Contribution. Several works have actually tackled +the limitations of the original Fed ML proposal, +entitled FedAvg, and proposed many optimization +approaches, +essentially +regarding +the +communication load, data exchange, and other +aspects, which can help to address directly or +indirectly the limited energy constraint. The +purpose of this paper is to conduct, to the best of +our knowledge, the first Systematic Mapping Study +(SMS) on Fed ML optimization for energy- +constrained devices. This SMS is tasked with i.) +counting and categorizing relevant primary +studies published in this topic based on five +research questions, ii.) analyzing and discussing +the results to provide a clear understanding of +recent improvements for the research community, +and iii.) assisting engineers in developing +innovative Fed ML solutions for IoT and mobile +devices. + +Contents. The remainder of this paper is +structured as follows: First, we present related +works and surveys in Section 2. Then we provide +some theoretical foundations through the original +FedAvg algorithm as well as the formulation of the +energy optimization problem in Section 3. In +Section 4, we present the method used to conduct +this study, including the paper selection and +filtering process, as well as the research questions +(RQs). + +Table 1 Related works +Year +Title +Type +Focus +2020 +A Systematic Literature Review +on +Federated +Machine +Learning: From A Software +Engineering +Perspective. [41] +SLR +Software +engineering +aspects +2020 +A Systematic Literature Review on +Federated +Learning: From A Model +Quality Perspective. [40] +SLR +Model +quality +2020 +Federated Learning in Mobile Edge +Networks: A +Comprehensive Survey.[38] +Survey +General +2020 +Federated Learning: A Survey +on +Enabling +Technologies, +Protocols, and Applications. [1] +Survey +Applications +2020 +A Review of Privacy-preserving +Federated Learning for +the Internet-of-Things. [6] +Survey +Privacy + + +In Section 5, we answer them and analyze the +results obtained from the studied papers. We +follow +up +with +a +discussion +and +some +recommendations for research directions in +Section 6. Section 7 exhibits some threats to the +validity of our study. Finally, in Section 8, we +conclude and outline some possible future works. +2 Related works +In this section, we present three surveys and +two literature reviews that have been identified as +being related to this work (see Table 1).[41] +presented a systematic literature review on +Federated Learning, from a software engineering +perspective, where they covered the Federated +Learning system in general, with a focus on the +software +development +aspects +and +general +challenges for real applications. [40], on the other +hand, conducted a systematic literature review on +Federated +Learning +from +a +model +quality +perspective, where they studied the methods for +improving the quality of the Fed ML model and +data. Additionally, the authors compared the +model between federated and non-federated +learning on the same data. Furthermore, [38] +presented a survey on Federated Learning for +mobile edge networks, in which they investigated +the characteristics and limitations of good +performance, resource allocations, communication +costs, and data privacy concerns.Moreover, [1] +presented +a +FedML +survey +on +enabling +technologies, protocols, and applications. They + +provided the most relevant protocols, platforms, +and real-life use-cases of Federated Learning to +enable data scientists to build better privacy- +preserving solutions for industries; they also +explored the challenges and advantages of Fed ML +for real-life applications. Finally, [5] presented a +survey on federated learning from a privacy +preservation angle. +Although these surveys and SLRs are excellent, +we think that our study tackles some aspects that +were not directly addressed by them. They do not +focus on the energy factor in the optimization of +federated learning, except for [38], where it is not +thoroughly tackled. We attempt to shed light on +power consumption aspects in FedML for the IoT. +As reported by Cisco in [9], IoT connections will +represent more than half (14.6 billion) of all global +connected devices and connections (28.5 billion) +by 2022, showing their increasing pervasiveness in +human lives [14]. Our personal use of smart +phones and watches, which need frequent and +sometimes bothersome recharging, is also a +practical witness to this concern. Finally, there is +also the particular case of wireless sensor +networks that can be deployed in hostile +environments with no possibility at all of energy +replenishment. +3 Background +In this section, we talk about the global Federated +Learning process, the FedAvg algorithm, the +energy consumption problem, and some other +background information. +3.1 Federated Learning +Federated Machine Learning is the process of +developing +accurate +models +on +large-scale +distributed systems made up of small devices by +combining their computation power and local data +[46].The goal is to solve a class of problems that +cannot be solved by a single central computer, such +as those involving users' personal data, real-time +computing, and on-device artificial intelligence +[32]. + +FedML is based on a distributed architecture that +involves several nodes performing training tasks on +their local data and exchanging their model’s +parameters with a central server. The server then +builds, from local models, a global aggregated model, +which is equivalent to a trained model on all nodes' +consolidated data. In the case of FedAvg, the global +model Wg is built as a weighted average of the local +models Wi (see equation 1). +Wg  = ∑ ni +n Wi +i + (1) +The optimization of the global objective function f +can be expressed as the optimization of the +average of local objective functions fi for all +participating nodes i = 1,...,ni, as given by the +equation 2 [54]. +minwf(w) = minw +1 +n ∑ fi(w) +i + (2) +where: +𝑓𝑖(𝑤) ≔ +1 +𝑘 ∑ +𝑙(𝑤, ξ) +ξ∈𝐷𝑖 + +(3) +fi is defined as an average of the local loss +function l, for each node i, on its local sample +points, Di = ξi1,··· ,ξim for i ∈ [n], where Di is the local +data set of the node i, composed of m data points; ξi +and w are the model parameters. +𝑚𝑖𝑛𝑤𝑓(𝑤) ≔ 𝑚𝑖𝑛𝑤 +1 +𝑛𝑘 ∑ +𝑙(𝑤, ξ) +ξ∈𝐷 + (4) +Finally, to solve equation 4, a gradient descent +method is used by each node to minimize the loss +li over its local training data Di, and eventually the +aggregated model Wg will minimize the global +objective function. +3.1.1 Federated Learning pseudo- +algorithm +Algorithm 1 shows the idea behind Federated +Averaging (FedAvg), proposed by [46] for Fed +ML . + + + + + + + + + + + + + + + +end + +/* Run on server*/ + initialize w0; +for each round t = 1,2, ... do +m ← +max(C.K,1); + end + +/* Run on client k*/ +Function ClientUpdate(k,w): B ← (split Pk +into batches of size B); for each local +epoch i from 1 to E do +for batch b ∈ B do +w ← w − µ∇l(w,b); +end +return w to server; +end + +Algorithm 1: FedAvg pseudo-algorithm +The notations employed in the algorithm are +explained underneath. +• C Fraction of selected clients in each round +• K Total number of clients +• m Number of randomly selected clients for each round +• St Set of clients for each round +• wt Global model parameters at round t +• +Received model parameters from client k at round t +• nk Number of data points of client k +• n Total number of data points of all clients +• Pk Local data-set of client k +• B Local data-set mini batch size to use for client training +• B Set of data-set mini batches for local training +• E Number of training passes performed by each client before +sending the update to the server. +• µ Learning rate +• l Loss function +• w Local model parameters +As shown in the aforementioned algorithm, the +server initiates the model’s parameters w0, then, +for each round, it determines the number m of +participant clients to choose for training as a +fraction C of K total clients. The subset of devices St +is determined randomly, and then each client +device k receives the model’s parameters wt from +the server to perform the training on its respective + +Fig. 1 Federated Learning global schema +local data set Pk. This training process performs a +split of the local data into small batches of size B, +and a number of E local epoch runs to train the +local model. Finally, all selected clients compute an +update of the parameters w, then send it back to +the server, which averages them to get the new +global model parameters wt+1. This round is +repeated as many times as determined by the +server to reach the target performance. +3.1.2 Federated Learning process +Fed ML architecture is composed of the client nodes +and the central server. The server receives the +computed updates from client devices and performs +an aggregation operation to build the global model. +It is then improved continuously, by running +additional iterations on the nodes, to train their local +models, until obtaining the desirable results. + +Figure 1 globally shows the components +involved in the Fed ML architecture, as well as the +stages of the FedAvg algorithm execution. In each +round of the training, the following operations are +performed: +S t ← randomsetofmclients); +( +for eachclient k ∈ S t inparallel do +w t +1 ← ClientUpdate ( k,w t ) ; +w t +1 ← P K +k =1 +n k +n w k +t +1 ; +end + +1-Initialization +4-Updates +aggregation +2-Local model +5- Global model +update +update +3- Updates sent to server +6-New iteration +Uplink +Downlink +Updatedlocalmodel +JpdatedGlobalmodel +Local training +Newiteration1. Definition of model’s structure, random +initiation of parameters and selection of +participating devices: the central server must +define the parameters E, C and B prior to start +of the training, and it must select a subset of +clients to participate in each round. +2. Model Update on local data: each selected +client computes an update of the global model, +by running local training iterations as many +times as defined by the central server. +3. Transmission of Local Model updates to +server: each participating device sends the +computed update of the model. +4. Aggregation of all received model updates: +the server aggregates the received updates in +such a way that builds a global model. +5. Sharing the updated global model with the +devices. +3.1.3 Heterogeneity +Very often, in real applications, the participant +nodes in the FL have uneven resources and +training data, we refer to this by system +heterogeneity +and +statistical +heterogeneity +respectively [35]. +System heterogeneity +During the collaborative training of the global +model, different nodes have different capacities +(e.g., CPU, Battery, Memory, Bandwidth). As result, +if we ignore this fact, the convergence will be very +slow, and the weak clients will exhaust their +resources before the end of the training, resulting +in bad model performance. +Statistical heterogeneity +When FedAvg was first proposed by [46], it was +based on the assumption of independent and +identically-distributed (iid) data across nodes, +which guarantees a theoretical solution for the +equation 4, regarding balanced local data-sets Di, +by using the gradient descent optimization +method. However, this assumption cannot be held +for the majority of distributed data on IoT and +users’ devices; this is a big limiting factor facing the +deployment of Fed ML in real-world scenarios [76]. +In fact, the majority of works published on this +topic display good results for iid data and poor +ones for non-iid setups, which is shown by a bad +impact on the global model’s performance and the +required time and energy for the training [76]. This +substantial problem has driven several teams to +develop techniques to adapt the original federated +learning algorithm to both types of heterogeneity +[11, 34, 70]. +3.2 Energy consumption formulation +The main goal of FedML optimization for energy- +constrained devices is to minimize the functional +energy consumption of the nodes while building a +good global model. In general, a wireless device's +total energy consumption ET can be divided into +three major parts: Enet, Ec, and Esys (Equation 5). +𝐸𝑇 = 𝐸𝑛𝑒𝑡 + 𝐸𝑐 + 𝐸𝑠𝑦𝑠 +(5) +Enet is the energy consumed by the device for +communications with other devices or the server +for update exchanges. Ec is the energy consumed +by the device’s local processing unit and memory +to accomplish the training computations. Esys is the +energy +consumed +by +the +general +system +operations of the device, which are not related to +its participation in the Federated training. +Note that Esys is generally small and negligible +compared to the total amount used in IoT [45]. In +addition, it is not specific to the problems +considered in this study, so we omit it from this +formulation. +Moreover, communications generally consume +more energy than processing, for an equivalent +amount of operations (this justifies multiple +aggregation approaches before data transmission). +Equation 6 gives the amount of energy consumed +by network communication, expressed by a set of +parameters related to our context. +𝐸𝑛𝑒𝑡 +≃ +∑ +𝑁𝑇𝑏𝑖𝑡 +𝑖 +𝑃𝑇 +𝑅𝑇 +𝑁𝑇 +𝑖=1 ++ ∑ +𝑁𝑅𝑏𝑖𝑡 +𝑖 +𝑃𝑅 +𝑅𝑅 +𝑁𝑅 +𝑖=1 ++ 𝑐 (6) + +NT and NR are, respectively, the number of +transmitted and received updates by the device. PT +and PR are the transceiver power at transmission +and reception, respectively. RT and RR are bit rates +for transmission and reception, respectively. + +and + are the number of bits transmitted and +received, respectively, in a given update i, and c is +amount of energy consumed by irrelevant factors +such as channel noise, transmission errors, etc. +If PR = PT = P and RR = RT = R, the equation 6 can +be simplified into equation 7: + +𝐸𝑛𝑒𝑡 +≃ +(∑ +𝑁𝑇𝑏𝑖𝑡 +𝑖 +𝑁𝑇 +𝑖=1 ++ ∑ +𝑁𝑅𝑏𝑖𝑡 +𝑖 +𝑁𝑅 +𝑖=1 +) +𝑃 +𝑅 + 𝑐1 (7) + +Moreover, the energy consumption by local +computations +on +each +client +device +is +approximated by the equation 8. +𝐸𝑐 +≃ +∑ +𝑇𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔 +𝑖 +𝑁𝑟𝑜𝑢𝑛𝑑 +𝑖=1 +× 𝑃𝑐 +𝑖 +(8) +Where + is the consumed power per training time +unit at round i, Ttraining is the duration of +computation operation, and Nround is the number of +operations to run by a given device. +If Ttraining and Pc are equivalent for all rounds on +a given device, the equation 8 can be simplified as : +𝐸𝑐 +≃ +𝑁𝑟𝑜𝑢𝑛𝑑𝑇𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔𝑃𝑐 +(9) +In summary, the approximated total energy +consumed by each client device (Equation 5) can +be expressed by equation 10. +𝐸𝑇 ≃ 𝑁𝑟𝑜𝑢𝑛𝑑𝑇𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔𝑃𝑐 + (∑ +𝑁𝑇𝑏𝑖𝑡 +𝑖 +𝑁𝑇 +𝑖=1 ++ ∑ +𝑁𝑅𝑏𝑖𝑡 +𝑖 +𝑁𝑅 +𝑖=1 +) +𝑃 +𝑅 (10) + +From the above energy formulation, we can +identify a list of parameters which impact the +energy consumption of the participant client +devices in Federated Learning: the number of +exchanged updates NT and NR, the number of bits in +each exchanged update NTbiti and NRbiti , the +transmission power P, the transmission bit rates R, +the duration of local training Ttraining, and the +number of local training rounds Nround. +3.3 Fed ML optimization parameters +Based on the established equations in the previous +section, together with the studied selected papers, +we identify a number of energy optimization +aspects. +Accordingly, in order to minimize the total energy +in equation 9, the optimization of the local training +tasks to accelerate the model convergence should +result in decreasing the number of federation +rounds Nround. Moreover, the training time duration +Ttraining will be improved if we reduce the trained +model’s +complexity, +which +impacts +energy +efficiency. Aggregating updates with the least cost, +by reducing the size of exchanged data with the +central server (i.e., decreasing NTbit and NRbit in +equation +7), +will +help +save +battery +life. +Furthermore, the frequency of model update +exchanges affects the total number of updates NT +and NR (equation 7), thus optimizing even more the +energy consumed in communications. More +optimization can also be achieved by making smart +use of the heterogeneous nodes’ computing +resources to participate in the training, in addition +to optimizing the client selection to balance the +load over the participant nodes and involve the +best ones for accelerated convergence. Finally, +decreasing the transmission power +P and +maximizing the bit rates R (equation 7) also helps +to reduce the total spent energy. +This analysis will help us later to classify the +different approaches and techniques proposed in +the literature, as we will see in subsection 5.3. + +Fig. 2 Our Search Process +4 Systematic Mapping Study +Process +This section describes the process followed +throughout this Systematic Mapping Study. + +Start +Automatic papers +Google +Science +Scholar +Direct +search +Categorizationof +papers +ENDAdditional material is available on the online +repository created for it 1. +Figure 2 illustrates the steps taken. After an +automatic search based on the defined keywords +and search string in the three common databases, +the first step consists of filtering relevant papers +based on their title. Then, we refined the selection +based on the abstract. We refined our search even +further by reading the full text.Finally, we added a +manual search step afterwards to spot any articles +that were not found the automatic way. Details +about each step of the workflow will be presented +in the upcoming paragraphs. +4.1 Papers selection +In order to obtain all relevant papers for our study, +we have queried three main databases (Google +Scholar, IEEE Explore, and ScienceDirect) by using +the search string in Listing 1, built mainly using the +following keywords: federated machine learning, +edge computing, on-device intelligence, energy, +and optimization. +Listing 1 search query +”(” Federated Machine Learning” OR ” +Federated Learning”) AND (”edge computing” OR +”on−device intelligence ”) AND ( energy OR power) +AND ( optimization OR optimal OR efficient OR +efficiency )” +Filtering papers. We filtered the initial search +results to keep only papers, that meet all the +following inclusion and exclusion criteria. +i. Inclusion criteria: +• Papers from 2016 to July 2021 +• Papers in the English language +• Papers which propose an optimization of +Federated Learning w.r.t. energy consumption, +using techniques including communication cost, +or training time reduction +• Papers which target the IoT or mobile devices in +general +ii. Exclusion criteria: + +1 https://gitlab.com/rachid-el-mokadem/fedmlsysrev +• Works on distributed machine learning with no +explicit application to federated learning on +resource-limited devices +• Similar works of the same authors +Manual searching. In order to cover the literature +as much as possible, another step was added to +look for potential papers that might have been +missed earlier: backward snowballing by looking +Table 2 Research questions + +RQ ID +Question +RQ1 +What is the publications tendency? +RQ2 +What network architectures are proposed? +RQ3 +How is the energy optimization achieved? +RQ4 +How is the optimization validated? +RQ5 +What are the reported optimization results ? + +at cited references in the selected papers. +Thereby, additional papers were added for a total +of 67 papers. In the remainder of this study, we +will refer to selected papers by identifiers, +attributed according to the chronological order of +the publication: P1, P2, up to P67. The list of all +papers, along with their classification, is depicted +in Table 5 in Appendix A. +4.2 Research questions +In order to analyze the literature and compare the +proposed techniques in a systematic way, we +define a set of research questions that will guide +our analysis (Table 2). RQ1 indicates the timeline +and sources of the papers; RQ2 presents the +network topology considered by each paper; RQ3 +examines the FedML energy optimization aspects +that are addressed by each paper; RQ4 presents +the experimentation setups used to validate the +approaches; and RQ5 measures the optimization +improvements of the experiments. + + + + +Fig. 3 Fed ML papers publication trend over time +5 Questions answering +In this section, we present the results analysis from +the study of the selected papers, arranged as +answers to the research questions defined in +subsection 4.2. +5.1 RQ1 - What is the publications +tendency +Answering this research question will account for +providing the number of publications evolution, +their distribution over the publishing venues, and +the nature of papers, as well as their influence on +the field of FedML. +The graph in Figure 3 shows the papers +publication trend over time. The growing number +of papers over the last 3 years is clear, with 33 +papers in only 2020. Given that the first paper from +[32] was published in 2016, we can clearly see the + +Fig. 4 Paper types distribution +big interest this subject is receiving from several +research teams around the world. + +The +majority +of +papers, +as shown in +Figure 4, were published in journals (≈36%) and +conferences(≈34%). This shows the growing +maturity of this subject and the engaged efforts by +the scientific community. We also have 20 out of 67 +(≈29%) papers published as pre-prints on the +ArXiv database, including 10 in 2020. This could be +justified by the fact that the subject is evolving +quickly, with fast feedbacks. We have also included +the non-peer reviewed papers of Konecny´, Jakub +et al. [32, 33], since they are considered the most +impactful in the subject, with 747 and 1733 +citations, respectively. The same team is behind +the seminal work on the FedML proposal [46]. +Furthermore, we consider the number of +citations for each paper, shown in Figure 5, to +measure their influence on the subject. It is obvious +that older papers tend to get more citations than +new +ones. +However, +it +does +provide +an +approximate idea of the paper’s scientific interest +for the community. From the graph, we notice +some spikes on a couple papers. For older papers +such as P1 through P8, this is somehow reasonable. +However, in the case of P15 ([57]) with 345 +citations, P19 ([8]) with 110 citations and P35 +([54]) with 145 citations, this definitely shows the +high impact of those papers. More details on the +techniques used by them in subsection 5.3 +5.2 +RQ2 - What network +architectures are proposed +Fig. 5 Number of citations per paper +In this question we consider the proposed network +architectures of the studied papers. This is +important to us, because the network topology has +an impact on the communication cost, and +therefore the energy consumption. +The architectures are as follows. +• Centralized: based on a central server to ensure +the communication and model’s parameters +exchange, between the participating devices. +This option is energy consuming, due to long +range communication between the devices and +the server, which requires higher transmission +power P (equation 7). It also suffers from a single +point of failure. + +IEEE (42) +■ArXiv (20) +■MDPI (2) +PMLR (2) +■SPRINGER (1) +35 +1 +30 +2 +2 +25 +10 +20 +15 +5 +10 +18 +5 +1 +10 +11 +1 +1 +0 +3 +2016 (1) +2017 (1) +2018 (4) +2019 (12) +2020 (33) +2021 (16)1800 +1600 +1400 +1200 +1000 +800 +600 +20029.85% +34.33% +Conferencepaper +Journalpaper +Preprint +35.82%• Decentralized: +based +on +node +to +node +communication without the need for a central +server. In this setup, the devices can save lot of +energy, +by +opting +for +short +range +communication between the nodes only [15]. +• Hybrid: this architecture is based on at least +three layers of devices, where intermediate ones +are placed between the central server and the +end devices. +Fig. 6 Number of papers by Network topology +The hybrid architecture is based on adding edge +servers between the main server and the end +devices. These intermediate devices can play +several roles, such as managing direct clients under +their control, which allows the offloading of the +central server and lowers the waiting time for +aggregating multiple received updates. In some +cases, this edge server can also be used to offload +the end devices from local update computing, by +periodically querying the training data from the +selected clients, doing the updates with a much +higher computing capacity and communicating +with the server, on behalf of the end nodes. As a +result, this architecture can allow a high energy +optimization on the devices, although posing some +threats to data privacy, especially when these edge +servers are not trustworthy, and the data is very +sensitive. +Figure 6 shows that the majority of papers (59 +out of 67) are based on a centralized setup, while 2 +papers have a fully decentralized one, and 6 +propose a hybrid architecture. The predominance +of the centralized scheme can be explained by the +influence of the architecture in the original paper +[46], which comes from Google. Moreover, the fully +decentralized scheme faces some algorithmic and +practical challenges to aggregate the models +without a central device [30]. +5.3 RQ3 - How is the energy +optimization achieved +Fig. 7 Federated Learning optimization techniques (recap) + +In this question, we analyze the techniques used by +the papers, to optimize the Fed ML. Our study +focuses on the power consumption reduction, so as +seen in subsection 3.2, all studied optimization +aspects are linked with the energy through +equation 10. We classified these techniques into +the +following +categories: +(1) +convergence +acceleration (2) data exchange optimization and +(3) client resource management. Figure 7 recaps +the different techniques. Table 3 presents the +optimization aspects addressed by each studied +paper. + +5.3.1 convergence acceleration +In federated machine learning, the training tasks +are performed by the client nodes to build a global + model under the orchestration of the central +server, during as many rounds as needed to reach +a good performance. In order to save the battery +life of the client devices, the total time to reach + +Updates +compression +Data +Exchange +Updates +Optimization +frequency +Logits +exchange +Local training +acceleration +FedML +Convergence +accelerating +Model pruning +Optimization +Optimized +averaging +Clients +selection +Client +Resource +Resource +optimization +Management +Transmission +settings +Adjusted +clients' +models3% +9% +Centralized +Hybrid +Decentralized +88%global model convergence can be reduced with +several approaches. +Local training acceleration +Many works have used different optimizations to +accelerate the local training, such as adaptive +learning rate [42], and Adam optimization method +[47]. +Equation 11 is used in the original version of +Federated Learning. w are the model weights, b is +the model bias, ∇ is the gradient of the loss function +l and µ is the learning rate. +In this version, the server defines a learning +rate parameter at the beginning, used to compute +the gradient descent steps in local training. +Opposed to that are the aforementioned methods, +which determine the best steps to take in order to +quickly achieve the convergence of the global +model. +𝑤 ← 𝑤 − 𝜇∇𝑙(𝑤, 𝑏) (11) + + +The benefit of these techniques is to decrease the +number of rounds Nround (equation 9) required for +the model convergence, and thus reduce the +energy consumption for the participant devices. +Accordingly, [47] proposed CE-FedAvg, which +improved how the nodes compute their local +updates by using the Adam method, known for its +improved learning rate, instead of SGD (used in the +original FedAvg algorithm). The weights’ update +method of the proposed algorithm, executed by +each client, is shown in equation 12. + +wk,mk,vk ← AdamSGD(wk,mk,vk) +(12) +Where wk are the model weights, mk is Adam’s +first moment, and vk is Adam’s second moment. +These parameters are used to compute the Adam +steps by averaging them over all received updates +or gradients and sending them back to the clients +in the next round of the training. +Feature augmentation is a technique used in +machine learning to improve training performance +in an unbalanced class distribution ([75]). +Similarly, in the context of Federated Learning, +FedFusion is an algorithm presented by [72] to +accelerate the global model’s training by using a +technique named Feature Fusion, which is based +on using a combination of the global model’s +feature space with the local model’s feature space +to train the local model. The global model is used +as a feature extractor, and then multiple types of +feature fusions are employed to efficiently +aggregate all of them. Additionally, [71] presented +a two-Stream model learning with Maximum Mean +Discrepancy (MMD), where the nodes training is +performed on two models, in parallel, both +initialized with the global model parameters, but +one of them (global model) is kept unchanged +during the training. An MMD loss is computed +between the output of the two models, which is +used to optimize the local one. This technique is +often employed with learning transfer and +knowledge distillation in standard machine +learning, and its adoption for Federated Learning +helps to accelerate the training and reduce the +communication cost. In essence, it consists in +constraining the local model training by the global +model parameters, to avoid that local models over- +fit their local data, thereby building a good global +model in lesser training rounds Nround. [4] used an +adaptive +dropout +schema +to +decrease the +convergence time by reducing the local model’s +complexity and number of trainable parameters. In +practice, each round a random sub-net wc of the +global model is sent to each participant client c, +then an activation score map M is used to track the +indexes A of the best sub-models to be reused in +the next rounds. +Model pruning +Model pruning is another technique widely used in +deep learning, which accelerates the training, by +reducing the number of model parameters, based +on training data. The reduction simplifies the +model, thereby decreasing the computation time +(Ttraining in equation 9), local training energy +consumption Ec, while keeping a good model +performance. [29] implemented an algorithm +named PruneFL where the pruning is performed +initially by a selected client on its local data. Then +the resulting smaller model is iteratively adapted +by the server in each round w.r.t. to the training +efficiency, by involving all clients updates, to +reconfigure it, through removing or adding back +some +parameters. +In +order +to +allow +the +reversibility of parameters adding and deleting, +the authors used a mask with zeros and ones for +removed and kept weights respectively. Similarly, +[69] proposed a structured model pruning +combined with weights quantization and selective +update, to accelerate the training and reduce the +computation cost on the devices. In particular, the + +authors used an l1 − norm based pruning of the +model weights with a variable ratio from 0 to 90%. +Optimized averaging +While original Federated Learning works by +gathering the local model updates, and simply +averaging them, several papers proposed to use +advanced averaging methods, allowing a fast +training convergence. Accordingly, [23] proposed +Federated Momentum (FedMom), a technique +with biased gradients that uses the momentum +method to update the global model, according to +equations 13 and 14: +𝑣𝑡+1 = 𝑤𝑡 − η ∑ 𝑛𝑘 +𝑛 +𝐾 +𝑘=1 +(𝑤𝑡 − 𝑤𝑡+1 +𝑘 ) (13) + + + +𝑤𝑡+1 = 𝑣𝑡+1 + β(𝑣𝑡+1 − 𝑣𝑡) (14) + + +Where vt is the average of the previous round’s +updates and beta β (often equal to 0.9) is the +parameter used to compute the moving average of +the updates, through time. On the other hand, [39] +used a hierarchical architecture by introducing L +edge servers between the central server and client +nodes. Each edge server has a subset s of clients +from which it aggregates the updates before +forwarding them to the main server. According to +the authors, this method reduces training time and +decreases node energy consumption. +5.3.2 Data exchange optimization +The global model is built by gathering and +aggregating the updates from the participant +nodes at the central server. The frequency of +exchanging the computed updates and their data +sizes are optimized by several works in order to +achieve +communication-efficient +federated +learning, which drastically saves the battery life of +the participant nodes without compromising the +global model’s performance. In FedAvg, the +aggregation of the local models is achieved +according to the following equation 15: +𝑤𝑡+1 ← ∑ 𝑛𝑘 +𝑛 𝑤𝑘 +𝑡+1 +𝐾 +𝑘 + (15) +Where wk are the learned weights at each node, +nk are the number of data points at each node, and +n the total number of data points for all K +participant clients. +Updates compression +A stated limitation of FedAvg [46] is that the +participant clients must upload the full computed +updates at each round of the training, which has an +impact on the power consumption of these devices. +The proposed optimization methods allow the +reduction of the data exchanged between the +nodes and the server while preserving the quality +of the global model. +In addition to ordinary data compression +algorithms used to encode the final updates with +lower amounts of bits, such as Huffman encoding +used by [7, 43], data size reduction is achieved by +several +other +methods, +such +as +update +quantization, sparsification, and sketching. The goal +of all these techniques is to reduce the amount of +bits per round NTbit, sent through the wireless +interface, +which +subsequently +decreases +significantly the energy usage for exchanging +model’s updates (equation 7). +The quantization of machine learning models is +based on using low float-point precision to +represent the model’s weights in order to reduce +the bit size ([2]). [68]proposed a method called +Federated Trained Ternary Quantization (FTTQ), +which reduces both upstream and downstream +traffic. +It +implements +a +layer-wise +weight +quantization with an adjustable threshold during +the training, which has the additional benefit of +reducing the training tasks’ energy budget. +Similarly, [27, 43, 47, 54] used quantization for +data size reduction, in most cases mixed with other +techniques. Furthermore, [27, 44] proposed an +adaptive schema for updating quantization to +achieve communication-efficient training. +Additionally, the sparsification of the global and +local models is used to compress the exchanged +data by eliminating the gradient values of the +computed update that are below a given threshold +and replacing them with zeros. This operation +results in a sparse model update that can be +encoded with a small number of bits in order to +optimize the communication cost and energy +budget. Accordingly, [57] proposed Sparse Ternary +Compression +(STC), +a +new +compression +framework created especially for the requirements +of Federated Learning on resource-limited devices. +STC extends the existing compression methods (in + +particular top-k sparsification [60]) to support +downstream +compression; +additionally, +the +authors combined sparsification with quantization +and +Golomb +encoding +to +achieve +better +optimization results. [20] developed an adaptive +gradient sparsification based on bidirectional top- +k gradient sparsification to reduce communication +costs in both directions between the server and the +nodes. The sparsification’s parameter k is +determined by the server as a trade-off between +communication and global model accuracy. +Moreover, [62] used a gradient sparsification with +gradient correction, in order to accumulate the +insignificant eliminated gradients and add them +lately to speed up the convergence of the model +training. +Other techniques used to this end are gradient +sketching [33, 55] and subsampling. The first one +is based on compressing the update with a data +structure named Count Sketch [61]. The second +one [33] involves clients, sending only a smaller +update derived from the computed one, by +randomly sampling their values. The server then +averages all received sub-sampled updates to get +an estimate of the global model parameters. +Additionally, +[52] +used +dual-side +low-rank +compression to reduce the size of the models in +both directions between the server and the nodes. +Finally, [36] used a layer-based parameter +selection in order to transfer only the important +parameters of each model’s layer. +Updates frequency +In the original Fed ML algorithm, clients send +updates at each iteration of model training, which +induces high communication costs and energy +consumption as the number of updates exchanged +with the server NT and NR increases. In order to +perform, under a given resource budget, [65] +proposed a control algorithm that determines the +best trade-off between local update and global +parameter +aggregation. +It +learns +the +data +distribution, and system characteristics along the +distributed training, then determines dynamically +the frequency of global model aggregation, with +respect to the resource constraints. Alternatively, +[8] presented a different method, which is based +on the model’s layer-wise frequency. It means that +important layers’ parameters are more frequently +exchanged than less important ones. The reason is +that the first layers of a deep model tend to learn +general features for different data sets, while the +deeper layers learn more particular ones. +Consequently, each node separates its model into +shallow layers’ weights wg and deep layers’ +weights ws, which are exchanged with the server +separately and asynchronously, under the control +of the server. It determines, for each client, the type +of weights to consider, and performs a temporally +weighted aggregation to give more importance to +the newest received models. [64] proposed +another method where the clients pull the global +model less frequently from the server (to reduce +down-link energy consumption) and compensate +the gap with local updates. +Logits exchange +Some works have chosen not to exchange the +updates with the server: only the outputs of the +trained local models, called logits, are sent to the +server +which +reduces +drastically +the +communication cost of the federation by many +orders of magnitude [58]; nevertheless, all clients +and the server must have shared public data +samples to compute and share their outputs. +In order to build a global model out of this +reduced data, the authors of [24, 26, 50, 58] used a +learning transfer technique called knowledge +distillation [59], where multiple teacher models +(local models) transfer their learning to a single +student model (the global model) [21]; In all cases, +the distillation task is performed by the server, +except for [24] where the sent logits are averaged +by the server and sent back to the clients to +perform the distillation themselves. In that case, +the communication cost is reduced in both +directions as the server does not send the whole +model to the nodes. +5.3.3 Clients resource management +Many +approaches +allow +client +devices +to +participate in model training, with optimal energy. +Client resources generally refer to CPU time, +memory and wireless bandwidth, which are often +related to energy consumption. Two of the most +used approaches for client resource management +affect (1) client participation and (2) transmission +settings. + +Clients selection +In the original FedML, participants were selected +randomly, each round, from the available nodes. +Subsequent analysis showed that this approach +yields poor model convergence and causes node +resources to be wasted ([49]). Accordingly, many +works have addressed this aspect, by adaptive and +optimal client selection, based on their available +resources and data in each round. [49] presented +FedCS, a Federated Learning algorithm with +optimized client selection, where the server starts +by selecting a random set of clients, then performs +a more informed selection using client resources +and the time taken to compute the updates. +Furthermore, [3] presented a Reinforcement +Learning scheme at the server, based on energy +units en, number of CPU cycles cn, and the amount +of data points used for training by each client, per- +round. +A server reward is then computed from these +values to help it find the best policies and actions +for efficient training with optimal resource usage. +In the same vein, [53] proposed a multicriteria +client selection model, named FedMCCS, that is +based on a discriminative selection of client +devices based on CPU, Energy, Memory and Time. +The server tracks these values along the training +by +an +auxiliary +data +exchange +of +requests/responses with the nodes. A linear +regression model is trained on these attributes to +predict whether a client has enough resources to +participate in training tasks. Moreover, [67] +proposed the selection of clients based on their +participation history, which impacts the global +model’s performance. Additionally, [56] proposed +a data imbalance aware selection of the +participants in each round, such that all data +categories must be covered at least once. This is +achieved by requesting a bit-mask η containing C +bits corresponding to the available data categories +from each node. The server then sorts these bit +masks in a decreasing order of the number of sets +and minimizes the required number of clients to +get all categories covered by the averaged updates. +Hybrid scheme +Other papers have proposed a hybrid scheme, +based on the architectures presented in our second +research question, to optimize the resources of the +client devices. [12] developed a self balancing +system based on mediator edge servers, gathering +near uniform data distribution subsets of clients, +and aggregating the trained models, before +sending them to the central server to build a global +one; Similarly, [39] achieved energy consumption +reduction by balancing the exchange of parameters +with L edge servers with respect to the training +time and communication budget where each edge +server incorporates a small number of clients [66] +used a hierarchical aggregation of the model +updates to overcome the communication overload +between the nodes and the server. Moreover, [25] +and [74] proposed a cloud-edge-client scheme +wherein the clients offload a part or all the training +tasks to the edge servers, which get portions of the +clients’ data for the training. This approach has +some flaws w.r.t. communication overhead and +privacy concerns for clients’ data, but it may be +relevant in some application-specific scenarios. +Transmission settings +Wireless transmission has a high energy cost for +mobile and IoT devices in general. As we saw in +equation 7, it is related to the transmission power +P and the bit rate R of the network interface. Many +papers have tackled the optimization of wireless +transmission +and +bandwidth +settings. +[48] +presented a technique for transmission power and +rate +optimization +for +energy-efficient +communication between the nodes and the central +server. They consider minimizing total energy +consumption as a joint optimization problem over +bit rate, transmission power, and CPU frequency +for local updates. On the other hand, [28] proposed +a fully decentralized communication between the +nodes, based on a Segmented Gossip protocol. In +this communication scheme, the workers segment +their updates into small fragments that are +separately shared with a subset of the participants, +who then relay them progressively to other +workers until all of them get everything. The peer +workers with faster bandwidth are chosen to pull +updates from. Moreover, [63] presented an +energy-aware worker scheduling algorithm: a +node monitors its energy consumption at each +round and, by comparing it with an adjustable +threshold, decides whether to participate in the +next training round or not. Subsequently, each +worker partitions its update into M segments, +which are transmitted separately on M sub- +channels with adjusted transmission power. + +Adaptive local models +Recently, [73] and [11] proposed to adapt the local +models to the nodes’ capabilities and data, in +opposition +to +the +state-of-the-art +FedAvg +algorithm, where all clients get the same model +architecture. In both works, an adapted model is +derived for each client as a sub-net of the global +model, with a smaller number of layers and +parameters. Then, different averaging methods are +used to aggregate the updates. These new methods +are heterogeneity and data imbalance aware and +allow an adaptive saving of energy used for +training and communications. +From Figure 8, we see that the data exchange +optimization category has the highest number of +papers +(32), +followed +by +client +resource +management +with +25 +papers, +and +finally +convergence acceleration (10 papers). Figure 9 +shows in more detail the optimization techniques +used by each + +Fig. 8 Papers count per optimization category + +Fig. 9 Papers count per optimization technique +paper. It was not surprising that Updates +Compression and Clients selection represented +most of the papers because the original FedAvg +algorithm had a substantial limitation on these +aspects. Additionally, this will have the highest +impact on devices’ energy preservation, knowing +that wireless communications use the biggest part +of the operational energy. The other techniques +are shared between the rest of the papers, with a +small advantage for Local training acceleration. +5.4 RQ4 - How is the optimization +validated +This +Research +Question +is +about +the +experimentation setup (models, data sets, and +testing platforms) used by different papers to +validate their proposed works. +The classification data in Table 4 shows that the +majority of papers considered only neural +networks (specifically convolutional ones) with +MNIST and CIFAR10 data sets for the experimental +part of their research. While this seems to restrict +validation, it can be explained by the ease of getting +these famous data sets and implementing NNs on +top of popular machine learning frameworks such +as PyTorch and Tensorflow. Some papers have +implemented additional validations on other ML +models such as Linear Regression, Logistic +Regression, +and +Support +Vector +Machines. +However, there is a substantial shortage of +validation results for non-neural network models, +which may exhibit lower complexity and therefore +lower resource consumption. +As for the experimentation platforms, different +papers +considered +different +numbers +of +participating nodes, from 2 up to 50000. The +majority of papers (52 out of 57) have used +emulated nodes on multi-GPU computers, while +only five papers (P7, P14, P17, P34, P58) have +performed experiments on real devices such as the +Raspberry Pi™and smartphones. Emulated devices +can give an insight into validation, but it is +important to have further results on real ones, in +real life scenarios, especially regarding wireless +communications, +energy +constraints, +and +computing power. The democratization of rapid +prototyping platforms in the industry and +academia (e.g., Arduino™, Raspberry Pi™ & +ESP31™) is another motivation for that. + +32 +25 +105.5 RQ5 - What are the reported +optimization results + +Fig. 10 Communication cost improvements (per paper) + +In this question, we list and compare the +optimization results obtained in the surveyed +papers. The numerical results are classified into +three categories: (1) Communication Cost, (2) +Convergence Time, (3) Energy Consumption. Each +paper quantified its optimization improvement +compared to the standard FedAvg [46] algorithm, +and the numerical results w.r.t. each category are +listed in the graphs. From Figure 10, we see a very +wide range of improvement values related to +global communication cost reduction, which goes +from 2x up to 320x. In Figure 12, we see +convergence time improvements going from 3% +up to 98%. As for the energy consumption, Figure +11 reports a range of improvement from 14% to +99%. + + Fig. 11 Energy consumption improvements (per paper) + + +Fig. 12 Convergence time improvements (per paper) +The convergence time and communication cost +optimization results are very encouraging, which is +consistent with the important interest of the +community in these two aspects (Figure 8). +Although +they +directly +impact +the +energy +consumption, note, that during our readings, +relatively few papers (8 out of 67) have evaluated +Table 3 Papers list by optimization techniques +Category +Optimization technique +Papers +Data exchange optimization +Updates compression +Updates frequency +P2 P15 P17 P19 P23 P45 P24 P26 P27 P35 P37 P39 P40 +P42 P64 P52 +P1 P7 P8 P20 P56 + +Logits exchange +P4 P10 P46 P49 +Client resource management +Clients selection +Hybrid scheme +P3 P5 P18 P25 P28 P31 P32 P38 P44 P47 P48 +P16 P29 P34 P36 P51 + +Transmission settings +P67 P22 P30 P57 + +Adaptive models +P54 P55 +Convergence acceleration +Local training acceleration +Model pruning +P6 P11 P12 P13 P33 P63 +P14 P58 + +Optimized model averaging +P21 + + +the optimization’s benefit directly on the energy +consumption, which is crucial to our study. +6 Discussion +In this section we discuss the Systematic Mapping +Study results, in the light of the previous analysis, +guided by the RQs in Section 5. For each Research +Issue (RI) presented, we provide (1) some +remarkable +limitations +related +to +Energy +constrained Fed ML, and (2) some improvement +directions and recommendations for the research +community. +6.1 RI1: Fully-decentralized scheme +In a centralized scheme based Fed ML, client nodes +exchange data during the training with a central +server, which is generally located in the cloud. +Consequently, the nodes have to use long-range +wireless communication to reach the server, which +implies high power consumption [15]. To +overcome this, we must take advantage of the +short-range communication between the nodes, +which is by far less power-intensive, to exchange +the updates using peer-to-peer communications. +The proposed approaches to implementing a +fully decentralized FedML induce an overload on +the resource-limited devices, caused by the +additional operations performed by the nodes to +compensate for the role of the central server. In +[28], the nodes have to also play the role of the +aggregating server, and [13] proposed a technique +where all nodes compute and exchange their +updates in a chain-like scheme (using some sort of +multi-hop +Table 4 Papers validation setups +Paper +ML model +Dataset +Number of nodes + + + + +P1 +RNN +Blog posts dataset +1000 +P2 +CNN - RNN +CIFAR10 - public post reddit +100 - 1024 +P3 +CNN +CIFAR10 FashionMNIST +1000 +P4 +CNN +MNIST +10 +P5 +RNN +- +3 +P6 +CNN : AlexNet +CIFAR10 - MNIST +2 – 100 +P7 +CNN +CIFAR10 - MNIST +5 (RaspberryPi) – 500 +P8 +- +- +50 +P9 +CNN - LSTM +MNIST +30 +P10 +ANN +MNIST +25 +P11 +CNN +MNIST - CIFAR10 +- +P12 +CNN +MNIST - CIFAR10 +- +P13 +CNN +CIFAR10 +20 +P14 +CNN: VGG11 - LeNet +FeMNIST +5 - 10 (RaspberryPi) +P15 +CNN: VGG11 - LSTM - Logistic +Regression +CIFAR10 - KWS dataset - FashionM- +NIST - MNIST +100 +P16 +CNN +EMNIST - CIFAR10 - CINIC10 +500 +P17 +CNN +MNIST CIFAR10 +10 - 40 (RaspberryPi) +P18 +Linear model +Random integer data +20 +P19 +CNN RNN +MNIST HAR +20 +P20 +ANN +MNIST +15 +P21 +Logistic Reg - CNN (ResNet18) +CIFAR10 +20 +P22 +CNN +CIFAR10 - MNIST - SVHN +20 +P23 +CNN +FEMNIST +50 +P25 +Regression model - CNN - SVM +Boston Housing dataset - MNIST - KDD +Cup’99 dataset +5 - 100 - 500 +P26 +SNN +MNIST-DVS dataset +2 +P27 +CNN (LeNet-5 – CifarNet – +DenseNet-121) +MNIST - CIFAR10 - ImageNet +64 +P28 +CNN +MNIST +50 +P29 +CNN +MNIST - CIFAR10 +50 +P30 +ANN +MNIST +50 +P31 +CNN +- +80 +P32 +CNN +Fashion/MNIST - CIFAR10 +100 +P33 +CNN +MNIST +3 +P35 +Linear Regression - CNN +MNIST - CIFAR10 +50 + +P36 +CNN +MNIST +500 Clients/10 Edge server +P37 +CNN - Linear Regression +MINST - California Housing dataset +10 +P39 +CNN (AlexNet) - Transformer +(GPT2-small) +CIFAR10 - PersonaChat + +10000 to 50000 +P40 +CNN - Logistic Reg +CIFAR10 +- +FashionMNIST +- +ment140 +Senti- +100 +P41 +CNN +CiFar10 - MINST + +10 +P42 +CNN MNIST +CIFAR10 + +1 to 64 +P43 +CNN (ResNet) +CiFar10 - CiFar100 + +16 +P44 +ANN +MNIST - FEMNIST + +100 +P46 +CNN +MNIST - CIFAR10 + +10 to 20 +P47 +CNN +FeMNIST - CiFar10 - CiFar100 + +100 +P49 +CNN - LSTM +MNIST - IMDb + +NA +P50 +CNN (ResNet) +CIFAR10 + +10 +P52 +CNN +MNIST - EMNIST + +20 +P53 +ANN - CNN +EMNIST + +50 - 1000 +P55 +CNN (ResNet) - Transformer +MNIST - CIFAR10 - WikiText-2 + +100 +P56 +CNN +MNIST + +30 +P57 +CNN +MNIST - CIFAR10 and SVHN + +20 +P58 +CNN +MNIST - CIFAR10 + +3 (Core i5 PCs) +P59 +CNN +MNIST + +10-18 +P60 +CNN +CiFar10 - FEMNIST - IMDB + +10 +P61 +CNN +CiFar10 + +10 +P63 +CNN +FEMNIST - Shakespare - Sentiment140 +- +P64 +CNN +MNIST – Cifar10 +12 +P65 +CNN +CiFar10 - FashionMNIST +8 +P66 +CNN +MNIST - FashionMNIST - CIFAR10 +- +P67 +CNN - LSTM +FEMNIST - Shakespeare dataset +- +communications). The limitation of the first +technique is the overhead tasks for the nodes to +play the server role, where the second one forces +the nodes to run all the time of the training: in both +cases, more energy and resources are required at +the node level +The hybrid scheme seems to overcome some of +these problems since it has a cloud-based central +server that is only used to manage the client +participation and selection, with very limited data +querying from the nodes, while keeping model +aggregation between the client devices. At the +same +time, +in +order +to +advocate +the +fullydecentralized scheme, there is a need for a +new theoretical framework that supports complete +decentralized model aggregation with convenient +energy and resource consumption. +6.2 RI2: Large models reduction +Some Fed ML models with large sizes and a big +number of trainable parameters (e.g., Deep Neural +Networks) require a computationally expensive +training [17] for energy constrained IoT devices. + +If we consider Fed ML as a bootstrap +aggregation [22] of the global model over different +distributed nodes’ data sets, we could reduce the +local model’s size and complexity to make the +training tasks easier for the devices. We could still +build a high-performance global model by +aggregating +(e.g., +by +majority +voting) +the +distributed models repeatedly. +Another possible solution lies in the Lottery +winning ticket hypothesis, elaborated by Frankle +and Carbin [16], which states that a dense neural +network contains a sparse sub network, that can be +trained to equivalent performance of the initial +network. By applying this technique in the context +of Federated Learning, the global model can be +drastically reduced in size and complexity, to +accelerate +the +training, +and +reduce +the +computation load over the nodes. The target +sparse model could be obtained from the global +model by Adaptive Iterative Pruning (AIP) [19] or +Neural Architecture Search (NAS) [51], performed +adaptively by the server based on model +performance and available client resources. +6.3 RI3: Energy-aware data +compression +Many papers proposed different techniques to +reduce the amount of data to be sent or received +from the server [33, 54, 57]. The compression +techniques used are: sketching, sparsification, +quantization, and data encoding. They have helped +to drastically reduce the communication costs for +the client devices. However, they add overhead +tasks to the nodes, resulting in memory and CPU +usage to compress, encode, and decode the +transmitted data. +Many works [37, 37] have revealed the benefit +of error-controlled lossy compression schemes on +the compression rate and computation efficiency. +We recommend studying an equivalent technique +adapted +to +Federated +Learning’s +update +compression in order to further reduce the +communication energy cost. +6.4 RI4: Heterogeneity aware +optimization +Nodes heterogeneity is a crucial issue for +Federated Learning in real world applications [35]. +As a result, this topic has drawn the attention of the +research community through several works [49, +70], which proposed different methods based on +discriminative participant nodes selection. They +only choose the devices that have both the +required resources and data for the training. While +this seems to solve the heterogeneity issue, it may +impact the model performance and convergence +time, by eliminating some devices with either one +of those. +In this case, it would be more profitable to +manage the devices in a way that takes advantage +of their data and computation capabilities +separately. Some nodes may participate with their +data, others with their computing capacity, and the +rest with both. To achieve this flexibility, some +devices may exchange raw data, extracted features, +or data labels. To preserve data privacy (one of the +Fed ML rationales) during these communications, +we may use a lightweight encryption technique +(e.g., based on elliptic curves) for node-node +communication or an homomorphic encryption +scheme [18] for untrustworthy node-to-node +relationships. +We could also suggest an approach to managing +heterogeneity that would be based on the +separation of client nodes into two groups. The +first one would contain powerful, resourceful +devices dedicated to training tasks, and the other +one would contain poor nodes for validation only +on their own data. The validation score may be +used as feedback to adaptively adjust model +aggregation parameters. +6.5 RI5: Results validation +The majority of studied papers have validated their +approach using emulated nodes on powerful +computers. Moreover, a substantial focus was +given +to +image +recognition +tasks +using +Convolutional Neural Networks (CNNs) and +common data sets such as MNIST and CIFAR10. +The choice of image data for validation can be + +explained by its sensitivity and significant size, +resulting in elevated communication costs thereby +justifying Fed ML usage. However, in IoT and also +on mobile devices, there are other types of +potential applications with different forms of data +and learning tasks (e.g., environmental quantities +such as temperature or humidity). +In order to validate the proposed techniques +and achieved results in a transparent and +replicable way, we underline the importance of +conducting an advanced testbed under real-world +conditions with real IoT or mobile devices and +diverse learning tasks. Moreover, we recommend +to build a standardized benchmark for Federated +Learning performance analysis, in order to allow +researchers from all over the world to validate +their works with real diverse data and real-life +scenarios. +6.6 RI6: Federated inference +The majority of the literature on collaborative +machine learning concentrated on the training +phase.Although the inference task is less expensive +in terms of energy and resources, we may need to +consider it in the FedML context with energy- +constrained +IoT +devices +to +collaboratively +compute predictions or classify events. This would +also be beneficial in the case of audio and video +processing, which involve large amounts of data +and models. Moreover, the importance of this topic +is apparent in the case where the correlation +between multiple nodes is required to classify or +predict a value. +In this way, we recommend working on a +collaborative inference framework for FedML that +allows the nodes to support each other to balance +the prediction or classification load instead of +relying on the server for this task. Again, +appropriate encryption mechanisms have to be +used to guarantee data privacy. +7 Threats to validity +Any +survey +or +systematic +mapping +study +(including ours) is likely to have some common +limitations [10], related to literature coverage and +biases in processing the studied items. In order to +reduce these threats as much as possible, we tried +to follow a well-defined process [31]. It started +with a thorough search of relevant papers in +different databases, leveraging search term +synonyms to get as many valid results as possible. +We manually filtered the papers in multiple stages: +using the title and the keywords, then reading the +abstract, and finally studying the full text. We have +repeated this process at least two times: at the +beginning, and after a couple of months. +However, since we worked with the resources +available at the time, there may be issues related to +search string choice, the data collection process, +research question choice, and time span. +Regarding the search string choice, although +we used clear keywords, there may be some +missed opportunities due to bad keyword +indexation. We have done a manual snowballing +from the earlier validated papers, which helped us +spot some missed articles by the automatic search +process. However, this might not be always +enough. + +Regarding the data collection process, each +article was reviewed (title, abstract, and full text) +by a single researcher, which might cause some +errors. This problem was partially solved by +discussions between us. +In relation to the choice of research questions, +despite our extensive discussions to be as +comprehensive and clear as possible, there could +be some aspects that were not covered. + +Regarding the time span, we covered the period +starting from the seminal paper's publication in +2016 until July 2021. Some interesting papers may +have been published after. +Finally, we hope to have more resources in the +future +to +address +the +previous +eventual +shortcomings as well as others that our fellow +researchers will kindly point out. +8 Conclusions and future works +Summary. In this paper, we presented the first +Systematic Mapping Study, to the best of our +knowledge, on Fed ML for Energy Constrained IoT + +devices. Through a reproducible Research Process, +we selected 67 papers related to the topic since the +publication of the founding paper by [46] and tried +to compensate for eventual biases by snowballing +and manual searches. +The results analysis was structured around 5 +Research Questions related to publications overall +tendency, Fed ML network architecture, and energy +optimization schemes (reported results and +validation). It appears that updates compression +and clients selection have had the highest focus in +the literature and yield interesting results in terms +of decreasing the communication cost (up to 320x), +convergence time (up to 98%) ; and energy +consumption (up to 99%). +From our analysis, we identified 6 Research +Issues with associated recommendations: fully +decentralized schemes, large model reduction, +energy-aware data compression, heterogeneity +exploitation, real-world results validation, and +federated inference. Recommendations include +methods, such as, global model size reduction and +efficient data compression schemes, to help reduce +the communication and computation costs for the +nodes. +To +efficiently +address +the +system +heterogeneity, we pointed towards an adaptive +and flexible management of the resource-limited +devices and involved them in the training. Finally, +we underline the need for a standard benchmark, +dedicated to a transparent and rigorous validation +of the results, with real world conditions and real +test-beds. +Future works. We plan to conduct a Systematic +Literature Review (SLR) on the specific topic of +fully decentralized Fed ML, which appears to be +very interesting. Indeed, it eliminates the single +point of failure and presents difficult challenges +related to aggregating updates without any focal +point. An SLR is dedicated to going in depth +regarding a specific question, as opposed to an +SMS, which broadly structures the field. 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Appendix +This appendix lists all papers included in our study, +tagged from P1 to P67 (chronological order). + + +Table 5 Papers list +Id +Paper title +Optimization technique + +P1 +Federated Optimization:Distributed Machine Learning for On-Device Intelligence +Data exchange optimization +P2 +Federated Learning: Strategies For Improving Communication Efficiency +Data exchange optimization +P3 +Client Selection for Federated Learning with Heterogeneous Resources in Mobile Edge +Clients resource Management +P4 +Communication-Efficient On-Device Machine Learning: Federated Distillation and +Augmentation under Non-IID Private Data +Data exchange optimization +P5 +Efficient Training Management for Mobile Crowd-Machine Learning: A Deep +Reinforcement Learning Approach +Clients resource Management +P6 +Two-Stream Federated Learning: Reduce the Communication Costs +Convergence acceleration +P7 +Adaptive Federated Learning in Resource Constrained Edge Computing Systems +Data exchange optimization +P8 +Federated Learning over Wireless Networks: Optimization Model Design and Analysis +Data exchange optimization +P9 +CMFL: Mitigating Communication Overhead for Federated Learning +Data exchange optimization +P10 +Distilling On-Device Intelligence at the Network Edge +Data exchange optimization +P11 +Federated Learning with Additional Mechanisms on-Clients to Reduce Communication Costs +Convergence acceleration +P12 +Towards Faster and Better Federated Learning: A Feature Fusion Approach +Convergence acceleration +P13 +On-Device Federated Learning via Second-Order Optimization with Over-the-Air Computation +Convergence acceleration +P14 +Model Pruning Enables Efficient Federated Learning on Edge Devices +Convergence acceleration +P15 +Robust and Communication-Efficient Federated Learning from Non-IID Data +Data exchange optimization +P16 +Astraea: Self-balancing Federated Learning for Improving Classification Accuracy of Mobile Deep +Learning Applications +Clients resource Management +P17 +Communication-Efficient Federated Learning for Wireless Edge Intelligence in IoT +Data exchange optimization +P18 +Performance Optimization of Federated Learning over Wireless Networks +Clients resource Management +P19 +Communication-Efficient Federated Deep Learning With Layerwise Asynchronous +Model Update and Temporally Weighted Aggregation +Data exchange optimization +P20 +Intermittent Pulling with Local Compensation for Communication-Efficient Federated Learning +Data exchange optimization +P21 +Faster On-Device Training Using New Federated Momentum Algorithm +Convergence acceleration +P22 +BACombo—Bandwidth-Aware Decentralized Federated Learning +Clients resource Management +P23 +Ternary Compression for Communication-Efficient Federated Learning +Data exchange optimization +P24 +Dynamic Sampling and Selective Masking for Communication-Efficient Federated Learning +Data exchange optimization +P25 +SAFA: a Semi-Asynchronous Protocol for Fast Federated Learning with Low Overhead +Clients resource Management +P26 +Federated Neuromorphic Learning of Spiking Neural Networks for Low-Power Edge +Intelligence +Data exchange optimization +P27 +Towards Communication-Efficient Federated Learning in the Internet of Things with Edge +Computing +Data exchange optimization +P28 +Energy-Efficient Radio Resource Allocation for Federated Edge Learning +Clients resource Management +P29 +Client-Edge-Cloud Hierarchical Federated Learning +Clients resource Management +P30 +Energy-Aware Analog Aggregation for Federated Learning with Redundant Data +Clients resource Management +P31 +Convergence Time Minimization of Federated Learning over Wireless Networks +Clients resource Management + +P32 +Optimizing Federated Learning on Non-IID Data with Reinforcement Learning +Clients resource Management +P33 +Accelerating Federated Learning via Momentum Gradient Descent +Convergence acceleration +P34 +Optimal User Selection for High-Performance and Stabilized Energy-Efficient Federated Learning +Platforms +Clients resource Management +P35 +FedPAQ: A Communication-Efficient Federated Learning Method with Periodic Averaging and +Quantization +Data exchange optimization +P36 +Accelerating Federated Learning over Reliability-Agnostic Clients in Mobile Edge Computing +Systems +Clients resource Management +P37 +Q-GADMM: Quantized Group Admm For Communication Efficient Decentralized Machine +Learning +Data exchange optimization +P38 +FedMCCS: Multi Criteria Client Selection Model for Optimal IoT Federated Learning +Clients resource Management +P39 +FetchSGD: Communication-Efficient Federated Learning with Sketching +Data exchange optimization +P40 +FedAT: A Communication-Efficient Federated Learning Method with Asynchronous Tiers under +Non-IID Data +Data exchange optimization + +P41 +Lazily Aggregated Quantized Gradient Innovation for Communication-Efficient Federated +Learning +Data exchange optimization +P42 +Toward Communication-Efficient Federated Learning in the Internet of Things With Edge +Computing +Data exchange optimization +P43 +Group Knowledge Transfer: Federated Learning of Large CNNs at the Edge +Clients resource Management +P44 +CatFedAvg: Optimizing Communication-efficiency and Classification Accuracy in Federated +Learning +Clients resource Management +P45 +Adaptive Gradient Sparsification for Efficient Federated Learning: An Online Learning Approach +Data exchange optimization +P46 +Communication-Efficient Federated Distillation +Data exchange optimization +P47 +A Trust and Energy-Aware Double Deep Reinforcement Learning Scheduling +Strategy for Federated Learning on IoT Devices +Clients resource Management +P48 +Device Scheduling for Energy-Efficient Federated Learning over Wireless Network +Based on TDMA Mode +Clients resource Management +P49 + Distillation-Based +Semi-Supervised +Federated +Learning +for +Communication- +Efficient Collaborative Training with Non-IID Private Data +Data exchange optimization +P50 +Time-Correlated Sparsification for Communication-Efficient Federated Learning +Data exchange optimization +P51 +Energy-Aware Resource Management for Federated Learning in Multi-Access Edge Computing +Systems +Clients resource Management +P52 +FEDZIP: A Compression Framework for Communication-Efficient Federated Learning +Data exchange optimization +P53 +FedProf: Optimizing Federated Learning with Dynamic Data Profiling +Clients resource Management +P54 +Toward Resource-Efficient Federated Learning in Mobile Edge Computing +Clients resource Management +P55 +HeteroFL: Computation and Communication Efficient Federated Learning for Heterogeneous +Clients +Clients resource Management +P56 +Wirelessly Powered Federated Edge Learning: Optimal Tradeoffs Between Convergence and +Power Transfer +Data exchange optimization +P57 +Gradient Statistics Aware Power Control for Over-the-Air Federated Learning in Fading Channels +Clients resource Management +P58 +Accelerating Federated Learning for IoT in BigData Analytics With Pruning, +Quantization andSelective Updating +Convergence acceleration +P59 +Communication Efficient Federated Learning with Adaptive Quantization +Data exchange optimization +P60 +Adaptive Batch Size for Federated Learning in Resource-Constrained Edge Computing +Convergence acceleration +P61 +Communication-Efficient Federated Learning with Dual-Side Low-Rank Compression +Data exchange optimization +P62 +Resource-Efficient Federated Learning with Hierarchical Aggregation in Edge Computing +Clients resource Management +P63 +Adaptive Federated Dropout: Improving Communication Efficiency and Generalization for +Federated Learning +Convergence acceleration +P64 +To Talk or to Work: Flexible Communication Compression for Energy Efficient +Federated Learning over Heterogeneous Mobile Edge Devices +Data exchange optimization + +P65 +Adaptive Quantization of Model Updates for Communication-Efficient Federated Learning +Data exchange optimization +P66 +COFEL: Communication-Efficient and Optimized Federated Learning with Local Differential +Privacy +Data exchange optimization +P67 +Energy-Efficient Federated Edge Learning with Joint Communication and Computation Design +Clients resource Management + + diff --git a/x9E2T4oBgHgl3EQfMAaP/content/tmp_files/load_file.txt b/x9E2T4oBgHgl3EQfMAaP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ffb68f840ef1526a4495746ac65a0e99e9a1c07 --- /dev/null +++ b/x9E2T4oBgHgl3EQfMAaP/content/tmp_files/load_file.txt @@ -0,0 +1,1701 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf,len=1700 +page_content='Federated Learning for Energy Constrained IoT devices: A systematic mapping study Rachid EL Mokadem, Yann Ben Maissa and Zineb El Akkaoui {elmokadem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='rachid, benmaissa, elakkaoui}@inpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='ma Telecommunications Systems, Networks and Services Lab, National Institute of Posts and Telecommunications, Rabat, 10587, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Abstract Federated Machine Learning (Fed ML) is a new distributed machine learning technique applied to collaboratively train a global model using clients’ local data without transmitting it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Nodes only send parameter updates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', weight updates in the case of neural networks), which are fused together by the server to build the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' By not divulging node data, Fed ML guarantees its confidentiality, a crucial aspect of network security, which enables it to be used in the context of data-sensitive Internet of Things (IoT) and mobile applications, such as smart geo-location and the smart grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, most IoT devices are particularly energy constrained, which raises the need to optimize the Fed ML process for efficient training tasks and optimized power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In this paper, we conduct, to the best of our knowledge, the first Systematic Mapping Study (SMS) on FedML optimization techniques for energy-constrained IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' From a total of more than 800 papers, we select 67 that satisfy our criteria and give a structured overview of the field using a set of carefully chosen research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, we attempt to provide an analysis of the energy-constrained Fed ML state of the art and try to outline some potential recommendations for the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Keywords: Federated Machine Learning, Energy Optimization, Internet of Things, Edge and Mobile Computing, On-device Intelligence 1 Introduction Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Machine learning (ML) has become an important and increasingly used paradigm in different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In the last decade, the IoT computer systems and their potential applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', smart cities, smart grids) have grown considerably, which would make them benefit from the capabilities of ML in such a large and complex context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Furthermore, widespread IoT adoption in industry and academia (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', via rapid prototyping platforms such as the Raspberry PI™ and Arduino™) raises expectations for data privacy preservation and efficient resource utilization in a wide range of critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Therefore, in light of ML limitations for distributed systems and sensitive data, Federated Machine Learning (Fed ML) was proposed by McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' in 2016 [46] to address these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The approach delegated model training tasks to client devices, which collaboratively built a global shared model that consolidated their respective local data learning while avoiding any private data from leaving its original device [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Since the seminal paper, Fed ML has become one of the ”hot topics” in ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' IoT and mobile devices have a major constraint related to energy sources, and as a result, the power consumption on these devices must be optimized for any assigned task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In particular, a machine learning algorithm is known to be a highly power-consuming multi-task process [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In a distributed ML setup, nodes must continually exchange data with a master node, which may drive up overhead costs for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' FedML attempts to solve this issue by limiting the exchanged data to the local model’s weights [46], trained by nodes, instead of voluminous raw data exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' At the same time, FedML still requires improvement to enable resolving further critical challenges related to IoT and mobile device characteristics, namely, the limited resources and energy constraints [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' As a consequence, several Fed ML works addressing these aspects have increasingly been proposed by the scientific community in the last few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In light of this evolving literature, there is a substantial need for a comprehensive study in order to provide a clear overview of energy optimization approaches and propose new research directions for the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Several works have actually tackled the limitations of the original Fed ML proposal, entitled FedAvg, and proposed many optimization approaches, essentially regarding the communication load, data exchange, and other aspects, which can help to address directly or indirectly the limited energy constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The purpose of this paper is to conduct, to the best of our knowledge, the first Systematic Mapping Study (SMS) on Fed ML optimization for energy- constrained devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This SMS is tasked with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=') counting and categorizing relevant primary studies published in this topic based on five research questions, ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=') analyzing and discussing the results to provide a clear understanding of recent improvements for the research community, and iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=') assisting engineers in developing innovative Fed ML solutions for IoT and mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The remainder of this paper is structured as follows: First, we present related works and surveys in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Then we provide some theoretical foundations through the original FedAvg algorithm as well as the formulation of the energy optimization problem in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In Section 4, we present the method used to conduct this study, including the paper selection and filtering process, as well as the research questions (RQs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Table 1 Related works Year Title Type Focus 2020 A Systematic Literature Review on Federated Machine Learning: From A Software Engineering Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [41] SLR Software engineering aspects 2020 A Systematic Literature Review on Federated Learning: From A Model Quality Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [40] SLR Model quality 2020 Federated Learning in Mobile Edge Networks: A Comprehensive Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [38] Survey General 2020 Federated Learning: A Survey on Enabling Technologies, Protocols, and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [1] Survey Applications 2020 A Review of Privacy-preserving Federated Learning for the Internet-of-Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [6] Survey Privacy In Section 5, we answer them and analyze the results obtained from the studied papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We follow up with a discussion and some recommendations for research directions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Section 7 exhibits some threats to the validity of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, in Section 8, we conclude and outline some possible future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 2 Related works In this section, we present three surveys and two literature reviews that have been identified as being related to this work (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [41] presented a systematic literature review on Federated Learning, from a software engineering perspective, where they covered the Federated Learning system in general, with a focus on the software development aspects and general challenges for real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [40], on the other hand, conducted a systematic literature review on Federated Learning from a model quality perspective, where they studied the methods for improving the quality of the Fed ML model and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Additionally, the authors compared the model between federated and non-federated learning on the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Furthermore, [38] presented a survey on Federated Learning for mobile edge networks, in which they investigated the characteristics and limitations of good performance, resource allocations, communication costs, and data privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Moreover, [1] presented a FedML survey on enabling technologies, protocols, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' They provided the most relevant protocols, platforms, and real-life use-cases of Federated Learning to enable data scientists to build better privacy- preserving solutions for industries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' they also explored the challenges and advantages of Fed ML for real-life applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, [5] presented a survey on federated learning from a privacy preservation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Although these surveys and SLRs are excellent, we think that our study tackles some aspects that were not directly addressed by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' They do not focus on the energy factor in the optimization of federated learning, except for [38], where it is not thoroughly tackled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We attempt to shed light on power consumption aspects in FedML for the IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' As reported by Cisco in [9], IoT connections will represent more than half (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='6 billion) of all global connected devices and connections (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='5 billion) by 2022, showing their increasing pervasiveness in human lives [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Our personal use of smart phones and watches, which need frequent and sometimes bothersome recharging, is also a practical witness to this concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, there is also the particular case of wireless sensor networks that can be deployed in hostile environments with no possibility at all of energy replenishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3 Background In this section, we talk about the global Federated Learning process, the FedAvg algorithm, the energy consumption problem, and some other background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1 Federated Learning Federated Machine Learning is the process of developing accurate models on large-scale distributed systems made up of small devices by combining their computation power and local data [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content="The goal is to solve a class of problems that cannot be solved by a single central computer, such as those involving users' personal data, real-time computing, and on-device artificial intelligence [32]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' FedML is based on a distributed architecture that involves several nodes performing training tasks on their local data and exchanging their model’s parameters with a central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=" The server then builds, from local models, a global aggregated model, which is equivalent to a trained model on all nodes' consolidated data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In the case of FedAvg, the global model Wg is built as a weighted average of the local models Wi (see equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Wg = ∑ ni n Wi i (1) The optimization of the global objective function f can be expressed as the optimization of the average of local objective functions fi for all participating nodes i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=',ni, as given by the equation 2 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' minwf(w) = minw 1 n ∑ fi(w) i (2) where: 𝑓𝑖(𝑤) ≔ 1 𝑘 ∑ 𝑙(𝑤, ξ) ξ∈𝐷𝑖 (3) fi is defined as an average of the local loss function l, for each node i, on its local sample points, Di = ξi1,··· ,ξim for i ∈ [n], where Di is the local data set of the node i, composed of m data points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' ξi and w are the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 𝑚𝑖𝑛𝑤𝑓(𝑤) ≔ 𝑚𝑖𝑛𝑤 1 𝑛𝑘 ∑ 𝑙(𝑤, ξ) ξ∈𝐷 (4) Finally, to solve equation 4, a gradient descent method is used by each node to minimize the loss li over its local training data Di, and eventually the aggregated model Wg will minimize the global objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1 Federated Learning pseudo- algorithm Algorithm 1 shows the idea behind Federated Averaging (FedAvg), proposed by [46] for Fed ML .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' end /* Run on server*/ initialize w0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' for each round t = 1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' do m ← max(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='K,1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' end /* Run on client k*/ Function ClientUpdate(k,w): B ← (split Pk into batches of size B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' for each local epoch i from 1 to E do for batch b ∈ B do w ← w − µ∇l(w,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' end return w to server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' end Algorithm 1: FedAvg pseudo-algorithm The notations employed in the algorithm are explained underneath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='C Fraction of selected clients in each round ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='K Total number of clients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='m Number of randomly selected clients for each round ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='St Set of clients for each round ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='wt Global model parameters at round t Received model parameters from client k at round t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='nk Number of data points of client k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='n Total number of data points of all clients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Pk Local data-set of client k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='B Local data-set mini batch size to use for client training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='B Set of data-set mini batches for local training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='E Number of training passes performed by each client before ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='sending the update to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' µ Learning rate l Loss function w Local model parameters As shown in the aforementioned algorithm, the server initiates the model’s parameters w0, then, for each round, it determines the number m of participant clients to choose for training as a fraction C of K total clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The subset of devices St is determined randomly, and then each client device k receives the model’s parameters wt from the server to perform the training on its respective Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 1 Federated Learning global schema local data set Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This training process performs a split of the local data into small batches of size B, and a number of E local epoch runs to train the local model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, all selected clients compute an update of the parameters w, then send it back to the server, which averages them to get the new global model parameters wt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This round is repeated as many times as determined by the server to reach the target performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2 Federated Learning process Fed ML architecture is composed of the client nodes and the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The server receives the computed updates from client devices and performs an aggregation operation to build the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It is then improved continuously, by running additional iterations on the nodes, to train their local models, until obtaining the desirable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Figure 1 globally shows the components involved in the Fed ML architecture, as well as the stages of the FedAvg algorithm execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In each round of the training, the following operations are performed: S t ← randomsetofmclients);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' ( for eachclient k ∈ S t inparallel do w t +1 ← ClientUpdate ( k,w t ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' w t +1 ← P K k =1 n k n w k t +1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' end 1-Initialization 4-Updates aggregation 2-Local model 5- Global model update update 3- Updates sent to server 6-New iteration Uplink Downlink Updatedlocalmodel JpdatedGlobalmodel Local training Newiteration1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Definition of model’s structure, random initiation of parameters and selection of participating devices: the central server must define the parameters E, C and B prior to start of the training, and it must select a subset of clients to participate in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Model Update on local data: each selected client computes an update of the global model, by running local training iterations as many times as defined by the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Transmission of Local Model updates to server: each participating device sends the computed update of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Aggregation of all received model updates: the server aggregates the received updates in such a way that builds a global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Sharing the updated global model with the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3 Heterogeneity Very often, in real applications, the participant nodes in the FL have uneven resources and training data, we refer to this by system heterogeneity and statistical heterogeneity respectively [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' System heterogeneity During the collaborative training of the global model, different nodes have different capacities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', CPU, Battery, Memory, Bandwidth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' As result, if we ignore this fact, the convergence will be very slow, and the weak clients will exhaust their resources before the end of the training, resulting in bad model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Statistical heterogeneity When FedAvg was first proposed by [46], it was based on the assumption of independent and identically-distributed (iid) data across nodes, which guarantees a theoretical solution for the equation 4, regarding balanced local data-sets Di, by using the gradient descent optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, this assumption cannot be held for the majority of distributed data on IoT and users’ devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' this is a big limiting factor facing the deployment of Fed ML in real-world scenarios [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In fact, the majority of works published on this topic display good results for iid data and poor ones for non-iid setups, which is shown by a bad impact on the global model’s performance and the required time and energy for the training [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This substantial problem has driven several teams to develop techniques to adapt the original federated learning algorithm to both types of heterogeneity [11, 34, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2 Energy consumption formulation The main goal of FedML optimization for energy- constrained devices is to minimize the functional energy consumption of the nodes while building a good global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=" In general, a wireless device's total energy consumption ET can be divided into three major parts: Enet, Ec, and Esys (Equation 5)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 𝐸𝑇 = 𝐸𝑛𝑒𝑡 + 𝐸𝑐 + 𝐸𝑠𝑦𝑠 (5) Enet is the energy consumed by the device for communications with other devices or the server for update exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Ec is the energy consumed by the device’s local processing unit and memory to accomplish the training computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Esys is the energy consumed by the general system operations of the device, which are not related to its participation in the Federated training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Note that Esys is generally small and negligible compared to the total amount used in IoT [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In addition, it is not specific to the problems considered in this study, so we omit it from this formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, communications generally consume more energy than processing, for an equivalent amount of operations (this justifies multiple aggregation approaches before data transmission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Equation 6 gives the amount of energy consumed by network communication, expressed by a set of parameters related to our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 𝐸𝑛𝑒𝑡 ≃ ∑ 𝑁𝑇𝑏𝑖𝑡 𝑖 𝑃𝑇 𝑅𝑇 𝑁𝑇 𝑖=1 + ∑ 𝑁𝑅𝑏𝑖𝑡 𝑖 𝑃𝑅 𝑅𝑅 𝑁𝑅 𝑖=1 + 𝑐 (6) NT and NR are, respectively, the number of transmitted and received updates by the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' PT and PR are the transceiver power at transmission and reception, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RT and RR are bit rates for transmission and reception, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' and are the number of bits transmitted and received, respectively, in a given update i, and c is amount of energy consumed by irrelevant factors such as channel noise, transmission errors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' If PR = PT = P and RR = RT = R, the equation 6 can be simplified into equation 7: 𝐸𝑛𝑒𝑡 ≃ (∑ 𝑁𝑇𝑏𝑖𝑡 𝑖 𝑁𝑇 𝑖=1 + ∑ 𝑁𝑅𝑏𝑖𝑡 𝑖 𝑁𝑅 𝑖=1 ) 𝑃 𝑅 + 𝑐1 (7) Moreover, the energy consumption by local computations on each client device is approximated by the equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 𝐸𝑐 ≃ ∑ 𝑇𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔 𝑖 𝑁𝑟𝑜𝑢𝑛𝑑 𝑖=1 × 𝑃𝑐 𝑖 (8) Where is the consumed power per training time unit at round i, Ttraining is the duration of computation operation, and Nround is the number of operations to run by a given device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' If Ttraining and Pc are equivalent for all rounds on a given device, the equation 8 can be simplified as : 𝐸𝑐 ≃ 𝑁𝑟𝑜𝑢𝑛𝑑𝑇𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔𝑃𝑐 (9) In summary, the approximated total energy consumed by each client device (Equation 5) can be expressed by equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 𝐸𝑇 ≃ 𝑁𝑟𝑜𝑢𝑛𝑑𝑇𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔𝑃𝑐 + (∑ 𝑁𝑇𝑏𝑖𝑡 𝑖 𝑁𝑇 𝑖=1 + ∑ 𝑁𝑅𝑏𝑖𝑡 𝑖 𝑁𝑅 𝑖=1 ) 𝑃 𝑅 (10) From the above energy formulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' we can identify a list of parameters which impact the energy consumption of the participant client devices in Federated Learning: the number of exchanged updates NT and NR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' the number of bits in each exchanged update NTbiti and NRbiti ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' the transmission power P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' the transmission bit rates R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' the duration of local training Ttraining,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' and the number of local training rounds Nround.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3 Fed ML optimization parameters Based on the established equations in the previous section, together with the studied selected papers, we identify a number of energy optimization aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Accordingly, in order to minimize the total energy in equation 9, the optimization of the local training tasks to accelerate the model convergence should result in decreasing the number of federation rounds Nround.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, the training time duration Ttraining will be improved if we reduce the trained model’s complexity, which impacts energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Aggregating updates with the least cost, by reducing the size of exchanged data with the central server (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', decreasing NTbit and NRbit in equation 7), will help save battery life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Furthermore, the frequency of model update exchanges affects the total number of updates NT and NR (equation 7), thus optimizing even more the energy consumed in communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' More optimization can also be achieved by making smart use of the heterogeneous nodes’ computing resources to participate in the training, in addition to optimizing the client selection to balance the load over the participant nodes and involve the best ones for accelerated convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, decreasing the transmission power P and maximizing the bit rates R (equation 7) also helps to reduce the total spent energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This analysis will help us later to classify the different approaches and techniques proposed in the literature, as we will see in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 2 Our Search Process 4 Systematic Mapping Study Process This section describes the process followed throughout this Systematic Mapping Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Start Automatic papers Google Science Scholar Direct search Categorizationof papers ENDAdditional material is available on the online repository created for it 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Figure 2 illustrates the steps taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' After an automatic search based on the defined keywords and search string in the three common databases, the first step consists of filtering relevant papers based on their title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Then, we refined the selection based on the abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We refined our search even further by reading the full text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Finally, we added a manual search step afterwards to spot any articles that were not found the automatic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Details about each step of the workflow will be presented in the upcoming paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1 Papers selection In order to obtain all relevant papers for our study, we have queried three main databases (Google Scholar, IEEE Explore, and ScienceDirect) by using the search string in Listing 1, built mainly using the following keywords: federated machine learning, edge computing, on-device intelligence, energy, and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Listing 1 search query ”(” Federated Machine Learning” OR ” Federated Learning”) AND (”edge computing” OR ”on−device intelligence ”) AND ( energy OR power) AND ( optimization OR optimal OR efficient OR efficiency )” Filtering papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We filtered the initial search results to keep only papers, that meet all the following inclusion and exclusion criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Inclusion criteria: Papers from 2016 to July 2021 Papers in the English language Papers which propose an optimization of Federated Learning w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' energy consumption, using techniques including communication cost, or training time reduction Papers which target the IoT or mobile devices in general ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Exclusion criteria: 1 https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='com/rachid-el-mokadem/fedmlsysrev Works on distributed machine learning with no explicit application to federated learning on resource-limited devices Similar works of the same authors Manual searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In order to cover the literature as much as possible, another step was added to look for potential papers that might have been missed earlier: backward snowballing by looking Table 2 Research questions RQ ID Question RQ1 What is the publications tendency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RQ2 What network architectures are proposed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RQ3 How is the energy optimization achieved?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RQ4 How is the optimization validated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RQ5 What are the reported optimization results ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' at cited references in the selected papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Thereby, additional papers were added for a total of 67 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In the remainder of this study, we will refer to selected papers by identifiers, attributed according to the chronological order of the publication: P1, P2, up to P67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The list of all papers, along with their classification, is depicted in Table 5 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2 Research questions In order to analyze the literature and compare the proposed techniques in a systematic way, we define a set of research questions that will guide our analysis (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RQ1 indicates the timeline and sources of the papers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RQ2 presents the network topology considered by each paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RQ3 examines the FedML energy optimization aspects that are addressed by each paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' RQ4 presents the experimentation setups used to validate the approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' and RQ5 measures the optimization improvements of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 3 Fed ML papers publication trend over time 5 Questions answering In this section, we present the results analysis from the study of the selected papers, arranged as answers to the research questions defined in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1 RQ1 - What is the publications tendency Answering this research question will account for providing the number of publications evolution, their distribution over the publishing venues, and the nature of papers, as well as their influence on the field of FedML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The graph in Figure 3 shows the papers publication trend over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The growing number of papers over the last 3 years is clear, with 33 papers in only 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Given that the first paper from [32] was published in 2016, we can clearly see the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 4 Paper types distribution big interest this subject is receiving from several research teams around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The majority of papers, as shown in Figure 4, were published in journals (≈36%) and conferences(≈34%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This shows the growing maturity of this subject and the engaged efforts by the scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We also have 20 out of 67 (≈29%) papers published as pre-prints on the ArXiv database, including 10 in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This could be justified by the fact that the subject is evolving quickly, with fast feedbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We have also included the non-peer reviewed papers of Konecny´, Jakub et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [32, 33], since they are considered the most impactful in the subject, with 747 and 1733 citations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The same team is behind the seminal work on the FedML proposal [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Furthermore, we consider the number of citations for each paper, shown in Figure 5, to measure their influence on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It is obvious that older papers tend to get more citations than new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, it does provide an approximate idea of the paper’s scientific interest for the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' From the graph, we notice some spikes on a couple papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' For older papers such as P1 through P8, this is somehow reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, in the case of P15 ([57]) with 345 citations, P19 ([8]) with 110 citations and P35 ([54]) with 145 citations, this definitely shows the high impact of those papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' More details on the techniques used by them in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2 RQ2 - What network architectures are proposed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 5 Number of citations per paper In this question we consider the proposed network architectures of the studied papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This is important to us, because the network topology has an impact on the communication cost, and therefore the energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The architectures are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Centralized: based on a central server to ensure the communication and model’s parameters exchange, between the participating devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This option is energy consuming, due to long range communication between the devices and the server, which requires higher transmission power P (equation 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It also suffers from a single point of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' IEEE (42) ■ArXiv (20) ■MDPI (2) PMLR (2) ■SPRINGER (1) 35 1 30 2 2 25 10 20 15 5 10 18 5 1 10 11 1 1 0 3 2016 (1) 2017 (1) 2018 (4) 2019 (12) 2020 (33) 2021 (16)1800 1600 1400 1200 1000 800 600 20029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='85% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='33% Conferencepaper Journalpaper Preprint 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='82%• Decentralized: based on node to node communication without the need for a central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In this setup, the devices can save lot of energy, by opting for short range communication between the nodes only [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Hybrid: this architecture is based on at least three layers of devices, where intermediate ones are placed between the central server and the end devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 6 Number of papers by Network topology The hybrid architecture is based on adding edge servers between the main server and the end devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' These intermediate devices can play several roles, such as managing direct clients under their control, which allows the offloading of the central server and lowers the waiting time for aggregating multiple received updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In some cases, this edge server can also be used to offload the end devices from local update computing, by periodically querying the training data from the selected clients, doing the updates with a much higher computing capacity and communicating with the server, on behalf of the end nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' As a result, this architecture can allow a high energy optimization on the devices, although posing some threats to data privacy, especially when these edge servers are not trustworthy, and the data is very sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Figure 6 shows that the majority of papers (59 out of 67) are based on a centralized setup, while 2 papers have a fully decentralized one, and 6 propose a hybrid architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The predominance of the centralized scheme can be explained by the influence of the architecture in the original paper [46], which comes from Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, the fully decentralized scheme faces some algorithmic and practical challenges to aggregate the models without a central device [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3 RQ3 - How is the energy optimization achieved Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 7 Federated Learning optimization techniques (recap) In this question, we analyze the techniques used by the papers, to optimize the Fed ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Our study focuses on the power consumption reduction, so as seen in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2, all studied optimization aspects are linked with the energy through equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We classified these techniques into the following categories: (1) convergence acceleration (2) data exchange optimization and (3) client resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Figure 7 recaps the different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Table 3 presents the optimization aspects addressed by each studied paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1 convergence acceleration In federated machine learning, the training tasks are performed by the client nodes to build a global model under the orchestration of the central server, during as many rounds as needed to reach a good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In order to save the battery life of the client devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=" the total time to reach Updates compression Data Exchange Updates Optimization frequency Logits exchange Local training acceleration FedML Convergence accelerating Model pruning Optimization Optimized averaging Clients selection Client Resource Resource optimization Management Transmission settings Adjusted clients' models3% 9% Centralized Hybrid Decentralized 88%global model convergence can be reduced with several approaches." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Local training acceleration Many works have used different optimizations to accelerate the local training, such as adaptive learning rate [42], and Adam optimization method [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Equation 11 is used in the original version of Federated Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' w are the model weights, b is the model bias, ∇ is the gradient of the loss function l and µ is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In this version, the server defines a learning rate parameter at the beginning, used to compute the gradient descent steps in local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Opposed to that are the aforementioned methods, which determine the best steps to take in order to quickly achieve the convergence of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 𝑤 ← 𝑤 − 𝜇∇𝑙(𝑤, 𝑏) (11) The benefit of these techniques is to decrease the number of rounds Nround (equation 9) required for the model convergence, and thus reduce the energy consumption for the participant devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Accordingly, [47] proposed CE-FedAvg, which improved how the nodes compute their local updates by using the Adam method, known for its improved learning rate, instead of SGD (used in the original FedAvg algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The weights’ update method of the proposed algorithm, executed by each client, is shown in equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' wk,mk,vk ← AdamSGD(wk,mk,vk) (12) Where wk are the model weights, mk is Adam’s first moment, and vk is Adam’s second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' These parameters are used to compute the Adam steps by averaging them over all received updates or gradients and sending them back to the clients in the next round of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Feature augmentation is a technique used in machine learning to improve training performance in an unbalanced class distribution ([75]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Similarly, in the context of Federated Learning, FedFusion is an algorithm presented by [72] to accelerate the global model’s training by using a technique named Feature Fusion, which is based on using a combination of the global model’s feature space with the local model’s feature space to train the local model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The global model is used as a feature extractor, and then multiple types of feature fusions are employed to efficiently aggregate all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Additionally, [71] presented a two-Stream model learning with Maximum Mean Discrepancy (MMD), where the nodes training is performed on two models, in parallel, both initialized with the global model parameters, but one of them (global model) is kept unchanged during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' An MMD loss is computed between the output of the two models, which is used to optimize the local one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This technique is often employed with learning transfer and knowledge distillation in standard machine learning, and its adoption for Federated Learning helps to accelerate the training and reduce the communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In essence, it consists in constraining the local model training by the global model parameters, to avoid that local models over- fit their local data, thereby building a good global model in lesser training rounds Nround.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [4] used an adaptive dropout schema to decrease the convergence time by reducing the local model’s complexity and number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In practice, each round a random sub-net wc of the global model is sent to each participant client c, then an activation score map M is used to track the indexes A of the best sub-models to be reused in the next rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Model pruning Model pruning is another technique widely used in deep learning, which accelerates the training, by reducing the number of model parameters, based on training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The reduction simplifies the model, thereby decreasing the computation time (Ttraining in equation 9), local training energy consumption Ec, while keeping a good model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [29] implemented an algorithm named PruneFL where the pruning is performed initially by a selected client on its local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Then the resulting smaller model is iteratively adapted by the server in each round w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' to the training efficiency, by involving all clients updates, to reconfigure it, through removing or adding back some parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In order to allow the reversibility of parameters adding and deleting, the authors used a mask with zeros and ones for removed and kept weights respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Similarly, [69] proposed a structured model pruning combined with weights quantization and selective update, to accelerate the training and reduce the computation cost on the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In particular, the authors used an l1 − norm based pruning of the model weights with a variable ratio from 0 to 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Optimized averaging While original Federated Learning works by gathering the local model updates, and simply averaging them, several papers proposed to use advanced averaging methods, allowing a fast training convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Accordingly, [23] proposed Federated Momentum (FedMom), a technique with biased gradients that uses the momentum method to update the global model, according to equations 13 and 14: 𝑣𝑡+1 = 𝑤𝑡 − η ∑ 𝑛𝑘 𝑛 𝐾 𝑘=1 (𝑤𝑡 − 𝑤𝑡+1 𝑘 ) (13) 𝑤𝑡+1 = 𝑣𝑡+1 + β(𝑣𝑡+1 − 𝑣𝑡) (14) Where vt is the average of the previous round’s updates and beta β (often equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='9) is the parameter used to compute the moving average of the updates, through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' On the other hand, [39] used a hierarchical architecture by introducing L edge servers between the central server and client nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Each edge server has a subset s of clients from which it aggregates the updates before forwarding them to the main server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' According to the authors, this method reduces training time and decreases node energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2 Data exchange optimization The global model is built by gathering and aggregating the updates from the participant nodes at the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The frequency of exchanging the computed updates and their data sizes are optimized by several works in order to achieve communication-efficient federated learning, which drastically saves the battery life of the participant nodes without compromising the global model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In FedAvg, the aggregation of the local models is achieved according to the following equation 15: 𝑤𝑡+1 ← ∑ 𝑛𝑘 𝑛 𝑤𝑘 𝑡+1 𝐾 𝑘 (15) Where wk are the learned weights at each node, nk are the number of data points at each node, and n the total number of data points for all K participant clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Updates compression A stated limitation of FedAvg [46] is that the participant clients must upload the full computed updates at each round of the training, which has an impact on the power consumption of these devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The proposed optimization methods allow the reduction of the data exchanged between the nodes and the server while preserving the quality of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In addition to ordinary data compression algorithms used to encode the final updates with lower amounts of bits, such as Huffman encoding used by [7, 43], data size reduction is achieved by several other methods, such as update quantization, sparsification, and sketching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The goal of all these techniques is to reduce the amount of bits per round NTbit, sent through the wireless interface, which subsequently decreases significantly the energy usage for exchanging model’s updates (equation 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The quantization of machine learning models is based on using low float-point precision to represent the model’s weights in order to reduce the bit size ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [68]proposed a method called Federated Trained Ternary Quantization (FTTQ), which reduces both upstream and downstream traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It implements a layer-wise weight quantization with an adjustable threshold during the training, which has the additional benefit of reducing the training tasks’ energy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Similarly, [27, 43, 47, 54] used quantization for data size reduction, in most cases mixed with other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Furthermore, [27, 44] proposed an adaptive schema for updating quantization to achieve communication-efficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Additionally, the sparsification of the global and local models is used to compress the exchanged data by eliminating the gradient values of the computed update that are below a given threshold and replacing them with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This operation results in a sparse model update that can be encoded with a small number of bits in order to optimize the communication cost and energy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Accordingly, [57] proposed Sparse Ternary Compression (STC), a new compression framework created especially for the requirements of Federated Learning on resource-limited devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' STC extends the existing compression methods (in particular top-k sparsification [60]) to support downstream compression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' additionally, the authors combined sparsification with quantization and Golomb encoding to achieve better optimization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [20] developed an adaptive gradient sparsification based on bidirectional top- k gradient sparsification to reduce communication costs in both directions between the server and the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The sparsification’s parameter k is determined by the server as a trade-off between communication and global model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, [62] used a gradient sparsification with gradient correction, in order to accumulate the insignificant eliminated gradients and add them lately to speed up the convergence of the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Other techniques used to this end are gradient sketching [33, 55] and subsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The first one is based on compressing the update with a data structure named Count Sketch [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The second one [33] involves clients, sending only a smaller update derived from the computed one, by randomly sampling their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The server then averages all received sub-sampled updates to get an estimate of the global model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Additionally, [52] used dual-side low-rank compression to reduce the size of the models in both directions between the server and the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, [36] used a layer-based parameter selection in order to transfer only the important parameters of each model’s layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Updates frequency In the original Fed ML algorithm, clients send updates at each iteration of model training, which induces high communication costs and energy consumption as the number of updates exchanged with the server NT and NR increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In order to perform, under a given resource budget, [65] proposed a control algorithm that determines the best trade-off between local update and global parameter aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It learns the data distribution, and system characteristics along the distributed training, then determines dynamically the frequency of global model aggregation, with respect to the resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Alternatively, [8] presented a different method, which is based on the model’s layer-wise frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It means that important layers’ parameters are more frequently exchanged than less important ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The reason is that the first layers of a deep model tend to learn general features for different data sets, while the deeper layers learn more particular ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Consequently, each node separates its model into shallow layers’ weights wg and deep layers’ weights ws, which are exchanged with the server separately and asynchronously, under the control of the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It determines, for each client, the type of weights to consider, and performs a temporally weighted aggregation to give more importance to the newest received models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [64] proposed another method where the clients pull the global model less frequently from the server (to reduce down-link energy consumption) and compensate the gap with local updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Logits exchange Some works have chosen not to exchange the updates with the server: only the outputs of the trained local models, called logits, are sent to the server which reduces drastically the communication cost of the federation by many orders of magnitude [58];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' nevertheless, all clients and the server must have shared public data samples to compute and share their outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In order to build a global model out of this reduced data, the authors of [24, 26, 50, 58] used a learning transfer technique called knowledge distillation [59], where multiple teacher models (local models) transfer their learning to a single student model (the global model) [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In all cases, the distillation task is performed by the server, except for [24] where the sent logits are averaged by the server and sent back to the clients to perform the distillation themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In that case, the communication cost is reduced in both directions as the server does not send the whole model to the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3 Clients resource management Many approaches allow client devices to participate in model training, with optimal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Client resources generally refer to CPU time, memory and wireless bandwidth, which are often related to energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Two of the most used approaches for client resource management affect (1) client participation and (2) transmission settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Clients selection In the original FedML, participants were selected randomly, each round, from the available nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Subsequent analysis showed that this approach yields poor model convergence and causes node resources to be wasted ([49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Accordingly, many works have addressed this aspect, by adaptive and optimal client selection, based on their available resources and data in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [49] presented FedCS, a Federated Learning algorithm with optimized client selection, where the server starts by selecting a random set of clients, then performs a more informed selection using client resources and the time taken to compute the updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Furthermore, [3] presented a Reinforcement Learning scheme at the server, based on energy units en, number of CPU cycles cn, and the amount of data points used for training by each client, per- round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' A server reward is then computed from these values to help it find the best policies and actions for efficient training with optimal resource usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In the same vein, [53] proposed a multicriteria client selection model, named FedMCCS, that is based on a discriminative selection of client devices based on CPU, Energy, Memory and Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The server tracks these values along the training by an auxiliary data exchange of requests/responses with the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' A linear regression model is trained on these attributes to predict whether a client has enough resources to participate in training tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, [67] proposed the selection of clients based on their participation history, which impacts the global model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Additionally, [56] proposed a data imbalance aware selection of the participants in each round, such that all data categories must be covered at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This is achieved by requesting a bit-mask η containing C bits corresponding to the available data categories from each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The server then sorts these bit masks in a decreasing order of the number of sets and minimizes the required number of clients to get all categories covered by the averaged updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Hybrid scheme Other papers have proposed a hybrid scheme, based on the architectures presented in our second research question, to optimize the resources of the client devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [12] developed a self balancing system based on mediator edge servers, gathering near uniform data distribution subsets of clients, and aggregating the trained models, before sending them to the central server to build a global one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Similarly, [39] achieved energy consumption reduction by balancing the exchange of parameters with L edge servers with respect to the training time and communication budget where each edge server incorporates a small number of clients [66] used a hierarchical aggregation of the model updates to overcome the communication overload between the nodes and the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, [25] and [74] proposed a cloud-edge-client scheme wherein the clients offload a part or all the training tasks to the edge servers, which get portions of the clients’ data for the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This approach has some flaws w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' communication overhead and privacy concerns for clients’ data, but it may be relevant in some application-specific scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Transmission settings Wireless transmission has a high energy cost for mobile and IoT devices in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' As we saw in equation 7, it is related to the transmission power P and the bit rate R of the network interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Many papers have tackled the optimization of wireless transmission and bandwidth settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' [48] presented a technique for transmission power and rate optimization for energy-efficient communication between the nodes and the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' They consider minimizing total energy consumption as a joint optimization problem over bit rate, transmission power, and CPU frequency for local updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' On the other hand, [28] proposed a fully decentralized communication between the nodes, based on a Segmented Gossip protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In this communication scheme, the workers segment their updates into small fragments that are separately shared with a subset of the participants, who then relay them progressively to other workers until all of them get everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The peer workers with faster bandwidth are chosen to pull updates from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, [63] presented an energy-aware worker scheduling algorithm: a node monitors its energy consumption at each round and, by comparing it with an adjustable threshold, decides whether to participate in the next training round or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Subsequently, each worker partitions its update into M segments, which are transmitted separately on M sub- channels with adjusted transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Adaptive local models Recently, [73] and [11] proposed to adapt the local models to the nodes’ capabilities and data, in opposition to the state-of-the-art FedAvg algorithm, where all clients get the same model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In both works, an adapted model is derived for each client as a sub-net of the global model, with a smaller number of layers and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Then, different averaging methods are used to aggregate the updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' These new methods are heterogeneity and data imbalance aware and allow an adaptive saving of energy used for training and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' From Figure 8, we see that the data exchange optimization category has the highest number of papers (32), followed by client resource management with 25 papers, and finally convergence acceleration (10 papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Figure 9 shows in more detail the optimization techniques used by each Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 8 Papers count per optimization category Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 9 Papers count per optimization technique paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It was not surprising that Updates Compression and Clients selection represented most of the papers because the original FedAvg algorithm had a substantial limitation on these aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Additionally, this will have the highest impact on devices’ energy preservation, knowing that wireless communications use the biggest part of the operational energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The other techniques are shared between the rest of the papers, with a small advantage for Local training acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='4 RQ4 - How is the optimization validated This Research Question is about the experimentation setup (models, data sets, and testing platforms) used by different papers to validate their proposed works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The classification data in Table 4 shows that the majority of papers considered only neural networks (specifically convolutional ones) with MNIST and CIFAR10 data sets for the experimental part of their research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' While this seems to restrict validation, it can be explained by the ease of getting these famous data sets and implementing NNs on top of popular machine learning frameworks such as PyTorch and Tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Some papers have implemented additional validations on other ML models such as Linear Regression, Logistic Regression, and Support Vector Machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, there is a substantial shortage of validation results for non-neural network models, which may exhibit lower complexity and therefore lower resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' As for the experimentation platforms, different papers considered different numbers of participating nodes, from 2 up to 50000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The majority of papers (52 out of 57) have used emulated nodes on multi-GPU computers, while only five papers (P7, P14, P17, P34, P58) have performed experiments on real devices such as the Raspberry Pi™and smartphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Emulated devices can give an insight into validation, but it is important to have further results on real ones, in real life scenarios, especially regarding wireless communications, energy constraints, and computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The democratization of rapid prototyping platforms in the industry and academia (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', Arduino™, Raspberry Pi™ & ESP31™) is another motivation for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 32 25 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='5 RQ5 - What are the reported optimization results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 10 Communication cost improvements (per paper) In this question, we list and compare the optimization results obtained in the surveyed papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The numerical results are classified into three categories: (1) Communication Cost, (2) Convergence Time, (3) Energy Consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Each paper quantified its optimization improvement compared to the standard FedAvg [46] algorithm, and the numerical results w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' each category are listed in the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' From Figure 10, we see a very wide range of improvement values related to global communication cost reduction, which goes from 2x up to 320x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In Figure 12, we see convergence time improvements going from 3% up to 98%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' As for the energy consumption, Figure 11 reports a range of improvement from 14% to 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 11 Energy consumption improvements (per paper) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 12 Convergence time improvements (per paper) The convergence time and communication cost optimization results are very encouraging, which is consistent with the important interest of the community in these two aspects (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Although they directly impact the energy consumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' note,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' that during our readings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='relatively few papers (8 out of 67) have evaluated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Table 3 Papers list by optimization techniques ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Category ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Optimization technique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Papers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Data exchange optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Updates compression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Updates frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P2 P15 P17 P19 P23 P45 P24 P26 P27 P35 P37 P39 P40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P42 P64 P52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P1 P7 P8 P20 P56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Logits exchange ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P4 P10 P46 P49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Client resource management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Clients selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Hybrid scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P3 P5 P18 P25 P28 P31 P32 P38 P44 P47 P48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P16 P29 P34 P36 P51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Transmission settings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P67 P22 P30 P57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Adaptive models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P54 P55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Convergence acceleration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Local training acceleration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Model pruning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P6 P11 P12 P13 P33 P63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P14 P58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Optimized model averaging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='the optimization’s benefit directly on the energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='consumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' which is crucial to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 6 Discussion In this section we discuss the Systematic Mapping Study results, in the light of the previous analysis, guided by the RQs in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' For each Research Issue (RI) presented, we provide (1) some remarkable limitations related to Energy constrained Fed ML, and (2) some improvement directions and recommendations for the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1 RI1: Fully-decentralized scheme In a centralized scheme based Fed ML, client nodes exchange data during the training with a central server, which is generally located in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Consequently, the nodes have to use long-range wireless communication to reach the server, which implies high power consumption [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' To overcome this, we must take advantage of the short-range communication between the nodes, which is by far less power-intensive, to exchange the updates using peer-to-peer communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The proposed approaches to implementing a fully decentralized FedML induce an overload on the resource-limited devices, caused by the additional operations performed by the nodes to compensate for the role of the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In [28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' the nodes have to also play the role of the aggregating server,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' and [13] proposed a technique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='where all nodes compute and exchange their ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='updates in a chain-like scheme (using some sort of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='multi-hop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Table 4 Papers validation setups ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Paper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='ML model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Number of nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='RNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Blog posts dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN - RNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CIFAR10 - public post reddit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='100 - 1024 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CIFAR10 FashionMNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='RNN 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN : AlexNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CIFAR10 - MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2 – 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CIFAR10 - MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='5 (RaspberryPi) – 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P8 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN - LSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='ANN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='MNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P11 ' 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CiFar100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN - LSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='MNIST - IMDb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='NA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN (ResNet) ' 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and SVHN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='MNIST - CIFAR10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3 (Core i5 PCs) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN ' 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+page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='MNIST – Cifar10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CiFar10 - FashionMNIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='MNIST - FashionMNIST - CIFAR10 P67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='CNN - LSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='FEMNIST - Shakespeare dataset communications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The limitation of the first technique is the overhead tasks for the nodes to play the server role,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' where the second one forces the nodes to run all the time of the training: in both cases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' more energy and resources are required at the node level The hybrid scheme seems to overcome some of these problems since it has a cloud-based central server that is only used to manage the client participation and selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' with very limited data querying from the nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' while keeping model aggregation between the client devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' At the same time, in order to advocate the fullydecentralized scheme, there is a need for a new theoretical framework that supports complete decentralized model aggregation with convenient energy and resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='2 RI2: Large models reduction Some Fed ML models with large sizes and a big number of trainable parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', Deep Neural Networks) require a computationally expensive training [17] for energy constrained IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' If we consider Fed ML as a bootstrap aggregation [22] of the global model over different distributed nodes’ data sets, we could reduce the local model’s size and complexity to make the training tasks easier for the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We could still build a high-performance global model by aggregating (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', by majority voting) the distributed models repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Another possible solution lies in the Lottery winning ticket hypothesis, elaborated by Frankle and Carbin [16], which states that a dense neural network contains a sparse sub network, that can be trained to equivalent performance of the initial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' By applying this technique in the context of Federated Learning, the global model can be drastically reduced in size and complexity, to accelerate the training, and reduce the computation load over the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The target sparse model could be obtained from the global model by Adaptive Iterative Pruning (AIP) [19] or Neural Architecture Search (NAS) [51], performed adaptively by the server based on model performance and available client resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='3 RI3: Energy-aware data compression Many papers proposed different techniques to reduce the amount of data to be sent or received from the server [33, 54, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The compression techniques used are: sketching, sparsification, quantization, and data encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' They have helped to drastically reduce the communication costs for the client devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, they add overhead tasks to the nodes, resulting in memory and CPU usage to compress, encode, and decode the transmitted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Many works [37, 37] have revealed the benefit of error-controlled lossy compression schemes on the compression rate and computation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We recommend studying an equivalent technique adapted to Federated Learning’s update compression in order to further reduce the communication energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='4 RI4: Heterogeneity aware optimization Nodes heterogeneity is a crucial issue for Federated Learning in real world applications [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' As a result, this topic has drawn the attention of the research community through several works [49, 70], which proposed different methods based on discriminative participant nodes selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' They only choose the devices that have both the required resources and data for the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' While this seems to solve the heterogeneity issue, it may impact the model performance and convergence time, by eliminating some devices with either one of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In this case, it would be more profitable to manage the devices in a way that takes advantage of their data and computation capabilities separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Some nodes may participate with their data, others with their computing capacity, and the rest with both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' To achieve this flexibility, some devices may exchange raw data, extracted features, or data labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' To preserve data privacy (one of the Fed ML rationales) during these communications, we may use a lightweight encryption technique (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', based on elliptic curves) for node-node communication or an homomorphic encryption scheme [18] for untrustworthy node-to-node relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We could also suggest an approach to managing heterogeneity that would be based on the separation of client nodes into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The first one would contain powerful, resourceful devices dedicated to training tasks, and the other one would contain poor nodes for validation only on their own data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The validation score may be used as feedback to adaptively adjust model aggregation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='5 RI5: Results validation The majority of studied papers have validated their approach using emulated nodes on powerful computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, a substantial focus was given to image recognition tasks using Convolutional Neural Networks (CNNs) and common data sets such as MNIST and CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The choice of image data for validation can be explained by its sensitivity and significant size, resulting in elevated communication costs thereby justifying Fed ML usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, in IoT and also on mobile devices, there are other types of potential applications with different forms of data and learning tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=', environmental quantities such as temperature or humidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In order to validate the proposed techniques and achieved results in a transparent and replicable way, we underline the importance of conducting an advanced testbed under real-world conditions with real IoT or mobile devices and diverse learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, we recommend to build a standardized benchmark for Federated Learning performance analysis, in order to allow researchers from all over the world to validate their works with real diverse data and real-life scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='6 RI6: Federated inference The majority of the literature on collaborative machine learning concentrated on the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Although the inference task is less expensive in terms of energy and resources, we may need to consider it in the FedML context with energy- constrained IoT devices to collaboratively compute predictions or classify events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This would also be beneficial in the case of audio and video processing, which involve large amounts of data and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Moreover, the importance of this topic is apparent in the case where the correlation between multiple nodes is required to classify or predict a value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In this way, we recommend working on a collaborative inference framework for FedML that allows the nodes to support each other to balance the prediction or classification load instead of relying on the server for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Again, appropriate encryption mechanisms have to be used to guarantee data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 7 Threats to validity Any survey or systematic mapping study (including ours) is likely to have some common limitations [10], related to literature coverage and biases in processing the studied items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In order to reduce these threats as much as possible, we tried to follow a well-defined process [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It started with a thorough search of relevant papers in different databases, leveraging search term synonyms to get as many valid results as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We manually filtered the papers in multiple stages: using the title and the keywords, then reading the abstract, and finally studying the full text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We have repeated this process at least two times: at the beginning, and after a couple of months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, since we worked with the resources available at the time, there may be issues related to search string choice, the data collection process, research question choice, and time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Regarding the search string choice, although we used clear keywords, there may be some missed opportunities due to bad keyword indexation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We have done a manual snowballing from the earlier validated papers, which helped us spot some missed articles by the automatic search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' However, this might not be always enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Regarding the data collection process, each article was reviewed (title, abstract, and full text) by a single researcher, which might cause some errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' This problem was partially solved by discussions between us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In relation to the choice of research questions, despite our extensive discussions to be as comprehensive and clear as possible, there could be some aspects that were not covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=" Regarding the time span, we covered the period starting from the seminal paper's publication in 2016 until July 2021." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Some interesting papers may have been published after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, we hope to have more resources in the future to address the previous eventual shortcomings as well as others that our fellow researchers will kindly point out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' 8 Conclusions and future works Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' In this paper, we presented the first Systematic Mapping Study, to the best of our knowledge, on Fed ML for Energy Constrained IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Through a reproducible Research Process, we selected 67 papers related to the topic since the publication of the founding paper by [46] and tried to compensate for eventual biases by snowballing and manual searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' The results analysis was structured around 5 Research Questions related to publications overall tendency, Fed ML network architecture, and energy optimization schemes (reported results and validation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' It appears that updates compression and clients selection have had the highest focus in the literature and yield interesting results in terms of decreasing the communication cost (up to 320x), convergence time (up to 98%) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' and energy consumption (up to 99%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' From our analysis, we identified 6 Research Issues with associated recommendations: fully decentralized schemes, large model reduction, energy-aware data compression, heterogeneity exploitation, real-world results validation, and federated inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Recommendations include methods, such as, global model size reduction and efficient data compression schemes, to help reduce the communication and computation costs for the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' To efficiently address the system heterogeneity, we pointed towards an adaptive and flexible management of the resource-limited devices and involved them in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Finally, we underline the need for a standard benchmark, dedicated to a transparent and rigorous validation of the results, with real world conditions and real test-beds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' We plan to conduct a Systematic Literature Review (SLR) on the specific topic of fully decentralized Fed ML, which appears to be very interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Indeed, it eliminates the single point of failure and presents difficult challenges related to aggregating updates without any focal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' An SLR is dedicated to going in depth regarding a specific question, as opposed to an SMS, which broadly structures the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Therefore, it is, in our opinion, the logical extension of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Aledhari, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Razzak, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Parizi, and F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='00582, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' Appendix This appendix lists all papers included in our study, tagged from P1 to P67 (chronological order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Table 5 Papers list ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Paper title ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Optimization technique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Federated Optimization:Distributed Machine Learning for On-Device Intelligence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Data exchange optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Federated Learning: Strategies For Improving Communication Efficiency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Data exchange optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Client Selection for Federated Learning with Heterogeneous Resources in Mobile Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Clients resource Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Communication-Efficient On-Device Machine Learning: Federated Distillation and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Augmentation under Non-IID Private Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Data exchange optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Efficient Training Management for Mobile Crowd-Machine Learning: A Deep ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Reinforcement Learning Approach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Clients resource Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Two-Stream Federated Learning: Reduce the Communication Costs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Convergence acceleration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Adaptive Federated Learning in Resource Constrained Edge Computing Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Data exchange optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Federated Learning over Wireless Networks: Optimization Model Design and Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='Data exchange optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E2T4oBgHgl3EQfMAaP/content/2301.03720v1.pdf'} +page_content='P9 ' 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a/yNE3T4oBgHgl3EQfPQkf/content/tmp_files/2301.04400v1.pdf.txt b/yNE3T4oBgHgl3EQfPQkf/content/tmp_files/2301.04400v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8fa44aa15d0192cdfe22d7c0782ee016226f337 --- /dev/null +++ b/yNE3T4oBgHgl3EQfPQkf/content/tmp_files/2301.04400v1.pdf.txt @@ -0,0 +1,1404 @@ +Resynthesis-based Attacks Against Logic Locking +Felipe Almeida†, Levent Aksoy†, Quang-Linh Nguyen‡, Sophie Dupuis‡, Marie-Lise Flottes‡ and Samuel Pagliarini† +†Department of Computer Systems, Tallinn University of Technology, Tallinn, Estonia +Email: {felipe.almeida, levent.aksoy, samuel.pagliarini}@taltech.ee +‡LIRMM, University of Montpellier, Montpellier, France +Email: {quang-linh.nguyen, sophie.dupuis, marie-lise.flottes}@lirmm.fr +Abstract—Logic locking has been a promising solution to many +hardware security threats, such as intellectual property infringe- +ment and overproduction. Due to the increased attention that +threats have received, many efficient specialized attacks against +logic locking have been introduced over the years. However, the +ability of an adversary to manipulate a locked netlist prior to +mounting an attack has not been investigated thoroughly. This +paper introduces a resynthesis-based strategy that utilizes the +strength of a commercial electronic design automation (EDA) +tool to reveal the vulnerabilities of a locked circuit. To do +so, in a pre-attack step, a locked netlist is resynthesized using +different synthesis parameters in a systematic way, leading to a +large number of functionally equivalent but structurally different +locked circuits. Then, under the oracle-less threat model, where +it is assumed that the adversary only possesses the locked circuit, +not the original circuit to query, a prominent attack is applied to +these generated netlists collectively, from which a large number +of key bits are deciphered. Nevertheless, this paper also describes +how the proposed oracle-less attack can be integrated with an +oracle-guided attack. The feasibility of the proposed approach +is demonstrated for several benchmarks, including remarkable +results for breaking a recently proposed provably secure logic +locking method and deciphering values of a large number of key +bits of the CSAW’19 circuits with very high accuracy. +Index Terms—Logic locking, resynthesis, EDA tools, oracle-less +and oracle-guided attacks. +I. INTRODUCTION +Due to the globalized integrated circuit (IC) supply chain, +serious security threats, such as hardware Trojans, piracy, +overbuilding, reverse engineering, and counterfeiting, have +emerged [1]. Many defense techniques, such as watermark- +ing [2], digital rights management [3], metering [4], and logic +locking [5], have been introduced over the years to deal +with these threats. Among those, logic locking stands out by +being a well-established technique and by offering protection +against a diverse array of adversaries [6]. Logic locking inserts +additional logic driven by key bits so that the circuit behaves +as expected only when the secret key is applied. +On the other hand, many efficient attacks have been in- +troduced to overcome the defenses built by logic locking [7]. +However, the impact of an electronic design automation (EDA) +tool on the manipulation of the locked netlist before per- +forming an attack has not been investigated thoroughly. In +This work has been partially conducted in the project “ICT programme” +which was supported by the European Union through the European Social +Fund. It was also partially supported by European Union’s Horizon 2020 +research and innovation programme under grant agreement No 952252 +(SAFEST). +this work, we explore if EDA tools can be used to make +a locked circuit vulnerable to existing logic locking attacks. +Thus, the main contributions of this work are three-fold: +(i) we introduce a resynthesis procedure that is a pre-attack +step, where functionally equivalent but structurally different +locked circuits are generated by resynthesizing the original +locked circuit using different optimization parameters and +delay constraints in order to create structural vulnerabilities +that can be exploited by existing attacks; (ii) we present an +oracle-less (OL) resynthesis-based attack, which applies the +prominent SCOPE attack [8] to these resynthesized circuits +and gathers all its solutions to discover the secret key; (iii) we +show that our OL attack can be combined with a traditional +oracle-guided (OG) attack for further improving the number of +correctly deciphered key bits. The last contribution is essential, +since we consider circuits from the CSAW’19 contest – these +circuits compound the use of two logic locking techniques at +the same time. +The main finding of this work is that the use of many +resynthesized locked circuits enables us to discover values of +more key bits, and even the whole key, when compared to a +single attack mounted on the original locked netlist. +The remainder of this paper is organized as follows: Sec- +tion II presents the background concepts and related work. +The resynthesis process and the proposed attacks are described +in Section III. Experimental results are given in Section IV. +Finally, Section V concludes the paper. +II. BACKGROUND +A. Logic Locking and Threat Models +The procedure of logic locking is applied at the gate level +in the IC design flow, as shown in Fig. 1. Note that the layout +of the locked circuit is sent to the foundry without revealing +the secret key. After the locked IC is produced and delivered +to the design house, the values of the secret key are stored in +a tamper-proof memory, before the functional IC is sent to the +market. +It is assumed that the gate-level netlist of the locked +circuit can be obtained directly by an untrusted foundry or +by reverse-engineering a functional IC obtained from the +open market. An adversary can also use the functional IC +programmed with the secret key as an oracle to apply inputs +and observe outputs. Thus, in logic locking, there are generally +two threat models: OL and OG. In the OL threat model, only +the gate-level netlist of the locked circuit is available to the +arXiv:2301.04400v1 [cs.CR] 11 Jan 2023 + +����� +��������� +����� +������� +������ +������� +�������� +��������� +������ +����������� +���� � +��������� +������ +�� +��� +���������� +���������� +�� +������� ������ +��������� ������ +��������� ������ +������� ������ +���������� +������ +������������� +Fig. 1. Conventional logic locking in the IC design flow (adapted from [6]). +Original Circuit +Locking Unit +inputs +key bits +output +Original Circuit +Restore Unit +inputs +key bits +output +Pertub Unit +Stripped Circuit +X +X +X +X Critical Point +(a) +(b) +Fig. 2. SAT-resilient logic locking methods: (a) SFLT; (b) DFLT. +adversary. The adversary has both the netlist of the locked +circuit and the functional IC in the OG threat model. +B. Related Work +After the introduction of random logic locking (RLL) using +XOR/XNOR gates in [9], earlier work focused on different +types of key gates, such as AND/OR, multiplexors, and look-up +tables, taking into account the hardware complexity of the +locked circuit [5]. However, the OG satisfiability (SAT)-based +attack [10] overcame all the defenses existing at that time. +Note that the SAT-based attack iteratively finds differentiating +input patterns (DIPs) that rule out wrong keys. To thwart the +SAT-based attack and its variants, circuits are locked using a +point function that sets a limit on the number of wrong keys +which a DIP can eliminate, forcing these attacks to explore an +exponential number of queries [6], [11]–[14]. +The SAT-resilient methods can be categorized into two +groups: single-flip locking technique (SFLT) and double-flip +locking technique (DFLT), as shown in Fig. 2. An SFLT +has only one critical point, which is responsible to corrupt +a protected output under a specific input pattern. In this +category, SARLock [15] adds a comparator and a masking +circuit connected with the original netlist in a way that it +generates a corruption on one input pattern. Anti-SAT [11] +utilizes two complementary AND gate trees, whose output is +merged with the original circuit. CASLock [12] is based on +the same concept of Anti-SAT, however it uses both AND and +OR gates. SKG-Lock [14] uses decoy key bits and provides +a tunable output corruption. Note that SFLTs are susceptible +to removal attacks [16]–[18]. If an attacker can identify this +single critical point, he/she can split the design into a recovered +netlist (original) and the locking unit. +A DFLT has two critical points, one that connects the +original netlist with a perturbation unit and another one that +connects the output of the stripped circuit with the restore +unit. Under this category, stripped functionality logic locking +(SFLL) [6], [13] initially corrupts an output based on an input +combination in the perturbation unit and then, corrects this +output only when the secret key is applied in the restore +unit. Note that a removal attack becomes inefficient for a +DFLT, since the original circuit is mixed with the perturbation +unit, even though it can easily identify the restore unit. +However, there exist efficient structural attacks developed for +DFLTs [19]–[22]. +Alternative locking techniques have also been introduced. +In [23], a technique, which has more than two critical points, +called the multi-flip locking technique (MFLT), was proposed. +However, it leads to a significant increase in area, power +dissipation, and delay when compared to other techniques. +Compound logic locking techniques were proposed to over- +come the main drawback of a SAT-resilient technique, i.e., +its low output corruptibility as can be observed in Fig. 2, +by locking a design using both low and high output corrupt- +ibility techniques, such as SFLL and RLL, respectively [24]. +Recently, efficient attacks have also been introduced against +compound logic locking [25], [26]. +Moreover, the OL attacks explore patterns in the structure +of a locked netlist using statistical analysis [8], [27], [28]. For +example, the SCOPE attack [8] is an unsupervised constant +propagation technique, which analyzes each key bit of the +locked design for critical features that can reveal its correct +value after it is assigned to logic 0 and 1 value. These critical +features include area, power dissipation, delay, and many +other circuit characteristics obtained by a synthesis tool. These +features are analyzed using linear regression and machine +learning based clustering. +III. PROPOSED RESYNTHESIS-BASED ATTACK +This section describes our resynthesis-based attack in detail. +We initially introduced the pre-attack stage, where the locked +circuit is resynthesized using different synthesis parameters, +leading to a large number of structurally different netlists with +the same functionality [29]. Then, we present the OL attack +that utilizes these resynthesized netlists in order to find the +secret key. Finally, in order to handle the compound logic +locking efficiently, we present its modified version, where our +proposed OL attack cooperates with an OG attack. +A. The Pre-attack Step: Resynthesis of the Locked Netlist +The locked circuit is synthesized multiple times using a +different script each time, where the synthesis parameters are +explored in a systematic way. We use the following parameters +to increase the number of resynthesized locked circuits: +Synthesis Effort: In a synthesis tool, logic optimizations can +be applied with different efforts at different synthesis stages. +This flexibility enables a designer to explore the trade-off +between the quality of results and run time. The following ef- +forts are considered at the given synthesis stage: low, medium, +and high at generic transformations (syn_gen); low, medium, +and high at mapping (syn_map); and low, medium, high, and +extreme at optimization (syn_opt). + +Delay Constraint: To meet performance targets, delay con- +straints are used to guide the synthesis tool. We initially +resynthesize the locked circuit without a delay constraint and +find the delay of its critical path, i.e., dcp. Then, in an interval +between 0 and dcp, d − 1 points, which are computed as +(dcp/d)i with 1 ≤ i ≤ d − 1, are set as delay constraints. +Note that d is set to 5 in order to generate a large number +of resynthesized circuits. Even though some delay constraints +are impossible to meet, the synthesis tool always generates a +netlist equivalent to the original one in terms of functionality. +Maximum Transition: The transition time of a net in a circuit +is defined as the longest time required for its driving pin to +change its logic value. We choose the maximum transition +value to be 5%, 10%, and 15% of the delay constraint for all +the nets in the locked circuit to explore different resynthesized +circuits. +Key Constraints: To direct the synthesis tool to work in- +tensively on the paths that contain the keyed logic, a delay +constraint, which is impossible to be satisfied, can also be +used. In this case, we force the delay between all key bits and +all primary outputs to be 1 ps. +Thus, the combination of parameters given above generates +3 × 3 × 4 × 5 × 3 × 2 = 1080 netlists. We eliminate the +resynthesized circuits with identical characteristics and keep +only the unique ones. Additionaly, we prevent the use of +XOR/XNOR gates, which can be problematic for the SCOPE +attack, during technology mapping. Note that our resynthesis +methodology aims to generate different versions of the locked +circuit, making it more vulnerable to existing attacks. Thus, +any existing attack, either OL or OG, may potentially benefit +from this pre-attack strategy to discover the secret key. +B. Attacks on the Resynthesized Netlists +Time-efficient attacks are chosen in order to handle a large +number of resynthesized circuits. In our OL resynthesis-based +attack, SCOPE [8] is used to predict the values of key bits. In +its modified version developed for compound logic locking, +a query attack is used to find the values of key bits in a +deterministic way. +1) Proposed OL Attack: SCOPE is applied to each resyn- +thesized locked circuit and a solution is found. Note that this +solution may return a logic 0, 1, or an unknown value for +a key bit. Then, the values of key bits deciphered for each +netlist are merged into a single solution that represents the +overall guess. To do so, for each key bit, ki with 1 ≤ i ≤ p, +where p denotes the number of key bits, we initially count the +number of solutions, where ki is deciphered as logic 0 and +1, denoted as dk0 +i and dk1 +i , respectively. Then, if dk0 +i > dk1 +i +or dk1 +i > dk0 +i , the value of ki is determined to be 0 or 1, +respectively. Otherwise, in the case of a tie, the value of ki is +decided to be unknown. +2) Proposed OG Attack: In order to handle a large number +of resynthesized netlists efficiently, we introduce a SAT-based +query attack, which can determine the actual values of indi- +vidual key bits. Note that traditional SAT-based attacks rather +attempt to find the whole secret key, which increases the +� +� +� +� +� +� +� +� +� +� +� +� +� +� +�� +�� +� +� +� +� +� +� +� +�� +�� +� +� +� +� +�� +�� +��� +��� +��� +� +Fig. 3. +(a) Majority circuit; (b) Locked majority circuit; (c) Constant +propagation on the locked majority circuit. +computational effort significantly. In this attack, we initially +find queries, i.e., values of inputs of the oracle circuit, using +two techniques. The first technique uses the ATPG tool Ata- +lanta [30] to find test patterns for the stuck-at-fault of each key +bit on the locked circuit and stores the values of the related +primary inputs as queries. The aim is to find input patterns +that can propagate each key bit to a primary output, making +it observable. The second technique finds queries randomly. +The aim is to find input patterns that may make multiple key +bits observable. In our experiments, we generate a total of 2p +queries, where p denotes the number of key bits. +Then, we describe the locked circuit in a conjunctive normal +form (CNF) formula C by expressing each gate in its CNF. +Each query is applied to the oracle and the values of primary +outputs are obtained. Then, the related input and output values +are assigned to the associated nets in the locked circuit, the +constant values of these nets are propagated, and the Boolean +equations including key bits are derived in a CNF formula E. +The SAT problem including the locked circuit in CNF, i.e., +C, is augmented with these equations, i.e., C = C ∧ E. After +all the queries are considered, the SAT problem C is solved +using a SAT solver and the values of key bits are determined. +Note that the locked circuit with the found values of key bits +behaves exactly the same as the oracle under the given queries, +but not under all possible input values. Hence, these key values +are not guaranteed to be the values of the secret key. +However, the value found for a key bit can be proved if it is +indeed equal to the actual value of the related bit in the secret +key using the concept of proof by contradiction. To do so, for +each key bit, the complement of its found value is added into +C and the SAT solver is run. If there exists no solution to C, +i.e., the SAT problem is unsatisfiable, the value of the related +key bit is proven to be the one in the found solution. +As a simple example, consider the majority circuit in +Fig. 3(a) and suppose that it is locked using XOR/XNOR +gates as given in Fig. 3(b). Assume that a query is found as +abc = 000 and thus, the value of its output f is obtained as 0 +using the oracle. After propagating these values on the locked +circuit as shown in Fig. 3(c), a Boolean equation k0 ∨ k1 = 0, +i.e., k0 ∧ k1 in CNF, is obtained. In the SAT solution, the key +bit values are found as k0k1 = 01. Note that these are the +proven key values since a SAT solver guarantees that there +exists no solution to the SAT problem C, which is extended +by either k0 = 1, i.e., k0 in CNF, or k1 = 0, i.e., k1 in CNF, +due to a conflict with the found Boolean equation, i.e., k0 ∧k1 +in CNF. + +TABLE I +DETAILS OF THE ISCAS’85 CIRCUITS. +Circuit +Original Netlist +Locked Netlist +p +Anti-SAT CASLock SFLL SKG-Lock +#in #out #gates +#gates +#gates +#gates +#gates +c2670 +157 64 +1193 +64 +1321 +1320 +1421 +1401 +c3540 +50 +22 +1669 +32 +1733 +1732 +1783 +1773 +c5315 +178 123 2307 +64 +2435 +2434 +2523 +2514 +c6288 +32 +32 +2416 +32 +2480 +2479 +2531 +2516 +c7552 +206 105 3512 +64 +3640 +3639 +3729 +3713 +The query attack is run on all the resynthesized circuits and +the proven values of key bits in each netlist are combined +into a single solution. Note that the query attack is developed +in Perl and is equipped with the incremental SAT solver +CaDiCaL [31]. +Finally, the solution of the OG resynthesis-based attack is +determined after merging the solution of the SCOPE attack +over all resynthesized circuits into that of the query attack on +all resynthesized circuits without changing the proven values +of key bits. +IV. EXPERIMENTAL RESULTS +This section initially presents the results of the proposed +OL resynthesis-based attack on the ISCAS’85 circuits [32] +and then, those of the OG resynthesis-based attack on the +CSAW’19 circuits [24] including compound logic locking. +A. Results on the ISCAS’85 Circuits +As the first experiment set, five ISCAS’85 circuits were +considered. Table I presents their details. For our exper- +iments, these circuits were locked by the Anti-SAT [11], +CASlock [12], SFLL [6], and SKG-Lock [14] techniques. +Note that while Anti-SAT and SFLL were taken from the +NEOS tool [33], SKG-Lock was provided by its developer, and +CASLock was implemented by ourselves. Table I also presents +details of the locked circuits. Note that the number of keys, +i.e., p, was determined based on the number of inputs and +overhead of the locking technique, and circuit characteristics, +i.e., the number of inputs, outputs, and gates, were taken from +the gate-level netlist. +Observe from Table I that all logic locking techniques lead +to circuits with a number of gates close to each other, whereas +the one locked by SFLL has a slightly large number of gates. +Besides, the overhead on the number of gates in circuits +locked by SFLL varies from 4.7% to 19.1% when compared +to original circuits. +In the following subsections, we present the results of the +resynthesis process and OL resynthesis-based attack, analyze +the impact of synthesis parameters on the performance of +the resynthesis process and SCOPE attack, and introduce +improvements to the run-time of the resynthesis process. +1) Resynthesis of the Locked ISCAS’85 Circuits: The resyn- +thesis is performed by Cadence Genus with a commercial +65 nm standard cell library, and the whole process is automated +in a Perl script. Table II presents the resynthesis results of +locked circuits. In this table, unique denotes the number of +unique locked netlists out of 1080 generated netlists and area, +delay, and power stand respectively for the average values of +total area in µm2, delay in the critical path in ps, and total +power dissipation in µW on the unique locked netlists. Finally, +time is the total run time of the resynthesis process. The +resynthesized netlists were generated on a computing server +with Intel Xeon processing units at 3.9 GHz and a total of +1 TB memory. +Observe from Table II that the number of unique netlists is +less than half of the total number of generated netlists, i.e., +540, except the c3540 circuit locked by SKG-Lock. Note that +Anti-SAT, CASLock, and SFLL lead to fewer unique netlists +when compared to SKG-Lock, which is mainly because the +logic added by these techniques is more compact than that +added by SKG-Lock, which uses a chain of AND gates. We +note that the synthesis tool consumes a large amount of time +to fulfill a delay constraint that is impossible to meet, such +as strict delay constraints and key constraints described in +Section III-A. Hence, the run-time of the resynthesis process +depends on the locked circuit and the logic locking technique, +and more importantly, if there exists enough room for the +synthesis tool to satisfy the constraints. +In order to illustrate the diversity of resynthesized netlists, +the c2670 circuit locked by SFLL is considered. Fig. 4 presents +the area, delay, and power dissipation of each unique netlist, +normalized by their average values given in Table II. Observe +that resynthesis generates circuits significantly different from +each other in terms of hardware complexity. The standard +deviation on area, delay, and power dissipation values of +all these netlists are computed as 1578, 235, and 4964, +respectively. Note also that in this figure, the netlists after +instance number 232 have a distinct profile, since they are +generated using key constraints described in Section III-A. +In order to illustrate the differences in the structure of +generated netlists, the c2670 circuit locked by SKG-Lock +is considered. Fig. 5 presents the graphs of two netlists +resynthesized using the same synthesis parameters, except for +the delay constraint. In this figure, red, green, and blue circles +denote the inputs, key bits, and outputs, respectively; the gray +triangles represent the gates. Observe that a small change in +the delay constraint can lead to a structurally different netlist, +where the difference between the number of gates and logic +levels is 599 and 12, respectively. +2) Attacks on the Locked ISCAS’85 Circuits: Table III +presents the results of the SCOPE attack on the original locked +netlists and those of OL resynthesis-based attack on the unique +locked netlists generated in the resynthesis process. In this +table, cdk and dk stand respectively for the number of correctly +deciphered key bits and the total number of deciphered key +bits and time is the total time required for the attack. The +attacks were also run on the same server used to resynthesize +the locked netlists. +Observe from Table III that the SCOPE attack is not entirely +successful on any of the original locked netlists. However, the +use of resynthesized netlists enables us to decipher the values +of a large number of key bits, and even the whole key, e.g., for +the c2670 and c3540 circuits locked by SKG-Lock. Note that + +TABLE II +RESULTS OF RESYNTHESIZED LOCKED ISCAS’85 CIRCUITS. +Technique +Details +c2670 +c3540 +c5315 +c6288 +c7552 +Anti-SAT +#unique +480 +537 +464 +498 +439 +area +2357 +2803 +4112 +7265 +5387 +delay +504 +818 +663 +2144 +694 +power +5518 +4934 +4297 +9403 +7479 +time +17h14m51s +1d05h56m12s +1d09h56m22s +3d20h50m46s +1d16h01m13s +CASLock +#unique +473 +449 +488 +410 +479 +area +2359 +3112 +4173 +7739 +5337 +delay +513 +874 +650 +2146 +676 +power +5170 +3304 +3852 +10693 +6765 +time +15h29m56s +1d11h02m52s +1d06h52m54s +4d03h12m29s +1d16h06m52s +SFLL +#unique +468 +484 +477 +523 +504 +area +2817 +3444 +4326 +7646 +5340 +delay +481 +870 +697 +2144 +604 +power +6189 +6337 +9053 +12115 +11320 +time +13h13m23s +1d47m51s +21h57m07s +2d22h15m07s +22h40m29s +SKG-Lock +#unique +521 +541 +507 +527 +521 +area +2673 +2773 +4646 +6293 +4774 +delay +936 +986 +782 +2093 +874 +power +3881 +3831 +8160 +7201 +7822 +time +22h22m01s +1d08h8m27s +1d03h56m15s +2d14h29m32s +1d04h19m + 0.5 + 1 + 1.5 + 2 + 2.5 + 0 + 50 + 100 + 150 + 200 + 250 + 300 + 350 + 400 + 450 +Normalized Area +Number of netlists + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 0 + 50 + 100 + 150 + 200 + 250 + 300 + 350 + 400 + 450 +Normalized Delay +Number of netlists + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 0 + 50 + 100 + 150 + 200 + 250 + 300 + 350 + 400 + 450 +Normalized Power +Number of netlists +(a) +(b) +(c) +Fig. 4. Normalized complexity of resynthesized netlists of the c2670 circuit locked by SFLL: (a) area; (b) delay; (c) power. +the SCOPE attack can decipher almost all of the key bits using +the resynthesized netlists locked by the SKG-Lock technique. +While the results on the netlists locked by SKG-Lock are all +correct, the ones on the netlists locked by Anti-SAT, CASLock, +and SFLL are slightly better than a random guess. The run time +of the SCOPE attack and our resynthesis-based attack depends +mainly on the number of gates and keys in the locked design. +To find the SAT resiliency of resynthesized locked circuits, +the SAT-based attack of [10] was run on 541 netlists of the +c3540 circuit locked by SKG-Lock with a time limit of 2 days. +This circuit was chosen since it has the smallest number of key +bits with the smallest number of gates. Note that the SAT-based +attack was not able to find the secret key of any resynthesized +locked netlists. This experiment indicates that the resynthesis +changes only the structure of the circuit as shown in Fig. 5, +but maintains its SAT resiliency. +3) Redundant Synthesis Runs: +Observe from Tables II +and III that the total run-time of the proposed attack is +dominated by the resynthesis process. However, it is possible +to reduce the time required to resynthesize the locked netlist +by removing redundant synthesis runs without sacrificing any +unique netlists. For example, it is observed that the high +value of the syn_gen parameter given in Section III-A can be +removed from the parameter list, since all possible synthesis +scripts including this parameter generate the same circuit +when this parameter is low or medium. Thus, the number of +generated circuits, i.e., 1080, reduces to 720. +4) Convergence on the Number of Deciphered Keys: It +is also observed that the number of key bits deciphered +by the SCOPE attack on all unique resynthesized netlists +can actually be obtained using a small number of netlists. +Fig. 6 presents the number of deciphered key bits along the +unique resynthesized netlists of the c2670 circuit locked by +SKG-Lock. Observe from this figure that although a large +number of unique netlists increases the quality of the SCOPE +attack, actually a small number of unique netlists, 147 in this +case, is sufficient to achieve the same result as when all 521 +unique netlists are considered. We note that a similar situation +was also observed on circuits locked by other techniques. +5) Promising Resynthesized Netlists: Moreover, it is ob- +served that the SCOPE attack is more successful on specific +resynthesized netlists. To find a set of synthesis parameters +that enables the SCOPE attack to decipher more key values, +we initially define two categories of netlists based on the slack +time of the design, i.e., the difference between the required +and arrived time in the critical path, as follows: i) netlists + +(a) +(b) +Fig. 5. Graphs of resynthesized netlists generated using a difference in the delay constraint dc: (a) dc is 990 ps; (b) dc is 496 ps. +TABLE III +RESULTS OF ATTACKS ON THE LOCKED ISCAS’85 CIRCUITS. +Circuit +Anti-SAT +CASLock +SFLL +SKG-Lock +SCOPE +Resynthesis +SCOPE +Resynthesis +SCOPE +Resynthesis +SCOPE +Resynthesis +cdk/dk time +cdk/dk +time +cdk/dk time +cdk/dk +time +cdk/dk time +cdk/dk +time +cdk/dk time +cdk/dk +time +c2670 +0/0 +4s +37/64 34m18s +0/0 +4s +35/64 33m47s +0/0 +4s +34/64 37m32s +32/32 +4s +64/64 44m37s +c3540 +0/0 +3s +17/32 21m27s +0/0 +3s +17/32 18m12s +0/0 +2s +19/32 21m29s +17/17 +2s +32/32 24m30s +c5315 +0/0 +5s +38/64 42m34s +0/0 +5s +30/64 43m54s +0/0 +5s +33/64 46m23s +32/32 +5s +62/62 52m06s +c6288 +0/0 +3s +18/32 29m08s +0/0 +3s +16/32 27m18s +0/0 +3s +16/31 33m19s +16/16 +3s +31/31 34m24s +c7552 +0/0 +6s +38/64 45m31s +0/0 +6s +47/64 49m13s +0/0 +6s +38/63 52m26s +32/32 +6s +61/61 56m45s + 30 + 35 + 40 + 45 + 50 + 55 + 60 + 65 + 0 + 100 + 200 + 300 + 400 + 500 +Number of deciphered keys +Number of netlists +Fig. 6. Convergence on the number of deciphered keys over the number of +resynthesized netlists in the SCOPE attack. +with a slack value less than or equal to 0; ii) netlists with a +slack value greater than 0. The slack value of a design gives +indeed a rough idea of the effort put in by the synthesis tool; +for the netlists in the first category, the synthesis tool works +extremely hard to meet the delay constraint, trying many logic +transformations and optimization techniques. +Then, the solutions of the SCOPE attack on all possible +1080 netlists are obtained and sorted based on the number of +deciphered key bits in descending order. The top 10% of these +sorted netlists are categorized based on their slack values. +Fig. 7 presents the results of this experiment on the circuits +locked by SKG-Lock. Observe that the netlists that enable the +SCOPE attack to decipher more key values generally have a +slack value less than or equal to 0. Thus, to generate such +circuits, one can easily add strict delay constraints or key +constraints as described in Section III-A. We note that a similar + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 +c2670 +c3540 +c5315 +c6288 +c7552 +Number of netlists +Circuit +slack ≤ 0 +slack > 0 +Fig. 7. Classification of resynthesized netlists based on their slack values on +promising solutions of SCOPE attack. +result was also observed on resynthesized netlists locked by +other techniques. +6) Structural Analysis: In order to improve the performance +of the resynthesis process, the logic cone, which is the locking +technique is applied on, can be extracted and resynthesized. +Note that the output of this logic cone is a single primary +output, while its inputs are primary inputs, but not necessarily +all the primary inputs of the locked design. Thus, the run- +time of the resynthesis process can be decreased, since the +logic cone has a small number of inputs, outputs, and gates +when compared to the whole locked circuit. +Table IV presents details on the resynthesis process on entire +locked circuits and logic cones when the circuits locked by +SFLL are used. Observe that the resynthesis process on a +logic cone generates less number of unique designs and takes +significantly less time without a significant loss on the solution + +TABLE IV +RESULTS OF THE RESYNTHESIS PROCESS ON ENTIRE CIRCUIT AND LOGIC +CONE. +Circuit +Entire Circuit +Logic Cone +#unique +time +cdk/dk +#unique +time +cdk/dk +c2670 +468 +13h13m23s +34/64 +319 +07h46m26s +34/64 +c3540 +484 +1d47m51s +19/32 +320 +06h29m35s +16/32 +c5315 +477 +21h57m14s +33/64 +313 +07h06m16s +32/64 +c6288 +523 +2d22h15m7s +16/31 +302 +06h20m57s +19/32 +c7552 +504 +22h40m29s +38/63 +279 +06h57m14s +38/63 +TABLE V +DETAILS OF THE LOCKED CSAW’19 CIRCUITS. +Circuit +Details +Number of keys +#in +#out +#gates +RLL +SFLL-rem +Total +small +522 +512 +15995 +40 +40 +80 +medium +767 +757 +24008 +60 +60 +120 +large +1452 +1445 +36584 +80 +80 +160 +bonus +892 +1746 +23004 +128 +128 +256 +quality when compared to the resynthesis process on the entire +circuit. We note that similar results were also observed on +circuits locked by other techniques. +B. Results on the CSAW’19 Circuits +As the second experiment set, we used the state-of-the- +art locked circuits from the CSAW’19 contest [24]. Details +of these circuits are given in Table V. Note that two logic +locking techniques – RLL [9] and SFLL-rem [13] – are applied +together to lock a circuit. +In the following two subsections, we present the results of +the resynthesis process and the resynthesis-based attack. +1) Resynthesis of the Locked CSAW’19 Circuits: Table VI +presents the resynthesis results of locked circuits. Observe that +the number of unique resynthesized netlists is larger than half +of the total number of generated netlists, i.e., 540. As the +hardware complexity of designs increases, the run-time of the +resynthesis process increases. We note that diverse netlists in +terms of complexity are obtained, e.g., the standard deviation +on area, delay, and power dissipation values of all the locked +netlists of the small circuit is computed as 8526, 1029, and +20074, respectively. +2) Attacks on the Locked CSAW’19 Circuits: Table VII +presents results of the attacks obtained, after they are applied +to the original locked netlist, denoted as OLN, and all unique +resynthesized netlists, denoted as URNs. In this table, prv +stands for the number of proven values of key bits. Note that +since the secret key is not publicly available, the cdk values +are omitted for the SCOPE and resynthesis-based attacks. +Observe from Table VII that the original SCOPE attack +could only decipher a small number of key bits, all of which +belongs to RLL, and the query attack can prove the values of a +large number of key bits, all of which again belong to RLL, on +the original locked circuits. Thus, the resynthesis-based attack +could only decipher the RLL key bits on the original locked +circuits. However, the use of resynthesized circuits makes the +SCOPE attack decipher more key bits that also belong to +SFLL-rem and makes the query attack prove the values of +TABLE VI +RESULTS OF RESYNTHESIZED LOCKED CSAW’19 CIRCUITS. +Circuit +unique +area +delay +power +time +small +557 +18935 +1631 +23571 +5d3h22m28s +medium +569 +26080 +1745 +31284 +6d12h24m16s +large +567 +31348 +1798 +24610 +5d21h42m10s +bonus +560 +20643 +1758 +19090 +4d14h44m29s +TABLE VII +RESULTS OF ATTACKS ON THE LOCKED CSAW’19 CIRCUITS. +Circuit-Netlist +SCOPE +Query +Resynthesis +dk +time +prv +time +dk +time +small - OLN +19 +20s +39 +1m21s +40 +1m41s +small - URNs +77 4h10m42s +40 1d10h4m37s +79 1d14h15m19s +medium - OLN +32 +41s +58 +6m37s +59 +7m18s +medium - URNs +117 8h33m56s +58 3d19h12m13s +120 4d3h46m9s +large - OLN +30 +1m7s +79 +6m19s +79 +7m26s +large - URNs +15212h56m15s +80 3d2h52m11s +159 3d15h48m26s +bonus - OLN +64 +1m46s +118 +3m2s +120 +4m48s +bonus - URNs +233 16h7m17s +1251d20h29m22s +252 2d12h36m39s +more key bits that belong to RLL. Thus, the resynthesis-based +attack could decipher almost all the values of the secret key, +proving almost all the values of the key bits of RLL. Note that +all the unknown key bits belong to SFLL-rem. Observe that +the run-time of attacks increases, as the number of gates and +key bits increases. +After the values of key bits of the CSAW’19 circuits were +determined, they were sent to the contest organizers for eval- +uation. Table VIII presents the results of the resynthesis-based +attack along with those of other techniques which participated +in the contest. +Observe from Table VIII that our proposed attack can +determine all the key bits of RLL correctly, even though there +are unproven key bits in the medium and bonus circuits as +shown in Table VII. This observation implies that the guesses +of the SCOPE attack on those key bits are actually correct. +Moreover, the proposed technique can decipher the key bits +of SFLL-rem with a number of deciphered key bits greater +than any other OL technique with a high accuracy. +V. CONCLUSIONS +This work has shown that EDA tools can be used to generate +variants of locked circuits that may be vulnerable to existing +logic locking attacks and such circuits can be generated using +a specific set of synthesis parameters. It was shown that the +run-time of the proposed technique can be improved using +a small number of resynthesized netlists without diminishing +its solution quality. Experimental results clearly indicated that +the use of many resynthesized circuits enables existing attacks +to decipher values of a large number of key bits with high +accuracy. Hence, the resynthesis of a locked circuit can be +utilized as a pre-attack step for many existing attacks in order +to improve their success rate. As future work, we plan to +consider other synthesis parameters, such as fanout, capaci- +tance limits, and wire loads, which enable synthesis tools to +generate different circuits. Also, we aim to incorporate other + +TABLE VIII +RESULTS OF ATTACKS ON THE LOCKED CSAW’19 CIRCUITS. +Approach +Attack Scenario +Circuit +small (40+40) +medium (60+60) +large (80+80) +bonus (128+128) +RLL +SFLL-rem +RLL +SFLL-rem +RLL +SFLL-rem +RLL +SFLL-rem +Key sensitization [34] +OG +40/40 +— +60/60 +— +80/80 +— +— +— +Hamming distance-based attack [24] +OG +30/30 +— +50/50 +— +72/72 +— +— +— +Automated analysis + SAT [24] +OG +11/18 +— +31/50 +— +10/34 +— +— +— +Sub-circuit SAT [24] +OG +17/17 +— +29/29 +— +— +— +— +— +Redundancy-based [27] +OL +28/28 +4/12 +35/35 +23/28 +45/45 +0/51 +66/66 +8/27 +Bit-flipping attack [35] +OG +40/40 +— +60/60 +— +80/80 +— +— +— +Topology guided attack [28] +OL +15/32 +— +19/50 +— +36/73 +— +75/108 +— +Resynthesis-based attack +OG +40/40 +20/39 +60/60 +29/60 +80/80 +35/79 +128/128 +55/124 +commercial and open source EDA tools into the resynthesis +process to generate different unique netlists. +ACKNOWLEDGMENT +The authors thank Nimisha Limaye for evaluating the keys +found by the proposed technique on the CSAW’19 bench- +marks. +REFERENCES +[1] M. 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Shamsi, “Netlist Encryption and Obfuscation Suite,” 2021. [Online]. +Available: https://bitbucket.org/kavehshm/neos/src/master/ +[34] J. Rajendran, Y. Pino, O. Sinanoglu, and R. Karri, “Security Analysis +of Logic Obfuscation,” in DAC, 2012, pp. 83–89. +[35] Y. Shen, A. Rezaei, and H. Zhou, “SAT-based Bit-Flipping Attack on +Logic Encryptions,” in DATE, 2018, pp. 629–632. + diff --git a/yNE3T4oBgHgl3EQfPQkf/content/tmp_files/load_file.txt b/yNE3T4oBgHgl3EQfPQkf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..755dfd19ada5dd709a011554ab936ee3fa967fa1 --- /dev/null +++ b/yNE3T4oBgHgl3EQfPQkf/content/tmp_files/load_file.txt @@ -0,0 +1,1010 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf,len=1009 +page_content='Resynthesis-based Attacks Against Logic Locking Felipe Almeida†, Levent Aksoy†, Quang-Linh Nguyen‡, Sophie Dupuis‡, Marie-Lise Flottes‡ and Samuel Pagliarini† †Department of Computer Systems, Tallinn University of Technology, Tallinn, Estonia Email: {felipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='almeida, levent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='aksoy, samuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='pagliarini}@taltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='ee ‡LIRMM, University of Montpellier, Montpellier, France Email: {quang-linh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='nguyen, sophie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='dupuis, marie-lise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='flottes}@lirmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='fr Abstract—Logic locking has been a promising solution to many hardware security threats, such as intellectual property infringe- ment and overproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Due to the increased attention that threats have received, many efficient specialized attacks against logic locking have been introduced over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, the ability of an adversary to manipulate a locked netlist prior to mounting an attack has not been investigated thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' This paper introduces a resynthesis-based strategy that utilizes the strength of a commercial electronic design automation (EDA) tool to reveal the vulnerabilities of a locked circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' To do so, in a pre-attack step, a locked netlist is resynthesized using different synthesis parameters in a systematic way, leading to a large number of functionally equivalent but structurally different locked circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Then, under the oracle-less threat model, where it is assumed that the adversary only possesses the locked circuit, not the original circuit to query, a prominent attack is applied to these generated netlists collectively, from which a large number of key bits are deciphered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Nevertheless, this paper also describes how the proposed oracle-less attack can be integrated with an oracle-guided attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The feasibility of the proposed approach is demonstrated for several benchmarks, including remarkable results for breaking a recently proposed provably secure logic locking method and deciphering values of a large number of key bits of the CSAW’19 circuits with very high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Index Terms—Logic locking, resynthesis, EDA tools, oracle-less and oracle-guided attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' INTRODUCTION Due to the globalized integrated circuit (IC) supply chain, serious security threats, such as hardware Trojans, piracy, overbuilding, reverse engineering, and counterfeiting, have emerged [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Many defense techniques, such as watermark- ing [2], digital rights management [3], metering [4], and logic locking [5], have been introduced over the years to deal with these threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Among those, logic locking stands out by being a well-established technique and by offering protection against a diverse array of adversaries [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Logic locking inserts additional logic driven by key bits so that the circuit behaves as expected only when the secret key is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' On the other hand, many efficient attacks have been in- troduced to overcome the defenses built by logic locking [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, the impact of an electronic design automation (EDA) tool on the manipulation of the locked netlist before per- forming an attack has not been investigated thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In This work has been partially conducted in the project “ICT programme” which was supported by the European Union through the European Social Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' It was also partially supported by European Union’s Horizon 2020 research and innovation programme under grant agreement No 952252 (SAFEST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' this work, we explore if EDA tools can be used to make a locked circuit vulnerable to existing logic locking attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, the main contributions of this work are three-fold: (i) we introduce a resynthesis procedure that is a pre-attack step, where functionally equivalent but structurally different locked circuits are generated by resynthesizing the original locked circuit using different optimization parameters and delay constraints in order to create structural vulnerabilities that can be exploited by existing attacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (ii) we present an oracle-less (OL) resynthesis-based attack, which applies the prominent SCOPE attack [8] to these resynthesized circuits and gathers all its solutions to discover the secret key;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (iii) we show that our OL attack can be combined with a traditional oracle-guided (OG) attack for further improving the number of correctly deciphered key bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The last contribution is essential, since we consider circuits from the CSAW’19 contest – these circuits compound the use of two logic locking techniques at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The main finding of this work is that the use of many resynthesized locked circuits enables us to discover values of more key bits, and even the whole key, when compared to a single attack mounted on the original locked netlist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The remainder of this paper is organized as follows: Sec- tion II presents the background concepts and related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The resynthesis process and the proposed attacks are described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Experimental results are given in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Finally, Section V concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' BACKGROUND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Logic Locking and Threat Models The procedure of logic locking is applied at the gate level in the IC design flow, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that the layout of the locked circuit is sent to the foundry without revealing the secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' After the locked IC is produced and delivered to the design house, the values of the secret key are stored in a tamper-proof memory, before the functional IC is sent to the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' It is assumed that the gate-level netlist of the locked circuit can be obtained directly by an untrusted foundry or by reverse-engineering a functional IC obtained from the open market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' An adversary can also use the functional IC programmed with the secret key as an oracle to apply inputs and observe outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, in logic locking, there are generally two threat models: OL and OG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In the OL threat model, only the gate-level netlist of the locked circuit is available to the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='04400v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='CR] 11 Jan 2023 ����� ��������� ����� ������� ������ ������� �������� ��������� ������ ����������� ���� � ��������� ������ �� ��� ���������� ���������� �� ������� ������ ��������� ������ ��������� ������ ������� ������ ���������� ������ ������������� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Conventional logic locking in the IC design flow (adapted from [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Original Circuit Locking Unit inputs key bits output Original Circuit Restore Unit inputs key bits output Pertub Unit Stripped Circuit X X X X Critical Point (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' SAT-resilient logic locking methods: (a) SFLT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (b) DFLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The adversary has both the netlist of the locked circuit and the functional IC in the OG threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Related Work After the introduction of random logic locking (RLL) using XOR/XNOR gates in [9], earlier work focused on different types of key gates, such as AND/OR, multiplexors, and look-up tables, taking into account the hardware complexity of the locked circuit [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, the OG satisfiability (SAT)-based attack [10] overcame all the defenses existing at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that the SAT-based attack iteratively finds differentiating input patterns (DIPs) that rule out wrong keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' To thwart the SAT-based attack and its variants, circuits are locked using a point function that sets a limit on the number of wrong keys which a DIP can eliminate, forcing these attacks to explore an exponential number of queries [6], [11]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The SAT-resilient methods can be categorized into two groups: single-flip locking technique (SFLT) and double-flip locking technique (DFLT), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' An SFLT has only one critical point, which is responsible to corrupt a protected output under a specific input pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In this category, SARLock [15] adds a comparator and a masking circuit connected with the original netlist in a way that it generates a corruption on one input pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Anti-SAT [11] utilizes two complementary AND gate trees, whose output is merged with the original circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' CASLock [12] is based on the same concept of Anti-SAT, however it uses both AND and OR gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' SKG-Lock [14] uses decoy key bits and provides a tunable output corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that SFLTs are susceptible to removal attacks [16]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' If an attacker can identify this single critical point, he/she can split the design into a recovered netlist (original) and the locking unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' A DFLT has two critical points, one that connects the original netlist with a perturbation unit and another one that connects the output of the stripped circuit with the restore unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Under this category, stripped functionality logic locking (SFLL) [6], [13] initially corrupts an output based on an input combination in the perturbation unit and then, corrects this output only when the secret key is applied in the restore unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that a removal attack becomes inefficient for a DFLT, since the original circuit is mixed with the perturbation unit, even though it can easily identify the restore unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, there exist efficient structural attacks developed for DFLTs [19]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Alternative locking techniques have also been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In [23], a technique, which has more than two critical points, called the multi-flip locking technique (MFLT), was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, it leads to a significant increase in area, power dissipation, and delay when compared to other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Compound logic locking techniques were proposed to over- come the main drawback of a SAT-resilient technique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', its low output corruptibility as can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 2, by locking a design using both low and high output corrupt- ibility techniques, such as SFLL and RLL, respectively [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Recently, efficient attacks have also been introduced against compound logic locking [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Moreover, the OL attacks explore patterns in the structure of a locked netlist using statistical analysis [8], [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' For example, the SCOPE attack [8] is an unsupervised constant propagation technique, which analyzes each key bit of the locked design for critical features that can reveal its correct value after it is assigned to logic 0 and 1 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' These critical features include area, power dissipation, delay, and many other circuit characteristics obtained by a synthesis tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' These features are analyzed using linear regression and machine learning based clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' PROPOSED RESYNTHESIS-BASED ATTACK This section describes our resynthesis-based attack in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We initially introduced the pre-attack stage, where the locked circuit is resynthesized using different synthesis parameters, leading to a large number of structurally different netlists with the same functionality [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Then, we present the OL attack that utilizes these resynthesized netlists in order to find the secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Finally, in order to handle the compound logic locking efficiently, we present its modified version, where our proposed OL attack cooperates with an OG attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The Pre-attack Step: Resynthesis of the Locked Netlist The locked circuit is synthesized multiple times using a different script each time, where the synthesis parameters are explored in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We use the following parameters to increase the number of resynthesized locked circuits: Synthesis Effort: In a synthesis tool, logic optimizations can be applied with different efforts at different synthesis stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' This flexibility enables a designer to explore the trade-off between the quality of results and run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The following ef- forts are considered at the given synthesis stage: low, medium, and high at generic transformations (syn_gen);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' low, medium, and high at mapping (syn_map);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' and low, medium, high, and extreme at optimization (syn_opt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Delay Constraint: To meet performance targets, delay con- straints are used to guide the synthesis tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We initially resynthesize the locked circuit without a delay constraint and find the delay of its critical path, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', dcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Then, in an interval between 0 and dcp, d − 1 points, which are computed as (dcp/d)i with 1 ≤ i ≤ d − 1, are set as delay constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that d is set to 5 in order to generate a large number of resynthesized circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Even though some delay constraints are impossible to meet, the synthesis tool always generates a netlist equivalent to the original one in terms of functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Maximum Transition: The transition time of a net in a circuit is defined as the longest time required for its driving pin to change its logic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We choose the maximum transition value to be 5%, 10%, and 15% of the delay constraint for all the nets in the locked circuit to explore different resynthesized circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Key Constraints: To direct the synthesis tool to work in- tensively on the paths that contain the keyed logic, a delay constraint, which is impossible to be satisfied, can also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In this case, we force the delay between all key bits and all primary outputs to be 1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, the combination of parameters given above generates 3 × 3 × 4 × 5 × 3 × 2 = 1080 netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We eliminate the resynthesized circuits with identical characteristics and keep only the unique ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Additionaly, we prevent the use of XOR/XNOR gates, which can be problematic for the SCOPE attack, during technology mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that our resynthesis methodology aims to generate different versions of the locked circuit, making it more vulnerable to existing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, any existing attack, either OL or OG, may potentially benefit from this pre-attack strategy to discover the secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Attacks on the Resynthesized Netlists Time-efficient attacks are chosen in order to handle a large number of resynthesized circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In our OL resynthesis-based attack, SCOPE [8] is used to predict the values of key bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In its modified version developed for compound logic locking, a query attack is used to find the values of key bits in a deterministic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 1) Proposed OL Attack: SCOPE is applied to each resyn- thesized locked circuit and a solution is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that this solution may return a logic 0, 1, or an unknown value for a key bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Then, the values of key bits deciphered for each netlist are merged into a single solution that represents the overall guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' To do so, for each key bit, ki with 1 ≤ i ≤ p, where p denotes the number of key bits, we initially count the number of solutions, where ki is deciphered as logic 0 and 1, denoted as dk0 i and dk1 i , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Then, if dk0 i > dk1 i or dk1 i > dk0 i , the value of ki is determined to be 0 or 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Otherwise, in the case of a tie, the value of ki is decided to be unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 2) Proposed OG Attack: In order to handle a large number of resynthesized netlists efficiently, we introduce a SAT-based query attack, which can determine the actual values of indi- vidual key bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that traditional SAT-based attacks rather attempt to find the whole secret key, which increases the � � � � � � � � � � � � � � �� �� � � � � � � � �� �� � � � � �� �� ��� ��� ��� � Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (a) Majority circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (b) Locked majority circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (c) Constant propagation on the locked majority circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' computational effort significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In this attack, we initially find queries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', values of inputs of the oracle circuit, using two techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The first technique uses the ATPG tool Ata- lanta [30] to find test patterns for the stuck-at-fault of each key bit on the locked circuit and stores the values of the related primary inputs as queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The aim is to find input patterns that can propagate each key bit to a primary output, making it observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The second technique finds queries randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The aim is to find input patterns that may make multiple key bits observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In our experiments, we generate a total of 2p queries, where p denotes the number of key bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Then, we describe the locked circuit in a conjunctive normal form (CNF) formula C by expressing each gate in its CNF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Each query is applied to the oracle and the values of primary outputs are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Then, the related input and output values are assigned to the associated nets in the locked circuit, the constant values of these nets are propagated, and the Boolean equations including key bits are derived in a CNF formula E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The SAT problem including the locked circuit in CNF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', C, is augmented with these equations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', C = C ∧ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' After all the queries are considered, the SAT problem C is solved using a SAT solver and the values of key bits are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that the locked circuit with the found values of key bits behaves exactly the same as the oracle under the given queries, but not under all possible input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Hence, these key values are not guaranteed to be the values of the secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, the value found for a key bit can be proved if it is indeed equal to the actual value of the related bit in the secret key using the concept of proof by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' To do so, for each key bit, the complement of its found value is added into C and the SAT solver is run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' If there exists no solution to C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', the SAT problem is unsatisfiable, the value of the related key bit is proven to be the one in the found solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' As a simple example, consider the majority circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 3(a) and suppose that it is locked using XOR/XNOR gates as given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Assume that a query is found as abc = 000 and thus, the value of its output f is obtained as 0 using the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' After propagating these values on the locked circuit as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 3(c), a Boolean equation k0 ∨ k1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', k0 ∧ k1 in CNF, is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In the SAT solution, the key bit values are found as k0k1 = 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that these are the proven key values since a SAT solver guarantees that there exists no solution to the SAT problem C, which is extended by either k0 = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', k0 in CNF, or k1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', k1 in CNF, due to a conflict with the found Boolean equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', k0 ∧k1 in CNF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' TABLE I DETAILS OF THE ISCAS’85 CIRCUITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Circuit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Original Netlist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Locked Netlist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Anti-SAT CASLock SFLL SKG-Lock ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='#in #out #gates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='#gates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='#gates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='#gates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='#gates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='c2670 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='157 64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='1193 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='2531 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='2516 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='c7552 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='206 105 3512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='3640 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='3639 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='3729 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='3713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='The query attack is run on all the resynthesized circuits and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='the proven values of key bits in each netlist are combined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='into a single solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that the query attack is developed in Perl and is equipped with the incremental SAT solver CaDiCaL [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Finally, the solution of the OG resynthesis-based attack is determined after merging the solution of the SCOPE attack over all resynthesized circuits into that of the query attack on all resynthesized circuits without changing the proven values of key bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' EXPERIMENTAL RESULTS This section initially presents the results of the proposed OL resynthesis-based attack on the ISCAS’85 circuits [32] and then, those of the OG resynthesis-based attack on the CSAW’19 circuits [24] including compound logic locking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Results on the ISCAS’85 Circuits As the first experiment set, five ISCAS’85 circuits were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Table I presents their details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' For our exper- iments, these circuits were locked by the Anti-SAT [11], CASlock [12], SFLL [6], and SKG-Lock [14] techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that while Anti-SAT and SFLL were taken from the NEOS tool [33], SKG-Lock was provided by its developer, and CASLock was implemented by ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Table I also presents details of the locked circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that the number of keys, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', p, was determined based on the number of inputs and overhead of the locking technique, and circuit characteristics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', the number of inputs, outputs, and gates, were taken from the gate-level netlist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe from Table I that all logic locking techniques lead to circuits with a number of gates close to each other, whereas the one locked by SFLL has a slightly large number of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Besides, the overhead on the number of gates in circuits locked by SFLL varies from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='7% to 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='1% when compared to original circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In the following subsections, we present the results of the resynthesis process and OL resynthesis-based attack, analyze the impact of synthesis parameters on the performance of the resynthesis process and SCOPE attack, and introduce improvements to the run-time of the resynthesis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 1) Resynthesis of the Locked ISCAS’85 Circuits: The resyn- thesis is performed by Cadence Genus with a commercial 65 nm standard cell library, and the whole process is automated in a Perl script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Table II presents the resynthesis results of locked circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In this table, unique denotes the number of unique locked netlists out of 1080 generated netlists and area, delay, and power stand respectively for the average values of total area in µm2, delay in the critical path in ps, and total power dissipation in µW on the unique locked netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Finally, time is the total run time of the resynthesis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The resynthesized netlists were generated on a computing server with Intel Xeon processing units at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='9 GHz and a total of 1 TB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe from Table II that the number of unique netlists is less than half of the total number of generated netlists, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', 540, except the c3540 circuit locked by SKG-Lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that Anti-SAT, CASLock, and SFLL lead to fewer unique netlists when compared to SKG-Lock, which is mainly because the logic added by these techniques is more compact than that added by SKG-Lock, which uses a chain of AND gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We note that the synthesis tool consumes a large amount of time to fulfill a delay constraint that is impossible to meet, such as strict delay constraints and key constraints described in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Hence, the run-time of the resynthesis process depends on the locked circuit and the logic locking technique, and more importantly, if there exists enough room for the synthesis tool to satisfy the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In order to illustrate the diversity of resynthesized netlists, the c2670 circuit locked by SFLL is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 4 presents the area, delay, and power dissipation of each unique netlist, normalized by their average values given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe that resynthesis generates circuits significantly different from each other in terms of hardware complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The standard deviation on area, delay, and power dissipation values of all these netlists are computed as 1578, 235, and 4964, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note also that in this figure, the netlists after instance number 232 have a distinct profile, since they are generated using key constraints described in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In order to illustrate the differences in the structure of generated netlists, the c2670 circuit locked by SKG-Lock is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 5 presents the graphs of two netlists resynthesized using the same synthesis parameters, except for the delay constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In this figure, red, green, and blue circles denote the inputs, key bits, and outputs, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' the gray triangles represent the gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe that a small change in the delay constraint can lead to a structurally different netlist, where the difference between the number of gates and logic levels is 599 and 12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 2) Attacks on the Locked ISCAS’85 Circuits: Table III presents the results of the SCOPE attack on the original locked netlists and those of OL resynthesis-based attack on the unique locked netlists generated in the resynthesis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In this table, cdk and dk stand respectively for the number of correctly deciphered key bits and the total number of deciphered key bits and time is the total time required for the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The attacks were also run on the same server used to resynthesize the locked netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe from Table III that the SCOPE attack is not entirely successful on any of the original locked netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, the use of resynthesized netlists enables us to decipher the values of a large number of key bits, and even the whole key, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', for the c2670 and c3540 circuits locked by SKG-Lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that TABLE II RESULTS OF RESYNTHESIZED LOCKED ISCAS’85 CIRCUITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Technique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Details ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='c2670 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='c3540 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='c5315 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='c6288 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='c7552 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Anti-SAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='#unique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='480 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='537 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='464 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='498 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='439 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='area ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 0 50 100 150 200 250 300 350 400 450 Normalized Area Number of netlists 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 0 50 100 150 200 250 300 350 400 450 Normalized Delay Number of netlists 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='5 0 50 100 150 200 250 300 350 400 450 Normalized Power Number of netlists (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Normalized complexity of resynthesized netlists of the c2670 circuit locked by SFLL: (a) area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (b) delay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (c) power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' the SCOPE attack can decipher almost all of the key bits using the resynthesized netlists locked by the SKG-Lock technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' While the results on the netlists locked by SKG-Lock are all correct, the ones on the netlists locked by Anti-SAT, CASLock, and SFLL are slightly better than a random guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The run time of the SCOPE attack and our resynthesis-based attack depends mainly on the number of gates and keys in the locked design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' To find the SAT resiliency of resynthesized locked circuits, the SAT-based attack of [10] was run on 541 netlists of the c3540 circuit locked by SKG-Lock with a time limit of 2 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' This circuit was chosen since it has the smallest number of key bits with the smallest number of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that the SAT-based attack was not able to find the secret key of any resynthesized locked netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' This experiment indicates that the resynthesis changes only the structure of the circuit as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 5, but maintains its SAT resiliency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 3) Redundant Synthesis Runs: Observe from Tables II and III that the total run-time of the proposed attack is dominated by the resynthesis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, it is possible to reduce the time required to resynthesize the locked netlist by removing redundant synthesis runs without sacrificing any unique netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' For example, it is observed that the high value of the syn_gen parameter given in Section III-A can be removed from the parameter list, since all possible synthesis scripts including this parameter generate the same circuit when this parameter is low or medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, the number of generated circuits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', 1080, reduces to 720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 4) Convergence on the Number of Deciphered Keys: It is also observed that the number of key bits deciphered by the SCOPE attack on all unique resynthesized netlists can actually be obtained using a small number of netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 6 presents the number of deciphered key bits along the unique resynthesized netlists of the c2670 circuit locked by SKG-Lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe from this figure that although a large number of unique netlists increases the quality of the SCOPE attack, actually a small number of unique netlists, 147 in this case, is sufficient to achieve the same result as when all 521 unique netlists are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We note that a similar situation was also observed on circuits locked by other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 5) Promising Resynthesized Netlists: Moreover, it is ob- served that the SCOPE attack is more successful on specific resynthesized netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' To find a set of synthesis parameters that enables the SCOPE attack to decipher more key values, we initially define two categories of netlists based on the slack time of the design, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', the difference between the required and arrived time in the critical path, as follows: i) netlists (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Graphs of resynthesized netlists generated using a difference in the delay constraint dc: (a) dc is 990 ps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' (b) dc is 496 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' TABLE III RESULTS OF ATTACKS ON THE LOCKED ISCAS’85 CIRCUITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Circuit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Anti-SAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='CASLock ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SFLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SKG-Lock ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SCOPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Resynthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SCOPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Resynthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SCOPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Resynthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SCOPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Resynthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='cdk/dk time ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='cdk/dk time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='cdk/dk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='c2670 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='0/0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='4s ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Number of deciphered keys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Number of netlists ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Convergence on the number of deciphered keys over the number of resynthesized netlists in the SCOPE attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' with a slack value less than or equal to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' ii) netlists with a slack value greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The slack value of a design gives indeed a rough idea of the effort put in by the synthesis tool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' for the netlists in the first category, the synthesis tool works extremely hard to meet the delay constraint, trying many logic transformations and optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Then, the solutions of the SCOPE attack on all possible 1080 netlists are obtained and sorted based on the number of deciphered key bits in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' The top 10% of these sorted netlists are categorized based on their slack values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 7 presents the results of this experiment on the circuits locked by SKG-Lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe that the netlists that enable the SCOPE attack to decipher more key values generally have a slack value less than or equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, to generate such circuits, one can easily add strict delay constraints or key constraints as described in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We note that a similar 0 5 10 15 20 25 30 35 40 45 c2670 c3540 c5315 c6288 c7552 Number of netlists Circuit slack ≤ 0 slack > 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Classification of resynthesized netlists based on their slack values on promising solutions of SCOPE attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' result was also observed on resynthesized netlists locked by other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 6) Structural Analysis: In order to improve the performance of the resynthesis process, the logic cone, which is the locking technique is applied on, can be extracted and resynthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that the output of this logic cone is a single primary output, while its inputs are primary inputs, but not necessarily all the primary inputs of the locked design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, the run- time of the resynthesis process can be decreased, since the logic cone has a small number of inputs, outputs, and gates when compared to the whole locked circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Table IV presents details on the resynthesis process on entire locked circuits and logic cones when the circuits locked by SFLL are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe that the resynthesis process on a logic cone generates less number of unique designs and takes significantly less time without a significant loss on the solution TABLE IV RESULTS OF THE RESYNTHESIS PROCESS ON ENTIRE CIRCUIT AND LOGIC CONE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Circuit Entire Circuit Logic Cone #unique time cdk/dk #unique time cdk/dk c2670 468 13h13m23s 34/64 319 07h46m26s 34/64 c3540 484 1d47m51s 19/32 320 06h29m35s 16/32 c5315 477 21h57m14s 33/64 313 07h06m16s 32/64 c6288 523 2d22h15m7s 16/31 302 06h20m57s 19/32 c7552 504 22h40m29s 38/63 279 06h57m14s 38/63 TABLE V DETAILS OF THE LOCKED CSAW’19 CIRCUITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Circuit Details Number of keys #in #out #gates RLL SFLL-rem Total small 522 512 15995 40 40 80 medium 767 757 24008 60 60 120 large 1452 1445 36584 80 80 160 bonus 892 1746 23004 128 128 256 quality when compared to the resynthesis process on the entire circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We note that similar results were also observed on circuits locked by other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Results on the CSAW’19 Circuits As the second experiment set, we used the state-of-the- art locked circuits from the CSAW’19 contest [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Details of these circuits are given in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that two logic locking techniques – RLL [9] and SFLL-rem [13] – are applied together to lock a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In the following two subsections, we present the results of the resynthesis process and the resynthesis-based attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 1) Resynthesis of the Locked CSAW’19 Circuits: Table VI presents the resynthesis results of locked circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe that the number of unique resynthesized netlists is larger than half of the total number of generated netlists, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', 540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' As the hardware complexity of designs increases, the run-time of the resynthesis process increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' We note that diverse netlists in terms of complexity are obtained, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=', the standard deviation on area, delay, and power dissipation values of all the locked netlists of the small circuit is computed as 8526, 1029, and 20074, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' 2) Attacks on the Locked CSAW’19 Circuits: Table VII presents results of the attacks obtained, after they are applied to the original locked netlist, denoted as OLN, and all unique resynthesized netlists, denoted as URNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' In this table, prv stands for the number of proven values of key bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that since the secret key is not publicly available, the cdk values are omitted for the SCOPE and resynthesis-based attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe from Table VII that the original SCOPE attack could only decipher a small number of key bits, all of which belongs to RLL, and the query attack can prove the values of a large number of key bits, all of which again belong to RLL, on the original locked circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, the resynthesis-based attack could only decipher the RLL key bits on the original locked circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' However, the use of resynthesized circuits makes the SCOPE attack decipher more key bits that also belong to SFLL-rem and makes the query attack prove the values of TABLE VI RESULTS OF RESYNTHESIZED LOCKED CSAW’19 CIRCUITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Circuit unique area delay power time small 557 18935 1631 23571 5d3h22m28s medium 569 26080 1745 31284 6d12h24m16s large 567 31348 1798 24610 5d21h42m10s bonus 560 20643 1758 19090 4d14h44m29s TABLE VII RESULTS OF ATTACKS ON THE LOCKED CSAW’19 CIRCUITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Circuit-Netlist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SCOPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Query ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Resynthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='dk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='prv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} 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+page_content='4m48s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='bonus - URNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='233 16h7m17s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='1251d20h29m22s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='252 2d12h36m39s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='more key bits that belong to RLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Thus, the resynthesis-based attack could decipher almost all the values of the secret key, proving almost all the values of the key bits of RLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Note that all the unknown key bits belong to SFLL-rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe that the run-time of attacks increases, as the number of gates and key bits increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' After the values of key bits of the CSAW’19 circuits were determined, they were sent to the contest organizers for eval- uation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Table VIII presents the results of the resynthesis-based attack along with those of other techniques which participated in the contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Observe from Table VIII that our proposed attack can determine all the key bits of RLL correctly, even though there are unproven key bits in the medium and bonus circuits as shown in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' This observation implies that the guesses of the SCOPE attack on those key bits are actually correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Moreover, the proposed technique can decipher the key bits of SFLL-rem with a number of deciphered key bits greater than any other OL technique with a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' CONCLUSIONS This work has shown that EDA tools can be used to generate variants of locked circuits that may be vulnerable to existing logic locking attacks and such circuits can be generated using a specific set of synthesis parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' It was shown that the run-time of the proposed technique can be improved using a small number of resynthesized netlists without diminishing its solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Experimental results clearly indicated that the use of many resynthesized circuits enables existing attacks to decipher values of a large number of key bits with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Hence, the resynthesis of a locked circuit can be utilized as a pre-attack step for many existing attacks in order to improve their success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' As future work, we plan to consider other synthesis parameters, such as fanout, capaci- tance limits, and wire loads, which enable synthesis tools to generate different circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Also, we aim to incorporate other TABLE VIII RESULTS OF ATTACKS ON THE LOCKED CSAW’19 CIRCUITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Approach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Attack Scenario ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Circuit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='small (40+40) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='medium (60+60) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='large (80+80) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='bonus (128+128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='RLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SFLL-rem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='RLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SFLL-rem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='RLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SFLL-rem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='RLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='SFLL-rem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Key sensitization [34] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='OG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='40/40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='60/60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='80/80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Hamming distance-based attack [24] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='OG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='30/30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='50/50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='72/72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Automated analysis + SAT [24] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='OG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='11/18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='31/50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='10/34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Sub-circuit SAT [24] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='OG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='17/17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='29/29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Redundancy-based [27] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='OL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='28/28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='4/12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='35/35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='23/28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='45/45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='0/51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='66/66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='8/27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Bit-flipping attack [35] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='OG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='40/40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='60/60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='80/80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Topology guided attack [28] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='OL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='15/32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='19/50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='36/73 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='75/108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='Resynthesis-based attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='OG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='40/40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='20/39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='60/60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='29/60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='80/80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='35/79 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='128/128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='55/124 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='commercial and open source EDA tools into the resynthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content='process to generate different unique netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors thank Nimisha Limaye for evaluating the keys found by the proposed technique on the CSAW’19 bench- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Rostami, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Koushanfar, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQfPQkf/content/2301.04400v1.pdf'} +page_content=' Karri, “A Primer on Hardware Security: Models, Methods, and Metrics,” Proceedings of the IEEE, vol.' metadata={'source': 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0000000000000000000000000000000000000000..5695fbd9f99b8e754c2834e4cc5272735c55a6a4 --- /dev/null +++ b/zNE4T4oBgHgl3EQfZAwg/content/tmp_files/2301.05052v1.pdf.txt @@ -0,0 +1,1659 @@ +arXiv:2301.05052v1 [math.AG] 12 Jan 2023 +MATRIX FACTORIZATION FOR QUASI-HOMOGENEOUS +SINGULARITIES +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +Abstract. Given an isolated, quasi-homogeneous singularity X we prove that there is a group +isomorphism between the group of rank one reflexive sheaves on X and the free abelian group +generated by C∗-divisors, modulo linear equivalence. When dim(X) = 2 we reduce the problem +of finding matrix factorizations of arbitrary reflexive OX-modules to the same question on rank +one reflexive sheaves. We then enumerate the matrix factorizations of all rank one reflexive +sheaves. As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules. +1. Introduction +Let X ⊂ Cn be an integral, normal hypersurface defined by an equation F ∈ C[[X1, . . . , Xn]]. +Recall, matrix factorizations of F are pairs of square matrices (M1, M2) of the same rank such +that the products M1.M2 and M2.M1 equals F times an identity matrix. Eisenbud [9] showed +that there is a one-to-one correspondence between (reduced) matrix factorizations of F and max- +imal Cohen-Macaulay OX-modules without free direct summands. Matrix factorization plays a +central role in singularity theory. Using matrix factorization, Kn¨orrer [17] and Buchweitz-Greuel- +Schreyer [6] proved that isolated hypersurface singularities of finite Cohen-Macaulay represen- +tation type are exactly the simple ones. In the early 2000s, Kapustin [16], and Orlov [20–22] +showed that matrix factorizations can be applied to study Landau-Ginzburg models appearing +in string theory, and to the study of Kontsevich’s homological mirror symmetry. In particu- +lar, by the work of Orlov there exists an equivalence between the bounded derived category +Db(X) and the homotopy category of matrix factorizations of F. In general, the first category +is hard to compute. Thus, producing concrete families of matrix factorizations can be one way +of understanding Db(X). +Unfortunately, there are no “good” algorithms to obtain matrix factorizations. As a result +concrete examples of matrix factorizations are rather limited in the literature. For example, +Buchweitz, Eisenbud and Herzog [5] proved that for Fn(X1, . . . , Xn) = X2 +1 +· · ·+Xn +n with n ≥ 8 +the smallest size of a matrix factorization is bounded below by 2 +n−2 +2 +× 2 +n−2 +2 . In particular for +F16 the smallest matrix factorization is of size 128 × 128. Crisler and Diveris [8] produced an +algorithm to produce matrix factorization for the polynomial Fn only for n ≤ 8. By studying the +polynomial F16 they notice that their algorithm fails and it is impossible to fix it. Laza, Pfister +and Popescu [18] computed all the matrix factorization associated to rank one reflexive sheaves +over the surface defined by F3. Baciu [3] computed all the matrix factorizations associated to +rank two graded Ulrich modules on the hypersurface defined by X3 +1 +X2 +1X3−X2X3. Etingof and +Ginzburg [11] produced a family of matrix factorizations for the family of hypersurfaces gives +by the polynomial X3 +1 + X3 +2 + X3 +3 + τX1X2X3 as τ varies over non-zero complex numbers. Ros +Camacho and Newton [25,26] computed concrete matrix factorizations for exceptional unimodal +Date: January 13, 2023. +2020 Mathematics Subject Classification. Primary: 13C14, 14J17, 32S25, 14E16. +Key words and phrases. Matrix factorization, Maximal Cohen-Macaulay modules, Quasi-homogeneous singu- +larities, McKay correspondence, C∗-curves, cusp singularities. +1 + +2 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +hypersurface singularities. The goal of this article is to generalize some of these results to any +isolated, quasi-homogeneous hypersurface singularity (upto topologically trivial deformations). +Let (X, x) be an isolated, quasi-homogeneous hypersurface singularity of dimension 2. This +means that there exist integers (ω1, ω2, ω3, d) such that the defining equation F satisfies: +F(λω1X1, λω2X2, λω3X3) = λdF(X1, X2, X3), for all λ ∈ C∗. +The integers ω1, ω2, ω3 are called the weights of the hypersurface. Note first that every maximal +Cohen-Macaulay module M on X sits in an exact sequence with 4 terms. +Besides M the +remaining three terms are a trivial bundle, a skyscraper sheaf supported on the singular point +x and a rank one reflexive sheaf L, which we will call the determinant of M (Theorem 3.5). +Projective resolutions of skyscraper sheaves are well-understood. Moreover, to obtain projective +resolutions of short exact sequences, one simply needs to determine the projective resolution of +two of the three terms (satisfying the obvious compatibility conditions). As a result, finding +the matrix factorization corresponding to M reduces to determining the matrix factorization +corresponding to its determinant L. We first classify all such rank one reflexive sheaves. Denote +by Ref(1)(X) the group of all reflexive rank one sheaves on X (see §3.4 for the group structure) +and by D(X) the free abelian group generated by classes of C∗-curves (i.e., curves that are +invariant under the natural C∗-action on X, see §3.2), modulo linear equivalence. We prove: +Theorem 1.1. Any integral curve D in X is either a C∗-curve or is CI-linked (see Definition +3.2) to a C∗-curve. Moreover, there is an isomorphism of abelian groups: +D(X) → Ref(1)(X) sending D ∈ D(X) to i∗ OX∗(D ∩ X∗), +(1.1) +where X∗ := X\{x} is the regular locus in X and i : X∗ → X is the open immersion. +See Theorems 3.3 and 3.4 for a more general statement that holds in any dimension of X. This +can be viewed as a McKay-type correspondence where the left hand side of the correspondence +(1.1) parameterizes geometric objects namely C∗-divisors and the right hand side parameterizes +algebraic objects namely rank one reflexive sheaves. +In arbitrary rank, there is a 1 − 1 correspondence between maximal Cohen-Macaulay OX- +modules and rank one Cohen-Macaulay OX-modules supported on divisors (see Proposition 2.3). +This correspondence associates to a rank r maximal Cohen-Macaulay OX-modules M along with +a general choice of r sections, its degeneracy module. The advantage of this correspondence is +that one can obtain the matrix factorization of M from a projective resolution of the associated +degeneracy module (Theorem 4.1). The latter is an easier problem. We use this idea in the +proof of Theorem 1.2 below. +By Theorem 1.1 above, rank one maximal Cohen-Macaulay modules are generated (via ten- +sor product) by those arising from integral C∗-curves. As a result, maximal Cohen-Macaulay +modules associated to non-singular C∗-curves are of particular interest. We call such modules +generalized Wunram modules (see §4.2). We give an explicit description of the matrix factoriza- +tion corresponding to rank one generalized Wunram modules in Theorem 1.2 below. Note that, +X contains a non-singular C∗-curve if and only if (upto reparametrization) one of the weights of +X is one. Recall, Orlik and Wagreich [19] and Arnold [1] classified isolated quasi-homogeneous +surface singularities, upto topologically trivial deformations (see table in §4.4). Correspond- +ing to the types of singularities mentioned in this table we derive the following list of matrix +factorizations: + +MATRIX FACTORIZATION +3 +Theorem 1.2. Let X be a quasi-homogeneous singularity of weight (1, ω2, ω3) listed in Table +1 in §4.4 below. Given positive integers n, m and complex numbers c1, c2, denote by: +S(c1,c2,n,m)(Z1, Z2) := +m +� +j=1 +Z(j−1)n +1 +Zm−j +2 +cj−1 +1 +cjn +2 +Then, the matrix factorization associated to any rank one generalized Wunram module on X is +a pair of 2 × 2 matrices (adj(A), A) where A is a matrix of the form A := (mi,j) for i, j ∈ {1, 2} +with +m1,1 = Xω2 +1 b − X2aω2, m1,2 = X3aω3 − Xω3 +1 c and +the entries m2,1, m2,2 are given by the following table where the first column enumerates the +various singularity types from Table 1: +Type +m2,1 +m2,2 +Ip,q,r +S(c,a,ω3,r)(X1, X3) +S(b,a,ω2,q)(X1, X2) +IIp,q,r +bXω2 +1 +aω2 S(c,a,ω3,r)(X1, X3) +S(b,a,ω2,q)(X1, X2)+ ++Xr +3S(b,a,ω2,1)(X1, X2) +IIIp,q,r +bXω2 +1 +aω2 S(c,a,ω3,r)(X1, X3) ++ bqXqω2 +1 +aqω2 S(c,a,ω3,1)(X1, X3) +X3S(b,a,ω2,q)(X1, X2)+ ++Xr +3S(b,a,ω2,1)(X1, X2) +IVp,q,r +XS(c,a,ω3,r)(X1, X3) + bqXqω2 +1 +aqω2 S(c,a,ω3,1)(X1, X3) +X3S(b,a,ω2,q)(X1, X2) +Vp,q,r +XS(c,a,ω3,r)(X1, X3) + bqXqω2 +1 +aqω2 S(c,a,ω3,1)(X1, X3) +X3S(b,a,ω2,q)(X1, X2)+ ++Xp +1S(b,a,ω2,1)(X1, X2) +VIp,q,r,b2,b3 +XS(c,a,ω3,r)(X1, X3) ++ bb2 Xb2ω2 +1 +ab2ω2 +S(c,a,ω3,b3)(X1, X3) +XS(b,a,ω2,q)(X1, X2)+ ++Xb3 +3 S(b,a,ω2,b2)(X1, X2) +VIIp,q,r,b2,b3 +XS(c,a,ω3,r)(X1, X3) ++ bb2 Xb2ω2 +1 +ab2ω2 +S(c,a,ω3,b3)(X1, X3) +XS(b,a,ω2,q)(X1, X2)+ ++Xb3 +3 S(b,a,ω2,b2)(X1, X2)+ ++Xp +1S(b,a,ω2,1)(X1, X2) +where (a, b, c) varies over all points in X with a ̸= 0. +This result will be proved in §4.5. The remaining case a = 0 is treated in Remark 4.5. Note +that this also gives explicit families of matrix factorizations parameterized by points on X. +Our computation recovers the matrix factorizations obtained by Laza, Pfister and Popescu [18]. +As a consequence of Theorem 1.2 above we prove special cases of a conjecture of Etingof and +Ginzburg [11, Conjecture 3.6.8]: +Conjecture. Let F be the free tensor algebra with basis X1, X2, X3, Φ ∈ F/[F, F], A(Φ) := +F/⟨⟨∂iΦ⟩⟩i=1,2,3 and for a central element Ψ not a zero divisor in A(Φ) denote by B(Φ, Ψ) := +A(Φ)/⟨⟨Ψ⟩⟩. To any point module P (see [2, Definition 3.8]) over the algebra B(Φ, Ψ) one can +naturally associate a matrix factorization M(P) = (M+, M−). +Using Theorem 1.2 we prove: +Theorem 1.3. Take Φ := X1X2X3 − X2X1X3. +Then, for suitable choices of Ψ the above +conjecture holds true i.e., to any point module P over the algebra B(Φ, Ψ) one can naturally +associate a matrix factorization. +See Theorem 4.6 for the precise statement. Note that, the choices of Ψ in the above theorem +will correspond to quasi-homogeneous polynomials. +In Section 5.1 we study the case of cusp singularities. In the workshop of Singularities at Ober- +wolfach 2021, Prof. Duco van Straten asked a question to the second author on the construction +of matrix factorizations for cusp singularities. We obtain a partial answer to his question, in + +4 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +Theorem 5.1. In particular, we produce families of matrix factorizations for families of cusp +singularities. By fixing some numbers, this theorem also recovers the cubic studied by Etingof +and Ginzburg [11]. In Section 5.2 we study matrix factorization of non-isolated singularities and +generalize a result of Baciu [3]. +2. Preliminaries +In this section, we recall the notion of matrix factorization of hypersurface singularities. We +observe how this relates to the space of maximal Cohen-Macaulay modules (Theorem 2.1). +Finally, we recall basics on degeneracy modules (Proposition 2.3). This gives us a new approach +to studying matrix factorizations, which will be used in later sections for explicit computations. +2.1. Setup. Fix an integer n ≥ 3. Let X be an integral, normal hypersurface in Cn. Denote +by OCn := C[[X1, ..., Xn]] and F ∈ OCn defining the hypersurface X and OX := OCn /(F) the +associated coordinate ring. Note that X may have non-isolated singularities. +2.2. Matrix factorization. A matrix factorization of F is an ordered pair of m × m-matrices +(Φ, Ψ) with entries in OCn such that the matrix multiplication satisfies: +Φ · Ψ = F · Idm, +Ψ · Φ = F · Idm, +where Idm is the m × m identity matrix. The matrix factorization is reduced if and only if +Im(Φ : O⊕m +Cn → O⊕m +Cn ) ⊂ m O⊕m +Cn +and +Im(Ψ : O⊕m +Cn → O⊕m +Cn ) ⊂ m O⊕m +Cn , +where m is the maximal ideal of OCn. Recall, the following classical result on matrix factorization: +Theorem 2.1. There is a one-to-one correspondence between: +(1) equivalence classes of reduced matrix factorizations of F. +(2) isomorphism classes of non-trivial periodic minimal free resolutions of OX-modules of +periodicity two. +(3) maximal Cohen-Macaulay OX-modules without free summands. +Proof. See [9, Corollary 6.3] for a proof. +□ +In this article, we will exploit the equivalence between (1) and (3) in Theorem 2.1. So, we +briefly recall how one associates a matrix factorization of F to a maximal Cohen-Macaulay +module without free summands. Let M be a maximal Cohen-Macaulay OX-module without +free summands. This implies that the depth of M equals the dimension of X, which is n − 1. +By the Auslander-Buchsbaum formula, this means as an OCn-module, the projective dimension +of M equals 1. This implies we have a short exact sequence of the form +0 → O⊕b +Cn +Φ−→ O⊕a +Cn +(m1,...,ma) +−−−−−−−→ M → 0 +(2.1) +where mi ∈ M and the standard basis element ei ∈ O⊕a +Cn maps to mi for 1 ≤ i ≤ a. Since M +is supported on X, we have a = b. Then the morphism Φ is simply given by an a × a-matrix +with entries in OCn. Suppose that this is a minimal resolution of M. Since M is annihilated by +F, for every 1 ≤ i ≤ a, Fei ∈ O⊕a +Cn maps to zero in M. By the exactness of (2.1), there exists +Ψ(ei) ∈ O⊕a +Cn such that Φ ◦ Ψ(ei) = Fei. In other words, there exists an a × a-matrix Ψ with +entries in OCn such that Φ · Ψ = F · Ida. Therefore, (Ψ, Φ) is a matrix factorization of F. + +MATRIX FACTORIZATION +5 +2.3. Degeneracy module. Let M be a maximal Cohen-Macaulay OX-module of rank, say r. +Given an r-tuple of sections s := (s1, ..., sr) of M, the associated degeneracy locus is the zero +locus of the section s1 ∧ s2 ∧ ... ∧ sr ∈ ∧rM i.e., the locus of points where the r-tuple of sections +is linearly dependant. Consider the morphism +s : O⊕r +X → M, +sending a standard basis vector ei of O⊕r +X to si. Denote by Cs the cokernel of the morphism s. +Note that, the support of Cs is the associated degeneracy locus. For a general choice of r-sections +s := (s1, ..., sr) the associated degeneracy locus Supp(Cs) is reduced and Cohen-Macaulay of +codimension 1 (see [10, p. 431]). Furthermore, by the genericity of the r-tuple of sections, the +locus where r − 1 of the r-sections are linearly dependant is of codimension 2 (see [10, Lemma +5.2]). This implies that the cokernel Cs is supported on a reduced Cohen-Macaulay subvariety +of codimension 1 in X and is of rank 1 over its support. The cokernel Cs will be called the +degeneracy module associated to the r-tuple of sections s := (s1, ..., sr). +This motivates the +following definition: +Definition 2.2. We will call an r-tuple of sections s := (s1, ..., sr) of M weakly general if the +cokernel Cs of the induced morphism s is supported on a reduced Cohen-Macaulay subvariety +of X of codimension 1 and is a rank 1, Cohen-Macaulay OX-module over Supp(Cs). +2.4. Dualizing degeneracy modules. Let M be a maximal Cohen-Macaulay OX-module of +rank r and s := (s1, ..., sr) be an r-tuple of weakly general sections of M. By definition, we have +a short exact sequence of the form: +0 → O⊕r +X +s−→ M → Cs → 0, +(2.2) +for some Cohen-Macaulay OX-module Cs supported on a reduced Cohen-Macaulay subvariety +in X and is of rank 1 over its support. Dualizing this exact sequence, we get +0 → M∨ → O⊕r +X +s′ +−→ Ext1 +X(Cs, OX) → 0, +(2.3) +where the surjectivity on the right follows from the vanishing of Ext1 +X(M, OX) (see [4, Theorem +3.3.10]). Throughout this article, we shall denote As := Ext1 +X(Cs, OX). Note that, dualizing +(2.3) and using Ext1 +X(As, OX) ∼= Cs and M∨∨ ∼= M (see [4, Theorem 3.3.10]), we get back the +exact sequence (2.2). This implies: +Proposition 2.3. There is a 1 − 1 correspondence between pairs: + + + + + + + +(M, s) where M is a MCM +OX -module of rank r and +s := (s1, ..., sr) is an r-tuple +of weakly general sections of M + + + + + + + +←→ + + + + + + + +(As, s′) where As is a CM OX -module +supported on a CM subvariety of +codimension one in X and of rank 1 +over the support and generated by s′ + + + + + + + +where the bijection follows from (2.2) and (2.3). +Proof. See [12] for a detailed proof. +□ +Definition 2.4. Given a pair (M, s) with M a maximal Cohen-Macaulay OX-module of rank +r and s an r-tuple of weakly general sections of M, we will call the corresponding pair (As, s′) +as in Proposition 2.3, the degenerate pair associated to (M, s). +3. McKay-type correspondence for quasi-homogeneous singularities +Quasi-homogeneous hypersurface singularities are generalizations of homogeneous singulari- +ties. We study C∗-divisors contained in such hypersurfaces. We observe that every effective, +integral divisor is either a C∗-divisor or is CI-linked (in the sense of Definition 3.2) to a C∗-divisor + +6 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +(Theorem 3.3). Using this we observe that there is a 1 − 1 correspondence between C∗-divisors +(modulo linear equivalence) and rank one reflexive sheaves on a quasi-homogeneous hypersurface +(Theorem 3.4). Furthermore, if the dimension of the hypersurface is two, then we can express +every maximal Cohen-Macaulay modules solely in terms of the ideal sheaves of C∗-curves and +certain skyscraper sheaves (Theorem 3.5). +3.1. Quasi-homogeneous hypersurfaces. A polynomial F ∈ C[[X1, X2, ..., Xn]] is called +quasi-homogeneous if there exists positive integers (ω1, ω2, ..., ωn, d) such that for any λ ∈ C∗, +we have F(λX1, λX2, ..., λXn) = λdF(X1, ...., Xn). The hypersurface X defined by F is called a +quasi-homogeneous hypersurface with weights ω := (ω1, ω2, ..., ωn). Note that, there is a natural +C∗-action on X: +C∗ × X → X sending (λ, (x1, ..., xn)) �→ (λω1x1, λω2x2, ..., λωnxn). +Throughout this section we assume that X has only isolated singularity at the origin 0. Denote +by Pω +X∗ the quotient of X∗ := X\{0} by the C∗-action. Consider the resulting quotient map: +πX : X∗ → Pω +X∗. +(3.1) +3.2. C∗-divisors. Let X be a quasi-homogeneous hypersurface of dimension n with weights +ω := (ω1, ω2, ..., ωn, d). +Note that, given a closed point (a1, a2, ..., an) ∈ X, the associated +C∗-curve is the parametric curve given by: +n : C∗ → X sending λ �→ (λω1a1, λω2a2, ...λωnan). +We will denote by [a1, a2, ..., an] the corresponding point on Pω +X∗. Clearly, the fiber over [a1, ..., an] +to the morphism πX is an integral curve and n is the normalization map for the fiber. This +implies that the preimage under πX of an integral divisor in Pω +X∗ is irreducible. +An integral divisor D in X∗ is called a C∗-divisor if there exists an integral Weil divisor D′ +in Pω +X∗ such that D ∼= π−1 +X (D′)red, where πX is as in (3.1). An integral divisor in X is called +C∗-divisor if it is the closure of an integral C∗-divisor on X∗. Denote by D(X) the free abelian +group generated by integral C∗-divisors in X, modulo linear equivalence. Elements of D(X) will +be called C∗-divisors on X. +3.3. Liaisons and residual divisors. Let (X, 0) be an isolated, quasi-homogeneous hypersur- +face singularity with weights ω := (ω1, ω2, ...., ωn, d). Consider the quotient map πX as in (3.1) +from the regular locus of X to quotient by the C∗-action. +Definition 3.1. An integral divisor D ⊂ X is called horizontal if the composition +D\{0} ⊂ X\{0} +πX +−−→ Pω +X∗ +is dominant. +Definition 3.2. Two distinct divisors D, E are called CI-linked if there exists a polynomial +g ∈ C[[X1, X2, ..., Xn]] such that D ∪ E = Z(g) ∩ X, where Z(g) denotes the zero locus of g. +Moreover, if D and E are CI-linked then we call D residual to E (and vice versa, E is residual +to D). This terminology is inspired by the classical theory of liaisons (see [23]). +Theorem 3.3. Let D ⊂ X be an integral horizontal divisor. Then, there exist a C∗-divisor +E ⊂ X such that D is CI-linked to E. +Proof. Consider the quotient map πX from X∗ to Pω +X∗ as in (3.1). By the theorem on generic +smoothness, there exists an open dense affine subscheme U ⊂ Pω +X∗ such that the resulting +morphism from π−1 +X (U) to U is smooth. Since πX is an affine morphism and U is affine, we have +π−1 +X (U) is affine and non-singular. This implies Pic(π−1 +X (U)) = 0. As D is an integral horizontal + +MATRIX FACTORIZATION +7 +divisor, UD := D ∩ π−1 +X (D) is a non-empty Cartier divisor in π−1 +X (U). Since Pic(π−1 +X (U)) = 0, +the ideal sheaf IUD is simply f. Oπ−1 +X (U) for some f ∈ Oπ−1 +X (U). By [14, Lemma II.5.3], there +exists a regular function �f ∈ OX such that Z( �f) ∩ π−1 +X (U) = Z(f) ∩ π−1 +X (U). This implies that +the zero locus Z( �f) of �f is of the form +Z( �f) = Z(f) ∪ E +(3.2) +where Z(f) is the closure in X of the zero locus of f and E is a divisor lying in the complement +X\π−1 +X (U). Since E is a divisor and does not intersect π−1 +X (U), the scheme-theoretic image +πX(E) of E in Pω +X∗ does not intersect U. Since the fibers of πX are irreducible and of dimension +one, we conclude by the fiber dimension theorem that E ∼= π−1 +X (E′) for some divisor E′ in Pω +X∗. +In particular, E is a C∗-divisor. Moreover, as D is integral and agrees with Z(f) over π−1 +X (U), +we have Z(f) = D. By (3.2), this means D is CI-linked to a C∗-divisor E. This proves the +theorem. +□ +3.4. Rank one correspondence. Denote by Ref(1)(X) the space of reflexive rank one sheaves +on X. Let i : X∗ → X be the natural inclusion. Recall, every reflexive sheaf of rank one on +a regular variety is invertible (see [15, Proposition 1.9]). Moreover, every reflexive sheaf on X +arises as the pushforward via i of a reflexive sheaf on X∗ (see [15, Proposition 1.6]). This means +that under pushforward by i, +i∗ : Pic(X∗) → Ref(1)(X) sending L to i∗L +is an isomorphism. The group operation on Pic(X∗) induces one on Ref(1)(X), namely +M.N := i∗(i∗M ⊗OX∗ i∗N) and M∨ := i∗((i∗M)∨). +Theorem 3.4. The morphism +φ : D(X) → Ref(1)(X) +sending a C∗-divisor D to the reflexive sheaf i∗(OX∗(D ∩ X∗)) is an isomorphism of abelian +groups. +Proof. Clearly, this is a group homomorphism. Moreover, as U is integral and regular, D(X) is +contained in Pic(U). Since Ref(1)(X) is isomorphic to Pic(U) as argued above, this means the +morphism φ is injective. So it remains to check that φ is surjective. +Consider M ∈ Ref(1)(X). Note that, the restriction M|X∗ is a reflexive sheaf. Since X∗ is +regular, this implies M|X∗ is an invertible sheaf. In other words, +M|X∗ ∼= OX∗(D∗) +for some divisor D∗ on X∗. Write D∗ = � +i aiDi as a linear combination of integral divisors +Di. If Di is not horizontal, then by the fiber dimension theorem the scheme theoretic image +Ei of πX|Di is a divisor in Pω +X∗. Since Di is integral, Ei is irreducible. This implies π−1 +X (Ei) +is irreducible (see §3.2). Hence, Di = π−1 +X (Ei)red. In other words, Di is a C∗-divisor. If Di is +horizontal, then by Theorem 3.3 there exists a C∗-curve Dc +i such that Di is linearly equivalent +to −Dc +i. Therefore, D∗ is linearly equivalent to a divisor obtained as a linear combination of +C∗-divisors. This proves surjectivity of φ and hence the theorem. +□ +3.5. Dimension two case. Let (X, x) be an isolated, quasi-homogeneous hypersurface singu- +larity. Suppose that dim X = 2. Denote by kx the skyscraper sheaf over the singular point x of +a one dimensional vector space. + +8 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +Theorem 3.5. Let M be a maximal Cohen-Macaulay OX-module of rank, say r. Then, for a +general choice of r sections (s1, ..., sr) of M, we have an exact sequence of the form +0 → O⊕r−1 +X +(s1,...,sr−1) +−−−−−−−→ M → L → k⊕m +x +→ 0 +(3.3) +for some non-negative integer m and L is a reflexive sheaf on X of rank 1 i.e., L ∈ Ref(1)(X). +In particular, if C denotes the support of the cokernel of the morphism (s1, ..., sr), then L is the +dual of the ideal sheaf of C in X. +Proof. Denote by A the cokernel of the morphism +(s1, ..., sr) : O⊕r +X → M. +(3.4) +Note that, A is a Cohen-Macaulay module supported in dimension 1 and of rank one over its +support. Denote by C the support of A and A′ := Ext1 +X(A, OX). Dualizing (3.4), we then have +the following diagram of short exact sequences: +0 +✲ IC|X +✲ OX +✲ OC +✲ 0 +0 +✲ M∨ +❄ +✲ O⊕r +X +p1 +❄ (t1,...,tr)✲ A′ +p2 +❄ +✲ 0 +where the morphism p2 sends 1 to t1 and p1 sends 1 to the standard basis element e1 ∈ O⊕r +X . +Then, the cokernel of p1 is isomorphic O⊕r−1 +X +. Since A′ is Cohen-Macaulay, the morphism p2 +is injective (the section t1 is torsion-free over C). By Bertini-type theorem (see [24, p. 434]), +C\{x} is non-singular. Since any torsion-free sheaf on an affine non-singular curve is trivial, we +conclude A′ is isomorphic to OC over X∗. Taking t1 = 1 ∈ Γ(OC), we observe that the cokernel +of p2 is of the form k⊕m +x +for some non-negative integer m. Using Snake lemma, we get the exact +sequence: +0 → IC|X → M∨ → O⊕r−1 +X +→ k⊕m +x +→ 0 +Dualizing this sequence and applying [4, Theorem 3.3.10], gives us the exact sequence (3.3). +This proves the theorem. +□ +4. Matrix factorization using degeneracy modules +Matrix factorization of maximal Cohen-Macaulay modules is hard. However, one can instead +study resolutions of the associated degeneracy modules. This is a slightly easier problem. We +obtain matrix factorizations using this idea (see Theorem 4.1 and Corollary 4.3). We then apply +this to enumerate the matrix factorization of all Cohen-Macaulay modules arising from C∗-curves +in quasi-homogeneous surfaces (see Theorem 1.2 stated in the introduction and proved in §4.5). +4.1. Matrix factorization via degeneracy modules. Let M be a maximal Cohen-Macaulay +OX-module of rank r with no free direct summand (i.e., does not contain OX as a direct +summand). Let s be an r-tuple of weakly general sections of M. Let (As, s′) be the associated +degenerate pair. Since As is a Cohen-Macaulay OX-module supported in a dimension n − 2 +subvariety in Cn, the depth of As is n − 2. By the Auslander-Buchsbaum formula this implies +the projective dimension of As is 2. Then starting with s′ the pair induces an exact sequence of +the form: +0 → O⊕a +Cn +A +−→ O⊕b +Cn +B +−→ O⊕r +Cn +s′ +−→ As → 0 +(4.1) + +MATRIX FACTORIZATION +9 +where A (resp. B) is induced by a b × a (resp. r × b) matrix with entries in OCn, which we will +also denote by A (resp. B) for simplicity of notation. In particular, +Ae(a) +i += +b +� +j=1 +ajie(b) +j +and Be(b) +i += +r +� +j=1 +bjie(r) +j , +where {e(t) +i }t +i=1 is the standard basis of the free OCn-module O⊕t +Cn for t ∈ {r, a, b}. We show: +Theorem 4.1. Denote by K the OCn-submodule of O⊕b +Cn consisting of all m ∈ O⊕b +Cn such that +Bm ∈ I⊕r +X . Then, +(1) K is isomorphic to O⊕r +Cn, as OCn-modules, +(2) fix an isomorphism as in (1) from O⊕r +Cn to K given by a b × r-matrix +A′ : O⊕r +Cn +∼ +−→ K ⊂ O⊕b +Cn . +Then, (upto change of basis of O⊕r +Cn) the composition +O⊕r +Cn +A′ +−→ O⊕b +Cn +B +−→ O⊕r +Cn coincides with FIdr×r : O⊕r +Cn → O⊕r +Cn, +where F ∈ OCn defines X, +(3) the matrix factorization associated to M is of the form +� +adj(A|A′)T , (A|A′)T � +, where +(−)T denotes transpose of the matrix and adj(−) denotes the adjoint of the matrix. +Before we prove the theorem, note that by the exact sequence (4.1), we have b = r + a (the +support of As is of codimension 2 in Cn). Then, the matrix (A|A′) is a b × b-matrix. +Proof. Comparing the exact sequences (2.3) and (4.1), we get the following diagram of exact +sequences: +0 +✲ O⊕a +Cn +A ✲ O⊕b +Cn +B ✲ O⊕r +Cn +s′ +✲ As +✲ 0 +⟲ +⟲ +0 +✲ M∨ +ρ′ +❄ +✲ O⊕r +X +ρ +❄ +s′ +✲ As +id +❄ +✲ 0 +where the vertical morphism ρ is the natural restriction morphism and the first vertical morphism +ρ′ is induced by the universal property of kernel. Since the last two vertical arrows are surjective +then by a simple diagram chase (using the injectivity of the morphism from M∨ to O⊕r +X ) we +conclude that morphism ρ′ from O⊕b +Cn to M∨ is surjective. Note that, ρ sits in the short exact +sequence: +0 → O⊕r +Cn +F Idr×r +−−−−→ O⊕r +Cn +ρ−→ O⊕r +X → 0. +Using the Snake lemma applied to the above diagram of exact sequence, this gives us the +following exact sequence: +0 → O⊕a +Cn ⊕ O⊕r +Cn +(A|A′) +−−−−→ O⊕b +Cn +ρ′ +−→ M∨ → 0 +(4.2) +where the composition +O⊕r +Cn +A′ +−→ O⊕b +Cn +B +−→ O⊕r +Cn coincides with FIdr×r : O⊕r +Cn → O⊕r +Cn . +This proves parts (1) and (2) of the theorem (identify K with the image of A′). As mentioned +above b = r + a. Dualizing (4.2), we get the exact sequence: +0 → O⊕b +Cn +(A|A′)T +−−−−−→ O⊕b +Cn → Ext1 +Cn(M∨, OCn) → 0. +(4.3) + +10 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +Since F annihilates M∨ (as M∨ is supported in X), we have by [4, Lemma 1.2.4] +Ext1 +Cn(M∨, OCn) ∼= HomCn(M∨, OX) ∼= HomX(M∨, OX), +where the last isomorphism follows from adjunction of Hom-functor. Since M is a maximal +Cohen-Macaulay OX-module, it is in particular reflexive. Therefore, the double dual M∨∨ of M +is isomorphic to M. Hence, Ext1 +Cn(M∨, OCn) ∼= M and (4.3) gives a projective resolution of M. +In other words, +� +adj(A|A′)T , (A|A′)T � +is a matrix factorization. This proves the theorem. +□ +4.2. Generalized Wunram modules. Following [12], a maximal Cohen-Macaulay OX-module +M of rank 1 is called generalized Wunram if for a general choice of section s of M, the cokernel +of the natural morphism from OX to M, defined by multiplication with s, is isomorphic to OD +for a non-singular subvariety D ⊂ X of codimension 1. +4.3. Projective resolution of the degeneracy module. Let M be a rank one generalized +Wunram module, s ∈ M a general section and D be the associated degeneracy locus. +In +particular, we have a short exact sequence of the form: +0 → OX +.s +−→ M → OD → 0, +(4.4) +where D is a non-singular subvariety in X of codimension 1. Dualizing this exact sequence we +get a short exact sequence of the form: +0 → M∨ → OX → Ext1 +X(OD, OX) → 0 +(4.5) +Note that, Ext1 +X(OD, OX) is a Cohen-Macaulay OX-module supported on D and is of rank one +over its support. Now, a rank one maximal Cohen-Macaulay module over a smooth affine variety +is trivial. Hence, Ext1 +X(OD, OX) ∼= OD. We now produce a projective resolution of OD. Since +D is non-singular there exists f, g ∈ OCn such that the ideal of D (in Cn) is generated by f and +g (regular local rings are complete intersection rings). We then have the Koszul resolution: +Proposition 4.2. The projective resolution of OD is given by +0 → OCn +A +−→ O⊕2 +Cn +B +−→ OCn → OD → 0 +where Ae := −fe1 + ge2, Be1 := g and Be2 = f with e (resp. {e1, e2}) the standard basis of +OCn (resp. O⊕2 +Cn). +Corollary 4.3. Let X be a normal hypersurface singularity (not necessarily isolated) of any +dimension. Let M be a rank one generalized Wunram module, s ∈ M a general section and D +the degeneracy locus associated to the pair (M, s), which is non-singular as M is generalized +Wunram of rank one. Then, the matrix factorization associated to M is the pair (adj(C), C) +where C is the matrix +C := +� +−f +g +h1 +h2 +� +f, g ∈ OCn defines the non-singular variety D in Cn and X is defined by a regular function of +the form F := h1g + h2f (as D ⊂ X we have F ∈ (f, g)). +Proof. Translating into the notations of Theorem 4.1, we have a = 1, b = 2 and r = 1. The +morphisms A and B are defined in Proposition 4.2. We now need to compute K and A′ from +Theorem 4.1. Recall, +K = {a1e1 + a2e2|a1g + a2f ∈ IX} where +IX is the ideal of X in Cn generated by, say F. Of course, since D ⊂ X, there exists h1, h2 ∈ OCn +such that F = h1g + h2f. In other words, h1e1 + h2e2 ∈ K. We claim that K is generated as an + +MATRIX FACTORIZATION +11 +OCn-module by h1e1 + h2e2. Indeed, since K ∼= OCn (Theorem 4.1), it is generated by a single +element, say h′ +1e1 + h′ +2e2 ∈ O⊕2 +Cn. Then, there exists λ ∈ OCn such that +λ(h′ +1e1 + h′ +2e2) = h1e1 + h2e2. +Applying the OCn-linear morphism B, we have +λB(h′ +1e1 + h′ +2e2) = B(λ(h′ +1e1 + h′ +2e2)) = B(h1e1 + h2e2) = F. +(4.6) +Since h′ +1e1 + h′ +2e2 ∈ K, we have B(h′ +1e1 + h′ +2e2) = λ′F for some λ′ ∈ OCn. Substituting in (4.6) +this implies λλ′ = 1 i.e., λ is a unit in OCn. This proves our claim that K is generated as an +OCn-module by h1e1 + h2e2. Then, we can take the morphism +A′ : OCn +∼ +−→ K ⊂ O⊕2 +Cn sending 1 to h1e1 + h2e2. +This satisfies the condition that the composition B ◦ A′ = F × Id. By Theorem 4.1 the matrix +factorization of M is of the form +� +adj(A|A′)T , (A|A′)T � +where +(A|A′) = +� +−f +h1 +g +h2 +� +, so (A|A′)T = +� +−f +g +h1 +h2 +� +This proves the corollary. +□ +4.4. Matrix factorization for topological trivial deformations. Orlik and Wagreich [19] +and Arnold [1] showed that an isolated, quasi-homogeneous surface singularity can be can be +deformed into one of the following seven classes below keeping the link differentially constant +Type +Defining polynomial +Ip,q,r +F(X1, X2, X3) := Xp +1 + Xq +2 + Xr +3 +IIp,q,r +F(X1, X2, X3) := Xp +1 + Xq +2 + X2Xr +3 with q > 1 +IIIp,q,r +F(X1, X2, X3) := Xp +1 + X3Xq +2 + X2Xr +3 with q > 1 and r > 1 +IVp,q,r +F(X1, X2, X3) := Xp +1 + X3Xq +2 + X1Xr +3 with p > 1 +Vp,q,r +F(X1, X2, X3) := X2Xp +1 + X3Xq +2 + X1Xr +3 = 0 +VIp,q,r,b2,b3 +F(X1, X2, X3) := Xp +1 + X1Xq +2 + X1Xr +3 + Xb2 +2 Xb3 +3 with (p − 1)(qb3 + rb2) = pqr +VIIp,q,r,b2,b3 F(X1, X2, X3) := X2Xp +1 + X1Xq +2 + X1Xr +3 + Xb2 +2 Xb3 +3 with (p − 1)(qb3 + rb2) = +r(pq − 1) +Table 1. Quasi-homogeneous singularity types +Xu and Yau [27] proved that the above deformation is in fact a topological trivial deforma- +tion. Furthermore, the topological type of quasi-homogeneous singularities determine and is +determined by its weights. +We now use Corollary 4.3 to produce the matrix factorizations +corresponding to all rank one generalized Wunram modules. +4.5. Proof of Theorem 1.2. Given c1, c2 ∈ C and k, n, m ∈ Z>0 denote by +G(c1,c2,n)(Z1, Z2) := c1Zn +1 − cn +2Z2. +and S(c1,c2,n,m)(Z1, Z2) defined in Theorem 1.2. Note that, +Zk +3 G(c1,c2,n)(Z1, Z2)S(c1,c2,n,m)(Z1, Z2) = Zk +3 +�cm +1 Zmn +1 +cmn +2 +− Zm +2 +� +. +(4.7) +Let X be a quasi-homogeneous surface singularity defined by a quasi-homogeneous polynomial +F(X1, X2, X3) from the list in Table 1 above. By assumption, the weights of X is (1, ω2, ω3). +Take a point (a, b, c) ∈ X with a ̸= 0. The associated C∗-curve, denoted Wa,b,c, is given by the +following parametrization: +n: C∗ → X such that λ �→ (aλ, bλω2, cλω3). + +12 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +Note that, Wa,b,c is the zero locus (in C3) of the polynomials +G(b,a,ω2)(X1, X2) = Xω2 +1 b − X2aω2 +and +G(c,a,ω3)(X1, X3) = Xω3 +1 c − X3aω3. +By Corollary 4.3 we only need to find h1, h2 ∈ C[X1, X2, X3] such that +F = G(c,a,ω3)(X1, X3)h1 + G(b,a,ω2)(X1, X2)h2. +Type Ip,q,r: In this case F = Xp +1 + Xq +2 + Xr +3. By equation (4.7), +G(b,a,ω2)(X1, X2)S(b,a,ω2,q)(X1, X2)+G(c,a,ω3)(X1, X3)S(c,a,ω3,r)(X1, X3) = bqXqω2 +1 +aqω2 +−Xq +2+crXrω3 +1 +arω3 +−Xr +3. +As F is quasi-homogeneous we have p = pω1 = qω2 = rω3. Moreover, as (a, b, c) ∈ X, we have +ap + bq + cr = 0. Therefore, +bqXqω2 +1 +aqω2 ++ crXrω3 +1 +arω3 += Xp +1 +� bq +aqω2 + +cr +arω3 +� += Xp +1 +�bq + cr +ap +� += −Xp +1. +Thus, G(b,a,ω2)(X1, X2)S(b,a,ω2,q)(X1, X2) + G(c,a,ω3)(X1, X3)S(c,a,ω3,r)(X1, X3) = −F. In partic- +ular, h1 := S(c,a,ω3,r)(X1, X3) and h2 := S(b,a,ω2,q)(X1, X2). This prove the matrix factorization +in this case. +Type IIp,q,r: In this case F = Xp +1 + Xq +2 + X2Xr +3. By equation (4.7), +G(b,a,ω2)(X1, X2) +� +S(b,a,ω2,q)(X1, X2) + Xr +3S(b,a,ω2,1)(X1, X2) +� ++ bXω2 +1 +aω2 G(c,a,ω3)(X1, X3)S(c,a,ω3,r)(X1, X3) += bqXqω2 +1 +aqω2 +− Xq +2 + Xr +3 +bXω2 +1 +aω2 +− X2Xr +3 + +�bXω2 +1 +aω2 +� crXrω3 +1 +arω3 +− +�bXω2 +1 +aω2 +� +Xr +3 += bqXqω2 +1 +aqω2 +− Xq +2 − X2Xr +3 + bcrXrω3+ω2 +1 +arω3+ω2 +− Xr +3. +Arguing as before (F is quasi-homogeneous), we have +bqXqω2 +1 +aqω2 ++ bcrXrω3+ω2 +1 +arω3+ω2 += Xp +1 +�bq + bcr +ap +� += −Xp +1. +Therefore (use p = pω1 = qω2 = rω3 + ω2), +G(b,a,ω2)(X1, X2) +� +S(b,a,ω2,q)(X1, X2) + Xr +3S(b,a,ω2,1)(X1, X2) +� ++ bXω2 +1 +aω2 G(c,a,ω3)(X1, X3)S(c,a,ω3,r)(X1, X3) += −Xq +2 − X2Xr +3 − Xp +1. +This gives the matrix factorization in this case. +Type IIIp,q,r: In this case F = Xp +1 + X3Xq +2 + X2Xr +3. Arguing as before, we have using (4.7), +G(b,a,ω2)(X1, X2) +� +X3S(b,a,ω2,q)(X1, X2) + Xr +3S(b,a,ω2,1)(X1, X2) +� ++ G(c,a,ω3)(X1, X3) +�bXω2 +1 +aω2 S(c,a,ω3,r)(X1, X3) + bqXqω2 +1 +aqω2 +S(c,a,ω3,1)(X1, X3) +� += bqX3Xqω2 +1 +aqω2 +− X3Xq +2 + Xr +3 +bXω2 +1 +aω2 +− X2Xr +3 + bcrXrω3+ω2 +1 +arω3+ω2 +− +�bXω2 +1 +aω2 +� +Xr +3 + bqcXω3+qω2 +1 +aω3+qω2 +− bqXqω2 +1 +aqω2 +X3 += −X3Xq +2 − X2Xr +3 + bcrXrω3+ω2 +1 +arω3+ω2 ++ bqcXω3+qω2 +1 +aω3+qω2 +and +bcrXrω3+ω2 +1 +arω3+ω2 ++ bqcXω3+qω2 +1 +aω3+qω2 += Xp +1 +�bcr + bqc +ap +� += −Xp +1. +(use p = pω1 = qω2 + ω3 = rω3 + ω2 for the last equality). This proves the matrix factorization +in this case. + +MATRIX FACTORIZATION +13 +Type IVp,q,r: In this case F = Xp +1 + X3Xq +2 + X1Xr +3. Arguing as before, using (4.7) we have +G(b,a,ω2)(X1, X2) +� +X3S(b,a,ω2,q)(X1, X2) +� ++ G(c,a,ω3)(X1, X3) +� +XS(c,a,ω3,r)(X1, X3) + bqXqω2 +1 +aqω2 +S(c,a,ω3,1)(X1, X3) +� += −X3Xq +2 + crXrω3+1 +1 +arω3 +− X1Xr +3 + bqcXω3+qω2 +1 +aω3+qω2 +and +crXrω3+1 +1 +arω3 ++ bqcXω3+qω2 +1 +aω3+qω2 += Xp +1 +�acr + bqc +ap +� += −Xp +1. +(use p = pω1 = qω2 + ω3 = rω3 + ω1 = rω3 + 1 for the last equality). This proves the matrix +factorization in this case. +Type Vp,q,r: In this case F(X1, X2, X3) = X2Xp +1 + X3Xq +2 + X1Xr +3. Arguing as before using +equation (4.7) we have, +G(b,a,ω2)(X1, X2) +� +X3S(b,a,ω2,q)(X1, X2) + Xp +1S(b,a,ω2,1)(X1, X2) +� ++ G(c,a,ω3)(X1, X3) +� +XS(c,a,ω3,r)(X1, X3) + bqXqω2 +1 +aqω2 +S(c,a,ω3,1)(X1, X3) +� += −X3Xq +2 + bXω2+p +1 +aω2 +− X2Xp +1 + crXrω3+1 +1 +arω3 +− X1Xr +3 + bqcXω3+qω2 +1 +aω3+qω2 +and +bXω2+p +1 +aω2 ++ crXrω3+1 +1 +arω3 ++ bqcXω3+qω2 +1 +aω3+qω2 += Xω2+p +1 +�bap + acr + bqc +aω2+p +� += 0. +(use p + ω2 = qω2 + ω3 = rω3 + ω1 = rω3 + 1 for the last equality). This proves the matrix +factorization in this case. +Type VIp,q,r,b2,b3: In this case F = Xp +1 + X1Xq +2 + X1Xr +3 + Xb2 +2 Xb3 +3 . Arguing as before using +(4.7) we have +G(b,a,ω2)(X1, X2) +� +XS(b,a,ω2,q)(X1, X2) + Xb3 +3 S(b,a,ω2,b2)(X1, X2) +� ++ G(c,a,ω3)(X1, X3) +� +XS(c,a,ω3,r)(X1, X3) + bb2Xb2ω2 +1 +ab2ω2 +S(c,a,ω3,b3)(X1, X3) +� += bqXqω2+1 +1 +aqω2 +− X1Xq +2 + Xb3 +3 +bb2Xb2ω2 +1 +ab2ω2 +− Xb2 +2 Xb3 +3 + crXrω3+1 +1 +arω3 +− X1Xr +3 + bb2Xb2ω2 +1 +ab2ω2 +� +cb3Xω3b3 +1 +aω3b3 +− Xb3 +3 +� += bqXqω2+1 +1 +aqω2 +− X1Xq +2 − Xb2 +2 Xb3 +3 + crXrω3+1 +1 +arω3 +− X1Xr +3 + cb3bb2Xb2ω2+ω3b3 +1 +ab2ω2+ω3b3 +and +bqXqω2+1 +1 +aqω2 ++ crXrω3+1 +1 +arω3 ++ cb3bb2Xb2ω2+ω3b3 +1 +ab2ω2+ω3b3 += Xp +1 +� +abq +aqω2+1 + +acr +arω3+1 + +cb3bb2 +ab2ω2+ω3b3 +� += −Xp +1 +(use p = 1 + qω2 = rω3 + 1 = b2ω2 + b3ω3 for the last equality). +This proves the matrix +factorization in this case. + +14 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +Type VIIp,q,r,b2,b3: In this case F = X2Xp +1 + X1Xq +2 + X1Xr +3 + Xb2 +2 Xb3 +3 . Arguing as before +using (4.7) we have, +G(b,a,ω2)(X1, X2) +� +XS(b,a,ω2,q)(X1, X2) + Xb3 +3 S(b,a,ω2,b2)(X1, X2) + Xp +1S(b,a,ω2,1)(X1, X2) +� ++ G(c,a,ω3)(X1, X3) +� +XS(c,a,ω3,r)(X1, X3) + bb2Xb2ω2 +1 +ab2ω2 +S(c,a,ω3,b3)(X1, X3) +� += bqXqω2+1 +1 +aqω2 +− X1Xq +2 + Xb3 +3 +bb2Xb2ω2 +1 +ab2ω2 +− Xb2 +2 Xb3 +3 + bXω2+p +1 +aω2 +− X2Xp +1 + crXrω3+1 +1 +arω3 +− X1Xr +3 + bb2Xb2ω2 +1 +ab2ω2 +� +cb3Xω3b3 +1 +aω3b3 +− Xb3 +3 +� += bqXqω2+1 +1 +aqω2 +− X1Xq +2 − Xb2 +2 Xb3 +3 + bXω2+p +1 +aω2 +− X2Xp +1 + crXrω3+1 +1 +arω3 +− X1Xr +3 + cb3bb2Xb2ω2+ω3b3 +1 +ab2ω2+ω3b3 +. +Moreover, using ω2 + p = 1 + qω2 = rω3 + 1 = b2ω2 + b3ω3 we have +bqXqω2+1 +1 +aqω2 ++ bXω2+p +1 +aω2 ++ crXrω3+1 +1 +arω3 ++ cb3bb2Xb2ω2+ω3b3 +1 +ab2ω2+ω3b3 += Xqω2+1 +1 +�abq + apb + acr + cb3bb2 +aqω2+1 +� += 0 +This proves the matrix factorization in this case and hence the theorem. +□ +Remark 4.4. Notice that in the case of F = X3 +1 + X3 +2 + X3 +3, our computation recovers the +matrix factorization computed by Laza, Pfister and Popescu [18]. +Remark 4.5. For the sake of completeness we now consider the case when a = 0. For simplicity +we consider the polynomial of type Ip,q,r, the remaining cases follow similarly. To fix notation, +F = Xp +1 + Xq +2 + Xr +3 with weights (ω1, ω2, ω3) and V is the surface defined by F. Let (a, b, c) ∈ +V (p, q, r) with a = 0. Since the point (0, b, c) is different from the origin and it is a zero of F, +thus b and c are both non-zero. The C∗-curve, denoted Wb,c, associated to the point (0, b, c) is +given by the parametrization +n: C∗ → X such that λ �→ (0, bλω2, cλω3). +This C∗-curve is smooth if and only if ω2 = 1 or ω3 = 1 (upto reparametrization). Without +loss of generality suppose that ω2 = 1. Under this assumption the C∗-curve given by the point +(0, b, c) is cut out by the polynomials +f = X1 +and +G(c,b,ω3)(X2, X3) = cXω3 +2 +− bω3X3. +By equation (4.7), +X1(−Xp−1 +1 +) + G(c,b,ω3)(X2, X3)S(c,b,ω3,r)(X2, X3) = −Xp +1 + crXrω3 +2 +brω3 +− Xr +3. +By assumption, a = 0 and bq + cr = 0. Therefore, +−Xp +1 + crXrω3 +2 +brω3 +− Xr +3 = −Xp +1 − Xq +2 − Xr +3. +Let M be the maximal Cohen-Macaulay module corresponding to the degeneracy locus Wb,c +(see Proposition 2.3). Using Corollary 4.3, we conclude that the matrix factorization for M is: +� +−X +bω3X3 − cXω3 +2 +S(c,b,ω3,r)(X2, X3) +Xp−1 +1 +� +. + +MATRIX FACTORIZATION +15 +4.6. Conjecture of Etingof-Ginzburg. Take Φ := X1X2X3 − X2X1X3. +Then, A(Φ) = +C[X1, X2, X3] (see [13, Example 1.3.3]). We prove: +Theorem 4.6. Let Ψ ∈ A(Ψ) be one of polynomials mentioned in Table 1 such that one of the +weights is one. Then, to any point module on B(Φ, Ψ) one can naturally associate a matrix +factorization. +Proof. Denote by X the hypersurface defined by Ψ. Consider a point (a, b, c) ∈ X with a ̸= 0. +Denote by k(a, b, c) the residue field associated to the point (a, b, c). Note that, k(a, b, c) is a +point module. Then, by Theorem 1.2 we naturally associate to the point module P := k(a, b, c) +a matrix factorization M(P) = (M(P)+, M(P)−). Moreover, every point module is a direct +sum of copies of such residue fields i.e., any point module P is of the form: +P := +� +i∈I +k(ai, bi, ci)⊕mi, where (ai, bi, ci) ∈ X∗ and mi > 0. +Denote by Pi the point module k(ai, bi, ci) and by M(Pi) := (M(Pi)+, M(Pi)−) the correspond- +ing matrix factorization. Denote by M(P)+ (resp. M(P)−) the matrix with diagonal entries +mi-copies of M(Pi)+ (resp. +M(Pi)−) as i varies along the entries in I. +Then, the matrix +factorization associated to P is M(P) := (M(P)+, M(P)−). This proves the theorem. +□ +5. More examples: cusps and non-isolated singularities +In this section we obtain the matrix factorization for certain cusp singularities and non-isolated +singularities. +5.1. Cusp singularities. Let +F(X1, X2, X3) = X(r−2)q +1 ++ Xq +2 + Xr +3 + τX1X2X3, +with τ ∈ C∗ and r ≥ 3. Denote by X the surface defined by F. Let ω ∈ C such that ωr−1 = 1/τ. +Take a point (a, b, c) ∈ C3 different from the origin such that +aq(r−2) + bq = 0 and c(cr−1 + ab) = 0. +(5.1) +Consider the C∗-curve, denoted by Wa,b,c, given by the parametrization: +n: C∗ → X such that λ �→ (aλω, bλr−2ωr−2, cλ). +Note that, the morphism n indeed maps to X because +F(n(λ)) = (aλω)(r−2)q + (bλr−2ωr−2)q + (cλ)r + +1 +ωr−1 (aλω)(bλr−2ωr−2)(cλ) += (λω)(r−2)q � +a(r−2)q + bq� ++ λr (cr + abc) = 0 +where the last equality follows from (5.1). Let Ma,b,c be the maximal Cohen-Macaulay OX- +module associated to the degeneracy locus Wa,b,c (see Proposition 2.3). We prove: +Theorem 5.1. The matrix factorization associated to Ma,b,c is given by +� +G(c,aω,1)(X1, X3) +−G(b,a,r−2)(X1, X2) +S(b,a,r−2,q)(X1,X2) + cX2 +1 +aωr S(b,a,r−2,1)(X1,X2) +S(c,aω,1,r)(X1, X3) + X1X2 +ωr−1 S(c,aω,1,1)(X1, X3) +� +, +where G(c1,c2,n)(Z1, Z2) := c1Zn +1 − cn +2Z2 and +S(c1,c2,n,m)(Z1, Z2) := +m +� +j=1 +Z(j−1)n +1 +Zm−j +2 +cj−1 +1 +cjn +2 +. + +16 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +Proof. Note that the curve Wa,b,c is cut out by the polynomials: +G(c,aω,1)(X1, X3) = cX1 − aωX3 +and +G(b,a,r−2)(X1, X2) = bXr−2 +1 +− ar−2X2. +Using (4.7), we have +G(c,aω,1)(X1, X3) +� +S(c,aω,1,r)(X1, X3) + X1X2 +ωr−1 S(c,aω,1,1)(X1, X3) +� ++ G(b,a,r−2)(X1, X2) +� +S(b,a,r−2,q)(X1,X2) + cX2 +1 +aωr S(b,a,r−2,1)(X1,X2) +� += crXr +1 +arωr − Xr +3 − X1X2X3 +ωr−1 ++ bqXq(r−2) +1 +aq(r−2) +− Xq +2 + cbXr +1 +ar−1ωr and +crXr +1 +arωr + cbXr +1 +ar−1ωr = 0 and bqXq(r−2) +1 +aq(r−2) += −Xq(r−2) +1 +. +where the equalities in the last line follows from the hypothesis (5.1). Using Corollary 4.3 we +conclude that the matrix factorization associated to Ma,b,c is as given in the statement of the +theorem. This proves the theorem. +□ +Remark 5.2. Note that: +(1) If we assume q = r = 3, then F is the cubic polynomial studied by Etingof and +Ginzburg [11]. +(2) If we impose the inequality r < q(r − 2), then F is a cusp singularity of type T(r−2)q,q,r +(see [7, Theorem 7.10]). +5.2. Non-isolated singularities. Our next application is to show how to generalize the con- +struction of Baciu [3]. Consider the homogeneous polynomial +F = X4 +1 + X3 +1X3 − X4 +2X3. +In this case the singular locus is the line +Xsing = {(0, 0, z) ∈ C3 | z ∈ C}. +Let (a, b, 1) ∈ C3 \ {0} such that F(a, b, 1) = 0. The C∗-curve given by the point (a, b, 1) is the +zero locus of the ideal given by +f = X1 − X3a +and +g = X2 − X3b. +Let h1 = X3 +1 + X2 +1X2 + aX2 +1X3 + aX1X2X3 + a2X1X2 +3 + a2X2X2 +3 + a3X3 +3 and +h2 = X2 +2X3 + bX2X2 +3 + (a3 + b2)X3 +3. +We then have the corresponding matrix factorizations of F: +M(a, b, c) = +� +f +g +−h2 +h1 +� +. +Notice that h1 and h2 can be rewritten as: +h1 =X1(X2 +1 + X1X2 + aX1X3 + aX2X3) + X3(a2X1X3 + a2X2X3 + a3X2 +3), +h2 =X3(X2 +2 + bX2X3 + (a3 + b2)X2 +3). +Therefore, the following matrices (also parameterized by the points (a : b : 1) ∈ X∗) are matrix +factorizations of F: +M(a, b, c; 3) = + + +0 +−f +g +X +−X2 +2 − bX2X3 − (a3 + b2)X2 +3 +−a2X1X3 − a3X2 +3 +X3 +0 +X2 +1 + X1X2 + aX1X3 + aX2X − aX2X3 + + . + +MATRIX FACTORIZATION +17 +Acknowledgement +We thank Prof. +Javier F. de Bobadilla and Prof. +Duco van Straten for their interest in +this problem and helpful comments. +The first author is funded by EPSRC grant number +EP/T019379/1. +The second author is funded by OTKA 126683 and Lend¨ulet 30001. +The +second author thanks CIRM, Luminy, for its hospitality and for providing a perfect work envi- +ronment. He also thanks Prof. Javier F. de Bobadilla, the 2021 semester 2 Jean-Morlet Chair, +for the invitation. +References +[1] V. I. Arnol’d. Normal forms of functions in neighbourhoods of degenerate critical points. Russ. Math. Surv., +29(2):10–50, 1974. +[2] M. Artin, J. Tate, and M. Van den Bergh. Some algebras associated to automorphisms of elliptic curves. In +The Grothendieck Festschrift, pages 33–85. Springer, 2007. +[3] C. Baciu. Rank two Ulrich modules over the affine cone of the simple node. Analele S¸tiint¸ifice ale Universit˘at¸ii +“Ovidius” Constant¸a. Seria: Matematic˘a, 15(1):15–32, 2007. +[4] W. Bruns and H. J. Herzog. Cohen-Macaulay rings. Cambridge University Press, 1998. +[5] R.-O. Buchweitz, D. Eisenbud, and J. Herzog. Cohen-Macaulay modules on quadrics. With an appendix by +Ragnar-Olaf Buchweitz: The comparison theorem (p. 96- 116). Singularities, representation of algebras, and +vector bundles, Proc. Symp., Lambrecht/Pfalz/FRG 1985, Lect. 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Cambridge University +Press, 2016. +[11] P. Etingof and V. Ginzburg. Noncommutative del Pezzo surfaces and Calabi-Yau algebras. Journal of the +European Mathematical Society (JEMS), 12(6):1371–1416, 2010. +[12] J. Fern´andez de Bobadilla and A. Romano-Vel´azquez. Reflexive modules on normal Gorenstein Stein surfaces, +their deformations and moduli. arXiv preprint arXiv:1812.06543, 2018. +[13] V. Ginzburg. Calabi-Yau algebras. arXiv preprint math/0612139, 2006. +[14] R. Hartshorne. Algebraic Geometry. Graduate text in Mathematics-52. Springer-Verlag, 1977. +[15] R. Hartshorne. Stable reflexive sheaves. Mathematische Annalen, 254(2):121–176, 1980. +[16] A. Kapustin and Y. Li. D-Branes in Landau-Ginzburg Models and Algebraic Geometry. J. High Energy +Phys., 312, 10 2002. +[17] H. Kn¨orrer. Cohen-Macaulay modules on hypersurface singularities. I. Inventiones Mathematicae, 88:153–164, +1987. +[18] R. Laza, G. Pfister, and D. Popescu. Maximal Cohen-Macaulay modules over the cone of an elliptic curve. +Journal of Algebra, 253(2):209–236, 2002. +[19] P. Orlik and P. Wagreich. Isolated singularities of algebraic surfaces with C∗ action. Ann. Math. (2), 93:205– +228, 1971. +[20] D. Orlov. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. In Algebraic +geometry. Methods, relations, and applications. Collected papers. Dedicated to the memory of Andrei Niko- +laevich Tyurin., pages 227–248. Moscow: Maik Nauka/Interperiodica, 2004. +[21] D. Orlov. Derived categories of coherent sheaves and triangulated categories of singularities. In Algebra, +arithmetic, and geometry, pages 503–531. Springer, 2009. +[22] D. Orlov. Landau-Ginzburg models, D-branes and mirror symmetry. Matem´atica Contemporˆanea, 41:75–112, +2012. +[23] C. Peskine and L. Szpiro. Liaison des vari´et´es alg´ebriques. i. Inventiones mathematicae, 26(4):271–302, 1974. + +18 +ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ +[24] P. Pragacz. Enumerative geometry of degeneracy loci. In Annales scientifiques de l’ ´Ecole Normale Sup´erieure, +volume 21, pages 413–454, 1988. +[25] A. Ros Camacho and R. Newton. Orbifold autoequivalent exceptional unimodal singularities. 07 2016. +[26] A. Ros Camacho and R. Newton. Strangely dual orbifold equivalence. I. Journal of Singularities, 14:34–51, +2016. +[27] Y. Xu and S.-T. Yau. Classification of topological types of isolated quasihomogeneous two- dimensional +hypersurface singularities. Manuscr. Math., 64(4):445–469, 1989. +School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road, +S3 7RH, UK +Email address: a.dan@sheffield.ac.uk +Alfr´ed R´enyi Institute Of Mathematics, Hungarian Academy Of Sciences, Re´altanoda Utca 13-15, +H-1053, Budapest, Hungary +Universidad Nacional Aut´onoma de M´exico Avenida Universidad s/n, Colonia Lomas de Chamilpa +CP 62210, Cuernavaca, Morelos Mexico +Email address: agustin@renyi.hu, agustin.romano@im.unam.mx + diff --git a/zNE4T4oBgHgl3EQfZAwg/content/tmp_files/load_file.txt b/zNE4T4oBgHgl3EQfZAwg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5857811dcf906dde8f378b83dd02946220654117 --- /dev/null +++ b/zNE4T4oBgHgl3EQfZAwg/content/tmp_files/load_file.txt @@ -0,0 +1,1167 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf,len=1166 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='05052v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='AG] 12 Jan 2023 MATRIX FACTORIZATION FOR QUASI-HOMOGENEOUS SINGULARITIES ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Given an isolated, quasi-homogeneous singularity X we prove that there is a group isomorphism between the group of rank one reflexive sheaves on X and the free abelian group generated by C∗-divisors, modulo linear equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' When dim(X) = 2 we reduce the problem of finding matrix factorizations of arbitrary reflexive OX-modules to the same question on rank one reflexive sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We then enumerate the matrix factorizations of all rank one reflexive sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Introduction Let X ⊂ Cn be an integral, normal hypersurface defined by an equation F ∈ C[[X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' , Xn]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Recall, matrix factorizations of F are pairs of square matrices (M1, M2) of the same rank such that the products M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='M2 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='M1 equals F times an identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Eisenbud [9] showed that there is a one-to-one correspondence between (reduced) matrix factorizations of F and max- imal Cohen-Macaulay OX-modules without free direct summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Matrix factorization plays a central role in singularity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Using matrix factorization, Kn¨orrer [17] and Buchweitz-Greuel- Schreyer [6] proved that isolated hypersurface singularities of finite Cohen-Macaulay represen- tation type are exactly the simple ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In the early 2000s, Kapustin [16], and Orlov [20–22] showed that matrix factorizations can be applied to study Landau-Ginzburg models appearing in string theory, and to the study of Kontsevich’s homological mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In particu- lar, by the work of Orlov there exists an equivalence between the bounded derived category Db(X) and the homotopy category of matrix factorizations of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In general, the first category is hard to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Thus, producing concrete families of matrix factorizations can be one way of understanding Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Unfortunately, there are no “good” algorithms to obtain matrix factorizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' As a result concrete examples of matrix factorizations are rather limited in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' For example, Buchweitz, Eisenbud and Herzog [5] proved that for Fn(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' , Xn) = X2 1 +· · ·+Xn n with n ≥ 8 the smallest size of a matrix factorization is bounded below by 2 n−2 2 × 2 n−2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In particular for F16 the smallest matrix factorization is of size 128 × 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Crisler and Diveris [8] produced an algorithm to produce matrix factorization for the polynomial Fn only for n ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By studying the polynomial F16 they notice that their algorithm fails and it is impossible to fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Laza, Pfister and Popescu [18] computed all the matrix factorization associated to rank one reflexive sheaves over the surface defined by F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Baciu [3] computed all the matrix factorizations associated to rank two graded Ulrich modules on the hypersurface defined by X3 1 +X2 1X3−X2X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Etingof and Ginzburg [11] produced a family of matrix factorizations for the family of hypersurfaces gives by the polynomial X3 1 + X3 2 + X3 3 + τX1X2X3 as τ varies over non-zero complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Ros Camacho and Newton [25,26] computed concrete matrix factorizations for exceptional unimodal Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Primary: 13C14, 14J17, 32S25, 14E16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Matrix factorization, Maximal Cohen-Macaulay modules, Quasi-homogeneous singu- larities, McKay correspondence, C∗-curves, cusp singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 1 2 ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ hypersurface singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The goal of this article is to generalize some of these results to any isolated, quasi-homogeneous hypersurface singularity (upto topologically trivial deformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let (X, x) be an isolated, quasi-homogeneous hypersurface singularity of dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This means that there exist integers (ω1, ω2, ω3, d) such that the defining equation F satisfies: F(λω1X1, λω2X2, λω3X3) = λdF(X1, X2, X3), for all λ ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The integers ω1, ω2, ω3 are called the weights of the hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note first that every maximal Cohen-Macaulay module M on X sits in an exact sequence with 4 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Besides M the remaining three terms are a trivial bundle, a skyscraper sheaf supported on the singular point x and a rank one reflexive sheaf L, which we will call the determinant of M (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Projective resolutions of skyscraper sheaves are well-understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, to obtain projective resolutions of short exact sequences, one simply needs to determine the projective resolution of two of the three terms (satisfying the obvious compatibility conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' As a result, finding the matrix factorization corresponding to M reduces to determining the matrix factorization corresponding to its determinant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We first classify all such rank one reflexive sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by Ref(1)(X) the group of all reflexive rank one sheaves on X (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4 for the group structure) and by D(X) the free abelian group generated by classes of C∗-curves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', curves that are invariant under the natural C∗-action on X, see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2), modulo linear equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We prove: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Any integral curve D in X is either a C∗-curve or is CI-linked (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2) to a C∗-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, there is an isomorphism of abelian groups: D(X) → Ref(1)(X) sending D ∈ D(X) to i∗ OX∗(D ∩ X∗), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1) where X∗ := X\\{x} is the regular locus in X and i : X∗ → X is the open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' See Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4 for a more general statement that holds in any dimension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This can be viewed as a McKay-type correspondence where the left hand side of the correspondence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1) parameterizes geometric objects namely C∗-divisors and the right hand side parameterizes algebraic objects namely rank one reflexive sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In arbitrary rank, there is a 1 − 1 correspondence between maximal Cohen-Macaulay OX- modules and rank one Cohen-Macaulay OX-modules supported on divisors (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This correspondence associates to a rank r maximal Cohen-Macaulay OX-modules M along with a general choice of r sections, its degeneracy module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The advantage of this correspondence is that one can obtain the matrix factorization of M from a projective resolution of the associated degeneracy module (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The latter is an easier problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We use this idea in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 above, rank one maximal Cohen-Macaulay modules are generated (via ten- sor product) by those arising from integral C∗-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' As a result, maximal Cohen-Macaulay modules associated to non-singular C∗-curves are of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We call such modules generalized Wunram modules (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We give an explicit description of the matrix factoriza- tion corresponding to rank one generalized Wunram modules in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, X contains a non-singular C∗-curve if and only if (upto reparametrization) one of the weights of X is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Recall, Orlik and Wagreich [19] and Arnold [1] classified isolated quasi-homogeneous surface singularities, upto topologically trivial deformations (see table in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Correspond- ing to the types of singularities mentioned in this table we derive the following list of matrix factorizations: MATRIX FACTORIZATION 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let X be a quasi-homogeneous singularity of weight (1, ω2, ω3) listed in Table 1 in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Given positive integers n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' m and complex numbers c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' denote by: S(c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='m)(Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Z2) := m � j=1 Z(j−1)n 1 Zm−j 2 cj−1 1 cjn 2 Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' the matrix factorization associated to any rank one generalized Wunram module on X is a pair of 2 × 2 matrices (adj(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' A) where A is a matrix of the form A := (mi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='j) for i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' j ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 2} with m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 = Xω2 1 b − X2aω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 = X3aω3 − Xω3 1 c and the entries m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 are given by the following table where the first column enumerates the various singularity types from Table 1: Type m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 Ip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r S(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) IIp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r bXω2 1 aω2 S(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2)+ +Xr 3S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) IIIp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r bXω2 1 aω2 S(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) + bqXqω2 1 aqω2 S(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) X3S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q)(X1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) where (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' c) varies over all points in X with a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This result will be proved in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The remaining case a = 0 is treated in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that this also gives explicit families of matrix factorizations parameterized by points on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Our computation recovers the matrix factorizations obtained by Laza, Pfister and Popescu [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' As a consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 above we prove special cases of a conjecture of Etingof and Ginzburg [11, Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='8]: Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let F be the free tensor algebra with basis X1, X2, X3, Φ ∈ F/[F, F], A(Φ) := F/⟨⟨∂iΦ⟩⟩i=1,2,3 and for a central element Ψ not a zero divisor in A(Φ) denote by B(Φ, Ψ) := A(Φ)/⟨⟨Ψ⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' To any point module P (see [2, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='8]) over the algebra B(Φ, Ψ) one can naturally associate a matrix factorization M(P) = (M+, M−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 we prove: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Take Φ := X1X2X3 − X2X1X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, for suitable choices of Ψ the above conjecture holds true i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', to any point module P over the algebra B(Φ, Ψ) one can naturally associate a matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' See Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='6 for the precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, the choices of Ψ in the above theorem will correspond to quasi-homogeneous polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 we study the case of cusp singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In the workshop of Singularities at Ober- wolfach 2021, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Duco van Straten asked a question to the second author on the construction of matrix factorizations for cusp singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We obtain a partial answer to his question, in 4 ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In particular, we produce families of matrix factorizations for families of cusp singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By fixing some numbers, this theorem also recovers the cubic studied by Etingof and Ginzburg [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 we study matrix factorization of non-isolated singularities and generalize a result of Baciu [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Preliminaries In this section, we recall the notion of matrix factorization of hypersurface singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We observe how this relates to the space of maximal Cohen-Macaulay modules (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Finally, we recall basics on degeneracy modules (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This gives us a new approach to studying matrix factorizations, which will be used in later sections for explicit computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Fix an integer n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let X be an integral, normal hypersurface in Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by OCn := C[[X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', Xn]] and F ∈ OCn defining the hypersurface X and OX := OCn /(F) the associated coordinate ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that X may have non-isolated singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' A matrix factorization of F is an ordered pair of m × m-matrices (Φ, Ψ) with entries in OCn such that the matrix multiplication satisfies: Φ · Ψ = F · Idm, Ψ · Φ = F · Idm, where Idm is the m × m identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The matrix factorization is reduced if and only if Im(Φ : O⊕m Cn → O⊕m Cn ) ⊂ m O⊕m Cn and Im(Ψ : O⊕m Cn → O⊕m Cn ) ⊂ m O⊕m Cn , where m is the maximal ideal of OCn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Recall, the following classical result on matrix factorization: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' There is a one-to-one correspondence between: (1) equivalence classes of reduced matrix factorizations of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (2) isomorphism classes of non-trivial periodic minimal free resolutions of OX-modules of periodicity two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (3) maximal Cohen-Macaulay OX-modules without free summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' See [9, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3] for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ In this article, we will exploit the equivalence between (1) and (3) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' So, we briefly recall how one associates a matrix factorization of F to a maximal Cohen-Macaulay module without free summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let M be a maximal Cohen-Macaulay OX-module without free summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This implies that the depth of M equals the dimension of X, which is n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By the Auslander-Buchsbaum formula, this means as an OCn-module, the projective dimension of M equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This implies we have a short exact sequence of the form 0 → O⊕b Cn Φ−→ O⊕a Cn (m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=',ma) −−−−−−−→ M → 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1) where mi ∈ M and the standard basis element ei ∈ O⊕a Cn maps to mi for 1 ≤ i ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since M is supported on X, we have a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then the morphism Φ is simply given by an a × a-matrix with entries in OCn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Suppose that this is a minimal resolution of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since M is annihilated by F, for every 1 ≤ i ≤ a, Fei ∈ O⊕a Cn maps to zero in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By the exactness of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1), there exists Ψ(ei) ∈ O⊕a Cn such that Φ ◦ Ψ(ei) = Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In other words, there exists an a × a-matrix Ψ with entries in OCn such that Φ · Ψ = F · Ida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Therefore, (Ψ, Φ) is a matrix factorization of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' MATRIX FACTORIZATION 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Degeneracy module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let M be a maximal Cohen-Macaulay OX-module of rank, say r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Given an r-tuple of sections s := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr) of M, the associated degeneracy locus is the zero locus of the section s1 ∧ s2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' ∧ sr ∈ ∧rM i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', the locus of points where the r-tuple of sections is linearly dependant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Consider the morphism s : O⊕r X → M, sending a standard basis vector ei of O⊕r X to si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by Cs the cokernel of the morphism s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, the support of Cs is the associated degeneracy locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' For a general choice of r-sections s := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr) the associated degeneracy locus Supp(Cs) is reduced and Cohen-Macaulay of codimension 1 (see [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 431]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Furthermore, by the genericity of the r-tuple of sections, the locus where r − 1 of the r-sections are linearly dependant is of codimension 2 (see [10, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This implies that the cokernel Cs is supported on a reduced Cohen-Macaulay subvariety of codimension 1 in X and is of rank 1 over its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The cokernel Cs will be called the degeneracy module associated to the r-tuple of sections s := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This motivates the following definition: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We will call an r-tuple of sections s := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr) of M weakly general if the cokernel Cs of the induced morphism s is supported on a reduced Cohen-Macaulay subvariety of X of codimension 1 and is a rank 1, Cohen-Macaulay OX-module over Supp(Cs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Dualizing degeneracy modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let M be a maximal Cohen-Macaulay OX-module of rank r and s := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr) be an r-tuple of weakly general sections of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By definition, we have a short exact sequence of the form: 0 → O⊕r X s−→ M → Cs → 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2) for some Cohen-Macaulay OX-module Cs supported on a reduced Cohen-Macaulay subvariety in X and is of rank 1 over its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Dualizing this exact sequence, we get 0 → M∨ → O⊕r X s′ −→ Ext1 X(Cs, OX) → 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3) where the surjectivity on the right follows from the vanishing of Ext1 X(M, OX) (see [4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Throughout this article, we shall denote As := Ext1 X(Cs, OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, dualizing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3) and using Ext1 X(As, OX) ∼= Cs and M∨∨ ∼= M (see [4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='10]), we get back the exact sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This implies: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' There is a 1 − 1 correspondence between pairs: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 (M, s) where M is a MCM OX -module of rank r and s := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr) is an r-tuple of weakly general sections of M \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe ←→ \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 (As, s′) where As is a CM OX -module supported on a CM subvariety of codimension one in X and of rank 1 over the support and generated by s′ \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe where the bijection follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' See [12] for a detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Given a pair (M, s) with M a maximal Cohen-Macaulay OX-module of rank r and s an r-tuple of weakly general sections of M, we will call the corresponding pair (As, s′) as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3, the degenerate pair associated to (M, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' McKay-type correspondence for quasi-homogeneous singularities Quasi-homogeneous hypersurface singularities are generalizations of homogeneous singulari- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We study C∗-divisors contained in such hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We observe that every effective, integral divisor is either a C∗-divisor or is CI-linked (in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2) to a C∗-divisor 6 ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Using this we observe that there is a 1 − 1 correspondence between C∗-divisors (modulo linear equivalence) and rank one reflexive sheaves on a quasi-homogeneous hypersurface (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Furthermore, if the dimension of the hypersurface is two, then we can express every maximal Cohen-Macaulay modules solely in terms of the ideal sheaves of C∗-curves and certain skyscraper sheaves (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Quasi-homogeneous hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' A polynomial F ∈ C[[X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', Xn]] is called quasi-homogeneous if there exists positive integers (ω1, ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', ωn, d) such that for any λ ∈ C∗, we have F(λX1, λX2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', λXn) = λdF(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='., Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The hypersurface X defined by F is called a quasi-homogeneous hypersurface with weights ω := (ω1, ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, there is a natural C∗-action on X: C∗ × X → X sending (λ, (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', xn)) �→ (λω1x1, λω2x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', λωnxn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Throughout this section we assume that X has only isolated singularity at the origin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by Pω X∗ the quotient of X∗ := X\\{0} by the C∗-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Consider the resulting quotient map: πX : X∗ → Pω X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' C∗-divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let X be a quasi-homogeneous hypersurface of dimension n with weights ω := (ω1, ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', ωn, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, given a closed point (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', an) ∈ X, the associated C∗-curve is the parametric curve given by: n : C∗ → X sending λ �→ (λω1a1, λω2a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='λωnan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We will denote by [a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', an] the corresponding point on Pω X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Clearly, the fiber over [a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', an] to the morphism πX is an integral curve and n is the normalization map for the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This implies that the preimage under πX of an integral divisor in Pω X∗ is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' An integral divisor D in X∗ is called a C∗-divisor if there exists an integral Weil divisor D′ in Pω X∗ such that D ∼= π−1 X (D′)red, where πX is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' An integral divisor in X is called C∗-divisor if it is the closure of an integral C∗-divisor on X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by D(X) the free abelian group generated by integral C∗-divisors in X, modulo linear equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Elements of D(X) will be called C∗-divisors on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Liaisons and residual divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let (X, 0) be an isolated, quasi-homogeneous hypersur- face singularity with weights ω := (ω1, ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='., ωn, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Consider the quotient map πX as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1) from the regular locus of X to quotient by the C∗-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' An integral divisor D ⊂ X is called horizontal if the composition D\\{0} ⊂ X\\{0} πX −−→ Pω X∗ is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Two distinct divisors D, E are called CI-linked if there exists a polynomial g ∈ C[[X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', Xn]] such that D ∪ E = Z(g) ∩ X, where Z(g) denotes the zero locus of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, if D and E are CI-linked then we call D residual to E (and vice versa, E is residual to D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This terminology is inspired by the classical theory of liaisons (see [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let D ⊂ X be an integral horizontal divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, there exist a C∗-divisor E ⊂ X such that D is CI-linked to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Consider the quotient map πX from X∗ to Pω X∗ as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By the theorem on generic smoothness, there exists an open dense affine subscheme U ⊂ Pω X∗ such that the resulting morphism from π−1 X (U) to U is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since πX is an affine morphism and U is affine, we have π−1 X (U) is affine and non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This implies Pic(π−1 X (U)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' As D is an integral horizontal MATRIX FACTORIZATION 7 divisor, UD := D ∩ π−1 X (D) is a non-empty Cartier divisor in π−1 X (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since Pic(π−1 X (U)) = 0, the ideal sheaf IUD is simply f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Oπ−1 X (U) for some f ∈ Oπ−1 X (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By [14, Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3], there exists a regular function �f ∈ OX such that Z( �f) ∩ π−1 X (U) = Z(f) ∩ π−1 X (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This implies that the zero locus Z( �f) of �f is of the form Z( �f) = Z(f) ∪ E (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2) where Z(f) is the closure in X of the zero locus of f and E is a divisor lying in the complement X\\π−1 X (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since E is a divisor and does not intersect π−1 X (U), the scheme-theoretic image πX(E) of E in Pω X∗ does not intersect U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since the fibers of πX are irreducible and of dimension one, we conclude by the fiber dimension theorem that E ∼= π−1 X (E′) for some divisor E′ in Pω X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In particular, E is a C∗-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, as D is integral and agrees with Z(f) over π−1 X (U), we have Z(f) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2), this means D is CI-linked to a C∗-divisor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Rank one correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by Ref(1)(X) the space of reflexive rank one sheaves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let i : X∗ → X be the natural inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Recall, every reflexive sheaf of rank one on a regular variety is invertible (see [15, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, every reflexive sheaf on X arises as the pushforward via i of a reflexive sheaf on X∗ (see [15, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This means that under pushforward by i, i∗ : Pic(X∗) → Ref(1)(X) sending L to i∗L is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The group operation on Pic(X∗) induces one on Ref(1)(X), namely M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='N := i∗(i∗M ⊗OX∗ i∗N) and M∨ := i∗((i∗M)∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The morphism φ : D(X) → Ref(1)(X) sending a C∗-divisor D to the reflexive sheaf i∗(OX∗(D ∩ X∗)) is an isomorphism of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Clearly, this is a group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, as U is integral and regular, D(X) is contained in Pic(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since Ref(1)(X) is isomorphic to Pic(U) as argued above, this means the morphism φ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' So it remains to check that φ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Consider M ∈ Ref(1)(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, the restriction M|X∗ is a reflexive sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since X∗ is regular, this implies M|X∗ is an invertible sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In other words, M|X∗ ∼= OX∗(D∗) for some divisor D∗ on X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Write D∗ = � i aiDi as a linear combination of integral divisors Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' If Di is not horizontal, then by the fiber dimension theorem the scheme theoretic image Ei of πX|Di is a divisor in Pω X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since Di is integral, Ei is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This implies π−1 X (Ei) is irreducible (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Hence, Di = π−1 X (Ei)red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In other words, Di is a C∗-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' If Di is horizontal, then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 there exists a C∗-curve Dc i such that Di is linearly equivalent to −Dc i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Therefore, D∗ is linearly equivalent to a divisor obtained as a linear combination of C∗-divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves surjectivity of φ and hence the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Dimension two case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let (X, x) be an isolated, quasi-homogeneous hypersurface singu- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Suppose that dim X = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by kx the skyscraper sheaf over the singular point x of a one dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 8 ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let M be a maximal Cohen-Macaulay OX-module of rank, say r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, for a general choice of r sections (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr) of M, we have an exact sequence of the form 0 → O⊕r−1 X (s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=',sr−1) −−−−−−−→ M → L → k⊕m x → 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3) for some non-negative integer m and L is a reflexive sheaf on X of rank 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', L ∈ Ref(1)(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In particular, if C denotes the support of the cokernel of the morphism (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr), then L is the dual of the ideal sheaf of C in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by A the cokernel of the morphism (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', sr) : O⊕r X → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4) Note that, A is a Cohen-Macaulay module supported in dimension 1 and of rank one over its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by C the support of A and A′ := Ext1 X(A, OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Dualizing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4), we then have the following diagram of short exact sequences: 0 ✲ IC|X ✲ OX ✲ OC ✲ 0 0 ✲ M∨ ❄ ✲ O⊕r X p1 ❄ (t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=',tr)✲ A′ p2 ❄ ✲ 0 where the morphism p2 sends 1 to t1 and p1 sends 1 to the standard basis element e1 ∈ O⊕r X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, the cokernel of p1 is isomorphic O⊕r−1 X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since A′ is Cohen-Macaulay, the morphism p2 is injective (the section t1 is torsion-free over C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By Bertini-type theorem (see [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 434]), C\\{x} is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since any torsion-free sheaf on an affine non-singular curve is trivial, we conclude A′ is isomorphic to OC over X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Taking t1 = 1 ∈ Γ(OC), we observe that the cokernel of p2 is of the form k⊕m x for some non-negative integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Using Snake lemma, we get the exact sequence: 0 → IC|X → M∨ → O⊕r−1 X → k⊕m x → 0 Dualizing this sequence and applying [4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='10], gives us the exact sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Matrix factorization using degeneracy modules Matrix factorization of maximal Cohen-Macaulay modules is hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' However, one can instead study resolutions of the associated degeneracy modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This is a slightly easier problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We obtain matrix factorizations using this idea (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We then apply this to enumerate the matrix factorization of all Cohen-Macaulay modules arising from C∗-curves in quasi-homogeneous surfaces (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 stated in the introduction and proved in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Matrix factorization via degeneracy modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let M be a maximal Cohen-Macaulay OX-module of rank r with no free direct summand (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', does not contain OX as a direct summand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let s be an r-tuple of weakly general sections of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let (As, s′) be the associated degenerate pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since As is a Cohen-Macaulay OX-module supported in a dimension n − 2 subvariety in Cn, the depth of As is n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By the Auslander-Buchsbaum formula this implies the projective dimension of As is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then starting with s′ the pair induces an exact sequence of the form: 0 → O⊕a Cn A −→ O⊕b Cn B −→ O⊕r Cn s′ −→ As → 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1) MATRIX FACTORIZATION 9 where A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' B) is induced by a b × a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' r × b) matrix with entries in OCn, which we will also denote by A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' B) for simplicity of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In particular, Ae(a) i = b � j=1 ajie(b) j and Be(b) i = r � j=1 bjie(r) j , where {e(t) i }t i=1 is the standard basis of the free OCn-module O⊕t Cn for t ∈ {r, a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We show: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by K the OCn-submodule of O⊕b Cn consisting of all m ∈ O⊕b Cn such that Bm ∈ I⊕r X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, (1) K is isomorphic to O⊕r Cn, as OCn-modules, (2) fix an isomorphism as in (1) from O⊕r Cn to K given by a b × r-matrix A′ : O⊕r Cn ∼ −→ K ⊂ O⊕b Cn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, (upto change of basis of O⊕r Cn) the composition O⊕r Cn A′ −→ O⊕b Cn B −→ O⊕r Cn coincides with FIdr×r : O⊕r Cn → O⊕r Cn, where F ∈ OCn defines X, (3) the matrix factorization associated to M is of the form � adj(A|A′)T , (A|A′)T � , where (−)T denotes transpose of the matrix and adj(−) denotes the adjoint of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Before we prove the theorem, note that by the exact sequence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1), we have b = r + a (the support of As is of codimension 2 in Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, the matrix (A|A′) is a b × b-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Comparing the exact sequences (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1), we get the following diagram of exact sequences: 0 ✲ O⊕a Cn A ✲ O⊕b Cn B ✲ O⊕r Cn s′ ✲ As ✲ 0 ⟲ ⟲ 0 ✲ M∨ ρ′ ❄ ✲ O⊕r X ρ ❄ s′ ✲ As id ❄ ✲ 0 where the vertical morphism ρ is the natural restriction morphism and the first vertical morphism ρ′ is induced by the universal property of kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since the last two vertical arrows are surjective then by a simple diagram chase (using the injectivity of the morphism from M∨ to O⊕r X ) we conclude that morphism ρ′ from O⊕b Cn to M∨ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, ρ sits in the short exact sequence: 0 → O⊕r Cn F Idr×r −−−−→ O⊕r Cn ρ−→ O⊕r X → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Using the Snake lemma applied to the above diagram of exact sequence, this gives us the following exact sequence: 0 → O⊕a Cn ⊕ O⊕r Cn (A|A′) −−−−→ O⊕b Cn ρ′ −→ M∨ → 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2) where the composition O⊕r Cn A′ −→ O⊕b Cn B −→ O⊕r Cn coincides with FIdr×r : O⊕r Cn → O⊕r Cn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves parts (1) and (2) of the theorem (identify K with the image of A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' As mentioned above b = r + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Dualizing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2), we get the exact sequence: 0 → O⊕b Cn (A|A′)T −−−−−→ O⊕b Cn → Ext1 Cn(M∨, OCn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3) 10 ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ Since F annihilates M∨ (as M∨ is supported in X), we have by [4, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4] Ext1 Cn(M∨, OCn) ∼= HomCn(M∨, OX) ∼= HomX(M∨, OX), where the last isomorphism follows from adjunction of Hom-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since M is a maximal Cohen-Macaulay OX-module, it is in particular reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Therefore, the double dual M∨∨ of M is isomorphic to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Hence, Ext1 Cn(M∨, OCn) ∼= M and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3) gives a projective resolution of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In other words, � adj(A|A′)T , (A|A′)T � is a matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Generalized Wunram modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Following [12], a maximal Cohen-Macaulay OX-module M of rank 1 is called generalized Wunram if for a general choice of section s of M, the cokernel of the natural morphism from OX to M, defined by multiplication with s, is isomorphic to OD for a non-singular subvariety D ⊂ X of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Projective resolution of the degeneracy module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let M be a rank one generalized Wunram module, s ∈ M a general section and D be the associated degeneracy locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In particular, we have a short exact sequence of the form: 0 → OX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='s −→ M → OD → 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4) where D is a non-singular subvariety in X of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Dualizing this exact sequence we get a short exact sequence of the form: 0 → M∨ → OX → Ext1 X(OD, OX) → 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5) Note that, Ext1 X(OD, OX) is a Cohen-Macaulay OX-module supported on D and is of rank one over its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Now, a rank one maximal Cohen-Macaulay module over a smooth affine variety is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Hence, Ext1 X(OD, OX) ∼= OD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We now produce a projective resolution of OD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since D is non-singular there exists f, g ∈ OCn such that the ideal of D (in Cn) is generated by f and g (regular local rings are complete intersection rings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We then have the Koszul resolution: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The projective resolution of OD is given by 0 → OCn A −→ O⊕2 Cn B −→ OCn → OD → 0 where Ae := −fe1 + ge2, Be1 := g and Be2 = f with e (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' {e1, e2}) the standard basis of OCn (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' O⊕2 Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let X be a normal hypersurface singularity (not necessarily isolated) of any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let M be a rank one generalized Wunram module, s ∈ M a general section and D the degeneracy locus associated to the pair (M, s), which is non-singular as M is generalized Wunram of rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, the matrix factorization associated to M is the pair (adj(C), C) where C is the matrix C := � −f g h1 h2 � f, g ∈ OCn defines the non-singular variety D in Cn and X is defined by a regular function of the form F := h1g + h2f (as D ⊂ X we have F ∈ (f, g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Translating into the notations of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1, we have a = 1, b = 2 and r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The morphisms A and B are defined in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We now need to compute K and A′ from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Recall, K = {a1e1 + a2e2|a1g + a2f ∈ IX} where IX is the ideal of X in Cn generated by, say F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Of course, since D ⊂ X, there exists h1, h2 ∈ OCn such that F = h1g + h2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In other words, h1e1 + h2e2 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We claim that K is generated as an MATRIX FACTORIZATION 11 OCn-module by h1e1 + h2e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Indeed, since K ∼= OCn (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1), it is generated by a single element, say h′ 1e1 + h′ 2e2 ∈ O⊕2 Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, there exists λ ∈ OCn such that λ(h′ 1e1 + h′ 2e2) = h1e1 + h2e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Applying the OCn-linear morphism B, we have λB(h′ 1e1 + h′ 2e2) = B(λ(h′ 1e1 + h′ 2e2)) = B(h1e1 + h2e2) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='6) Since h′ 1e1 + h′ 2e2 ∈ K, we have B(h′ 1e1 + h′ 2e2) = λ′F for some λ′ ∈ OCn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Substituting in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='6) this implies λλ′ = 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', λ is a unit in OCn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves our claim that K is generated as an OCn-module by h1e1 + h2e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, we can take the morphism A′ : OCn ∼ −→ K ⊂ O⊕2 Cn sending 1 to h1e1 + h2e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This satisfies the condition that the composition B ◦ A′ = F × Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 the matrix factorization of M is of the form � adj(A|A′)T , (A|A′)T � where (A|A′) = � −f h1 g h2 � , so (A|A′)T = � −f g h1 h2 � This proves the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Matrix factorization for topological trivial deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Orlik and Wagreich [19] and Arnold [1] showed that an isolated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' quasi-homogeneous surface singularity can be can be deformed into one of the following seven classes below keeping the link differentially constant Type Defining polynomial Ip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r F(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) := Xp 1 + Xq 2 + Xr 3 IIp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r F(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) := Xp 1 + Xq 2 + X2Xr 3 with q > 1 IIIp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r F(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) := Xp 1 + X3Xq 2 + X2Xr 3 with q > 1 and r > 1 IVp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r F(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) := Xp 1 + X3Xq 2 + X1Xr 3 with p > 1 Vp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r F(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) := X2Xp 1 + X3Xq 2 + X1Xr 3 = 0 VIp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='b3 F(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) := Xp 1 + X1Xq 2 + X1Xr 3 + Xb2 2 Xb3 3 with (p − 1)(qb3 + rb2) = pqr VIIp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='b3 F(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) := X2Xp 1 + X1Xq 2 + X1Xr 3 + Xb2 2 Xb3 3 with (p − 1)(qb3 + rb2) = r(pq − 1) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Quasi-homogeneous singularity types Xu and Yau [27] proved that the above deformation is in fact a topological trivial deforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Furthermore, the topological type of quasi-homogeneous singularities determine and is determined by its weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We now use Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 to produce the matrix factorizations corresponding to all rank one generalized Wunram modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Given c1, c2 ∈ C and k, n, m ∈ Z>0 denote by G(c1,c2,n)(Z1, Z2) := c1Zn 1 − cn 2Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' and S(c1,c2,n,m)(Z1, Z2) defined in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, Zk 3 G(c1,c2,n)(Z1, Z2)S(c1,c2,n,m)(Z1, Z2) = Zk 3 �cm 1 Zmn 1 cmn 2 − Zm 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7) Let X be a quasi-homogeneous surface singularity defined by a quasi-homogeneous polynomial F(X1, X2, X3) from the list in Table 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By assumption, the weights of X is (1, ω2, ω3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Take a point (a, b, c) ∈ X with a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The associated C∗-curve, denoted Wa,b,c, is given by the following parametrization: n: C∗ → X such that λ �→ (aλ, bλω2, cλω3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 12 ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ Note that, Wa,b,c is the zero locus (in C3) of the polynomials G(b,a,ω2)(X1, X2) = Xω2 1 b − X2aω2 and G(c,a,ω3)(X1, X3) = Xω3 1 c − X3aω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 we only need to find h1, h2 ∈ C[X1, X2, X3] such that F = G(c,a,ω3)(X1, X3)h1 + G(b,a,ω2)(X1, X2)h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Type Ip,q,r: In this case F = Xp 1 + Xq 2 + Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7), G(b,a,ω2)(X1, X2)S(b,a,ω2,q)(X1, X2)+G(c,a,ω3)(X1, X3)S(c,a,ω3,r)(X1, X3) = bqXqω2 1 aqω2 −Xq 2+crXrω3 1 arω3 −Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' As F is quasi-homogeneous we have p = pω1 = qω2 = rω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, as (a, b, c) ∈ X, we have ap + bq + cr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Therefore, bqXqω2 1 aqω2 + crXrω3 1 arω3 = Xp 1 � bq aqω2 + cr arω3 � = Xp 1 �bq + cr ap � = −Xp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Thus, G(b,a,ω2)(X1, X2)S(b,a,ω2,q)(X1, X2) + G(c,a,ω3)(X1, X3)S(c,a,ω3,r)(X1, X3) = −F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In partic- ular, h1 := S(c,a,ω3,r)(X1, X3) and h2 := S(b,a,ω2,q)(X1, X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This prove the matrix factorization in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Type IIp,q,r: In this case F = Xp 1 + Xq 2 + X2Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7), G(b,a,ω2)(X1, X2) � S(b,a,ω2,q)(X1, X2) + Xr 3S(b,a,ω2,1)(X1, X2) � + bXω2 1 aω2 G(c,a,ω3)(X1, X3)S(c,a,ω3,r)(X1, X3) = bqXqω2 1 aqω2 − Xq 2 + Xr 3 bXω2 1 aω2 − X2Xr 3 + �bXω2 1 aω2 � crXrω3 1 arω3 − �bXω2 1 aω2 � Xr 3 = bqXqω2 1 aqω2 − Xq 2 − X2Xr 3 + bcrXrω3+ω2 1 arω3+ω2 − Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Arguing as before (F is quasi-homogeneous), we have bqXqω2 1 aqω2 + bcrXrω3+ω2 1 arω3+ω2 = Xp 1 �bq + bcr ap � = −Xp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Therefore (use p = pω1 = qω2 = rω3 + ω2), G(b,a,ω2)(X1, X2) � S(b,a,ω2,q)(X1, X2) + Xr 3S(b,a,ω2,1)(X1, X2) � + bXω2 1 aω2 G(c,a,ω3)(X1, X3)S(c,a,ω3,r)(X1, X3) = −Xq 2 − X2Xr 3 − Xp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This gives the matrix factorization in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Type IIIp,q,r: In this case F = Xp 1 + X3Xq 2 + X2Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Arguing as before, we have using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' G(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) � X3S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) + Xr 3S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) � + G(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) �bXω2 1 aω2 S(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) + bqXqω2 1 aqω2 S(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) � = bqX3Xqω2 1 aqω2 − X3Xq 2 + Xr 3 bXω2 1 aω2 − X2Xr 3 + bcrXrω3+ω2 1 arω3+ω2 − �bXω2 1 aω2 � Xr 3 + bqcXω3+qω2 1 aω3+qω2 − bqXqω2 1 aqω2 X3 = −X3Xq 2 − X2Xr 3 + bcrXrω3+ω2 1 arω3+ω2 + bqcXω3+qω2 1 aω3+qω2 and bcrXrω3+ω2 1 arω3+ω2 + bqcXω3+qω2 1 aω3+qω2 = Xp 1 �bcr + bqc ap � = −Xp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (use p = pω1 = qω2 + ω3 = rω3 + ω2 for the last equality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the matrix factorization in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' MATRIX FACTORIZATION 13 Type IVp,q,r: In this case F = Xp 1 + X3Xq 2 + X1Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Arguing as before, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7) we have G(b,a,ω2)(X1, X2) � X3S(b,a,ω2,q)(X1, X2) � + G(c,a,ω3)(X1, X3) � XS(c,a,ω3,r)(X1, X3) + bqXqω2 1 aqω2 S(c,a,ω3,1)(X1, X3) � = −X3Xq 2 + crXrω3+1 1 arω3 − X1Xr 3 + bqcXω3+qω2 1 aω3+qω2 and crXrω3+1 1 arω3 + bqcXω3+qω2 1 aω3+qω2 = Xp 1 �acr + bqc ap � = −Xp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (use p = pω1 = qω2 + ω3 = rω3 + ω1 = rω3 + 1 for the last equality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the matrix factorization in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Type Vp,q,r: In this case F(X1, X2, X3) = X2Xp 1 + X3Xq 2 + X1Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Arguing as before using equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7) we have, G(b,a,ω2)(X1, X2) � X3S(b,a,ω2,q)(X1, X2) + Xp 1S(b,a,ω2,1)(X1, X2) � + G(c,a,ω3)(X1, X3) � XS(c,a,ω3,r)(X1, X3) + bqXqω2 1 aqω2 S(c,a,ω3,1)(X1, X3) � = −X3Xq 2 + bXω2+p 1 aω2 − X2Xp 1 + crXrω3+1 1 arω3 − X1Xr 3 + bqcXω3+qω2 1 aω3+qω2 and bXω2+p 1 aω2 + crXrω3+1 1 arω3 + bqcXω3+qω2 1 aω3+qω2 = Xω2+p 1 �bap + acr + bqc aω2+p � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (use p + ω2 = qω2 + ω3 = rω3 + ω1 = rω3 + 1 for the last equality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the matrix factorization in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Type VIp,q,r,b2,b3: In this case F = Xp 1 + X1Xq 2 + X1Xr 3 + Xb2 2 Xb3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Arguing as before using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7) we have G(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) � XS(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) + Xb3 3 S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='b2)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) � + G(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) � XS(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) + bb2Xb2ω2 1 ab2ω2 S(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='b3)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='= bqXqω2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='aqω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='− X1Xq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 + Xb3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='bb2Xb2ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ab2ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='− Xb2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 Xb3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 + crXrω3+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='arω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='− X1Xr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 + bb2Xb2ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ab2ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='cb3Xω3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='aω3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='− Xb3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='= bqXqω2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='aqω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='− X1Xq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 − Xb2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 Xb3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 + crXrω3+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='arω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='− X1Xr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 + cb3bb2Xb2ω2+ω3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ab2ω2+ω3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='bqXqω2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='aqω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='+ crXrω3+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='arω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='+ cb3bb2Xb2ω2+ω3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ab2ω2+ω3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='= Xp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='abq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='aqω2+1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='acr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='arω3+1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='cb3bb2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ab2ω2+ω3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='= −Xp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='(use p = 1 + qω2 = rω3 + 1 = b2ω2 + b3ω3 for the last equality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the matrix factorization in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 14 ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ Type VIIp,q,r,b2,b3: In this case F = X2Xp 1 + X1Xq 2 + X1Xr 3 + Xb2 2 Xb3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Arguing as before using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7) we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' G(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) � XS(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='q)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) + Xb3 3 S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='b2)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) + Xp 1S(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X2) � + G(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) � XS(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='r)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) + bb2Xb2ω2 1 ab2ω2 S(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='b3)(X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' X3) � = bqXqω2+1 1 aqω2 − X1Xq 2 + Xb3 3 bb2Xb2ω2 1 ab2ω2 − Xb2 2 Xb3 3 + bXω2+p 1 aω2 − X2Xp 1 + crXrω3+1 1 arω3 − X1Xr 3 + bb2Xb2ω2 1 ab2ω2 � cb3Xω3b3 1 aω3b3 − Xb3 3 � = bqXqω2+1 1 aqω2 − X1Xq 2 − Xb2 2 Xb3 3 + bXω2+p 1 aω2 − X2Xp 1 + crXrω3+1 1 arω3 − X1Xr 3 + cb3bb2Xb2ω2+ω3b3 1 ab2ω2+ω3b3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, using ω2 + p = 1 + qω2 = rω3 + 1 = b2ω2 + b3ω3 we have bqXqω2+1 1 aqω2 + bXω2+p 1 aω2 + crXrω3+1 1 arω3 + cb3bb2Xb2ω2+ω3b3 1 ab2ω2+ω3b3 = Xqω2+1 1 �abq + apb + acr + cb3bb2 aqω2+1 � = 0 This proves the matrix factorization in this case and hence the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Notice that in the case of F = X3 1 + X3 2 + X3 3, our computation recovers the matrix factorization computed by Laza, Pfister and Popescu [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' For the sake of completeness we now consider the case when a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' For simplicity we consider the polynomial of type Ip,q,r, the remaining cases follow similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' To fix notation, F = Xp 1 + Xq 2 + Xr 3 with weights (ω1, ω2, ω3) and V is the surface defined by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let (a, b, c) ∈ V (p, q, r) with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Since the point (0, b, c) is different from the origin and it is a zero of F, thus b and c are both non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The C∗-curve, denoted Wb,c, associated to the point (0, b, c) is given by the parametrization n: C∗ → X such that λ �→ (0, bλω2, cλω3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This C∗-curve is smooth if and only if ω2 = 1 or ω3 = 1 (upto reparametrization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Without loss of generality suppose that ω2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Under this assumption the C∗-curve given by the point (0, b, c) is cut out by the polynomials f = X1 and G(c,b,ω3)(X2, X3) = cXω3 2 − bω3X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7), X1(−Xp−1 1 ) + G(c,b,ω3)(X2, X3)S(c,b,ω3,r)(X2, X3) = −Xp 1 + crXrω3 2 brω3 − Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' By assumption, a = 0 and bq + cr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Therefore, −Xp 1 + crXrω3 2 brω3 − Xr 3 = −Xp 1 − Xq 2 − Xr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let M be the maximal Cohen-Macaulay module corresponding to the degeneracy locus Wb,c (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3, we conclude that the matrix factorization for M is: � −X bω3X3 − cXω3 2 S(c,b,ω3,r)(X2, X3) Xp−1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' MATRIX FACTORIZATION 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Conjecture of Etingof-Ginzburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Take Φ := X1X2X3 − X2X1X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, A(Φ) = C[X1, X2, X3] (see [13, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We prove: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let Ψ ∈ A(Ψ) be one of polynomials mentioned in Table 1 such that one of the weights is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, to any point module on B(Φ, Ψ) one can naturally associate a matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by X the hypersurface defined by Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Consider a point (a, b, c) ∈ X with a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by k(a, b, c) the residue field associated to the point (a, b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, k(a, b, c) is a point module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2 we naturally associate to the point module P := k(a, b, c) a matrix factorization M(P) = (M(P)+, M(P)−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Moreover, every point module is a direct sum of copies of such residue fields i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', any point module P is of the form: P := � i∈I k(ai, bi, ci)⊕mi, where (ai, bi, ci) ∈ X∗ and mi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by Pi the point module k(ai, bi, ci) and by M(Pi) := (M(Pi)+, M(Pi)−) the correspond- ing matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by M(P)+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' M(P)−) the matrix with diagonal entries mi-copies of M(Pi)+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' M(Pi)−) as i varies along the entries in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Then, the matrix factorization associated to P is M(P) := (M(P)+, M(P)−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' More examples: cusps and non-isolated singularities In this section we obtain the matrix factorization for certain cusp singularities and non-isolated singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Cusp singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let F(X1, X2, X3) = X(r−2)q 1 + Xq 2 + Xr 3 + τX1X2X3, with τ ∈ C∗ and r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Denote by X the surface defined by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let ω ∈ C such that ωr−1 = 1/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Take a point (a, b, c) ∈ C3 different from the origin such that aq(r−2) + bq = 0 and c(cr−1 + ab) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1) Consider the C∗-curve, denoted by Wa,b,c, given by the parametrization: n: C∗ → X such that λ �→ (aλω, bλr−2ωr−2, cλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that, the morphism n indeed maps to X because F(n(λ)) = (aλω)(r−2)q + (bλr−2ωr−2)q + (cλ)r + 1 ωr−1 (aλω)(bλr−2ωr−2)(cλ) = (λω)(r−2)q � a(r−2)q + bq� + λr (cr + abc) = 0 where the last equality follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let Ma,b,c be the maximal Cohen-Macaulay OX- module associated to the degeneracy locus Wa,b,c (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We prove: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The matrix factorization associated to Ma,b,c is given by � G(c,aω,1)(X1, X3) −G(b,a,r−2)(X1, X2) S(b,a,r−2,q)(X1,X2) + cX2 1 aωr S(b,a,r−2,1)(X1,X2) S(c,aω,1,r)(X1, X3) + X1X2 ωr−1 S(c,aω,1,1)(X1, X3) � , where G(c1,c2,n)(Z1, Z2) := c1Zn 1 − cn 2Z2 and S(c1,c2,n,m)(Z1, Z2) := m � j=1 Z(j−1)n 1 Zm−j 2 cj−1 1 cjn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 16 ANANYO DAN AND AGUST´IN ROMANO-VEL´AZQUEZ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that the curve Wa,b,c is cut out by the polynomials: G(c,aω,1)(X1, X3) = cX1 − aωX3 and G(b,a,r−2)(X1, X2) = bXr−2 1 − ar−2X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='7), we have G(c,aω,1)(X1, X3) � S(c,aω,1,r)(X1, X3) + X1X2 ωr−1 S(c,aω,1,1)(X1, X3) � + G(b,a,r−2)(X1, X2) � S(b,a,r−2,q)(X1,X2) + cX2 1 aωr S(b,a,r−2,1)(X1,X2) � = crXr 1 arωr − Xr 3 − X1X2X3 ωr−1 + bqXq(r−2) 1 aq(r−2) − Xq 2 + cbXr 1 ar−1ωr and crXr 1 arωr + cbXr 1 ar−1ωr = 0 and bqXq(r−2) 1 aq(r−2) = −Xq(r−2) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' where the equalities in the last line follows from the hypothesis (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='3 we conclude that the matrix factorization associated to Ma,b,c is as given in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Note that: (1) If we assume q = r = 3, then F is the cubic polynomial studied by Etingof and Ginzburg [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' (2) If we impose the inequality r < q(r − 2), then F is a cusp singularity of type T(r−2)q,q,r (see [7, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Non-isolated singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Our next application is to show how to generalize the con- struction of Baciu [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Consider the homogeneous polynomial F = X4 1 + X3 1X3 − X4 2X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In this case the singular locus is the line Xsing = {(0, 0, z) ∈ C3 | z ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let (a, b, 1) ∈ C3 \\ {0} such that F(a, b, 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The C∗-curve given by the point (a, b, 1) is the zero locus of the ideal given by f = X1 − X3a and g = X2 − X3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Let h1 = X3 1 + X2 1X2 + aX2 1X3 + aX1X2X3 + a2X1X2 3 + a2X2X2 3 + a3X3 3 and h2 = X2 2X3 + bX2X2 3 + (a3 + b2)X3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' We then have the corresponding matrix factorizations of F: M(a, b, c) = � f g −h2 h1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Notice that h1 and h2 can be rewritten as: h1 =X1(X2 1 + X1X2 + aX1X3 + aX2X3) + X3(a2X1X3 + a2X2X3 + a3X2 3), h2 =X3(X2 2 + bX2X3 + (a3 + b2)X2 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Therefore, the following matrices (also parameterized by the points (a : b : 1) ∈ X∗) are matrix factorizations of F: M(a, b, c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' 3) = \uf8eb \uf8ed 0 −f g X −X2 2 − bX2X3 − (a3 + b2)X2 3 −a2X1X3 − a3X2 3 X3 0 X2 1 + X1X2 + aX1X3 + aX2X − aX2X3 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' MATRIX FACTORIZATION 17 Acknowledgement We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Javier F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' de Bobadilla and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Duco van Straten for their interest in this problem and helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The first author is funded by EPSRC grant number EP/T019379/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The second author is funded by OTKA 126683 and Lend¨ulet 30001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' The second author thanks CIRM, Luminy, for its hospitality and for providing a perfect work envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' He also thanks Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Javier F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' de Bobadilla, the 2021 semester 2 Jean-Morlet Chair, for the invitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Arnol’d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Normal forms of functions in neighbourhoods of degenerate 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+page_content=' Tate, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Van den Bergh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Some algebras associated to automorphisms of elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' In The Grothendieck Festschrift, pages 33–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Springer, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Baciu.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Manuscr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=', 64(4):445–469, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content=' School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road, S3 7RH, UK Email address: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='dan@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='uk Alfr´ed R´enyi Institute Of Mathematics, Hungarian Academy Of Sciences, Re´altanoda Utca 13-15, H-1053, Budapest, Hungary Universidad Nacional Aut´onoma de M´exico Avenida Universidad s/n, Colonia Lomas de Chamilpa CP 62210, Cuernavaca, Morelos Mexico Email address: agustin@renyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='hu, agustin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='romano@im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfZAwg/content/2301.05052v1.pdf'} +page_content='mx' metadata={'source': 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ANDREAS FISCHER +PROFESSEUR DES UNIVERSITES, +Haute école d'ingéniérie et d'archi. +Rapporteur du jury +M. HAROLD MOUCHERE +PROFESSEUR DES UNIVERSITES, +UNIVERSITE NANTES +Rapporteur du jury +M. CHRISTOPHER KERMORVANT +, +Membre du jury +MME LAURENCE LIKFORMAN-SULEM +PROFESSEUR ASSOCIE, TELECOM +PARISTECH +Membre du jury +MME CAROLINE PETITJEAN +PROFESSEUR DES UNIVERSITES, +Université de Rouen Normandie +Membre du jury +M. THIERRY PAQUET +PROFESSEUR DES UNIVERSITES, +Université de Rouen Normandie +Directeur de thèse +Thèse dirigée par THIERRY PAQUET (Laboratoire d'Informatique, du Traitement +de l'Information et des Systèmes) + +UNIVERSITE +DE ROUEN +MANMIIE +R E M E R C I E M E N T S +Je tiens, en premier lieu, à remercier Thierry Paquet, qui a accepté d’encadrer et de +superviser ma thèse, pour son expérience et ses précieux conseils. Je tiens également à +remercier mon co-encadrant de thèse, Christopher Kermorvant, pour l’opportunité qu’il m’a +offert en me proposant ce sujet, et pour m’avoir encadrée et soutenue durant ces trois années. +Leur patience mais aussi leurs expériences ont permis des discussions enrichissantes et très +intéressantes. +Je remercie les membres de mon jury d’avoir accepté d’évaluer mes travaux de thèse. Tout +d’abord, mes deux rapporteurs Harold Mouchère et Andreas Fischer, pour le temps consacré +à la lecture de ce manuscrit et leurs commentaires avisés. Merci également à Laurence +Likforman et Caroline Petitjean d’avoir accepté de faire partie du jury de thèse. +Merci à Teklia de m’avoir permis de réaliser ma thèse dans les meilleures conditions, en +m’encourageant à publier et en valorisant mon travail en l’intégrant dans de nombreux +projets. J’aimerais également remercier toutes les personnes de l’entreprise avec qui j’ai eu +la chance de collaborer. Je tiens tout particulièrement à remercier l’équipe de recherche : +Martin, Marie-Laurence, Blanche, Chaza et Solène pour nos rencontres et discussions très +inspirantes. Merci à toute l’équipe de Grenoble pour leur gentillesse et leur soutien, et plus +spécialement à Bastien, avec qui j’ai pu échanger sur de nombreux points techniques, pour +son aide et sa patience. +Je me dois aussi d’être très reconnaissante envers mon laboratoire, le LITIS, pour m’avoir +offert tout le confort et le soutien matériel nécessaire au bon déroulement de ma thèse et plus +particulièrement les membres de l’équipe Apprentissage pour leur accueil. Un chaleureux +merci à Denis avec qui j’ai eu la chance de partager mon bureau pendant ces trois dernières +années mais également de longues et captivantes discussions toujours remplies de joie. +Sur un plan plus personnel, je tiens à remercier toutes les personnes présentes durant ces +trois années. Tout d’abord, mes parents et mes frères pour les nombreuses distractions qu’ils +m’ont apportées mais aussi leur soutien. Merci également à ma belle famille pour leur présence +et leurs relectures. Enfin, un grand merci à Quentin pour m’avoir soutenue et écoutée jour +après jour. +iii + + +R É S U M É +Qu’ils soient historiques ou modernes, imprimés ou manuscrits, les documents constituent +un ensemble précieux d’informations souvent difficilement accessible dans leur forme +originale. La transformation de ces documents en documents digitaux est désormais possible +grâce à leur numérisation et à l’extraction automatique de leurs contenus. Cette extraction +nécessite la détection de différents éléments tels que les lignes de texte, éléments cruciaux +afin d’obtenir la transcription du texte présent dans les images. Bien que de nombreuses +méthodes aient été proposées pour détecter ces éléments, l’analyse de la structure des +documents reste un problème difficile : les modèles proposés souffrent de difficultés à +généraliser à de nouvelles données et à des structures plus complexes, et ils nécessitent de +nombreux exemples d’apprentissage. +Dans cette thèse, nous étudions différentes tâches liées à l’analyse de la mise en page de do- +cuments telles que la détection de lignes de texte, la séparation en actes ou encore la détection +du support d’écriture (page). Ainsi, nous proposons deux modèles fondés sur des réseaux de +neurones profonds suivant deux approches différentes. Les réseaux neuronaux ont démontré +de bonnes capacités d’apprentissage dans de nombreux domaines d’application et notamment +dans la détection d’objets. Récemment, de nouveaux types de réseaux neuronaux ont vu le +jour, les réseaux à base de Transformers. Ceux-ci permettent de traiter plus efficacement les +tâches de prédiction séquence-à-séquence telles que la traduction de texte. Leur adaptation +aux tâches de vision a rapidement suscité l’engouement grâce à leurs performances élevées et +leur capacité à produire des résultats séquentiels et structurés. +Notre objectif est de proposer un modèle permettant de détecter les objets en tenant +compte des difficultés liées au traitement de documents, notamment le nombre restreint +de données d’entraînement disponibles. De plus, les systèmes existants peuvent présenter +des temps de traitement longs qui peuvent entraîner des coûts financiers importants et +des impacts écologiques négatifs. Dans un cadre industriel, l’utilisation de tels systèmes ne +semble pas appropriée, il est donc nécessaire de proposer des modèles plus parcimonieux en +termes de nombre de paramètres afin d’obtenir des temps d’entraînement et d’inférence plus +réduits. +Dans cette optique, nous proposons un modèle de détection niveau pixel et un second +modèle de détection niveau objet. Nous commençons par proposer un modèle de détection +comportant peu de paramètres, rapide en prédiction, et qui permet d’obtenir des masques de +prédiction précis à partir d’un nombre réduit de données d’apprentissage. Le pré-entraînement +de ce modèle sur différents jeux de données annotés a permis d’obtenir des gains significatifs de +performances. Ces résultats nous ont donc conduits à mettre en place une stratégie de collecte +et d’uniformisation de nombreux jeux de données, utilisés afin d’entraîner un modèle unique +de détection de lignes démontrant de grandes capacités de généralisation à des documents +hors échantillon. +Nous proposons également un modèle de détection à base de Transformers. La conception +d’un tel modèle a nécessité de redéfinir la tâche de détection d’objets dans les images de +documents et à en étudier différentes modélisations. Suite à cette étude, nous proposons +une stratégie de détection d’objets consistant à prédire séquentiellement les coordonnées +des rectangles englobant les objets grâce à une classification pixel. Cette stratégie permet +d’obtenir un modèle comportant peu de paramètres et rapide en inférence. Les expériences +v + +préliminaires de détection de lignes de texte montrent des bonnes performances. +Enfin, dans un cadre industriel, de nouvelles données non annotées sont souvent disponibles. +Ainsi, dans le cas de l’adaptation d’un modèle à ces nouvelles données, on s’attend à fournir au +système le minimum de nouveaux exemples annotés. Le choix des exemples pertinents pour +l’annotation manuelle est donc crucial pour permettre une adaptation réussie. Il est donc +nécessaire que les systèmes effectuent la tâche finale tout en évaluant automatiquement leur +confiance quant à leurs décisions. Ainsi, les décisions moins confiantes peuvent être soumises +à un opérateur humain pour une annotation manuelle, tandis que les décisions plus confiantes +sont conservées telles quelles pour fournir une annotation automatique. +À cet égard, nous proposons des estimateurs de confiance issus d’approches différentes +pour la détection d’objets dans des images de documents. La première approche proposée est +inspirée de la méthode de Monte Carlo et consiste à construire des estimations de confiance +en utilisant la méthode du dropout au moment du test. Notre seconde proposition consiste à +construire un système dédié indépendant, entraîné à prédire une estimation de confiance de- +puis une seule prédiction pendant l’inférence. Nous montrons que ces estimateurs permettent +de réduire fortement la quantité de données annotées tout en optimisant les performances. +vi + +A B S T R A C T +Whether they are historical or modern, printed or handwritten, documents constitute a +valuable collection of information that is usually difficult to access. The transformation of +these documents into digital documents is now possible through their digitization and the +automatic extraction of their contents. This extraction requires the detection of different +elements such as text lines, which are essential to obtain the transcription of the image’s +textual contents. Although many methods have been proposed to detect these elements, the +analysis of document structure remains a difficult problem : the proposed models suffer from +difficulties in generalizing to new data and more complex structures, and they require many +training examples. +In this thesis, we study multiple tasks related to document layout analysis such as the +detection of text lines, the splitting into acts or the detection of the writing support (page). +Thus, we propose two deep neural models following two different approaches. Neural networks +have shown good learning capabilities in many application domains, and in particular in +object detection. Recently, new types of neural networks have emerged, the Transformer- +based networks. These systems allow processing more efficiently sequence-to-sequence tasks +such as text translation. Their adaptation to vision tasks has quickly become popular thanks +to their high performance and their ability to produce sequential and structured outputs. +We aim at proposing a model for object detection that considers the difficulties associated +with document processing, including the limited amount of training data available. Moreover, +existing systems can have long processing times that can result in significant financial costs +and negative ecological impacts. In an industrial setting, the use of such systems does not +seem appropriate, so it is necessary to propose more parsimonious models in terms of number +of parameters to obtain reduced training and inference times. +In this respect, we propose a pixel-level detection model and a second object-level detec- +tion model. We first propose a detection model with few parameters, fast in prediction, and +which can obtain accurate prediction masks from a reduced number of training data. The +pre-training of this model on different annotated datasets allowed us to obtain significant per- +formance gains. These results led us to implement a strategy of collection and uniformization +of many datasets, which are used to train a single line detection model that demonstrates +high generalization capabilities to out-of-sample documents. +We also propose a Transformer-based detection model. The design of such a model +required redefining the task of object detection in document images and to study different +approaches. Following this study, we propose an object detection strategy consisting +in sequentially predicting the coordinates of the objects enclosing rectangles through a +pixel classification. This strategy allows obtaining a fast model with only few parameters. +Preliminary experiments on text line detection show good performances. +Finally, in an industrial setting, new non-annotated data are often available. Thus, in the +case of a model adaptation to this new data, it is expected to provide the system as few +new annotated samples as possible. The selection of relevant samples for manual annotation +is therefore crucial to enable successful adaptation. Thus, it is necessary for the systems +to perform the final task while automatically assessing their confidence about their own +vii + +decisions. This way, less confident decisions can be submitted to a human operator for manual +annotation, while more confident decisions are kept as is to provide an automatic annotation. +For this purpose, we propose confidence estimators from different approaches for object +detection in document images. The first proposed approach is inspired by the Monte Carlo +method and consists in building confidence estimates using the dropout method at test time. +Our second proposal consists in building an independent dedicated system, trained to predict +a confidence estimate with a single prediction during inference. We show that these estimators +greatly reduce the amount of annotated data while optimizing the performances. +viii + +TA B L E D E S M AT I È R E S +Remerciements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +iii +Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +v +Liste des figures +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +xii +Liste des tableaux +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +xv +Liste des focus +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii +Liste des algorithmes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii +Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +xix +Acronymes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +xxi +1 +I N T R O D U C T I O N +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1.1 +Contexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1.1.1 +Analyse de la mise en page +. . . . . . . . . . . . . . . . . . . . . . . . +2 +1.1.2 +Reconnaissance de texte . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +1.2 +Cadre de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.3 +Objectifs et contributions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.4 +Organisation du manuscrit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2 +É TAT D E L’ A RT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.1 +Détection d’objets dans des images de documents . . . . . . . . . . . . . . . . +9 +2.1.1 +Méthodes ad hoc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2.1.2 +Méthodes par apprentissage profond . . . . . . . . . . . . . . . . . . . +13 +2.2 +Estimation de la confiance des objets détectés . . . . . . . . . . . . . . . . . . +38 +3 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N . . . . . . +43 +3.1 +Jeux de données +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +3.2 +Annotation des données . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +3.3 +Métriques d’évaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +3.3.1 +Métriques basées sur les pixels +. . . . . . . . . . . . . . . . . . . . . . +51 +3.3.2 +Métriques orientées objets . . . . . . . . . . . . . . . . . . . . . . . . . +55 +3.3.3 +Métriques orientées vers la tâche finale . . . . . . . . . . . . . . . . . . +57 +4 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S . . . . . . . . . . +59 +4.1 +Présentation du problème . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +4.2 +Systèmes à l’état de l’art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +ix + +4.3 +Architecture du système proposé : Doc-UFCN . . . . . . . . . . . . . . . . . . +60 +4.3.1 +Encodeur +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +61 +4.3.2 +Décodeur +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +61 +4.3.3 +Détails d’implémentation +. . . . . . . . . . . . . . . . . . . . . . . . . +62 +4.4 +Expériences de détection de lignes de texte . . . . . . . . . . . . . . . . . . . . +63 +4.4.1 +Jeux de données +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +4.4.2 +Résultats et discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +4.4.3 +Impact du pré-entraînement . . . . . . . . . . . . . . . . . . . . . . . . +66 +4.4.4 +Étude ablative +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +68 +4.5 +Expériences de détection d’actes +. . . . . . . . . . . . . . . . . . . . . . . . . +71 +4.5.1 +Jeux de données +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +72 +4.5.2 +Approche proposée . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +4.5.3 +Résultats et discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . +76 +4.6 +Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +78 +5 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C- +T I O N D’ O B J E T S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +79 +5.1 +Uniformisation des annotations . . . . . . . . . . . . . . . . . . . . . . . . . . +81 +5.1.1 +Analyse des annotations . . . . . . . . . . . . . . . . . . . . . . . . . . +82 +5.2 +Comparaison des approches de détection . . . . . . . . . . . . . . . . . . . . . +84 +5.2.1 +Doc-UFCN +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +85 +5.2.2 +dhSegment +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +85 +5.2.3 +ARU-Net +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +85 +5.3 +Évaluation des détections +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +86 +5.3.1 +Métriques niveau pixel . . . . . . . . . . . . . . . . . . . . . . . . . . . +86 +5.3.2 +Métriques niveau objet . . . . . . . . . . . . . . . . . . . . . . . . . . . +90 +5.4 +Évaluation orientée vers la tâche de reconnaissance . . . . . . . . . . . . . . . +94 +5.4.1 +CER niveau page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +95 +5.4.2 +CER niveau ligne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +98 +5.5 +Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 +6 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S . . . . . . . . . . . . . . 103 +6.1 +Méthodes d’estimation de la confiance . . . . . . . . . . . . . . . . . . . . . . 104 +6.1.1 +Estimateur basé sur les probabilités a posteriori . . . . . . . . . . . . . 104 +6.1.2 +Estimateurs basés sur le dropout de Monte Carlo . . . . . . . . . . . . 105 +6.1.3 +Estimateur basé sur les statistiques d’objets . . . . . . . . . . . . . . . 106 +6.2 +Cadre expérimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 +6.2.1 +Jeux de données +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 +6.2.2 +Entraînement des systèmes de détection . . . . . . . . . . . . . . . . . 109 +6.2.3 +Entraînement des estimateurs de confiance . . . . . . . . . . . . . . . . 110 +6.3 +Résultats et discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 +6.3.1 +Nombre de prédictions avec dropout . . . . . . . . . . . . . . . . . . . 111 +6.3.2 +Performances des estimateurs en rejet +. . . . . . . . . . . . . . . . . . 112 +x + +6.3.3 +Apprentissage actif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 +6.4 +Stratégie d’entraînement : sélection et annotation des données . . . . . . . . . 116 +6.4.1 +Détection de pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 +6.4.2 +Détection de lignes de texte . . . . . . . . . . . . . . . . . . . . . . . . 119 +6.5 +Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 +7 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 121 +7.1 +Modélisation de la tâche de détection . . . . . . . . . . . . . . . . . . . . . . . 122 +7.1.1 +Modélisation de la position et de la forme des objets . . . . . . . . . . 123 +7.1.2 +Stratégie de prédiction des coordonnées : singleton vs n-uplet . . . . . 125 +7.1.3 +Stratégie de prédiction des coordonnées : classification vs régression . . 126 +7.1.4 +Stratégie de prédiction de la classe des objets . . . . . . . . . . . . . . 126 +7.2 +Architecture du système proposé : Doc2Seq +. . . . . . . . . . . . . . . . . . . 128 +7.2.1 +Encodeur Doc-UFCN +. . . . . . . . . . . . . . . . . . . . . . . . . . . 128 +7.2.2 +Encodage positionnel 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 130 +7.2.3 +Décodeur Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 +7.2.4 +Branche de classification . . . . . . . . . . . . . . . . . . . . . . . . . . 131 +7.3 +Détails d’implémentation et stratégies d’entraînement +. . . . . . . . . . . . . 131 +7.3.1 +Taille des images en entrée . . . . . . . . . . . . . . . . . . . . . . . . . 132 +7.3.2 +Augmentation de données . . . . . . . . . . . . . . . . . . . . . . . . . 132 +7.3.3 +Décodeur Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 +7.3.4 +Choix du meilleur modèle . . . . . . . . . . . . . . . . . . . . . . . . . 132 +7.4 +Expériences et résultats +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +7.4.1 +Jeu de données . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +7.4.2 +Entraînement des modèles de détection . . . . . . . . . . . . . . . . . . 133 +7.4.3 +Résultats et discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 +7.5 +Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 +8 +C O N C L U S I O N S E T P E R S P E C T I V E S +. . . . . . . . . . . . . . . . . . . . . . . . . 139 +8.1 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 +8.2 +Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 +B I B L I O G R A P H I E +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 +xi + +L I S T E D E S F I G U R E S +Figure 1.1 +Pages présentant des difficultés de traitement : pages arrachées, dé- +gradées et parties de pages manquantes. +. . . . . . . . . . . . . . . . +1 +Figure 1.2 +Chaîne de traitement standard impliquant une détection de lignes +de texte, une reconnaissance du texte manuscrit (HTR) suivi d’une +détection des entités nommées (NER). +. . . . . . . . . . . . . . . . . +2 +Figure 1.3 +Détection d’objets sur l’image de la page 17 recto du Livre d’heures +Horae ad usum Romanum, Bibliothèque nationale de France, Dépar- +tement des manuscrits, NAL 3111. . . . . . . . . . . . . . . . . . . . . +3 +Figure 2.1 +Architecture du modèle LeNet. . . . . . . . . . . . . . . . . . . . . . . +15 +Figure 2.2 +Architecture du modèle AlexNet. +. . . . . . . . . . . . . . . . . . . . +15 +Figure 2.3 +Architecture du modèle VGG-16. +. . . . . . . . . . . . . . . . . . . . +16 +Figure 2.4 +Architecture du modèle ResNet-34. . . . . . . . . . . . . . . . . . . . +16 +Figure 2.5 +Schéma d’une convolution 2D. . . . . . . . . . . . . . . . . . . . . . . +17 +Figure 2.6 +Système R-CNN. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +Figure 2.7 +Système YOLO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +Figure 2.8 +Architecture du modèle U-Net. . . . . . . . . . . . . . . . . . . . . . . +24 +Figure 2.9 +Schéma d’une convolution transposée 2D. . . . . . . . . . . . . . . . . +25 +Figure 2.10 +Schéma d’une convolution dilatée 2D. . . . . . . . . . . . . . . . . . . +25 +Figure 2.11 +Architecture du modèle dhSegment. . . . . . . . . . . . . . . . . . . . +27 +Figure 2.12 +Architecture du modèle de Yang. +. . . . . . . . . . . . . . . . . . . . +28 +Figure 2.13 +Architecture du modèle Transformer original. +. . . . . . . . . . . . . +33 +Figure 2.14 +Architecture du modèle Vision Transformer original. . . . . . . . . . . +36 +Figure 2.15 +Système Pix2Seq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +Figure 2.16 +Apprentissage actif. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +Figure 3.1 +Représentation des modélisations d’une ligne de texte proposées dans +la littérature. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +Figure 3.2 +Visualisation des différents taux de relâchement détectés dans les jeux +de données. Les taux de relâchement indiquent la quantité de fond +présent autour des pixels de texte dans les annotations. . . . . . . . . +50 +Figure 3.3 +Masques de segmentation comparés par Peskin. +. . . . . . . . . . . . +51 +Figure 3.4 +Deux détections de lignes différentes obtenues pour une même image +et obtenant les mêmes scores d’IoU et de F1. . . . . . . . . . . . . . . +53 +Figure 4.1 +Architecture du modèle Doc-UFCN. . . . . . . . . . . . . . . . . . . . +61 +Figure 4.2 +Détections de lignes obtenues par dhSegment et Doc-UFCN sur +l’image de la page 5 verso du Livre d’heures Horae +. . . . . . . . . . +65 +Figure 4.3 +Impact du pré-entraînement de Doc-UFCN, évalué sur les ensembles +de test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +68 +xii + +Figure 4.4 +Annotations manuelles pour la détection et la classification d’actes sur +les jeux de données Balsac et Himanis-Act. . . . . . . . . . . . . . . . +72 +Figure 4.5 +Chaîne de traitement proposée pour la détection et la classification +d’actes avec l’utilisation du contenu textuel. . . . . . . . . . . . . . . +73 +Figure 5.1 +Processus de génération d’annotations pour une image du jeu de don- +nées de Bozen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +83 +Figure 5.2 +Détections de lignes produites sur une image du jeu de données Horae. 88 +Figure 5.3 +Détections de lignes produites sur une image du jeu de données Bozen. 92 +Figure 5.4 +Détections de lignes produites par les modèles génériques et spécifiques +sur une image du jeu de données ScribbleLens. . . . . . . . . . . . . . +93 +Figure 5.5 +Détections de lignes produites par les modèles génériques Doc-UFCN +et dhSegment sur une image du jeu de données HOME-Alcar. +. . . . +94 +Figure 5.6 +Simulation des scores, pour deux prédictions avec et sans fusion, sur +une image du jeu de données HOME-NACR. . . . . . . . . . . . . . . +97 +Figure 5.7 +Résultats de reconnaissance niveau ligne obtenus par les modèles gé- +nériques Doc-UFCN, dhSegment et ARU-Net sans adaptation. . . . . +99 +Figure 6.1 +Deux images issues du jeu de données Horae avec leurs prédictions et +la variance pour N =10 prédictions avec dropout. . . . . . . . . . . . . 106 +Figure 6.2 +Courbes de rejet présentant l’évolution des performances du modèle de +détection de pages de référence sur l’ensemble de test Horae-test-300. +Courbes présentées pour les estimateurs DAP et DOV en fonction du +nombre de prédictions avec dropout N. +. . . . . . . . . . . . . . . . . 111 +Figure 6.3 +Courbes de rejet présentant l’évolution du score mAP en fonction +du taux de rejet. Les courbes présentent les résultats du modèle de +détection de pages de référence sur l’ensemble de test Horae-test-300. +112 +Figure 6.4 +Évolution des performances de détection de pages sur l’ensemble de +test Horae-test-300 pendant les itérations d’apprentissage actif. +. . . 114 +Figure 6.5 +Évolution des performances de détection de lignes de texte sur l’en- +semble de test du jeu de données Hugin-Munin pendant les itérations +d’apprentissage actif. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 +Figure 6.6 +Évolution des performances de détection de pages sur l’ensemble de +test Horae-test-300 pendant les itérations d’apprentissage actif pour +différentes stratégies de sélection de données. . . . . . . . . . . . . . . 117 +Figure 6.7 +Évolution des performances de détection de lignes de texte sur l’en- +semble de test du jeu de données Hugin-Munin pendant les itérations +d’apprentissage actif pour différentes stratégies de sélection de données.118 +Figure 7.1 +Représentation de différentes modélisations de la position et de la +forme des objets à détecter. Exemple pour la détection d’une ligne de +texte. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 +Figure 7.2 +Exemple de séquence à deux classes : paragraphe et ligne de texte. +L’ordre de prédiction préserve la hiérarchie des objets. +. . . . . . . . 127 +Figure 7.3 +Architecture du modèle Doc2Seq. . . . . . . . . . . . . . . . . . . . . 129 +xiii + +Figure 7.4 +Détections de lignes produites par le modèle Doc2Seq, sélectionné sur +les valeurs du CER, sur quatre images de l’ensemble de test du jeu de +données IAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 +xiv + +L I S T E D E S TA B L E A U X +Table 3.1 +Tableau récapitulatif des différents jeux de données utilisés pour la +détection de lignes de texte. +. . . . . . . . . . . . . . . . . . . . . . . +44 +Table 3.2 +Tableau récapitulatif du type d’annotation des différents jeux de don- +nées utilisés pour la détection de lignes de texte. . . . . . . . . . . . . +49 +Table 3.3 +Métriques d’évaluation utilisées dans les récents travaux liés à la dé- +tection d’objets dans les images de documents. . . . . . . . . . . . . . +52 +Table 4.1 +Statistiques des jeux de données utilisés pour la détection de lignes de +texte. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +Table 4.2 +Résultats obtenus par Doc-UFCN et dhSegment au niveau pixel. Ré- +sultats donnés sur les ensembles de test pour la tâche de détection de +lignes de texte. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +65 +Table 4.3 +Temps d’inférence rapportés pour Doc-UFCN et dhSegment calculés +sur les ensembles de test. . . . . . . . . . . . . . . . . . . . . . . . . . +66 +Table 4.4 +Résultats obtenus par Doc-UFCN et dhSegment au niveau pixel pour +la tâche de détection de lignes de texte. Les résultats montrent les +performances des modèles génériques sur les ensembles de test avec et +sans adaptation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +67 +Table 4.5 +Étude ablative de Doc-UFCN sur la détection de lignes de texte. . . . +69 +Table 4.6 +Impact du taux de dilatation dans les blocs d’encodeur de Doc-UFCN +sur la détection de lignes de texte. . . . . . . . . . . . . . . . . . . . . +70 +Table 4.7 +Impact de la taille des images en entrée de Doc-UFCN sur la détection +des lignes de texte. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +70 +Table 4.8 +Statistiques des jeux de données utilisés pour la détection d’actes. . . +72 +Table 4.9 +Résultats du modèle générique de détection de lignes de texte sur +l’ensemble de test du jeu de données Balsac. . . . . . . . . . . . . . . +74 +Table 4.10 +Résultats de reconnaissance de textes manuscrits sur les jeux de don- +nées Balsac et Himanis-GMV. . . . . . . . . . . . . . . . . . . . . . . +75 +Table 4.11 +Résultats de classification des lignes de texte sur les jeux de données +Balsac et Himanis-Act. . . . . . . . . . . . . . . . . . . . . . . . . . . +76 +Table 4.12 +Résultats de détection d’actes. . . . . . . . . . . . . . . . . . . . . . . +77 +Table 4.13 +Résultats obtenus par Doc-UFCN et le système de Prieto sur le jeu +de données Himanis-Act avec et sans l’information textuelle. . . . . . +78 +Table 5.1 +Statistiques des jeux de données utilisés pour la détection de lignes de +texte. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +81 +Table 5.2 +Comparaison des systèmes Doc-UFCN, dhSegment et ARU-Net. . . . +86 +Table 5.3 +Résultats au niveau pixel obtenus par les systèmes Doc-UFCN, dh- +Segment et ARU-Net sur les ensembles de test. +. . . . . . . . . . . . +87 +xv + +Table 5.4 +Résultats au niveau pixel obtenus par Doc-UFCN avec et sans unifor- +misation des labels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +89 +Table 5.5 +Résultats au niveau ligne obtenus par les systèmes Doc-UFCN, dh- +Segment et ARU-Net sur les ensembles de test. +. . . . . . . . . . . . +91 +Table 5.6 +Résultats au niveau ligne obtenus par Doc-UFCN avec et sans unifor- +misation des labels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +94 +Table 5.7 +Résultats de reconnaissance niveau page obtenus par les systèmes Doc- +UFCN, dhSegment et ARU-Net sur les ensembles de test. . . . . . . . +96 +Table 5.8 +Résultats de reconnaissance niveau page obtenus par Doc-UFCN avec +et sans uniformisation des labels. +. . . . . . . . . . . . . . . . . . . . +98 +Table 5.9 +Résultats de reconnaissance niveau ligne obtenus par les systèmes Doc- +UFCN, dhSegment et ARU-Net sur les ensembles de test. . . . . . . . +99 +Table 5.10 +Résultats de reconnaissance niveau ligne obtenus par Doc-UFCN avec +et sans uniformisation des labels. +. . . . . . . . . . . . . . . . . . . . 100 +Table 6.1 +Statistiques des jeux de données utilisés pour la détection de pages. . 108 +Table 6.2 +Résultats de détection de pages obtenus par le modèle de référence +entraîné sur le jeu de données READ-BAD* et évalué sur les jeux de +données READ-BAD* et Horae-test-300. . . . . . . . . . . . . . . . . 110 +Table 6.3 +Résultats de détection de lignes de texte obtenus par le modèle de +référence entraîné sur 19 jeux de données et évalué l’ensemble de test +du jeu de données Hugin-Munin. . . . . . . . . . . . . . . . . . . . . . 110 +Table 6.4 +Résultats des modèles de détection de pages sur l’ensemble de test +Horae-test-300 après apprentissage actif. +. . . . . . . . . . . . . . . . 114 +Table 6.5 +Résultats des modèles de détection de lignes de texte sur l’ensemble +de test du jeu de données Hugin-Munin après apprentissage actif. . . 116 +Table 6.6 +Résultats des modèles de détection de pages sur l’ensemble de test +Horae-test-300 après apprentissage actif et pour différentes stratégies +de sélection de données. . . . . . . . . . . . . . . . . . . . . . . . . . . 117 +Table 6.7 +Résultats des modèles de détection de lignes de texte sur l’ensemble +de test du jeu de données Hugin-Munin après apprentissage actif et +pour différentes stratégies de sélection de données. . . . . . . . . . . . 118 +Table 7.1 +Tableau récapitulatif de différentes modélisations de la position et +forme des objets à détecter. +. . . . . . . . . . . . . . . . . . . . . . . 124 +Table 7.2 +Stratégies de prédiction séquentielle des rectangles englobants. . . . . 125 +Table 7.3 +Statistiques du jeu de données IAM utilisé pour la détection de lignes +de texte. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +Table 7.4 +Résultats de reconnaissance de textes manuscrits sur le jeu de données +IAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 +Table 7.5 +Résultats des modèles de détection de lignes sur le jeu de données +IAM, donnés en fonction du critère de sélection. . . . . . . . . . . . . 135 +xvi + +L I S T E D E S F O C U S +Focus 2.1 +Architecture CNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +Focus 2.2 +Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +Focus 2.3 +Regroupement / Pooling . . . . . . . . . . . . . . . . . . . . . . . . . +17 +Focus 2.4 +Système R-CNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +Focus 2.5 +Système YOLO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +Focus 2.6 +Architecture FCN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +Focus 2.7 +Convolution transposée . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +Focus 2.8 +Convolution dilatée . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +Focus 2.9 +Système dhSegment . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +Focus 2.10 +Système de Yang et al. . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +Focus 2.11 +Architecture Transformer . . . . . . . . . . . . . . . . . . . . . . . . . +31 +Focus 2.12 +Attention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +Focus 2.13 +Encodage positionnel . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +Focus 2.14 +Architecture Vision Transformer . . . . . . . . . . . . . . . . . . . . . +35 +Focus 2.15 +Système Pix2Seq +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +Focus 2.16 +Apprentissage actif / Active learning +. . . . . . . . . . . . . . . . . . +40 +Focus 3.1 +Précision et rappel +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +Focus 3.2 +Intersection-sur-Union +. . . . . . . . . . . . . . . . . . . . . . . . . . +54 +Focus 3.3 +F1-score +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +Focus 3.4 +Précision moyenne / Average precision +. . . . . . . . . . . . . . . . . +56 +xvii + +L I S T E D E S A L G O R I T H M E S +Algorithme 5.1 +Calcul du CER@page . . . . . . . . . . . . . . . . . . . . . . . . . . . +95 +xviii + +P U B L I C AT I O N S +— A. +Hazem, +B. +Daille, +M.-L. +Bonhomme, +M. +Maarand, +M. +Boillet, +C. +Kermorvant et D. Stutzmann (mai 2020). « Books of Hours : the First Li- +turgical Corpus for Text Segmentation ». In : 12th Language Resources and Evaluation +Conference (LREC), p. 776-784 +— M. Boillet, C. Kermorvant et T. Paquet (jan. 2021a). « Multiple Document +Datasets Pre-training Improves Text Line Detection With Deep Neural Networks ». +In : 25th International Conference on Pattern Recognition (ICPR), p. 2134-2141 +— M. Boillet, M. Maarand, T. Paquet et C. Kermorvant (sept. 2021b). « Inclu- +ding Keyword Position in Image-Based Models for Act Segmentation of Historical +Registers ». In : 6th International Workshop on Historical Document Imaging and +Processing (HIP), 31–36 +— J.-L. Debezia, M. Boillet, C. Kermorvant et Q. Barral (sept. 2021). « Drilling +a Large Corpus of Document Images of Geological Information Extraction ». In : +Machine Learning and Principles and Practice of Knowledge Discovery in Database +(ECML PKDD), p. 525-530 +— M. Boillet, C. Kermorvant et T. Paquet (mars 2022b). « Robust Text Line +Detection in Historical Documents : Learning and Evaluation Methods ». In : Interna- +tional Journal on Document Analysis and Recognition (IJDAR), p. 1433-2825 +— M. Boillet, C. Kermorvant et T. Paquet (2022a). « Confidence Estimation for +Document Object Detection ». In : Submitted to Pattern Recognition Letters (PRL) +xix + + +A C R O N Y M E S +AP +Average Precision +CER +Character Error Rate +CNN +Convolutional Neural Network +DAP +Dropout Average Precision +DLA +Document Layout Analysis +DOV +Dropout Object Variance +FCN +Fully Convolutional Network +HTR +Handwritten Text Recognition +IoU +Intersection-over-Union +mAP +Mean Average Precision +mAP-RFR +Mean Average Precision - Random Forest Regressor +MLP +Multi-Layer Perceptron +NER +Named Entity Recognition +OCR +Optical Character Recognition +PCE +Posterior Probability-based Confidence Estimator +WER +Word Error Rate +xxi + + +1 +I N T R O D U C T I O N +1.1 +C O N T E X T E +Les documents historiques constituent un patrimoine précieux que les archives, biblio- +thèques et certaines entreprises cherchent à protéger, préserver et rendre accessible au plus +grand nombre. Après de nombreuses années de numérisation, des millions d’images de do- +cuments sont maintenant disponibles dans le monde entier. Le contenu de ces images est +cependant souvent compréhensible uniquement par des experts, qui travaillent à rendre ac- +cessible cette grande quantité de contenus afin de permettre aux chercheurs et au grand +public de travailler plus facilement et efficacement. Pour cela, un long et coûteux travail de +transcription manuelle des documents est souvent nécessaire. Afin de rendre cette tâche plus +efficace, de nombreuses institutions cherchent à automatiser ce processus. +Grâce aux nouvelles technologies, et notamment l’amélioration majeure des méthodes +d’apprentissage profond, il devient désormais possible de transformer automatiquement les +documents originaux en documents digitaux, qui peuvent facilement être lus, traduits ou +encore dans lesquels il est possible de faire des recherches avancées, tout en nécessitant une +quantité plus raisonnable de travail de transcription manuelle. Dans le même temps, ces +évolutions ouvrent de nouvelles perspectives de recherche à la communauté du traitement +de document en mettant en évidence des documents complexes pour lesquels les avancées +récentes restent encore insuffisantes. +Les différentes tâches liées au traitement automatique de documents numérisés telles que +l’analyse de la mise en page (Document Layout Analysis (DLA)) ou la reconnaissance de +texte (Handwritten Text Recognition (HTR)) sont des problématiques étudiées depuis de +nombreuses années. Des solutions industrielles existent déjà mais sont souvent limitées à +Figure 1.1 – Pages présentant des difficultés de traitement : pages arrachées, dégradées et parties de +pages manquantes. À gauche et au centre, images 4 et 141 du Cartulaire de la famille +de Boussac 1et, à droite, pages 14 verso et 19 recto du Livre d’heures du Vatican 09488. +1. https://bvmm.irht.cnrs.fr/resultRecherche/resultRecherche.php?COMPOSITION_ID=28605 +1 + +cimefcndpatewleSeae +135 +peaaarayteudeyortapalarafujponarcnaerofa +1262 +fugu.Ya +anaCetamacutzCagbrenaggepaAdond +anca +astBinrglootnaerovnuederla +nieeBeSonatCanaialey +6F116OC +RIGH +14V. +19r. +CopyrightBiblioteca Apostolica Vaticana +http://digi.vatlib.it/view/MSSVat.lat.9488/0016 +poweredbyAMLAD·NTTDATA2 +I N T R O D U C T I O N +D´etection +des lignes +Extraction +des lignes +HTR +NER +Figure 1.2 – Chaîne de traitement standard impliquant une détection de lignes de texte, une recon- +naissance du texte manuscrit (HTR) suivi d’une détection des entités nommées (NER). +des documents modernes ou simples (mise en page simple, documents non dégradés). Les +récentes avancées en Machine Learning et plus particulièrement en Deep Learning permettent +désormais de lever ces limitations et d’améliorer la qualité des traitements automatiques. Ces +méthodes nécessitent cependant une quantité importante de données annotées manuellement. +Pour les documents dont les mises en page sont simples et dont il est facile et rapide d’en +annoter de grandes quantités, le traitement automatique a obtenu de très bons résultats. Au +contraire, les jeux de données disponibles pour le traitement de documents plus complexes, +tels que des documents historiques, sont très réduits. Cela est principalement dû au fait +que les documents sont très variés, et donc coûteux à annoter manuellement. De plus, +comme montré sur la Figure 1.1, les conditions de conservation et de numérisation peuvent +mener à des manuscrits abîmés avec notamment des pages tâchées, arrachées ou dégradées. +Pour toutes ces raisons, de nombreuses recherches s’intéressent à améliorer le traitement +automatique de tels documents. +Avoir une version digitale d’un manuscrit historique permet, entre autres, de pouvoir faire +de la recherche par mots-clés ou de retrouver des noms de personnes ou encore des dates. +Pour parvenir à cela, plusieurs étapes sont appliquées à chaque page numérisée. Une chaîne +de traitement utilisée dans de nombreux projets est présentée sur la Figure 1.2. +1.1.1 +analyse de la mise en page +En entrée de la chaîne, nous disposons d’une image d’une page ou d’une double-page +d’un document numérisé. Une première étape réalisée consiste à analyser la mise en page +du document. L’objectif de ce premier module est d’identifier les diverses régions physiques +d’un document et leurs caractéristiques. Cela revient donc à détecter différents éléments sur +l’image tels que les blocs de texte, images, graphiques ou encore lignes de texte. Ces régions +ne s’excluent pas mutuellement et une région peut contenir d’autres types de régions. +En plus de ces entités physiques, des étiquettes fonctionnelles ou logiques telles que des +titres ou légendes peuvent être attribuées à certaines de ces régions. Le processus d’analyse + +Thauuc +tth +giniLn +B贝人 +Bleo +Bge +cloghLen +dmuhLe neuf fevrier mil neuf centLe +date +neuf fevrier mil neuf cent1.1 C O N T E X T E +3 +Figure 1.3 – Détection d’objets sur l’image de la page 17 recto du Livre d’heures Horae ad usum +Romanum, Bibliothèque nationale de France, Département des manuscrits, NAL 3111. +Source https://gallica.bnf.fr/. +de la structure et de la mise en page d’un document tente donc de décomposer l’image d’un +document donné en ces régions et de comprendre leurs rôles fonctionnels et leurs relations. +Dans de nombreux cas d’usage, l’analyse de la mise en page d’un document revient à +détecter les lignes de texte dans le but d’appliquer un reconnaisseur sur ces lignes. Cependant, +certaines études s’intéressent également à d’autres éléments tels que les miniatures et initiales +dans les livres d’heures (Boillet et al., 2019) (exemple Figure 1.3), les actes (Vézina et al., +2020) ou encore les tableaux de recensement (Constum et al., 2022). Ces tâches nécessitent +des traitements plus spécifiques. En effet, la détection d’actes est souvent accompagnée d’une +classification selon le type d’acte présent (baptême, mariage, décès). Il en est de même pour +les tableaux, qui peuvent être traités de différentes manières : détection des lignes uniquement +ou conjointement avec les colonnes ou encore détection des cellules. +1.1.2 +reconnaissance de texte +Une fois les lignes obtenues, elles subissent chacune un traitement menant à une version +digitale du texte manuscrit (HTR). Enfin, ce texte peut être conservé tel quel, traduit dans +une autre langue, ou encore traité afin d’obtenir les entités présentes dans le document +(Named Entity Recognition (NER)). Des recherches récentes commencent à proposer des +systèmes qui s’affranchissent de la détection des lignes de texte et permettent de transcrire +le texte de l’image complète (Bluche, 2016 ; Coquenet et al., 2022 ; Singh et al., 2021 ; +Yousef et al., 2020). Malgré des premiers résultats prometteurs, ces systèmes sont encore +limités à des documents simples ou avec une grande régularité de mise en page. + +tnma +lotme +ntabusndmoiti +hs.amauutulhsnrtus +eterpobantnarouh +omtouuaammMiniature +L'igne +Text +eterpobatntmaouh +Marge ornée4 +I N T R O D U C T I O N +Dans cette thèse, nous étudions les tâches liées à l’analyse de la mise en page telles que la +détection de lignes de texte, d’actes ou encore de pages. Nous nous concentrons sur l’applica- +tion de méthodes basées sur les réseaux de neurones profonds pour la détection d’objets dans +les images de documents, principalement historiques. De nombreux systèmes permettant de +résoudre ces différentes tâches ont été proposés dans la littérature (Ares Oliveira et al., +2018 ; Grüning et al., 2019 ; Mechi et al., 2021), cependant, ils sont souvent évalués uni- +quement sur la tâche de détection de lignes de texte, et sont difficilement généralisables à des +documents aux structures plus diverses. Ainsi, dans cette thèse, nous cherchons à développer +des modèles plus génériques et à réaliser des évaluations plus complètes. +De plus, les systèmes actuellement utilisés pour la détection d’objets dans les images de +scènes naturelles, tels que les modèles YOLO (Redmon et al., 2016 ; 2017 ; 2018) et R-CNN +(Girshick, 2015 ; Girshick et al., 2014 ; Ren et al., 2015), sont difficilement applicables aux +documents historiques. Une des raisons à cela est l’importante quantité de données annotées +qu’ils nécessitent pour être entraînés. Ainsi, il devient nécessaire de développer des systèmes +moins complexes en termes de nombre de paramètres, de combiner plusieurs bases, de recourir +au Transfer learning (Das et al., 2018) ou d’augmenter la quantité de données. Ces différents +points ont été étudiés durant la thèse et les conclusions seront présentées dans la suite. +Enfin, des systèmes à base de Transformers (Vaswani et al., 2017) ont commencé à être +proposés afin de résoudre plus efficacement les tâches liées aux problèmes séquence-à-séquence +telles que la traduction de texte. À la suite de cela, certains travaux ont adapté ces systèmes +aux tâches de vision et ont montré qu’ils permettent d’obtenir de très bonnes performances +pour la classification d’images (Dosovitskiy et al., 2021) ou la détection d’objets (Chen +et al., 2022). Nous nous sommes également intéressés à cette catégorie de systèmes, qui +permettent d’avoir des sorties structurées des objets prédits. +1.2 +C A D R E D E L A T H È S E +Cette thèse a été réalisée au sein de l’entreprise Teklia 2 dans le cadre d’une collaboration +avec le Laboratoire d’Informatique, de Traitement de l’Information et des Systèmes (LITIS) 3 +à l’Université de Rouen Normandie. +Teklia a été fondée en 2014 et est spécialisée dans la compréhension automatique de docu- +ments. L’entreprise travaille sur diverses applications telles que le traitement automatique de +documents historiques (livres d’heures, chartes), mais également le traitement de documents +plus récents comme des tableaux de recensement de la population française. Les activités +de recherche de l’entreprise s’inscrivent dans des projets de recherche français mais aussi +internationaux comme les projets HOME 4, HuginMunin 5 et Balsac 6. +2. https://teklia.com +3. https://www.litislab.fr +4. https://www.history-of-medieval-europe.eu +5. https://hugin-munin-project.github.io/ +6. https://balsac.uqac.ca + +1.3 O B J E C T I F S E T C O N T R I B U T I O N S +5 +L’entreprise possède également une équipe spécialisée dans le développement, qui réalise +à la fois des projets pour des clients, mais intègre également les résultats de l’équipe de +recherche dans des applications 7 et facilite le travail de recherche en développant, notamment +une plateforme d’annotation 8. +L’entreprise travaille sur de nombreux projets et les produits des travaux de recherche y +sont directement intégrés, et donc appliqués dans de réelles conditions industrielles. Pour +répondre aux demandes des projets, il est nécessaire d’avoir un détecteur d’objets robuste, +performant et rapide pour traiter de grandes quantités de documents. De plus, il n’est pas +rare que dans un projet il y ait peu, voire aucune donnée annotée. Il est donc également +nécessaire d’avoir un détecteur assez générique afin de traiter ces documents plus facilement +et d’estimer automatiquement la qualité des résultats fournis. +1.3 +O B J E C T I F S E T C O N T R I B U T I O N S +Les défis liés à la tâche de détection d’objets dans des images de documents sont nombreux, +d’autant plus dans un cadre dans lequel de nouvelles données sont souvent disponibles, tou- +jours plus variées et complexes. Les problématiques auxquelles nous cherchons à répondre +sont les suivantes : +— Comment détecter efficacement les objets présents dans des images de documents variés, +et à partir de peu d’exemples annotés manuellement ? +— Comment évaluer les modèles de détection pour représenter correctement la qualité des +objets prédits ainsi que leurs impacts sur les tâches suivantes ? +— Comment estimer la confiance d’un modèle de détection quant à la qualité de ses +prédictions ? +Pour répondre à toutes ces problématiques, plusieurs contributions ont été proposées durant +cette thèse. Elles nous permettent de proposer une étude complète de détection d’objets allant +de l’annotation manuelle à l’évaluation finale : +— Certains réseaux de neurones utilisés pour la détection d’objets fournissent un masque +de prédiction où chaque pixel appartient à une classe d’objet. Nous proposons un mo- +dèle de détection possédant peu de paramètres et rapide en inférence, produisant des +masques de prédiction très précis tout en nécessitant un nombre réduit de données +annotées. +— D’autres systèmes plus récents permettent de générer une sortie structurée des objets +détectés. Suivant cette idée, nous proposons un second modèle de détection qui montre +des performances encourageantes. +— Nous montrons que malgré une grande hétérogénéité entre les documents mais aussi +entre leurs annotations manuelles, l’entraînement de réseaux de neurones génériques +permet d’obtenir des modèles encore plus performants et applicables à de nouvelles +7. https://arkindex.teklia.com +8. https://callico.teklia.com + +6 +I N T R O D U C T I O N +données sans ré-entraînement. De plus, l’uniformisation des annotations entre les diffé- +rents jeux de données permet d’entraîner des modèles de meilleure qualité. +— Nous proposons d’utiliser des métriques d’évaluation qui sont davantage en accord avec +la tâche finale. En particulier, nous proposons des métriques liées à la reconnaissance +de texte afin d’évaluer les modèles de détection de lignes de texte. +— Les données annotées sont souvent disponibles en faible quantité. Ainsi, nous proposons +différents estimateurs de confiance et montrons, dans un cadre d’active learning, qu’ils +permettent d’obtenir des modèles de détection d’objets plus performants avec moins +d’exemples annotés. +1.4 +O R G A N I S AT I O N D U M A N U S C R I T +Cette thèse est composée, outre cette introduction, de six chapitres. +Chapitre 2 : État de l’art +Le chapitre 2 présente un aperçu de l’état de l’art dans plusieurs domaines. Une revue +des différentes approches de détection de lignes de texte et d’objets dans des images +est présentée, allant des premières méthodes de traitements d’images aux plus récents +systèmes établis à partir de réseaux neuronaux. De plus, les récentes méthodes combinant +l’utilisation de l’image et du texte pour la détection d’objets sont décrites. Enfin, nous y +présentons les techniques permettant d’estimer une confiance reflétant la qualité d’une pré- +diction, élément crucial lorsque les systèmes de détection sont utilisés en phase de production. +Chapitre 3 : Entraînement et évaluation des systèmes de détection +Nous présentons, dans le chapitre 3, une revue des différents jeux de données utilisés +pour la détection d’objets dans les images de documents. Par la suite, nous proposons une +étude des stratégies d’entraînement et d’évaluation utilisées par les systèmes récents avec, +notamment, le détail des métriques d’évaluation basées sur les pixels et sur les objets. +Chapitre 4 : Détection d’objets dans des images de documents +Dans le chapitre 4, nous introduisons une architecture simple, rapide et efficace, mise au +point afin de détecter des objets dans les images de documents au niveau pixel. La détection +est réalisée grâce à un réseau de neurones entièrement convolutif. Ce chapitre décrit les +détails d’architecture ainsi que les avantages de celle-ci par rapport aux systèmes existants. +Enfin, les résultats de différentes expérimentations sur les tâches de détection de lignes de +texte et d’actes y sont présentés et discutés. +Chapitre 5 : Entraînement et évaluation d’un modèle robuste de détection +d’objets +Le chapitre 5 propose une étude avancée des techniques d’entraînement et d’évaluation des +systèmes de détection d’objets. Il expose la grande hétérogénéité et les incohérences des an- +notations des différents jeux de données actuellement disponibles, et présente une technique + +1.4 O R G A N I S AT I O N D U M A N U S C R I T +7 +d’uniformisation des annotations mise au point durant la thèse. De plus, ce chapitre met en +lumière les limitations des métriques d’évaluation actuellement utilisées et détaille plusieurs +métriques que nous proposons afin de lever ces limitations. Les résultats d’entraînements +de modèles de détection de lignes de texte à grande capacité de généralisation sont enfin +présentés. +Chapitre 6 : Estimation de la confiance des prédictions +Nous proposons, dans le chapitre 6, différents estimateurs de confiance. Dans un premier +temps, des estimateurs basés sur le modèle de détection d’objets entraîné sont présentés. +Des estimateurs basés sur un apprentissage externe au détecteur sont ensuite détaillés. Une +étude comparative des différentes approches est menée sur deux tâches de détection de pages +et de lignes de texte. +Chapitre 7 : Détection séquentielle d’objets dans des images de documents +Le chapitre 7 présente une seconde architecture de détection d’objets, celle-ci étant établie +à partir de Transformers. Les détails de l’architecture sont présentés ainsi que la justification +des choix de conception. Des premiers résultats d’expérimentations sont également présentés. +Chapitre 8 : Conclusions et perspectives +Dans le chapitre 8, nous concluons sur l’ensemble des travaux proposés et énonçons des +pistes de recherche complémentaires. + + +2 +É TAT D E L’ A RT +Les recherches axées autour de la mise en place et de l’amélioration de modèles de +détection d’objets sont toujours très actives, et ont motivé un nombre croissant de travaux +ces dernières années du fait d’importantes avancées dans le domaine de l’apprentissage +automatique. Dans ce chapitre, nous présentons une étude des travaux liés à la détection +d’objets en évoquant les premiers systèmes permettant de séparer les blocs de texte du +fond des images ainsi que les méthodes les plus récentes basées sur des réseaux de neurones +profonds. De plus, nous passons en revue différents systèmes d’estimation de la qualité des +prédictions. +En section 2.1, nous décrivons les méthodes ad hoc de détection d’objets dans les images de +documents. Nous présentons ensuite les méthodes proposées à base d’apprentissage profond, +avec notamment les approches pixel dans la section 2.1.2 et celles à base de Transformers en +section 2.1.2. Enfin, la section 2.2 présente les travaux permettant d’estimer la qualité des +prédictions, peu de travaux ayant été publiés pour la tâche de détection d’objets. +2.1 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +La mise en page d’un document fait référence à la position physique et aux limites des +différentes régions dans l’image du document. Le processus d’analyse de la mise en page d’un +document vise à décomposer une image de document en une hiérarchie de régions, telles +que les figures, l’arrière-plan, les blocs de texte, les lignes de texte, les mots, les caractères, +etc. Depuis plusieurs années, différentes méthodes permettant de détecter des objets dans +des images de documents ont émergé. Ces différentes approches peuvent être divisées en +deux groupes : les méthodes ad hoc et les méthodes par apprentissage automatique. De plus, +dans chacun de ces deux groupes, il existe des algorithmes dits ascendants et descendants +(Namboodiri et al., 2007 ; Song et al., 2003). +Les algorithmes ascendants partent des plus petits composants d’un document (pixels ou +composantes connexes) et les regroupent de manière itérative pour former des régions plus +grandes telles que les caractères, qui sont ensuite regroupés en mots, lignes ou blocs de texte. +En revanche, les algorithmes descendants partent de l’image complète du document et la +divisent itérativement en sous-images pour former des régions de plus en plus petites. La +procédure de découpage s’arrête lorsqu’une certaine condition est vérifiée, les sous-images +obtenues à ce stade constituent les résultats finaux de la segmentation. En outre, il existe +également des approches hybrides qui utilisent une combinaison de stratégies ascendantes et +descendantes. +9 + +10 +É TAT D E L’ A RT +2.1.1 +méthodes ad hoc +Les approches ad hoc sont basées sur la combinaison de différentes techniques d’analyse +d’image telles que le regroupement, les profils de projection ou encore le filtrage. Elles sont +établies pour un type d’images donné et sont peu généralisables à un grand nombre et une +grande variété d’images de documents mais sont encore aujourd’hui utilisées (Eskenazi et +al., 2017). +Les premières méthodes ayant vu le jour permettaient de séparer les contenus textuels +des contenus graphiques d’une image sans nécessiter d’annotations manuelles. Parmi les algo- +rithmes descendants, le Run-Length Smoothing Algorithm (RLSA) (Wong et al., 1982) a été +proposé pour segmenter les pages de documents. Cet algorithme fonctionne sur des images bi- +naires dans lesquelles deux pixels noirs voisins éloignés d’une distance maximale donnée sont +fusionnés en une séquence continue de pixels noirs. Le RLSA est d’abord appliqué ligne par +ligne, puis colonne par colonne, et les deux bitmaps résultants sont combinés en appliquant +une opération logique "ET" à chaque position de pixel. L’inconvénient de cet algorithme est +qu’il ne peut être utilisé que pour extraire de petits blocs. Par la suite, la méthode du XY- +Cut (Nagy et al., 1984) a été proposée afin de détecter les blocs de texte dans des images en +niveaux de gris. Cette méthode consiste à utiliser une projection horizontale et verticale des +valeurs des niveaux de gris des pixels afin de trouver les espaces interlignes et intercolonnes. +Les projections sont faites de manière itérative menant à des objets homogènes. Cette tech- +nique permet d’obtenir une détection de grande qualité mais est limitée à des documents dont +la mise en page est simple. En effet, elle est incapable de prédire des objets corrects sur des +images dans lesquelles les lignes sont mal alignées, ou si le document est légèrement incliné. +Akindele et al. (1993) ont proposé une amélioration de cette méthode afin de corriger l’in- +clinaison des lignes, cependant, d’autres problèmes persistent tels que la difficulté du système +à traiter des documents comportant des illustrations. Ces méthodes semblent difficilement +applicables à des documents historiques qui ne sont pas que textuels et qui ont des mises en +page complexes. Pour résoudre le problème posé par les images de pages obliques, Pavlidis +et al. (1992) ont proposé une méthode basée sur les « flux blancs ». Ils émettent l’hypo- +thèse que les colonnes de texte d’une page contiennent un type unique de données (texte ou +illustration) et qu’elles sont suffisamment espacées pour être distinguées des autres espaces +tels que l’espacement entre les mots. La méthode identifie donc les larges espaces blancs afin +d’estimer l’angle d’inclinaison de la page puis de localiser les objets comme étant les régions +entre ces espaces. Cette méthode permet également de traiter des documents plus complexes +contenant, entre autres, des illustrations. +Les systèmes présentés ci-dessus permettent de traiter des documents ayant des mises +en page de type Manhattan. Il s’agit de pages ayant des composants de formes arbitraires +(rectangulaires) où les segments des blocs sont parallèles ou perpendiculaires les uns par +rapport aux autres. Dans cette thèse, nous nous intéressons principalement aux documents +historiques. Ces méthodes semblent donc difficilement applicables à de tels documents dont les +colonnes contiennent rarement des types uniques de données, comme montré sur les Figures +1.1 et 1.3, et dont les mises en page sont non-Manhattan. + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +11 +Les méthodes ascendantes permettent de traiter des documents beaucoup plus variés aux +mises en page complexes mais sont en général plus lentes. Une des premières méthodes, +présentée par Kise et al. (1998), se base sur le diagramme de Voronoi pour la segmentation +d’images de pages. Les auteurs détectent tout d’abord les points des bords des composantes +connexes et construisent un diagramme de Voronoi à partir de ces points. Les arêtes de +Voronoi détectées entre des caractères, mots et lignes de texte d’un même bloc sont ensuite +filtrées pour garder uniquement celles qui séparent les blocs du document. Un désavantage +à cette méthode est qu’elle segmente parfois les illustrations et les titres ayant des polices +d’écriture larges. O’Gorman (1993) a présenté DocStrum, qui repose sur un regroupement +des plus proches voisins appliqué aux composantes connexes. Par la suite, ces deux propo- +sitions ont été conjointement utilisées et améliorées par Agrawal et al. (2009) avec leur +système appelé Voronoi++. Il a été mis au point pour répondre au manque d’adaptation des +systèmes existants aux variations de taille, d’orientation et de distance des composants d’une +page. Au lieu d’utiliser des relations linéaires entre la distance et le rapport de surfaces des +composantes connexes, les auteurs montrent que la détermination dynamique de ces relations +et la combinaison des caractéristiques angulaires et des caractéristiques de voisinage, ces +dernières venant de l’approche de DocStrum, améliorent la précision. +Pour lever les limitations liées aux systèmes présentés ci-dessus telles que le temps de +traitement, d’autres méthodes ont émergées. Celles-ci sont basées sur des algorithmes plus +robustes face aux images de documents couleurs et en niveaux de gris. De plus, elles ne +sont plus limitées aux documents possédant des structures et contenus simples. Dans cette +optique, Coüasnon (2006) a conçu et publié un langage de grammaire de mise en page +appelé DMOS. Il permet de décrire une grande variété de mises en page et l’analyseur syn- +taxique associé reconnaît cette disposition dans une image. La grammaire permet également +d’associer une étiquette à chaque région. Par la suite, la méthode a été améliorée (Lemaitre +et al., 2008) en intégrant une approche multirésolution lui permettant de segmenter des +lettres manuscrites et d’identifier les lignes de texte dans des documents administratifs. +Dans la même idée, Shafait et al. (2008) ont proposé un autre algorithme de grammaire +basé sur une formulation probabiliste de la mise en page. L’utilisateur définit un ensemble +de coupes horizontales et verticales dont la position est définie de manière approximative. +Ensuite, pour chaque image, un ajustement probabiliste est effectué pour obtenir les régions +finales. Cet algorithme est capable de segmenter des mises en page serrées avec de faibles +marges. Bien que cette méthode ainsi que DMOS aient obtenu de très bonnes performances, +les systèmes reposent sur l’hypothèse que les documents à traiter ont une mise en page +homogène puisqu’ils nécessitent que l’utilisateur définisse des règles de mise en page. +D’autres systèmes ont ensuite été proposés afin de traiter des documents complexes et ne +nécessitant pas de modèle de mise en page prédéfini. Par exemple, Louloudis et al. (2007) +ont utilisé la transformée de Hough sur un ensemble de composantes connexes sélectionnées +pour extraire les lignes de texte. Cette approche, basée sur la transformée de Hough, n’est +adaptée qu’aux images de documents où les lignes ne sont pas incurvées. Journet et al. + +12 +É TAT D E L’ A RT +(2008) utilisent une approche ascendante basée sur les textures des images de documents. Ils +extraient cinq caractéristiques liées aux fréquences et orientations calculées à quatre résolu- +tions, ainsi chaque pixel de l’image possède 20 valeurs. Ils utilisent ensuite un algorithme +de groupement afin de regrouper les pixels correspondant à des zones homogènes. Ils ont +testé leur méthode sur des documents modernes et historiques, et ont souligné l’importance +d’une approche multirésolution pour réduire le bruit dans les techniques ascendantes. Dans +Shi et al. (2009), les auteurs proposent une technique appelée ALCM (Adaptive Local +Connectivity Map). Ils utilisent des filtres directionnels orientables pour détecter les lignes +de texte et appliquent des post-traitements heuristiques pour séparer les lignes connectées. +Cette méthode descendante a obtenu des résultats intéressants sur la détection de lignes de +texte, l’algorithme ayant été conçu pour résoudre les problèmes particulièrement complexes +observés dans les documents manuscrits, notamment les lignes de texte qui fluctuent, se +touchent ou se superposent. Par la suite, Erkilinc et al. (2012) ont proposé une méthode +de segmentation robuste face aux fonds et aux structures complexes. Cette approche permet +de résoudre un problème de détection à trois classes : texte, photographie et ligne. Tout +d’abord, l’image subit une étape de prétraitement qui consiste à réaliser un filtrage, une +conversion de l’espace couleur et une correction gamma. Les éléments sont ensuite détectés +grâce à plusieurs techniques telles que la transformée en ondelettes et le codage par plages. +Les objets détectés sont enfin combinés par un algorithme de K-moyennes. Cette méthode +de classification en blocs et en pixels a montré de bons résultats. Cependant, comme la +plupart des méthodes présentées ici, elle consiste en plusieurs opérations successives et est +coûteuse en temps. De plus, cette méthode ne permet de résoudre qu’un problème spécifique +avec trois classes très distinctes. Enfin, une autre méthode ascendante qui a obtenu de bons +résultats pour détecter les lignes de texte est décrite dans Ryu et al. (2014). L’approche est +basée sur les super-pixels pour obtenir des composantes connexes. Les auteurs définissent +une fonction de coût pour agréger les super-pixels en une ligne de texte. Cette méthode a +gagné la compétition de l’International Conference on Document Analysis and Recognition +(ICDAR) sur la détection des lignes de texte (Murdock et al., 2015). +Concernant les méthodes hybrides, un travail récent proposé par Tran et al. (2015) +utilise la méthode Multilevel Homogeneous Structure (MHS), et a remporté la compé- +tition de segmentation de documents complexes en 2015 (Antonacopoulos et al., +2015). La méthode implique à la fois l’analyse en composantes connexes et l’analyse des +espaces blancs (arrière-plan). Tout d’abord, l’image est binarisée puis les composantes +connexes sont détectées et celles considérées de manière fiable comme étant du bruit ou +des régions sans texte sont filtrées. Une classification multiniveaux est effectuée, basée +sur l’analyse des régions homogènes multiniveaux et des espaces blancs, pour identifier +toutes les composantes textuelles et non textuelles. Cette méthode a montré de bonnes +performances sur la compétition, notamment pour sa capacité à manquer très peu de régions. +Même si la plupart de ces méthodes ont obtenu de bons résultats sur un jeu de données +spécifique, elles doivent être affinées manuellement, ce qui est une tâche fastidieuse et dépend + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +13 +généralement de l’ensemble de données considéré. De plus, une fois mises en place, ces mé- +thodes sont souvent difficiles à maintenir et à améliorer. Enfin, la plupart des algorithmes +mentionnés ci-dessus ne créent pas de descriptions hiérarchiques ou ne permettent pas aux +utilisateurs de préciser des informations sur la structure du document. En outre, à part pour +les modèles à base de grammaire, ils ne fournissent pas de méthodes d’estimation des pa- +ramètres de l’algorithme à partir de données. En d’autres termes, ils ne sont pas dotés de +capacités d’apprentissage. +2.1.2 +méthodes par apprentissage profond +Pour répondre à ces difficultés, des méthodes basées sur un apprentissage ont été proposées +afin d’apprendre automatiquement la variabilité des documents à partir de données. Nos +travaux se positionnent dans ce cadre. +Les méthodes par apprentissage automatique sont actuellement principalement constituées +d’algorithmes de réseaux de neurones profonds. Ces algorithmes permettent à la fois d’ap- +prendre automatiquement les caractéristiques importantes des images et d’effectuer la tâche +requise. Ils ont tendance à être une « boîte noire » dont le fonctionnement est difficile à +expliquer, cependant ils permettent de traiter des documents complexes que les méthodes ad +hoc sont incapables de traiter. +Les approches par apprentissage profond ont obtenu de bons résultats dans de nombreux +domaines d’application (LeCun et al., 2015), ainsi, de nombreux travaux ont étudié leur utili- +sation pour la détection d’objets dans les images. Puisque de multiples recherches s’orientent +autour de la détection d’objets dans des images en général et non spécifiquement dans des +images de documents, la section suivante passe en revue quelques travaux dans ces domaines +connexes de détection d’objets et de textes dans des images de scènes naturelles. Dans le +domaine de la vision par ordinateur, la littérature sur la détection d’objets peut être divisée +en trois catégories principales : les systèmes basés sur la proposition de régions, l’estimation +de la position des boîtes englobantes par régression et la détection au niveau du pixel. +proposition de régions +Pour la tâche de détection de texte dans des images de scènes naturelles, les premiers +travaux basés sur des approches par apprentissage profond utilisent une méthode de fenêtre +glissante (Zhu et al., 2016). Des parties d’images sont d’abord extraites à l’aide d’une fenêtre +glissante, puis elles sont étiquetées grâce à un réseau de neurones profond. L’utilisation +d’une fenêtre glissante induit un temps de traitement élevé et limite le contexte qui peut être +utilisé pour prendre une décision. Pour limiter le temps de traitement, une solution consiste +à utiliser un prétraitement pour extraire les candidats et ensuite prendre une décision pour +chacun de ces candidats. C’est la méthode utilisée par Huang et al. (2014), qui extrait les +candidats grâce aux Maximally Stable Extremal Regions (MSER) et les classe à l’aide d’un +réseau de neurones convolutif (Convolutional Neural Network (CNN)) (LeCun et al., 1998). +L’architecture CNN est décrite à la fin de cette section, dans le Focus 2.1. + +14 +É TAT D E L’ A RT +De la même manière, l’idée d’extraire les candidats avant de les classer a été utilisée pour +la détection d’objets. Ces systèmes, basés sur la proposition de régions, consistent en trois +étapes consécutives. Tout d’abord, un ensemble de propositions de régions indépendantes +de la catégorie est généré. Ensuite, un CNN est appliqué sur ces régions pour extraire les +informations significatives, et un classificateur prédit la classe de chaque proposition de région. +Cette stratégie a été proposée pour la première fois par Girshick et al. (2014) avec leur +système R-CNN, détaillé dans le Focus 2.4, et appliquée aux images de scènes naturelles des +jeux de données VOC 2010-2012. Malgré le développement de systèmes plus avancés (Fast +R-CNN (Girshick, 2015), Faster R-CNN (Ren et al., 2015) et Zhong et al. (2017)), cette +méthode a été peu adoptée par la communauté du traitement d’images de documents. En effet, +ces systèmes sont bien adaptés aux images de scènes naturelles où seuls quelques objets sont +présents sur les images, contrairement aux images de documents qui contiennent de nombreux +objets de toutes tailles. De plus, malgré différentes améliorations qui ont permis d’accélérer +ces systèmes, ils restent complexes et peu efficients, d’où l’introduction des méthodes dites « +one stage » où l’on s’abstient de l’étape de proposition de régions. Certains de ces systèmes +sont présentés dans les paragraphes suivants. +Focus 2.1 – ARCHITECTURE CNN +Définition +Un réseau de neurones profond est une succession de plusieurs couches où chaque +couche est généralement composée d’une fonction paramétrée suivie d’une fonction +de non-linéarité (fonction d’activation), chaque couche calculant une nouvelle repré- +sentation de l’image d’entrée. Dans le cas d’un réseau neuronal convolutif (CNN), +les fonctions paramétrées sont des opérations de convolution, détaillées dans le +Focus 2.2. La partie convolutive d’un CNN permet d’extraire et de compresser les +caractéristiques de l’image d’entrée grâce à des couches de regroupement (pooling, +expliqué dans le Focus 2.3). +Avantages +— Un CNN est capable de capturer avec succès les dépendances spatiales d’une +image par l’application de filtres. L’architecture s’adapte au mieux à l’ensemble +des données grâce à la réduction du nombre de paramètres impliqués et à la +réutilisation des poids. +— L’architecture CNN rend possible l’apprentissage de modèles profonds avec re- +lativement peu de paramètres grâce au partage des poids entre les couches +convolutives. +— Chaque filtre d’une couche de convolution est appliqué à l’ensemble de l’image +d’entrée, ainsi le traitement d’une image par un CNN est invariant par transla- +tion. +— Comparés à d’autres algorithmes de traitement d’image, les CNN utilisent rela- +tivement peu de prétraitement. + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +15 +Inconvénients +— Pour entraîner un CNN, il est souvent nécessaire d’avoir de nombreuses données +annotées. +— Comme pour la plupart des systèmes à base de réseaux neuronaux, il peut être +coûteux en mémoire et en temps d’entraîner un CNN. +Exemples de systèmes de type CNN +— LeNet (LeCun et al. (1998)) : LeNet est la première architecture CNN. Il a été +développé en 1998 et a été appliqué avec succès à la tâche de reconnaissance +de chiffres manuscrits. L’architecture LeNet se compose de plusieurs couches +de convolution et de regroupement (pooling), suivies d’une partie entièrement +connectée. Le modèle comporte cinq couches de convolution suivies de deux +couches entièrement connectées. +Figure 2.1 – Schéma de l’architecture du modèle LeNet, issu de LeCun et al. (1998), pour +une image d’entrée de taille 32×32 pixels. +— AlexNet (Krizhevsky et al. (2012)) : AlexNet est l’architecture d’apprentis- +sage profond qui a popularisé le CNN. Le réseau AlexNet a une architecture très +comparable à celle de LeNet, mais est plus profond, plus grand et comporte des +couches convolutives empilées les unes sur les autres. AlexNet a été utilisé pour +remporter l’ImageNet Large Scale Visual Recognition Challenge (ILSVRC) en +2012. AlexNet est composé de cinq couches convolutives avec une combinaison +de couches de max-pooling, de trois couches entièrement connectées et de deux +couches de dropout. Le nombre total de paramètres dans cette architecture est +d’environ 60 millions. +Figure 2.2 – Schéma de l’architecture du modèle AlexNet, issu de Krizhevsky et al. +(2012), pour une image d’entrée de taille 224×224 pixels. Ici, deux cartes +graphiques sont utilisées, une traite la partie haute de l’image et l’autre la +partie basse. + +Cs: ft. maps 16@10xi10 +Ci: foature mapg +INPUT +S4: f. maps 16@5x5 +6@28x28 +32x82 +s2: t. mapg +cs: layer +t6: layer +120 +OUIPUT +6@iNx +84 +10 +F ull conneean +Ceuss en connectans +Convoutong +Subsamplng +Convolutlang +ubsamplng +Bull connectbn16 +É TAT D E L’ A RT +— VGG (Simonyan et al. (2015)) : VGGNet est un réseau CNN à 16 couches +comptant jusqu’à 95 millions de paramètres et entraîné sur plus d’un milliard +d’images (1000 classes). Il prend des images d’entrée de taille 224×224 pixels. +Il nécessite beaucoup de données d’entraînement, ce qui est la principale raison +pour laquelle les architectures telles que AlexNet fonctionnent mieux pour la +plupart des tâches de classification d’images où les images d’entrée ont une +taille comprise entre 100×100 pixels et 350×350 pixels. Le modèle VGG est +efficace et sert de base solide pour de nombreuses applications en raison de +son applicabilité à de nombreuses tâches, notamment la détection d’objets. Ses +représentations profondes des caractéristiques sont utilisées dans de nombreuses +architectures de réseaux neuronaux telles que YOLO (Redmon et al. (2016)). +Figure 2.3 – Schéma de l’architecture du modèle VGG-16 (Simonyan et al. (2015)) pour +une image d’entrée de taille 224×224 pixels. Schéma extrait de l’article de +Ferguson et al. (2017). +— ResNet (He et al. (2016)) : ResNet a été développé dans le cadre de la +compétition pour la tâche de classification de l’ILSVRC 2015. Le réseau +contient des connexions résiduelles en plus des couches habituelles d’un CNN. +Outre les tâches de classification d’images, ResNet a été utilisé avec succès pour +résoudre des problèmes de traitement du langage naturel comme la complétion +de phrases ou la compréhension automatique par l’équipe Microsoft Research +Asia en 2016 et 2017 respectivement. +Figure 2.4 – Schéma de l’architecture du modèle ResNet-34, issu de He et al. (2016). + +tr2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +17 +Focus 2.2 – CONVOLUTION +Définition +La couche de convolution est le bloc de base utilisé dans les réseaux dits convolutifs +(CNN). Une couche de convolution permet de générer une nouvelle représentation +de l’image d’entrée ou intermédiaire (en sortie de la couche précédente). Pour cela, +elle possède un ou plusieurs filtres de convolution qui traitent une portion limitée, +le champ réceptif, de l’image d’entrée. Chaque filtre est défini par un ensemble de +poids appris durant l’entraînement du modèle et analyse une caractéristique de +l’image d’entrée (caractéristique de couleur, de texture. . .). Pour cela, chaque filtre +est appliqué à chaque pixel de l’image, calculant une nouvelle représentation pour +chacun de ces pixels. Dans la plupart des cas, le filtre a une taille plus grande que +1 ce qui mène à utiliser du contexte, les pixels voisins, pour calculer la nouvelle +représentation du pixel. +Schéma d’une convolution 2D +La Figure 2.5 présente le schéma d’une convolution 2D avec X l’image d’entrée, W +le filtre et Y la nouvelle représentation de l’image. Dans cet exemple, le filtre W a +une taille 3×3, ce qui implique que, pour calculer la représentation d’un pixel, les +valeurs de ses huit pixels voisins sont prises en compte. +x0,0 +x1,0 +x2,0 +x3,0 +x0,1 +x1,1 +x2,1 +x3,1 +x0,2 +x1,2 +x2,2 +x3,2 +x0,3 +x1,3 +x2,3 +x3,3 +X +w0,0 +w1,0 +w2,0 +w0,1 +w1,1 +w2,1 +w0,2 +w1,2 +w2,2 +W +y1,1 +y0,1 +y1,0 +y0,0 +Y +Figure 2.5 – Schéma d’une convolution 2D avec X l’image d’entrée, W le filtre et Y la +nouvelle représentation de l’image. +Focus 2.3 – REGROUPEMENT / POOLING +Définition +Le pooling consiste à regrouper des représentations locales ou globales en résumant +les valeurs de plusieurs pixels en une seule valeur unique. Les couches de regroupe- +ment réduisent les dimensions des données en combinant plusieurs entrées, et ainsi +extraient les caractéristiques dominantes. Il s’agit d’opérations simples, non para- +métriques, telles qu’un min (min pooling), un max (max pooling), une somme ou +encore une moyenne (average pooling). Dans un CNN, ces couches permettent à la +fois de réduire la taille des images intermédiaires en résumant les caractéristiques +qu’elles contiennent, mais aussi d’avoir davantage de contexte puisque les pixels +voisins sont regroupés. + +18 +É TAT D E L’ A RT +Focus 2.4 – SYSTÈME R-CNN +Le système R-CNN a été proposé par Girshick et al. (2014) et permet de réaliser +de la détection d’objets sur des images à partir de propositions de régions. Un +ensemble de régions (environ 2 000) est tout d’abord généré grâce à un algorithme +de recherche sélective. Les caractéristiques importantes de chacune de ces régions +sont ensuite extraites par un CNN. Enfin, un SVM linéaire prédit la classe de chaque +région. +Bien que ce système ait été utilisé avec succès afin de détecter des objets dans les +images de scènes naturelles, il reste très lent car prend en moyenne 47 secondes +pour traiter une image. +Figure 2.6 – Schéma du système R-CNN, issu de Girshick et al. (2014). +régression de boîtes englobantes +La détection d’objets dans des images a également été réalisée à l’aide de modèles de +prédiction des coordonnées des boîtes englobantes. Ces systèmes, fondés sur un algorithme +de régression, ont été introduits pour la première fois par Erhan et al. (2014) qui ont pro- +posé la méthode MultiBox. Celle-ci effectue une régression directe des positions des boîtes +englobantes au lieu de s’appuyer sur des propositions d’objets. Ils utilisent un CNN comme ré- +gresseur pour directement prédire un nombre donné de coordonnées de boîtes et une confiance +pour chaque boîte correspondant à sa probabilité de contenir un objet d’intérêt. Il permet de +détecter un nombre variable d’objets superposés de la même classe, la taille des objets n’étant +pas limitée. Mais, lorsqu’il est nécessaire de détecter un grand nombre d’objets, le nombre +de paramètres du modèle augmente et une grande quantité de données est nécessaire pour +l’apprentissage. YOLO et SSD peuvent être considérés comme des variantes de ce concept. +Redmon et al. (2016) ont proposé le modèle YOLO (You Only Look Once). L’objectif de +YOLO était de détecter et de classifier les objets en un seul traitement et d’être plus rapide +que les méthodes R-CNN. L’image est d’abord divisée en une grille régulière, puis chaque +cellule de la grille prédit un nombre prédéfini de boîtes englobantes avec leurs confiances ainsi +que les probabilités de classe grâce à un seul réseau neuronal. Les détections finales sont les +boîtes ayant le score de confiance le plus important et la probabilité de la classe la plus élevée +dans cette boîte. Ce système est présenté dans le Focus 2.5. De multiples méthodes ont ensuite +étendu l’idée originelle de YOLO (Bochkovskiy et al., 2020 ; Redmon et al., 2017 ; 2018) +mais très peu ont été appliquées aux images de documents, probablement pour la même raison +que celle mentionnée ci-dessus : les images de documents contiennent trop d’objets à détecter. + +waped rcgion +aetopl ane? mo. +petson? yes. +CNN +: +tyontor? mo +1. Jput +2.. Hixtiract regjon +3. Compute +4. Classifty +image +proposals (-2k) +CNN feauncs +Tegjons2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +19 +D’un autre côté, Liu et al. (2016) ont proposé SSD (Single Shot MultiBox Detector). Le +système discrétise l’espace de sortie des boîtes englobantes en un ensemble de boîtes prédé- +finies par défaut avec différents rapports d’aspect et échelles par emplacement de carte de +caractéristiques. Au moment de la prédiction, le réseau génère des scores reflétant la présence +de chaque catégorie d’objets dans chaque boîte par défaut, et ajuste la boîte pour mieux cor- +respondre à la forme de l’objet. De plus, le réseau combine les prédictions de plusieurs cartes +de caractéristiques de différentes résolutions pour traiter des objets de différentes tailles. SSD +est simple par rapport aux méthodes qui nécessitent des propositions d’objets, car il élimine +la génération de propositions et les étapes ultérieures de ré-échantillonnage de pixels ou de +caractéristiques. Cette méthode a montré de meilleurs résultats que Faster-RCNN et YOLO +sur les données VOC 2017 tout en étant plus rapide. +Bien qu’elles aient montré de très bonnes performances sur des images de scènes naturelles +où peu d’objets sont à détecter, ces méthodes sont moins adaptées au traitement d’images +de documents où il y a souvent un grand nombre d’éléments à localiser. Certains travaux +ont tout de même adapté ces systèmes aux images de documents. +Pour la détection de lignes de texte, les premières contributions ont été présentées par +Moysset et al. (2016a) ; Moysset et al. (2016b). Dans Moysset et al. (2016a), les auteurs +proposent une approche basée sur MultiBox pour détecter les boîtes englobantes des lignes +de texte en utilisant des poids partagés afin de permettre au système d’être entraîné sur une +quantité de données annotées réduite. Comme les modèles YOLO et SSD, les sorties sont +attribuées à des régions locales de l’image. Cependant, le modèle est capable de prédire les +objets dans sa région de support, ou en dehors. +Moysset et al. (2016b) proposent l’utilisation d’un réseau neuronal Multi Dimensional +Long Short Term Memory (MDLSTM) combiné à des couches convolutives pour prédire une +boîte englobante autour d’une ligne. Ils traitent la tâche de détection de lignes de texte comme +étant un problème de régression, et prédisent les coordonnées des boîtes englobantes directe- +ment à partir des valeurs des pixels des images. Ils ont comparé deux stratégies de régression : +prédire directement les boîtes englobantes et prédire séparément les points inférieurs gauche +et supérieurs droit avant de les coupler. La seconde stratégie a montré une réelle amélioration +pour la tâche de détection sur les documents du jeu de données Maurdor (Oparin et al., +2014) mais est limitée aux lignes horizontales. +Malgré les améliorations apportées aux modèles de régression de boîtes, cette approche est +toujours limitée aux éléments horizontaux et ne permet pas une détection précise des lignes +de texte par exemple. C’est pour cela que les méthodes niveau pixel ont été proposées. +Focus 2.5 – SYSTÈME YOLO +Le système YOLO (You Only Look Once) a été proposé par Redmon et al. (2016) +et permet de réaliser de la régression de boîtes englobantes d’objets sur des images. +YOLO divise l’image d’entrée en une grille régulière. Chaque cellule de la grille +prédit un nombre prédéfini de boîtes de délimitation et des scores de confiance pour + +20 +É TAT D E L’ A RT +chacune de ces boîtes. Enfin, les boîtes ayant les scores de confiance les plus élevés +et les probabilités de classe les plus élevées dans ces boîtes sont considérées comme +détections finales. +YOLO est beaucoup plus rapide que R-CNN. Il montre cependant plus de +difficultés à détecter des objets proches et les petits objets. +Figure 2.7 – Schéma du système YOLO, issu de Redmon et al. (2016). +détection niveau pixel +La détection d’objets au niveau pixel est actuellement l’approche la plus utilisée pour le +traitement d’images de documents. De nombreux systèmes ont été proposés et c’est également +dans ce cadre que se positionnent nos principaux travaux de recherche. La grande majorité +de ces systèmes se base sur l’architecture Fully Convolutional Network (FCN), expliquée en +détail dans le Focus 2.6, en fin de cette section. +Les premiers FCN proposés étaient composés d’une succession de convolutions et de +couches de regroupement (pooling) permettant de résumer les caractéristiques importantes +de l’image d’entrée. Ces systèmes, tels que le VGG (Simonyan et al., 2015) et le ResNet +(He et al., 2016), étaient principalement utilisés pour les tâches de classification avec une +classe unique en sortie. +Cependant, pour la segmentation sémantique ou la détection d’objets, il est nécessaire +d’avoir également la position de la classe dans l’image, c’est-à-dire une classe pour chaque +pixel de l’image d’entrée. Afin d’obtenir une telle sortie, Ciresan et al. (2012) ont entraîné +un réseau utilisant une fenêtre glissante et prédisant une classe pour chaque pixel grâce +à une région locale autour du pixel (un patch). Bien qu’il ait montré de très bonnes +performances en gagnant notamment la compétition sur la segmentation de structures +neuronales (ISBI 2012), le principal inconvénient de ce système est qu’il est très lent à +traiter une image, le modèle étant appliqué à chaque patch. Pour pallier cela, Long et al. +(2015) ont proposé un FCN pixel-à-pixel pour la tâche de segmentation sémantique d’images. +Les auteurs ont proposé une modification de l’architecture FCN en ajoutant, après les +couches standards de convolution et de regroupement (étape d’encodage), une étape de +décodage constituée d’une succession de couches équivalentes à l’encodeur dans laquelle les +opérations de regroupement sont remplacées par des opérations d’upsampling, augmentant +la résolution de sortie. L’upsampling étant réalisé à l’aide de convolutions transposées +(Focus 2.7). De plus, afin d’avoir une localisation plus précise, les auteurs proposent de +combiner les caractéristiques calculées durant l’étape d’encodage à celles du décodage. +Montrant de très bonnes performances et un temps d’inférence raisonnable, de nombreux + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +21 +autres travaux similaires ont vu le jour. Une modification de ce système a été proposée par +Ronneberger et al. (2015) avec leur architecture U-Net. Les auteurs se sont concentrés +sur le décodeur, l’encodeur étant comparable aux FCN que nous avons présentés plus tôt. +Ils ont proposé d’utiliser des matrices de caractéristiques avec de nombreux canaux durant +l’étape de décodage afin de propager davantage de contexte aux couches finales. Appliquée +sur différentes tâches de segmentation d’images médicales, cette architecture a montré des +gains importants de performances par rapport aux méthodes existantes. +Dans le domaine de la détection de texte dans des images de scènes naturelles, Zhang +et al. (2016b) appliquent également un FCN. Tout d’abord, un FCN TextBlock est utilisé +pour détecter les localisations approximatives des lignes de texte, qui sont ensuite extraites +en tenant compte des informations locales des caractères. Enfin, un autre FCN est appliqué +pour rejeter les fausses lignes de texte détectées. +Pour le traitement de documents, les FCN ont également été largement utilisés. En effet, +l’intérêt porté à l’analyse des documents a été stimulé par les compétitions sur la détection +des lignes de texte (Murdock et al., 2015), la détection des lignes de base (Diem et al., 2017 ; +Diem et al., 2019) ou l’analyse de la mise en page (Antonacopoulos et al., 2011). La plu- +part de ces tâches ont été abordées au niveau pixel, et donc de nombreux systèmes de type +FCN ont été développés. Ainsi, dhSegment (Ares Oliveira et al., 2018) a été proposé. Il +s’agit d’un système complexe permettant de traiter des documents avec de nombreuses classes. +C’est un réseau avec une architecture proche du U-Net où l’encodeur est pré-entraîné sur des +images de scènes naturelles (ImageNet (Deng et al., 2009)). Dans la suite de cette thèse, nous +comparons certains de nos modèles à dhSegment, c’est pourquoi son architecture est détaillée +dans le Focus 2.9. Dans dhSegment, contrairement aux réseaux proposés par Long et al. +(2015) et Ronneberger et al. (2015) où l’upsampling était réalisé à l’aide de convolutions +transposées, la résolution de sortie est augmentée à l’aide d’interpolations bilinéaires, ce qui +permet d’avoir moins de paramètres à apprendre. Cette méthode a obtenu de bons résultats +sur diverses tâches de traitement de documents historiques, telles que l’analyse de mise en +page ou l’extraction de lignes de base, avec peu de données d’entraînement. De plus, malgré +un grand nombre de paramètres, le temps d’entraînement est considérablement réduit grâce +à l’encodeur pré-entraîné. D’autres systèmes similaires ont ensuite été proposés, la principale +différence entre leurs architectures étant la manière dont la résolution est augmentée dans le +décodeur. Barakat et al. (2018) ont proposé un réseau entièrement convolutif pour détecter +les lignes de texte. Leur proposition consiste à utiliser uniquement des cartes de caractéris- +tiques de bas niveau pendant l’étape de décodage, en les sur-échantillonnant plusieurs fois, +à l’aide de convolutions transposées, avant de les combiner. Cette architecture a donné de +bons résultats sur des pages manuscrites arabes mais nécessite des images d’entrée binarisées. +Mechi et al. (2019) ont présenté une architecture U-Net adaptative pour la détection de +lignes de texte. Leur proposition est de réduire le nombre de filtres (deux fois moins) dans +les convolutions de l’encodeur afin de diminuer la quantité de paramètres du modèle, et donc +le temps d’inférence ainsi que le sur-apprentissage, leur quantité de données annotées étant +faible. + +22 +É TAT D E L’ A RT +Grüning et al. (2019) ont proposé un système plus complexe composé de deux étapes +pour détecter les lignes de base dans les documents historiques. Tout d’abord, un réseau +de neurones hiérarchique (ARU-Net) est appliqué pour détecter les lignes de texte. Cet +ARU-Net est une version étendue de l’architecture U-Net (Ronneberger et al., 2015) : +d’une part, un réseau d’attention spatiale est incorporé pour traiter les différentes tailles de +caractères dans les pages ; d’autre part, des blocs résiduels sont ajoutés à l’architecture U-Net. +Cela permet d’entraîner des réseaux neuronaux plus profonds tout en obtenant de meilleurs +résultats. Ensuite, des traitements successifs sont appliqués pour regrouper les super-pixels +afin de construire les lignes de base. Les auteurs ont montré que leur méthode était capable +d’extraire des lignes de texte courbes. Cependant, de nombreuses étapes de post-traitement +ont été introduites dans la seconde phase. Mechi et al. (2021) ont également présenté une +méthode en deux étapes pour segmenter les lignes de texte dans des images de documents +historiques arabes ou latins. Tout d’abord, un FCN est utilisé pour segmenter la zone centrale +du texte. La seconde étape affine les résultats du FCN. Elle est basée sur une version modifiée +du RLSA pour extraire les lignes complètes du texte (y compris les composantes ascendantes +et descendantes). Des évaluations quantitatives et qualitatives sont rapportées sur un grand +nombre d’images de documents arabes et latins collectés à partir des archives nationales +tunisiennes ainsi que d’autres ensembles de données de référence. Cependant, ce système +nécessite une binarisation de l’image d’entrée. Dans Tensmeyer et al. (2017), les auteurs +présentent PageNet, un système mis au point pour identifier les pages dans des images de +documents. Les pages détectées sont ensuite extraites, ce qui permet de supprimer le bruit +induit par la numérisation des pages, et différents traitements d’analyse de la mise en page +peuvent être appliqués. Dans PageNet, un réseau entièrement convolutif obtient une segmen- +tation par pixel post-traitée afin d’extraire une région quadrilatérale. Celui-ci traite l’image +d’entrée à quatre résolutions. Le système est évalué sur différents jeux de données et les au- +teurs montrent que PageNet peut segmenter des documents superposés à d’autres documents. +Enfin, Yang et al. (2017) ont conçu un réseau multimodal entièrement convolutif pour +l’analyse de la mise en page de documents. Ils tirent parti du contenu textuel ainsi que +de l’apparence visuelle pour extraire les structures sémantiques des images de documents. +Cette méthode a montré des scores élevés d’Intersection-over-Union (IoU) (voir le Focus 3.2) +mais nécessite des annotations de données plus complexes. En effet, pour chaque image de +document, une image étiquetée pixel par pixel ainsi que son contenu textuel sont nécessaires. +Ils utilisent des convolutions dilatées dans l’encodeur afin d’avoir une information contextuelle +plus large et des résultats plus précis. Puisque dans la suite de cette thèse nous comparons +certains de nos modèles à ce système, nous détaillons son architecture dans le Focus 2.10. +Dans leurs travaux, Renton et al. (2018) ont également démontré les avantages d’utiliser de +telles convolutions, détaillées dans le Focus 2.8, par rapport à des convolutions standards. Leur +réseau entièrement convolutif est composé de convolutions dilatées successives qui augmentent +le champ réceptif. Elles sont suivies d’une dernière convolution standard qui produit les images +étiquetées. Dans notre méthode, nous tirons également profit de ces convolutions dilatées afin +d’avoir un champ réceptif assez grand pour détecter correctement les objets. + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +23 +Focus 2.6 – ARCHITECTURE FCN +Définition +Un réseau entièrement convolutif (FCN) est une extension d’un réseau neuronal +convolutif (CNN) qui ne contient aucune couche dense et accepte des entrées de +tailles variables. Il permet de faire de la prédiction spatiale dense, au niveau pixel, +de manière rapide et précise. Pour faire de la prédiction dense, il est souvent +composé d’un encodeur, résumant les caractéristiques importantes de l’image +d’entrée, et d’un décodeur, augmentant la résolution des cartes de caractéristiques +et prédisant des probabilités de classe pour chaque pixel d’entrée. +Avantages +— La suppression des couches denses d’un CNN permet de travailler avec des tailles +d’entrée variables car les couches convolutives ne nécessitent pas un nombre fixe +d’entrées. +— Éviter les couches denses réduit fortement le nombre de paramètres. +— Les FCN sont capables de conserver l’information spatiale et de produire une +description spatiale de l’image d’entrée. +Inconvénients +— L’utilisation d’un réseau de neurones convolutif induit l’utilisation de couches +de regroupement (pooling), qui réduisent la résolution d’entrée dans le but +d’augmenter le champ réceptif sans augmenter le nombre de paramètres. Pour +avoir un étiquetage au niveau des pixels d’une image d’entrée, la résolution de +sortie du réseau doit être augmentée soit à l’aide d’une interpolation, d’une +convolution transposée (Focus 2.7), d’une opération d’unpooling ou encore +d’une convolution dilatée (Focus 2.8). +Système de type FCN +Un des premiers systèmes de type FCN proposé pour la détection niveau pixel est +le U-Net (Ronneberger et al. (2015)). Il a montré de très bonnes performances +sur différentes tâches de segmentation d’images biomédicales avec très peu de don- +nées annotées. Sur la Figure 2.8, la partie gauche constitue l’encodeur, composé +de couches de convolutions et de max pooling. La partie droite est le décodeur, +constitué de convolutions standards et transposées. + +24 +É TAT D E L’ A RT +Figure 2.8 – Schéma de l’architecture du modèle U-Net, issu de Ronneberger et al. +(2015), pour une image d’entrée de taille 572×572 pixels. +Focus 2.7 – CONVOLUTION TRANSPOSÉE +Définition +La +couche +de +convolution +transposée +est +utilisée +pour +inverser +le +sous- +échantillonnage induit par les couches de convolution standard ou de regroupement +utilisées dans les réseaux convolutifs (Long et al., 2015 ; +Ronneberger et al., +2015). Le principe est d’avoir la couche inverse d’une couche de convolution +standard. Cette couche permet d’avoir une sortie de plus grande résolution en +représentant la valeur d’un pixel d’entrée sur plusieurs pixels de sortie. Cette couche +est souvent utilisée dans les réseaux suivant l’architecture encodeur-décodeur où la +sortie du réseau a la même taille que l’image d’entrée. +Avantages +— Les filtres de la couche de convolution transposée doivent être entraînés ce qui +permet au réseau d’être plus expressif. +Inconvénients +— Le réseau est plus profond et comporte plus de paramètres qu’un réseau +contenant uniquement des opérations d’upsampling sans paramètres entraînés. +Schéma d’une convolution transposée 2D +La Figure 2.9 présente le schéma d’une convolution transposée 2D avec X l’image +d’entrée, W le filtre et Y la nouvelle représentation de l’image. + +t conv 3x3, Relu ++ up-com 2x2 +→ co 1x12.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +25 +x1,1 +x0,1 +x1,0 +x0,0 +X +w2,2 +w1,2 +w0,2 +w2,1 +w1,1 +w0,1 +w2,0 +w1,0 +w0,0 +W +y3,3 +y2,3 +y1,3 +y0,3 +y3,2 +y2,2 +y1,2 +y0,2 +y3,1 +y2,1 +y1,1 +y0,1 +y3,0 +y2,0 +y1,0 +y0,0 +Y +Figure 2.9 – Schéma d’une convolution transposée 2D avec X l’image d’entrée, W le filtre +et Y la nouvelle représentation de l’image. +Focus 2.8 – CONVOLUTION DILATÉE +Définition +Une convolution dilatée suit le principe de base d’une convolution standard, +mais calcule les nouvelles représentations sur une plus grande fenêtre. Les pixels +considérés pour le calcul de la nouvelle représentation d’un pixel ne sont plus ses +voisins directs mais des voisins plus éloignés, l’écart étant défini par le taux de +dilatation. +Avantages +— L’utilisation de convolutions dilatées permet d’avoir un champ réceptif plus +grand, la nouvelle représentation d’un pixel considérant davantage de contexte, +sans augmenter le nombre de paramètres. +— Elle est souvent utilisée à la place des couches de regroupement, ce qui permet +de perdre moins d’informations qu’avec une fonction de min ou max. +Schéma d’une convolution dilatée 2D +La Figure 2.10 présente le schéma d’une convolution dilatée 2D avec un taux de +dilatation de 2 et X l’image d’entrée, W le filtre et Y la nouvelle représentation de +l’image. +x0,0 +x1,0 +x2,0 +x3,0 +x4,0 +x5,0 +x0,1 +x1,1 +x2,1 +x3,1 +x4,1 +x5,1 +x0,2 +x1,2 +x2,2 +x3,2 +x4,2 +x5,2 +x0,3 +x1,3 +x2,3 +x3,3 +x4,3 +x5,3 +x0,4 +x1,4 +x2,4 +x3,4 +x4,4 +x5,4 +x0,5 +x1,5 +x2,5 +x3,5 +x4,5 +x5,5 +X +w0,0 +w1,0 +w2,0 +w0,1 +w1,1 +w2,1 +w0,2 +w1,2 +w2,2 +W +y1,1 +y0,1 +y1,0 +y0,0 +Y +Figure 2.10 – Schéma d’une convolution dilatée 2D avec un taux de dilatation de 2 et X +l’image d’entrée, W le filtre et Y la nouvelle représentation de l’image. + +26 +É TAT D E L’ A RT +Focus 2.9 – SYSTÈME DHSEGMENT +dhSegment (Ares Oliveira et al., 2018) est un des systèmes de référence pour les +tâches d’analyse d’images de documents historiques. Il possède plusieurs avantages +comme le fait de pouvoir être entraîné avec peu de données d’entraînement et un +temps d’entraînement réduit. De plus, le code permettant d’entraîner et de tester +le modèle est open-source a. +C’est un modèle profond puisqu’il possède jusqu’à 2048 cartes de caractéristiques +et suit l’architecture encodeur-décodeur. Dans un premier temps, l’image d’entrée +est traitée par l’encodeur qui va résumer les caractéristiques importantes de +l’image dans une matrice de caractéristiques. Cette matrice est ensuite trai- +tée par le décodeur qui va générer une carte de probabilités de même taille que +l’image d’entrée. Enfin, une étape de post-traitement est réalisée afin notamment de +seuiller les probabilités des pixels et de supprimer les petites composantes connexes. +Encodeur +L’encodeur est principalement constitué d’un CNN pré-entraîné sur des images de +scènes naturelles de la base ImageNet (Deng et al., 2009) et représenté à gauche sur +la Figure 2.11. Ce CNN pré-entraîné suit l’architecture du réseau ResNet-50 (He +et al., 2016) mais a été légèrement modifié afin de réduire le nombre de paramètres, +et donc la mémoire requise pendant l’entraînement. Il est également possible de +remplacer ce ResNet-50 par un VGG-16 (Simonyan et al., 2015) ou un U-Net +(Ronneberger et al., 2015). +Cette partie pré-entrainée présente l’avantage de réduire considérablement le +nombre de paramètres à apprendre. En effet, le réseau possède 32,8 millions de +paramètres au total dont la plupart proviennent du CNN. Ainsi, seuls 9,36 millions +de paramètres restent à entraîner. Cela permet au réseau d’apprendre rapidement +et correctement sur un nombre restreint de données annotées. +Décodeur +Le décodeur est standard et consiste en une succession de cinq blocs de déconvolu- +tion composés d’une couche de convolution standard et d’une couche d’upscaling, et +d’une couche finale de convolution afin de générer une carte de probabilités. Cette +partie est entièrement apprise sur les données d’entrée. +Post-traitement +En sortie du décodeur, nous disposons, pour chaque pixel, des probabilités d’appar- +tenir aux différentes classes définies. Différentes techniques d’agrégation des résul- +tats au niveau pixel sont possibles afin de détecter les objets pour la tâche considérée. +Quatre principales techniques ont été implémentées et sont disponibles : +— Seuillage : permet d’assigner une classe aux pixels ayant une probabilité supé- +rieure à un seuil prédéfini ; +— Opérations de morphologie mathématique : opérations d’érosion, de dilatation, +d’ouverture et de fermeture afin de créer des objets plus plausibles ; +— Analyse des composantes connexes : permet de filtrer les petites composantes +connexes restantes après l’étape de seuillage ; +— Vectorisation des objets : transforme les régions détectées en un ensemble de +coordonnées. + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +27 +Schéma de l’architecture de dhSegment +S +S +64 +S +2 +256 +S +4 +512 +S +8 +1024 +512 +S +16 +2048 +512 512 +S +16 +|| +512 512 +S +8 +|| +256 +256 +S +4 +|| +128 +128 +S +2 +|| +64 +64 +S +|| +32 c +S +Convolution +Max pooling +Bottleneck +Bottleneck S/2 +Upscaling +|| +Concatenation +c +Number of classes +Figure 2.11 – Schéma de l’architecture du modèle dhSegment (Ares Oliveira et al. +(2018)). +a. https://github.com/dhlab-epfl/dhSegment +Focus 2.10 – SYSTÈME DE YANG ET AL. +Yang et al. (2017) ont proposé un réseau multimodal permettant de segmenter des +documents en se basant sur le contenu visuel et textuel de ceux-ci. L’utilisation des +textes permet d’assigner des classes spécifiques aux régions de texte en fonction de +leur rôle dans le document. Ainsi, dans l’article original, les classes considérées sont +les suivantes : fond, image, tableau, paragraphe, titre, liste et légende. Le système +a montré de bonnes performances sur des ensembles de données synthétiques et +réelles d’images de documents modernes. De plus, le code permettant d’entraîner +un modèle est open-source a. +Le modèle de Yang est un réseau multimodal entièrement convolutif (FCN) dont +l’architecture est présentée sur la Figure 2.12. La base de ce modèle suit une archi- +tecture encodeur-décodeur et est constituée de quatre modules : +— Un encodeur ; +— Un décodeur ; +— Un décodeur auxiliaire ; +— Un pont (intégration du contenu textuel). +Encodeur +L’encodeur est constitué de quatre blocs dilatés, chaque bloc comportant cinq +couches de convolutions dilatées, de taux de dilatation 1, 2, 4, 8 et 16, exécutées en +parallèle. L’avantage d’utiliser de telles convolutions est que le champ réceptif est + +28 +É TAT D E L’ A RT +plus grand, ce qui permet au modèle d’avoir davantage de contexte par rapport à +une convolution standard. +Décodeurs +Les deux décodeurs ont la même architecture avec trois blocs contenant une couche +de convolution suivie par une couche d’unpooling. Le premier décodeur est stan- +dard et vise à produire une carte de probabilités. Le décodeur auxiliaire est, quant +à lui, utilisé uniquement durant l’entraînement et cherche à reconstruire l’image +d’entrée. Il a été montré qu’une branche auxiliaire de reconstruction aide à générer +de meilleures représentations de l’image d’entrée, et donc améliore les performances +de la tâche principale, ici, la tâche de segmentation (Zhang et al., 2016a). +Contenu textuel +L’information textuelle est extraite grâce à un algorithme de reconnaissance +(Optical Character Recognition (OCR) ou HTR) de textes de la manière suivante. +L’algorithme de reconnaissance est appliqué au document. Pour chaque phrase +extraite du document, un embedding moyen est calculé à partir des embeddings des +mots de cette phrase. Enfin, une carte de caractéristiques est construite à partir de +ces embeddings : pour chaque phrase, les pixels du document initial lui appartenant +prennent la valeur de cet embedding. Les pixels n’appartenant à aucune phrase +prennent la valeur 0. Enfin, cette carte est concaténée à la carte de caractéristiques +visuelles avant la dernière convolution du décodeur principal. +Schéma de l’architecture du système de Yang et al. +W +H +32 +W +H +64 +W +2 +H +2 +128 +W +4 +H +4 +256 +128 +128 +W +4 +H +4 +|| +64 +64 +W +2 +H +2 +|| +32 +32 +W +H +|| +c +64 +64 +W +2 +H +2 +32 +32 +W +H +c +Reconstructed +input +Text +Embedding +Map +Segmentation +Convolution +Dilated block +Max pooling +Upscaling +Text embedding map +|| +Concatenation +c +Number of classes +Figure 2.12 – Schéma de l’architecture du modèle de Yang et al. (2017). +a. http://personal.psu.edu/xuy111/projects/cvpr2017_doc.html + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +29 +modèles séquentiels à base de transformers +Avant le développement des modèles à attention et des systèmes Transformers, les tâches +de traitement du langage et notamment de traduction étaient réalisées à l’aide de réseaux +encodeurs-décodeurs récurrents. L’encodeur est utilisé pour traiter la phrase d’entrée entière +et l’encoder dans un vecteur de contexte unique. Les couches du décodeur produisent ensuite, +à partir du vecteur de contexte, les mots de la phrase les uns après les autres. Le principal +inconvénient de cette approche provient du traitement de la phrase d’entrée qui est résumée +dans un unique vecteur de taille fixe. En effet, Cho et al. (2014) ont démontré que la perfor- +mance du modèle encodeur-décodeur se dégrade rapidement lorsque la longueur de la phrase +d’entrée augmente. Un autre problème est que le modèle n’a aucun moyen de donner plus +d’importance à certains des mots en entrée par rapport à d’autres lors de la traduction de la +phrase. C’est pour résoudre ces problèmes que l’attention (voir le Focus 2.12) a été introduite +par Bahdanau et al. (2015). Celle-ci permet de considérer tous les mots de la phrase d’en- +trée dans le vecteur de contexte, mais également d’accorder une importance relative à chacun +d’entre eux. Ainsi, lorsque le modèle génère une phrase, il recherche un ensemble de positions +dans les états cachés de l’encodeur, dans lesquels les informations les plus pertinentes sont +disponibles. +Dans cette même optique, des systèmes à base de réseaux Transformers ont été proposés +récemment afin de réaliser des tâches de détection en tenant compte de la séquentialité +entre les éléments prédits. L’architecture de ces systèmes est présentée dans le Focus 2.11. +Il s’agit de modèles reposants sur le même mécanisme d’attention, qui sélectionne les +caractéristiques pertinentes à chaque itération du processus de prédiction. Le système met +en œuvre une seconde attention qui tient compte des éléments précédemment prédits en +sortie, pour agir comme un modèle de langage. Les premiers systèmes ont été principalement +conçus pour le traitement automatique des langues sans utiliser ni récurrence ni convolutions. +Le premier système à base de Transformers (Vaswani et al., 2017) a été établi afin +de résoudre plus efficacement la tâche de traduction de texte. Les auteurs ont proposé +une architecture composée d’un encodeur suivi d’un décodeur, qui génère une séquence +de sortie, un élément à la fois. Le modèle est auto-régressif : il intègre les éléments +prédits précédemment comme entrée supplémentaire lors de la prédiction de l’élément +suivant. L’encodeur extrait les caractéristiques des données d’entrée grâce à un mécanisme +d’attention qui permet de considérer le contexte, ici, l’ensemble des mots de la séquence +d’entrée. Cet encodeur est constitué de six blocs successifs identiques, composés de deux +principaux éléments : une couche d’auto-attention et un réseau dit entièrement connecté +(feed-forward). L’auto-attention permet de représenter l’interdépendance des mots de la +séquence en entrée. Le décodeur permet de modéliser le langage de sortie. Il est également +composé de six blocs successifs identiques, chacun contenant une couche d’auto-attention, +un réseau entièrement connecté et une couche d’attention dite d’attention croisée. Cette +dernière permet au décodeur de réaliser l’attention entre la séquence d’entrée et celle de +sortie. Tous les réseaux entièrement connectés du modèle contiennent deux couches linéaires. +De plus, les séquences en entrée et en sortie sont additionnées à un encodage de position, + +30 +É TAT D E L’ A RT +détaillé dans le Focus 2.13, avant d’être respectivement traitées par les encodeur et décodeur. +Ces encodages de position permettent de garder l’ordre de la séquence durant l’ensemble +des traitements. Le Transformer a rapidement été largement utilisé car il a permis de +remplacer les couches récurrentes, jusqu’alors utilisées, par des couches d’attention tout en +conservant des performances similaires. De plus, les couches récurrentes jusqu’ici utilisées +empêchaient la parallélisation des calculs durant la phase d’entraînement. Cette récurrence +ayant été remplacée par ces fameuses couches Transformer non récurrentes, entraînées +par une stratégie dite de teacher forcing, les calculs peuvent être parallélisés et le temps +d’entraînement fortement réduit. Cette architecture est détaillée dans le Focus 2.11. +Au vu des résultats obtenus par ces systèmes, certains travaux les ont adaptés à des tâches +de vision. Ainsi, les Vision Transformers (ViT) ont été proposés. Les premiers travaux intro- +duisant les ViT ont été présentés par Dosovitskiy et al. (2021) et sont détaillés dans le +Focus 2.14. Ils interprètent une image en entrée de l’encodeur comme étant une séquence de +patchs. Ainsi, la représentation vectorielle d’un caractère dans une tâche de traduction est +ici remplacée par les valeurs des pixels d’un patch de l’image d’entrée mis à plat. Cette fois, +l’encodage de position correspond à la position du patch dans l’image. Puisqu’il est appliqué +à la tâche de classification d’images, qui ne nécessite pas de sortie séquentielle, seul l’encodeur +Transformer est intégré au système. Ensuite, les auteurs utilisent un simple Multi-Layer Per- +ceptron (MLP) chargé de prédire la classe de l’image. Ce système a obtenu des performances +à l’état de l’art sur différents ensembles de classification d’images. Cependant, le système +nécessite un pré-entraînement sur un nombre imposant de données, 303 millions d’images +pour leur meilleur modèle. +Plusieurs approches ont également été présentées afin d’appliquer les Transformers à la +détection d’objets. Dans le cadre d’images de scènes naturelles, DETR (DEtection TRansfor- +mer) a été proposé par Carion et al. (2020). Il s’agit d’un système hybride qui combine un +encodeur CNN suivi d’un encodeur et d’un décodeur Transformer produisant un ensemble +de boîtes englobantes. Le modèle est entrainé à prédire un nombre fixe de boîtes englobantes +ainsi que leurs classes. Leur modèle obtient des résultats semblables à Faster R-CNN sur les +images de scènes naturelles du jeu de données COCO 1, tout en obtenant de meilleurs résul- +tats sur les grands objets grâce à l’auto-attention. Il ne tire cependant pas profit de la capacité +de prédiction séquentielle permise par les Transformers. Chen et al. (2022) ont ensuite pro- +posé Pix2Seq, présenté dans le Focus 2.15, afin de traiter la détection de manière séquentielle +en prédisant, pour chaque objet, une séquence de coordonnées suivie de la classe de l’objet. +Les auteurs comparent différents encodeurs à base de convolutions et de Transformers, suivis +par un décodeur Transformer standard. Pix2Seq obtient des performances à l’état de l’art +sur l’ensemble de données de référence COCO en obtenant des valeurs de précision moyenne +(Average Precision (AP), voir le Focus 3.4) supérieures à celles obtenues par Faster-RCNN +(Ren et al., 2015) et DETR (Carion et al., 2020) tout en nécessitant moins de paramètres. +En effet, les résultats montrent que, pour tous les encodeurs, la détection est meilleure par +rapport aux systèmes Faster-RCNN et DETR avec des encodeurs comparables. Les résultats +1. https://cocodataset.org/ + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +31 +montrent également qu’utiliser un encodeur Transformer est préférable. Cependant, davan- +tage de données d’entraînement sont nécessaires puisque le modèle comporte beaucoup plus +de paramètres. +Toujours dans le domaine de la détection dans des images naturelles, certains travaux ont +été proposés afin de faire de la prédiction dense en augmentant la sortie du Transformer, et +donc d’avoir une sortie de même taille que l’image d’entrée. Ainsi, Zheng et al. (2020) ont +proposé un ViT (appelé SETR) où l’encodeur Transformer est suivi d’un décodeur composé +de convolutions réalisant l’augmentation d’échelle (upsampling). Ils ont obtenu les meilleurs +résultats sur différentes bases de segmentation d’images. Biswas et al. (2022) ont proposé un +modèle hybride CNN-Transformer, très comparable à SETR mais incluant un encodeur CNN +avant l’encodeur Transformer. Bien que ces méthodes aient montré des gains de performances +par rapport aux systèmes existants, elles ne tirent pas pleinement profit des Transformers +qui permettent d’avoir des sorties séquentielles et structurées. De plus, dans le domaine des +Transformers, il n’y a pas, à notre connaissance, de travaux permettant de traiter la tâche +de détection d’objets de manière séquentielle dans les images de documents. Quelques rares +travaux ont appliqué les Transformers aux images de documents, principalement pour la re- +connaissance de caractères niveau paragraphe ou page. Ainsi, dans les travaux de Coquenet +et al. (2022) et Singh et al. (2021), les auteurs ont proposé des modèles hybrides combinant +un encodeur CNN et un décodeur Transformer afin de prédire séquentiellement les caractères +du texte d’un paragraphe ou document. Le système proposé par Coquenet et al. (2022) +fournit également une structuration des résultats en générant des tags de mise en page dans +la séquence des caractères reconnus. Ce système est le premier à résoudre la tâche de recon- +naissance de texte pleine page sans segmentation. Il a obtenu des performances de même +ordre que les systèmes à l’état de l’art travaillant au niveau ligne. De leur côté, Rouhou +et al. (2022) utilisent un modèle hybride avec un encodeur CNN suivi d’un encodeur et déco- +deur Transformer pour traiter la tâche de reconnaissance d’entités nommées. Leur approche +consiste à créer une architecture qui reconnaît les textes et les entités nommées à partir +d’images de paragraphes. Ils utilisent des labels dits "visuels" correspondant aux caractères +du texte présent dans les images ainsi que des labels dits "contextuels" correspondant aux +entités nommées. Enfin, Kim et al. (2022) ont proposé DONUT, un modèle de compréhension +de documents sans OCR composé d’un encodeur et décodeur Transformer. DONUT obtient +de très bons résultats en termes de temps d’exécution et de précision sur diverses tâches telles +que la classification de documents, l’extraction d’informations et le Visual Question Answe- +ring. Il nécessite cependant un important pré-entraînement sur des milliers de documents +synthétiques. +Focus 2.11 – ARCHITECTURE TRANSFORMER +Définition +Un modèle à base de Transformer est un modèle permettant de réaliser un trai- +tement séquence-à-séquence. Il s’agit d’un modèle auto-régressif prédisant séquen- +tiellement les éléments et utilisant les éléments de la séquence d’entrée ainsi que + +32 +É TAT D E L’ A RT +les éléments prédits précédemment en sortie. Ce modèle repose sur un mécanisme +d’attention (présenté dans le Focus 2.12), qui permet de représenter les données +en utilisant le contexte et, notamment, les interdépendances entre les éléments des +séquences d’entrée et de sortie. +Les modèles Transformers initialement proposés suivent une architecture encodeur- +décodeur où l’encodeur génère une représentation de la séquence en entrée incluant +l’interdépendance des éléments de cette séquence ainsi que leurs positions dans la +séquence. Le décodeur génère une séquence de sortie grâce à la séquence d’entrée +encodée et les éléments précédemment prédits. +Avantages +— Dans un modèle à base de Transformer, les couches récurrentes d’un réseau à +attention ont été remplacées par des couches non récurrentes pour réaliser cette +attention, ce qui conduit à des temps d’entraînement réduits tout en conservant +des performances similaires. +— Par rapport à un réseau récurrent, ce modèle permet de mieux représenter les +dépendances entre les éléments de la séquence d’entrée grâce au mécanisme +d’attention, notamment pour des séquences longues, tout en conservant un +temps de traitement raisonnable. +Système Transformer +Le premier système à base de Transformer a été proposé pour la tâche de traduction +de texte (Vaswani et al., 2017). Il s’agit d’un modèle encodeur-décodeur entière- +ment basé sur l’attention dont l’architecture est présentée sur la Figure 2.13. Il a +dépassé les résultats à l’état de l’art sur des tâches de traduction anglais-allemand +et anglais-français. +Encodeur +Sur la Figure 2.13, la partie de gauche compose l’encodeur qui traite la sé- +quence d’entrée. Dans l’implémentation originale, l’encodeur comporte une +couche d’embedding de la séquence puis six blocs d’encodage Transformer. De +plus, un encodage de position est additionné à la représentation de la séquence +d’entrée avant les couches d’encodage. Celui-ci est réalisé à l’aide des fonctions +cosinus et sinus comme détaillé dans le Focus 2.13. Les couches dites de Multi- +Head Attention calculent les vecteurs d’auto-attention (voir Focus 2.12) et sont +suivies d’un réseau entièrement connecté composé de deux couches linéaires. +Décodeur +La partie de droite de la Figure 2.13 présente le décodeur. Il comporte une +couche d’embedding de la séquence de sortie suivie de six blocs de décodage +Transformer. Le même encodage de position utilisé dans l’encodeur est appli- +qué sur la séquence partielle de sortie courante, avant les couches de Transfor- +mer. Enfin, les couches Multi-Head Attention et le réseau entièrement connecté +sont similaires à ceux de l’encodeur. La seule différence concerne la seconde +couche d’attention qui prend en entrée la séquence d’entrée encodée ainsi que +la séquence de sortie encodée pour réaliser l’attention croisée. + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +33 +Figure 2.13 – Schéma de l’architecture du modèle Transformer original, issu de Vaswani +et al. (2017). +Focus 2.12 – ATTENTION +Définition +Le concept d’attention permet de considérer la corrélation entre les éléments de +deux séquences grâce à des coefficients d’attention calculés entre chaque élément +de chaque séquence. Une fonction d’attention peut être décrite comme la mise en +correspondance d’une requête (q) et d’un ensemble de paires clé-valeur (k-v) avec +une sortie. La sortie est calculée comme une somme pondérée des valeurs, le poids +attribué à chaque valeur étant calculé par une fonction de compatibilité de la requête +avec la clé correspondante. +Lorsque l’attention est réalisée sur une unique séquence, celle-ci est appelée auto- +attention. L’attention croisée fait elle référence au mécanisme d’attention standard, +appliqué sur deux séquences distinctes. +Dans le cas du traitement de la langue, le mécanisme d’attention permet de +déterminer les mots sur lesquels le modèle doit porter le plus d’attention pour +traiter la séquence. +Auto-attention +L’auto-attention, appelée self-attention, correspond au mécanisme d’attention ap- +pliqué à une seule séquence. Elle détermine donc l’interdépendance (ou l’auto- +corrélation) des éléments d’une même séquence entre-eux afin de lui associer une +représentation pertinente. + +Output +Probabilities +Softmax +Linear +Add & Norm +Feed +Forward +Add & Norm +Add & Norm +Multi-Head +Feed +Attention +Forward +Nx +分 +Add & Norm +Nx +Add & Norm +Masked +Multi-Head +Multi-Head +Attention +Attention +Positional +Positional +Encoding +Encoding +Input +Output +Embedding +Embedding +个 +Inputs +Outputs +(shifted right)34 +É TAT D E L’ A RT +Attention multi-têtes +Dans la multi-head attention, le calcul d’attention est réalisé en parallèle par plu- +sieurs blocs d’attention différents. Cela permet au modèle de considérer des informa- +tions provenant de différents sous-espaces de représentations à différentes positions. +Dans le cadre d’un modèle de traitement du langage, cela permet de caractériser +les mots vis-à-vis de différents points de vue ou rôles qu’ils occupent dans la phrase +tels que sujet, verbe ou encore complément. +Le vecteur de sortie correspond à la concaténation des vecteurs de sortie de chaque +tête. +Mise en oeuvre +— Pour calculer une sortie (vecteur d’attention), trois vecteurs pour chaque élé- +ment de la séquence d’entrée sont considérés : +— Vecteur requête q (query) ; +— Vecteur clé k (key) de dimension dk ; +— Vecteur valeur v (value) de dimension dv. +Les valeurs de chacun de ces vecteurs sont apprises pendant l’entraînement du +modèle Transformer. +— Pour chaque élément de la séquence d’entrée (requête q), les produits scalaires +avec l’ensemble des éléments de la seconde séquence (clés k) sont calculés, puis +divisés par la racine carrée de la dimension du vecteur k (dk). Cette division +assure la stabilité du gradient. +— Une opération softmax est ensuite appliquée à chaque sortie puis celle-ci est +multipliée par le vecteur valeur v correspondant. +— Enfin, le vecteur d’attention d’une requête q correspond à la somme des +vecteurs ainsi calculés. +Équation +En pratique, la fonction d’attention est calculée sur un ensemble de requêtes +simultanément, regroupées dans la matrice Q. Les clés et valeurs sont elles aussi +regroupées respectivement dans des matrices K et V. Les sorties sont calculées +comme suit : +Attention(Q, K, V ) = Softmax +�QKT +√dk +� +V +(2.1) +avec : +— dk : la dimension du vecteur clé k. +Focus 2.13 – ENCODAGE POSITIONNEL +Définition +Dans un Transformer, chaque élément de la séquence d’entrée (ou de sortie) est +traité simultanément dans la pile d’encodeurs (ou de décodeurs). Ainsi, le modèle +n’a pas connaissance de la position de chaque élément dans la séquence. C’est +pourquoi l’encodage positionnel est utilisé dans les réseaux à base de Transformers, + +2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +35 +afin de ne pas perdre l’ordre des éléments de la séquence d’entrée (ou de sortie) +lors de la propagation des informations dans le modèle. +Équation +Le premier encodage de position a été proposé par Vaswani et al. (2017). Il s’agit +d’un encodage fixe qui se base sur les fonctions cosinus et sinus, et est calculé comme +suit : +PE(pos, 2i) = sin(wi · pos) ∀i ∈ +� +0, dmodel +2 +� +PE(pos, 2i + 1) = cos(wi · pos) ∀i ∈ +� +0, dmodel +2 +� +(2.2) +avec : +wi = +1 +10000 +2i +dmodel +et : +— pos : la position de l’élément dans la séquence ; +— dmodel : la dimension d’encodage de l’élément. +Focus 2.14 – ARCHITECTURE VISION TRANSFORMER +Définition +Un Vision Transformer (ViT) est une adaptation de l’architecture Transformer +standard appliquée aux images. La séquence en entrée du système correspond à +une séquence de patchs de taille fixe de l’image originale, où la couche d’embedding +est remplacée par une projection linéaire des valeurs des patchs aplanis. Pour la +tâche de classification, l’encodeur est suivi d’un MLP standard produisant des +probabilités pour chaque classe considérée. Pour la détection d’objets, il est suivi +d’un décodeur convolutif semblable à ceux des FCN. +Avantages +— Par rapport aux CNN, les performances sont au moins aussi bonnes tout en +nécessitant moins de mémoire pour le traitement et en étant plus rapide en +inférence. +Inconvénients +— Le modèle nécessite un très grand nombre de données d’apprentissage ou une +étape de pré-entraînement afin d’obtenir des résultats satisfaisants. +Système Vision Transformer +Le premier Vision Transformer a été proposé pour la tâche de classification d’images +(Dosovitskiy et al., 2021). Le modèle comporte un encodeur Transformer suivi +d’un MLP. Il a obtenu des performances comparables aux systèmes CNN à l’état +de l’art, tout en nécessitant beaucoup moins de ressources pour l’entraînement. + +36 +É TAT D E L’ A RT +Figure 2.14 – Schéma de l’architecture du modèle Vision Transformer original pour la +classification d’images, issu de Dosovitskiy et al. (2021). +Focus 2.15 – SYSTÈME PIX2SEQ +Pix2Seq (Chen et al., 2022) est un des premiers systèmes à base de Transformers +proposé pour traiter la détection d’objets dans les images de scènes naturelles de +manière séquentielle. Le modèle obtient des performances supérieures à celles obte- +nues par les systèmes à l’état de l’art, tels que Faster-RCNN (Ren et al., 2015) et +DETR (Carion et al., 2020), tout en nécessitant moins de paramètres. +Le modèle est composé d’un encodeur suivi d’un décodeur Transformer. Les +auteurs ont comparé différents encodeurs à base de convolutions, de Transformers +ou des encodeurs hybrides, leurs expériences montrant les meilleures performances +avec un encodeur Transformer. Le décodeur est standard et comporte six couches +de décodeur Transformer. Celui-ci produit une séquence de coordonnées et de +classes représentant les objets détectés ainsi que leurs classes. +Modélisation de la détection +Le modèle Pix2Seq est entraîné à prédire séquentiellement chaque objet, une co- +ordonnée à la fois, de la manière suivante : ordonnée du point supérieur gauche, +abscisse du point supérieur gauche, ordonnée du point inférieur droit, abscisse du +point inférieur droit et classe de l’objet. Ainsi, un objet et sa classe sont détectés par +cinq valeurs prédites. De plus, les auteurs considèrent la détection comme une tâche +de classification en considérant une classe pour chaque valeur possible en ordonnée +et en abscisse. + +Class +Bird +MLP +Ball +Head +Car +IPatsh -- IPosd(dilm +161829st3*0 +[cLe5 s] cmbedldig +Linear Projection of Flattened Patches2.1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +37 +Schéma de l’architecture de Pix2Seq +Figure 2.15 – Schéma du système Pix2Seq, issu de Chen et al. (2022). +approches combinant image et texte +Les systèmes présentés en section 2.1.2 sont actuellement les plus utilisés pour la détection +d’objets dans les images de documents. Certaines recherches se sont également orientées +vers des approches combinant l’image et le texte du document afin de réaliser la tâche +de détection. Dans le cas de documents complexes, l’ajout du texte dans le processus de +détection peut aider à détecter et à classifier des éléments de plus haut niveau tels que des +actes (Prieto et al., 2020). Dans la tâche de Visual Document Understanding (Delteil +et al., 2022), les informations sont extraites à l’aide d’une combinaison des caractéristiques +textuelles et visuelles de l’image d’un document. Certaines propositions sont présentées dans +cette section pour les tâches de détection, mais aussi de pré-entraînement pour différentes +tâches de compréhension d’images de documents. Il est important de noter que pour la +plupart des systèmes présentés dans cette section, un reconnaisseur (HTR ou OCR) a été +entraîné au préalable afin d’extraire le contenu textuel des images de documents. +Yang et al. (2017) ont été parmi les premiers à proposer un réseau entièrement convolutif +multimodal pour extraire des structures sémantiques de documents modernes. Pour aider à +distinguer des classes similaires comme les paragraphes et les listes, ils incorporent des in- +formations textuelles à l’aide d’une carte d’intégration de texte concaténée avant la dernière +convolution du modèle. L’ajout de cette carte n’a pas montré d’amélioration significative dans +des conditions réelles d’utilisation. Suivant cette idée, Barman et al. (2021) ont proposé un +système capable de segmenter finement les journaux historiques et de gérer les variations de +mise en page dans le temps. Ils utilisent la même représentation textuelle que Yang et al. +(2017) mais sur des jetons produits par un processus OCR au lieu de phrases, ce qui est plus +réaliste. Ils ont montré que l’ajout des cartes d’intégration du texte au début du réseau donne +de meilleures performances. Certains travaux ont également étudié la combinaison de carac- +téristiques textuelles et visuelles pour classifier les pages des documents. Dans Wiedemann +et al. (2018), une combinaison de deux réseaux neuronaux convolutifs (CNN), l’un basé sur +des données textuelles et l’autre sur des numérisations d’images, est utilisée pour classifier les +pages. Les paramètres sont ensuite combinés et transmis à un perceptron multicouche pour +la classification finale. Cette combinaison a permis d’augmenter les performances par rapport +à un seul CNN basé uniquement sur le texte ou l’image. + +Encoder38 +É TAT D E L’ A RT +Pour les documents modernes, LayoutLM (Xu et al., 2020) a été proposé. Il s’agit d’une mé- +thode de pré-entraînement simple pour les tâches de compréhension d’images de documents +qui permet de modéliser conjointement les interactions entre le texte et les informations de +mise en page dans les documents numérisés. Tout d’abord, un processus de reconnaissance +complet est appliqué à l’image d’entrée afin de détecter les objets textuels et de reconnaître +l’ensemble des textes. Ensuite, les auteurs utilisent une combinaison de BERT (Devlin et +al., 2019), où l’information textuelle d’entrée est principalement représentée par des plon- +gements de mots, et des caractéristiques d’image données par Faster-RCNN (Ren et al., +2015). LayoutLM permet de modéliser conjointement les interactions entre le texte et les +informations de mise en page dans les documents numérisés et est ensuite utilisé comme +pré-entraînement pour un grand nombre de tâches de compréhension d’images de documents. +Ce système a montré des performances à l’état de l’art sur des documents commerciaux +numérisés, mais nécessite un nombre important de données d’apprentissage. Dans Li et al. +(2021), les auteurs présentent VTLayout, un système qui fusionne les caractéristiques visuelles +profondes, superficielles et textuelles des documents pour localiser et identifier les différents +blocs. Dans la première étape, le modèle Cascade Mask R-CNN est appliqué directement sur +l’image pour localiser tous les blocs du document. Dans la seconde étape, les caractéristiques +visuelles profondes, superficielles et textuelles sont extraites et fusionnées afin d’identifier les +classes de chaque bloc. Les caractéristiques textuelles sont extraites par PaddleOCR (Du +et al., 2020) puis transformées par une application de TF-IDF. Ce modèle a montré un gain +de performances de détection par rapport aux systèmes standards manquant, notamment, de +précision sur la classe de titre. +Prieto et al. (2020) ont également étudié le cas où l’aspect graphique des images n’est +pas suffisant pour segmenter les chartes médiévales en actes. Ils ne visent pas seulement à +détecter les actes mais cherchent également à les classifier comme début, milieu, fin d’acte +ou acte complet. Ils utilisent une carte d’indexation probabiliste pour construire des caracté- +ristiques supplémentaires basées sur le contenu textuel, puis les caractéristiques graphiques +et textuelles sont fusionnées afin d’obtenir une seule entrée pour le système de segmentation. +Ils montrent que l’ajout de contenu textuel peut faciliter la segmentation des actes, et que +l’ajout de connaissances préalables permet d’améliorer encore les performances, cependant, +leur méthode reste complexe à mettre en place. +2.2 +E S T I M AT I O N D E L A C O N F I A N C E D E S O B J E T S D É T E C T É S +Les réseaux de neurones obtiennent désormais des performances remarquables dans de nom- +breux domaines d’application. Cependant, leur utilisation pour des applications industrielles +exige qu’ils soient à la fois capables de fournir le résultat attendu tout en évaluant leur propre +certitude, ou incertitude, quant à cette décision. Ceci est particulièrement important pour les +applications critiques telles que celles liées aux images médicales ou à la conduite autonome +par exemple. +L’apprentissage actif (active learning, détaillé dans le Focus 2.16) (Lewis et al., 1995) est +une méthode d’apprentissage automatique itératif dans lequel l’algorithme d’apprentissage + +2.2 E S T I M AT I O N D E L A C O N F I A N C E D E S O B J E T S D É T E C T É S +39 +demande des données d’entraînement, celles jugées les plus pertinentes. Ces données sont +sélectionnées en fonction de la confiance de l’algorithme quant à ses propres décisions. +Les premières propositions consistaient à utiliser directement les probabilités a posteriori +du classifieur afin de sélectionner les exemples à annoter. Ainsi, les exemples ayant une +probabilité proche de 0,5 (uncertainty sampling) étaient sélectionnés pour l’itération suivante. +Les réseaux neuronaux de détection d’objets produisent également des probabilités qui +pourraient directement être utilisées comme estimations de confiance. Cependant, il a été +démontré que ces probabilités sont souvent des estimateurs trop confiants qui donnent une +confiance élevée même sur des prédictions erronées (Nguyen et al., 2015). Pour résoudre ce +problème, plusieurs études ont été menées afin de concevoir de meilleurs estimateurs. +Ainsi, toujours dans le cadre de l’apprentissage actif, l’une des premières approches +proposées pour sélectionner les échantillons à annoter manuellement était basée sur les +machines à vecteurs de support (SVM) linéaires. Dans cette optique, Tong et al. (2002) +ont proposé SVM Min Margin qui consiste à entraîner un SVM linéaire et à choisir les +échantillons étant les plus proches de la limite de décision. Une autre approche populaire +est l’échantillonnage d’incertitude (uncertainty sampling) (Settles et al., 2008) où les +échantillons menant à des prédictions avec une grande incertitude sont sélectionnés. Pour +quantifier l’incertitude, plusieurs mesures basées sur les probabilités a posteriori ont été +proposées, comme l’entropie ou le score de moindre confiance (Brust et al., 2019). +Pour modéliser l’incertitude des décisions des réseaux neuronaux, d’autres approches ont +été proposées, comme le dropout de Monte Carlo (Gal et al., 2016). Le dropout (Srivastava +et al., 2014) est une méthode de régularisation utilisée dans les réseaux neuronaux afin de +lutter contre le manque de généralisation des modèles. Il consiste à désactiver (mettre à 0) des +valeurs, choisies aléatoirement, de l’image en entrée d’une couche. Il est appliqué uniquement +durant la phase d’apprentissage et permet d’éviter le sur-apprentissage et la coadaptation, +chaque neurone devant apprendre indépendamment des autres. Dans le MC dropout, au lieu +de calculer une seule prédiction au moment du test, il est demandé au réseau de fournir +plusieurs prédictions avec dropout, dont la distribution est ensuite analysée pour dériver une +estimation de la confiance de la prédiction sans dropout. Cette technique, qui se rapproche +des modèles bayésiens par apprentissage profond, a été utilisée pour de nombreuses tâches. +Elle s’est souvent révélée efficace pour la classification afin de choisir les données à étiqueter +(Gal et al., 2017). Dans Dechesne et al. (2021), le MC dropout est utilisé pour estimer +l’incertitude de résultats de segmentation sémantique d’images. De plus, Moon et al. (2020) +utilisent le MC dropout comme technique de régularisation des probabilités de classe pour +obtenir un meilleur classement ordinal des prédictions. +D’autres travaux font appel à des modèles d’estimation de confiance profonds indépendants +du modèle de détection. Dans Granell et al. (2021), un réseau adversaire est entraîné +en parallèle du modèle de détection. Celui-ci est entraîné pour estimer la proximité des +prédictions avec la vérité du terrain. + +40 +É TAT D E L’ A RT +La plupart des travaux présentés ici se concentrent sur la tâche de classification. En effet, +malgré les nombreux travaux présentant de nouveaux systèmes de détection d’objets, il y +a très peu de travaux, dans la littérature, discutant l’estimation de la confiance pour cette +tâche. +Focus 2.16 – APPRENTISSAGE ACTIF / ACTIVE LEARNING +Définition +L’Active learning (ou apprentissage actif) (Lewis et al., 1995) est une méthode +d’apprentissage automatique qui permet à un algorithme d’interagir avec un oracle +durant le processus. Dans un cadre d’apprentissage classique, les données sont choi- +sies au préalable et imposées. En apprentissage actif, c’est l’algorithme d’apprentis- +sage qui demande les données jugées les plus pertinentes. +Le processus est itératif et s’arrête lorsqu’un critère de performances ou un nombre +défini de données annotées ou d’itérations est atteint. +Avantages +— L’utilisation de l’apprentissage actif permet de réduire fortement le coût d’an- +notation manuelle. +— Les performances du modèle final sont améliorées en comparaison avec un entraî- +nement classique puisque les données sont choisies afin d’optimiser les résultats. +Inconvénients +— Une fonction d’acquisition est nécessaire. Les fonctions d’acquisition permettent +d’associer une donnée à une valeur qui encode soit l’incertitude du modèle sur +cet exemple soit sa contribution dans l’ajustement du modèle. Plusieurs fonc- +tions ont été proposées dans le domaine de l’uncertainty sampling telles que +l’entropie, se basant directement sur les probabilités a posteriori, ou la dis- +tance des exemples par rapport à la limite de décision dans les SVM. D’autres +approches se basent sur la différence entre les résultats donnés par plusieurs +modèles (Query By Committee). +— Il est également nécessaire de définir une stratégie de sélection : quels exemples +seront utilisés pour l’itération suivante ? Certains travaux choisissent les +exemples où les probabilités a posteriori sont les plus hautes ou les plus basses. +Dans le cas des SVM, certains choisissent les exemples les plus proches, d’autres +les plus éloignés de la limite de décision. Il n’y a, à notre connaissance, pas de +consensus sur la stratégie à utiliser. +Exemple d’apprentissage actif +La Figure 2.16 présente un exemple d’apprentissage actif. Les paramètres du modèle +sont initialisés ou pré-entraînés sur l’ensemble d’apprentissage annoté. Le modèle est +appliqué aux exemples non annotés qui sont ensuite sélectionnés selon la stratégie +choisie. Les exemples sélectionnés sont annotés par un opérateur puis ajoutés à l’en- +semble d’entraînement. Un nouveau modèle est entraîné sur ce nouvel ensemble. Le +processus est répété jusqu’à ce que les conditions d’arrêt prédéfinies soient atteintes. + +2.2 E S T I M AT I O N D E L A C O N F I A N C E D E S O B J E T S D É T E C T É S +41 +Ensemble +d’entraˆınement +annot´e +Mod`ele d’apprentissage +profond +Entraˆınement +initial +Corpus +non annot´e +Pr´ediction +Oracle +S´election +Nouvel ensemble +d’entraˆınement +annot´e +Annotation +Entraˆınement +Figure 2.16 – Schéma présentant le processus d’apprentissage actif. + + +3 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S +D E D É T E C T I O N +La mise en place et l’amélioration de modèles de détection d’objets conduisent à explo- +rer différents axes de recherche. Bien que la majorité des travaux dans la littérature se +concentrent uniquement sur la proposition de nouvelles architectures, nous avons souhaité +nous intéresser à des études et des solutions plus complètes, en évoquant notamment les +problématiques liées aux annotations et évaluations. En effet, l’évaluation de la qualité +des algorithmes de détection ou de reconnaissance est cruciale dans la mise au point de +systèmes et leurs comparaisons. Elle nécessite donc l’utilisation de métriques appropriées. +Cependant, il faut également étudier les données annotées utilisées pendant l’entraînement et +l’évaluation. Si les annotations des données ne sont pas cohérentes avec la métrique utilisée, +la métrique ne peut pas refléter les performances réelles du modèle. +Dans ce chapitre, nous mettons tout d’abord en avant les problèmes liés aux annotations +des jeux de données. Dans une première section 3.1, nous présentons une étude des récents +jeux de données utilisés dans les systèmes à l’état de l’art, principalement pour la détection +de lignes de texte. Nous mettons ensuite en évidence, en section 3.2, les différentes règles +d’annotation manuelle, ainsi que les défis liés et les solutions proposées dans la littérature. +Par la suite, nous discutons, en section 3.3, des différentes métriques proposées et utilisées +dans la littérature afin d’évaluer et de comparer les systèmes de détection d’objets dans les +images de documents. +3.1 +J E U X D E D O N N É E S +Dans cette partie, nous présentons les jeux de données utilisés dans les systèmes récemment +proposés, notamment pour la détection de lignes de texte. Nous présentons également les jeux +de données privés que nous avons utilisés durant la thèse. Ces jeux de données sont détaillés +dans les paragraphes suivants, résumés dans la Table 3.1, et un exemple est montré sur la +Figure 3.2. +Nous nous focalisons sur la tâche de détection de lignes de texte car c’est une étape centrale +de l’analyse de la mise en page des documents puisqu’elle est nécessaire à la reconnaissance +de texte et qu’elle a un fort impact sur la qualité de la reconnaissance. De plus, c’est une +des tâches pour laquelle le type d’annotation et la définition même de la tâche peuvent être +très variables. Une étude plus générale sur les jeux de données historiques est proposée par +Nikolaidou et al. (2022). +43 + +44 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N +Table 3.1 – Tableau récapitulatif des différents jeux de données utilisés pour la détection de lignes de texte. Le symbole "–" indique une résolution ou date non +disponible. Pour chaque jeu de données, la colonne Taille indique la taille moyenne des images, calculée sur l’ensemble d’entraînement. +Jeu de données +Date +Images +Lignes +Langue(s) +Résolution (dpi) +Taille (pixels) +AN-Index† +– +34 +666 +Français +– +[1 949, 1 338] ± [796, 607] +Balsac +Anglais +Vézina et al. (2020) +1850 – 1916 +913 +45 685 +Français +– +[3 746, 2 671] ± [1 141, 627] +BNPP† +19e – 20e siècle +12 +1 281 +Français +– +[3 710, 5 103] ± [21, 86] +Bozen +Sánchez et al. (2016) +1470 – 1805 +450 +10 550 +Allemand +– +[3 524, 2 398] ± [22, 62] +cBAD2019 +Diverses +Diem et al. (2019) +– +3 021 +193 858 +européennes +Variable +[3 268, 2 751] ± [1 364, 1 504] +DIVA-HisDB +Italien, Latin +Simistira et al. (2016) +11e / 14e siècles +150 +12 808 +Allemand, Grec +600 +[5 493, 3 843] ± [709, 728] +HOME-Alcar +Stutzmann et al. (2021) +12e – 14e siècle +1 845 +136 206 +Latin +Variable +[3 850, 4 506] ± [949, 1 820] +Allemand +HOME-NACR +Latin +Boros et al. (2020) +1145 – 1491 +496 +7 614 +Tchèque +– +[4 499, 6 206] ± [1 639, 2 292] +Hugin-Munin +Maarand et al. (2022) +19e – 20e siècle +849 +23 732 +Norvégien +– +[3 998, 3 740] ± [1 405, 1 403] +Horae +Boillet et al. (2019) +14e – 15e siècle +573 +13 796 +Latin +Variable +[4 200, 4 648] ± [1 361, 2 112] +IAM +Marti et al. (2002) +1999 +1 539 +13 353 +Anglais +300 +[3 542, 2 479] ± [0.45, 0.33] +RASM +Clausner et al. (2018) +10e – 19e siècle +120 +2 619 +Arabe +400 +[7 674, 5 408] ± [1 384, 914] +READ +Diverses +Grüning et al. (2017) +1470 – 1930 +2 035 +132 124 +européennes +Variable +[3 966, 3 121] ± [1 350, 1 268] +ScribbleLens +Variable +Dolfing et al. (2020) +16e – 18e siècle +1 000 +28 255 +Néerlandais +150 – 300 +[3 519, 2 375] ± [665, 459] +† Jeux de données privés utilisés durant la thèse. + +3.1 J E U X D E D O N N É E S +45 +AN-Index – Ce premier jeu de données est composé de 34 images de documents des +instruments de recherche numérisés des Archives nationales françaises. Il s’agit d’une +base privée dont les documents sont rédigés en français. +Balsac – Depuis 50 ans, le projet BALSAC 1 construit une importante base de données +sur la population du Québec. Pour entraîner des modèles de traitement automatique +et ainsi aider l’intégration de millions d’enregistrements, un échantillon du corpus +contenant les actes de naissance, de mariage et de décès de la population québécoise +de 1850 à 1916 a été annoté. Le jeu de données Balsac (Vézina et al., 2020) consiste +donc en 913 images (pages simples ou doubles) extraites de 74 registres manuscrits. +Elles ont été annotées au niveau des actes et des lignes avec leurs transcriptions et +entités nommées. +BNPP – Ce jeu de données privé a été fourni par les Archives historiques de la banque +BNP Paribas 2. Il consiste en un échantillon de 12 images manuscrites extraites de cinq +registres scannés de procès-verbaux du Comptoir National d’Escompte de Paris. Elles +ont été sélectionnées parmi une centaine de registres rédigés en français entre le 19e et +le 20e siècle. +Bozen – Ce jeu de données (Sánchez et al., 2016) fait partie du projet READ et +consiste en 450 pages manuscrites annotées. Les pages sont extraites de documents de +la collection Ratsprotokolle écrits entre 1470 et 1805. Il est annoté au niveau des lignes +de texte avec leurs transcriptions. +cBAD2019 – Le jeu de données cBAD (Diem et al., 2019) est constitué de 3 021 images +de documents collectées dans sept archives européennes. Il a été utilisé lors de la +compétition cBAD à ICDAR2019 pour la détection des lignes de base. +DIVA-HisDB – DIVA-HisDB (Simistira et al., 2016) est une base de données qui +contient 150 images extraites de trois manuscrits médiévaux des 11e et 14e siècles. +Ces manuscrits ont été choisis pour la complexité de leurs mises en page avec du +texte principal, et des commentaires dans les marges et entre les lignes. Pour chaque +manuscrit, 50 images ont été sélectionnées et réparties en 20 images d’entraînement, +10 images de validation, 10 images de test et 10 autres images de test « privées ». +Chaque image a été annotée manuellement au niveau du pixel pour les classes de corps +de texte, décorations et commentaires. +HOME-Alcar – Le jeu de données HOME-Alcar (Stutzmann et al., 2021) contient 17 +cartulaires, recueils des chartes et des actes juridiques produits entre le 12e et le 14e +siècle et écrits en latin. Les images ont été annotées au niveau des lignes de texte avec +leurs transcriptions. +1. https://balsac.uqac.ca/ +2. https://history.bnpparibas/ + +46 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N +HOME-NACR – Le jeu de données HOME-NACR (Boros et al., 2020) est composé +de 496 chartes médiévales sélectionnées parmi 43 000 chartes numérisées provenant +des archives de la Couronne de Bohême et des archives des monastères. Elles ont +été rédigées de 1145 à 1491 en allemand, latin et tchèque du début de l’ère moderne. +Les chartes ont été annotées au niveau des lignes avec leurs transcriptions et entités +nommées. +Hugin-Munin – La base de données Hugin-Munin (Maarand et al., 2022) est constituée +de pages provenant de correspondances et de journaux intimes de 12 artistes norvégiens +écrits de 1820 à 1950. Les documents ont été annotés au niveau des lignes avec leurs +transcriptions correspondantes. La base comporte 691 images d’entraînement, 85 de +validation et 73 de test. +Horae – Durant le projet de recherche Hours : Recognition, Analysis, Edition (HORAE) +(Stutzmann et al., 2019), un jeu de données a été créé (Boillet et al., 2019) et +consiste en 573 images annotées de livres d’heures. Elles ont été sélectionnées parmi +500 manuscrits car elles représentent la variété des mises en page et des contenus. +Les images ont été annotées à différents niveaux et avec différentes classes : page, +paragraphe, ligne, miniature, initiale (simple, ornée ou illustrée), marge (ornée ou +illustrée), ornementations et rubriques. +IAM – Le jeu de données IAM (Marti et al., 2002) a été créé en 1999 et contient 1 539 +images de documents. Chaque image comporte une page avec un texte imprimé extrait +du corpus Lancaster - Oslo/Bergen corpus (LOB), puis ce même texte écrit à la main. +Les textes datent de 1961 et sont très divers : fictions, écrits scientifiques ou encore +textes traitant de religion. +RASM – La compétition RASM 2018 (Clausner et al., 2018) visait la reconnaissance +de manuscrits historiques en arabe. Un ensemble de 15 images de pages à une colonne +a été utilisé pour l’entraînement et 85 pour évaluer les tâches de détection de lignes de +texte, segmentation de pages et reconnaissance d’écriture manuscrite. Au total, le jeu +contient 120 images extraites parmi une collection de manuscrits scientifiques arabes. +READ-BAD – Ce jeu de données (Grüning et al., 2017) contient de 2 035 images de +documents écrits entre 1470 et 1930 et extraits de neuf archives européennes. Les +données sont très variées avec des registres paroissiaux, des procès-verbaux ou encore +des tables de recensement. Ce jeu a été utilisé lors de la compétition sur la détection +de lignes de base cBAD : ICDAR2017 (Diem et al., 2017). Le jeu de données est divisé +en sous-ensembles simples et complexes dépendant de la complexité de mise en page +des documents. + +3.2 A N N O TAT I O N D E S D O N N É E S +47 +ScribbleLens – Le jeu de données ScribbleLens (Dolfing et al., 2020) contient 1 000 +images de pages de documents néerlandais datant du début de l’ère moderne, tels que +des journaux de bord de navires et des journaux de bord quotidiens produits entre le +16e et le 18e siècle. Les manuscrits consistent en des voyages écrits par des capitaines +et des commerçants de la Vereenigde Oost-indische Company (VOC). L’ensemble de +test est composé de 21 images annotées et transcrites au niveau de la ligne. +Nous observons que de nombreux jeux de données ont été présentés pour la détection de +lignes de texte. Il est important de noter que ces jeux de données ont été annotés à l’aide de +différents outils et pour différentes tâches : détection de lignes de texte ou de lignes de base. +Nous détaillons les différents types d’annotations dans la section suivante 3.2. +3.2 +A N N O TAT I O N D E S D O N N É E S +Bien que la tâche de détection de lignes de texte soit assez triviale dans le cas de documents +imprimés, dans le cas de documents manuscrits de nombreux aspects peuvent venir perturber +la bonne détection des lignes. En effet, il n’est pas rare que des lignes se chevauchent ou que +des initiales soient de taille très différente comparé au corps de texte. De plus, la qualité de +numérisation et les possibles dégradations liées à la conservation peuvent rendre cette tâche +d’autant plus complexe. +Un autre défi avec la détection de lignes de texte concerne la définition de ce qu’est une +ligne de texte. Dans la littérature, une ligne de texte a été définie de plusieurs manières, +comme présenté sur la Figure 3.1 (Mechi et al., 2021 ; Renton et al., 2018). Tout d’abord, +elle peut être définie uniquement par sa ligne de base (Grüning et al., 2017), qui correspond +à une ligne virtuelle soulignant la plupart des caractères tandis que les descendants restent +en dessous. Dans ce cas il est nécessaire d’estimer la hauteur de la ligne pour appliquer +un reconnaisseur d’écriture. Elle a également été définie comme étant une boîte englobante +Rectangle +englobant +Ascendant +Descendant +Polygone +englobant +X-height +Pixels +Ligne +de base +Figure 3.1 – Représentation des modélisations d’une ligne de texte proposées dans la littérature. + +48 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N +(rectangle ou polygone) incluant tous les ascendants et descendants (Moysset et al., 2015), +comme un ensemble de pixels appartenant au contenu textuel (Simistira et al., 2016 ; Vo +et al., 2016) ou encore s’appuyant sur la hauteur en X (X-height). Il s’agit de la bande de +base de la ligne sans les ascendants et descendants. +Définir une ligne de texte par sa bande de base présente de nombreux avantages par +rapport aux autres représentations. Tout d’abord, elle représente bien les interlignes même +lorsque les lignes se chevauchent en raison des ascendants ou des descendants, contrairement +à une représentation par boîte englobante incapable de séparer les lignes qui se chevauchent. +De plus, l’utilisation de la représentation pixel ou de la ligne de base nécessite un post- +traitement avant de pouvoir être transmise à un reconnaisseur, contrairement à la bande de +base qui semble plus appropriée à fournir une entrée convenable pour les reconnaisseurs de +texte. +Nous résumons, dans la Table 3.2, les détails d’annotations de chaque jeu de données pré- +senté en section 3.1. Dans cette Table, nous indiquons comment les lignes ont été annotées : +ligne de base, bande de base, polygone englobant, quadrilatère (ou polygone simple) englo- +bant et rectangle englobant. Nous indiquons également le taux de relâchement des annotations +par rapport aux pixels de texte. Nous définissons ce taux comme étant la quantité de fond +présent autour des pixels de texte dans les annotations (autres que la ligne de base). Les co- +lonnes Intersections et Source présentent respectivement la quantité de chevauchements +entre les annotations, et la manière dont les annotations ont été obtenues (manuellement, +semi-automatiquement ou automatiquement). La dernière colonne indique la présence de +transcription des lignes de texte. Cette Table montre explicitement que les annotations entre +les différents jeux de données sont très variables. En effet, il n’y a aucun consensus sur la +définition même d’une ligne de texte ni sur la forme que les annotations doivent avoir ou la +quantité de fond à intégrer dans les polygones et boîtes englobants. De plus, il n’y a aucune +étude, à notre connaissance, comparant les différentes annotations possibles et évaluant leurs +impacts sur les résultats de reconnaissance de texte finaux. +La Figure 3.2 présente une image de chaque jeu de données associée à son taux de +relâchement. +Les problèmes liés aux annotations des jeux de données ont également été peu étudiés +dans la littérature. En effet, Barakat et al. (2018) montrent les problèmes liés aux lignes +de texte qui se touchent et se superposent dans leur jeu de données mais ne traitent pas ces +différents problèmes. Quelques rares travaux en discutent et proposent quelques solutions. +Par exemple, face à des boîtes englobantes qui se touchent, Melnikov et al. (2020) ont +suggéré de supprimer les ascendants et les descendants des lignes de texte en réduisant la +hauteur des boîtes annotées de 30 % en haut et en bas. Ils ont ensuite redimensionné les +polygones à la résolution d’entrée du modèle pour entraîner le système. Même si cette méthode +s’est avérée efficace pour réduire le biais d’étiquetage de l’annotation, certains problèmes +subsistent lorsqu’il s’agit de lignes verticales et inclinées. Dans le même esprit, Peskin et al. +(2020) ont proposé différents masques d’annotation (voir Figure 3.3) pour la détection et + +3.2 A N N O TAT I O N D E S D O N N É E S +49 +Table 3.2 – Tableau récapitulatif du type d’annotation des différents jeux de données utilisés pour la détection de lignes de texte. La colonne Relâchement +indique la quantité de fond présent dans les annotations (important, moyen ou faible). La colonne Intersections indique si des lignes se chevauchent. +La colonne Source indique comment les annotations ont été obtenues : manuellement, semi-automatiquement ou automatiquement. La colonne Texte +indique la présence de transcriptions des lignes de texte. +Jeu de données +Ligne +Bande +Polygone +Quadrilatère +Rectangle +Relâchement +Intersections +Source +Texte +de base +de base +englobant +englobant +englobant +AN-Index† +✓ +✓ +Important +Rares +Manuelle +Balsac +Vézina et al. (2020) +✓ +✓ +Moyen +Rares +Manuelle +✓ +BNPP† +✓ +Moyen +Non +Manuelle +✓ +Bozen +Semi- +Sánchez et al. (2016) +✓ +✓ +✓ +Important +Oui +automatique +✓ +cBAD2019 +Diem et al. (2019) +✓ +✓ +✓ +Variable +Oui +Manuelle +DIVA-HisDB +Semi- +Simistira et al. (2016) +✓ +✓ +Faible +Non +automatique +HOME-Alcar +Stutzmann et al. (2021) +✓ +Important +Oui +Automatique +✓ +HOME-NACR +Boros et al. (2020) +✓ +Important +Rares +Manuelle +✓ +Hugin-Munin +Semi +Maarand et al. (2022) +✓ +✓ +Variable +Oui +automatique +✓ +Horae +Boillet et al. (2019) +✓ +Moyen +Non +Manuelle +✓ +IAM +Marti et al. (2002) +✓ +Moyen +Rares +Automatique +✓ +RASM +Clausner et al. (2018) +✓ +Faible +Rares +Manuelle +✓ +READ +Grüning et al. (2017) +✓ +✓ +✓ +Variable +Oui +Manuelle +ScribbleLens +Dolfing et al. (2020) +✓ +Important +Oui +Automatique +✓ +† Jeux de données privés utilisés durant la thèse. + +50 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N +HOME-Alcar +ScribbleLens +RASM +Bozen +IAM +Horae +HOME-NACR +AN-Index +Balsac +BNPP +cBAD +READ-BAD +Hugin-Munin +DIVA-HisDB +Relˆachement +important +Relˆachement +moyen +Relˆachement +faible +Figure 3.2 – Visualisation des différents taux de relâchement détectés dans les jeux de données. Les taux de relâchement indiquent la quantité de fond présent +autour des pixels de texte dans les annotations. + +31 +3c. +paeus Jeeun-die.cuos sharmm +Allacpmoa.er +a osull non cuffioyecas pstaaumtoa.o +Queng5olitayeonennateim umaneque c +cehrerephetnnaum +eroitee S faeecm poulersfoa +suiplepefn hoc ircam Loghootalye Ligaed +Cg +emoh +fiusoim nmlonem Louicg inlren f-em mcin +nwmme. +wnuo. +ana +nfimeonem piestatoum.sincar cice:pame +prefeires +ceae nfpedus.Olhe +is +2oamFaem +jemfaar- BAcmmenerim. +allmeimu +uosnm +Amcefmonum.fporefecesLiacas +jalels mane.aecrueyuose +nmndFeernun +mles. +.omeneelo +Lts.nommac +wnc +Chcoblitum saj-may itusallag-satue +mono +cem Cnicne.pcipiencos mmaentf-Afef-umSmt +Letues mles.fanemereisius fu.coainfl +ametoum.Aatbae +ncrufaemax +ng Silcie. +ylpino Eheliuramo nr +ccalte +-asupmoir-eeim.swie.cjuos6i +mahe +ek +ncuen +Arolles +gehaiert +velek +kranl +ancksedgin +Arere +wytrrehhs +Zagenasy +D27 +dwele +Leel012029 +S3SentenceDatabase +N06-169 +as of now, apart from a few sacks of gold dust." He winked at his partners, They all +watched us as we ate the beans, Then when we'd finished and I'd rolled a cigarette the +man called Shorty said, "You were saying when you cane in that 2romep'n happeued +last night." +heyes +hieeLobr +EHeAEL +eofForheederedLerolleekacroaek +A +Csemepnhepperedlasr +eh +Seaemat +YYYYYYYYCY +ooemceetal +Do0somrKODAKColorControlPatches +White +Kodaloul Jean +nhmulevue +tntchomcle +mahup d +aRochelh +0 691mkh13.10 +737 +mReclu +1318 +Melens174 +Tbanguay Sreat +KUC +essawmel +agenaSlasell +utle +Coli +ltapigaliot +Speilt房 +nenahkwbwtuleM +Cam +3 +M +Guhellu +340GrimmNr.Ms30 +ldeeLalt +Best +00 +Culpi-oitmneedeansparem +Cpadntagmuoam/e +uebemgeecodnuabnth +duetmnabrammem/monaadathemm +Caogpuemh-mmtnediczombu-dntcedmc +mampatmatongndemde +aogtimdagumdcapoeo +idnoumtit-AfatimnoncamnabmusT +mtaurmafmaommbnaaxpdae +odeayapdooneilmommn +Qcodecmmemmagpuconb +taggmonnepunoabitronuopuya +doptim-etogmamgprahgtiumdopunm3.3 M É T R I Q U E S D’ É VA L U AT I O N +51 +Figure 3.3 – Masques de segmentation comparés par Peskin et al. (2020) : A. petites marques cen- +trales, B. marques centrales plus grandes, C. petite marque centrale avec un contour +d’un pixel, D. petite marque centrale avec un contour de deux pixels. Schéma issu de +Peskin et al. (2020). +la classification de formes géométriques à partir d’images en niveaux de gris. Ils ont suggéré +d’annoter les objets (cercles, rectangles et triangles) de quatre façons : avec de petites marques +centrales, avec de grandes marques centrales, avec de petites marques centrales et un contour +d’un pixel, et avec de petites marques centrales et un contour de deux pixels. Concernant +les problèmes de localisation, ils ont montré que les petites marques centrales donnent les +meilleures performances. Cependant, de meilleurs résultats de classification sont obtenus avec +les petites marques centrales avec contours. Par conséquent, trouver l’annotation la plus +adaptée n’est pas un problème trivial et les discussions sont toujours ouvertes, mais nous +proposons, dans le chapitre 5, une solution pour l’unification des différents types d’annotations +pour entraîner des modèles de détection de lignes de texte. +3.3 +M É T R I Q U E S D’ É VA L U AT I O N +En plus des problèmes liés aux différents types d’annotations se pose le problème de re- +cherche de métriques appropriées pour évaluer et comparer correctement les résultats de +détection. En effet, afin d’évaluer la qualité d’un modèle, une métrique d’évaluation doit être +définie et calculée. Cette métrique doit avoir un sens vis-à-vis des données utilisées et de +l’application future des résultats obtenus. Les différentes métriques permettant d’évaluer un +modèle de détection d’objets peuvent être regroupées en plusieurs catégories : les métriques +basées sur les pixels, les métriques objets et les métriques orientées vers la tâche finale. Celles- +ci sont détaillées dans les paragraphes suivants. +De plus, Hemery et al. (2010) ont étudié les propriétés clés qu’une métrique doit avoir pour +une tâche de localisation. À partir de l’analyse de 33 métriques existantes, ils ont établi les +plus appropriées pour cette tâche. En suivant cette idée, nous montrons, dans cette section, +que les principales métriques actuellement utilisées ne sont pas suffisantes pour évaluer et +comparer les modèles de détection d’objets. +3.3.1 +métriques basées sur les pixels +Les métriques calculées au niveau des pixels sont principalement basées sur l’intersection +entre les pixels d’une région prédite et ceux d’une région annotée manuellement. +Comme le montre la Table 3.3, la majorité des systèmes de détection existants sont évalués +à l’aide de métriques pixel. Les mesures de précision et de rappel, détaillées dans le Focus 3.1, + +52 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N +Table 3.3 – Métriques d’évaluation utilisées dans les récents travaux liés à la détection d’objets dans +les images de documents. P et R représentent respectivement les métriques précision et +rappel. Les métriques R@.85 et mAP@.65 représentent respectivement le rappel pour un +seuil d’IoU de 0,85 et la précision moyenne pour un seuil d’IoU de 0,65. +Système +Pixel +Objet +IoU +P/R +F1 +P/R +R@.85/.95 +mAP@.65 +mAP +Tensmeyer et al. (2017) +✓ +Yang et al. (2017) +✓ +✓ +Barakat et al. (2018) +✓ +✓ +Renton et al. (2018) +✓ +✓ +dhSegment +Ares Oliveira et al. (2018) +✓ +✓ +✓ +✓ +Mechi et al. (2019) +✓ +✓ +✓ +Tarride et al. (2019) +✓ +✓ +✓ +✓ +Soullard et al. (2020) +✓ +Melnikov et al. (2020) +✓ +✓ +Mechi et al. (2021) +✓ +✓ +sont largement utilisées, ainsi que l’Intersection-sur-Union (IoU) (Focus 3.2) et le score F1 +(Focus 3.3). Dans le cas d’une détection à plusieurs classes et afin de calculer une unique valeur +d’évaluation, les métriques sont calculées pour chaque classe et la moyenne arithmétique +des valeurs obtenues est calculée. Alberti et al. (2017) ont d’ailleurs développé un outil +permettant d’évaluer des modèles à partir des images de vérité terrain et des prédictions. +Cet outil permet de calculer différentes valeurs dont l’IoU, mais également de visualiser les +résultats. +Ces métriques sont basées sur le nombre de pixels correctement prédits. Cependant, elles +ne donnent aucune information sur le nombre d’objets correctement prédits et manqués ou +divisés. Ces métriques sont également biaisées. En effet, comme le montre la Figure 3.4, +plusieurs prédictions de qualités différentes peuvent être caractérisées par les mêmes valeurs +d’IoU et de F1. En effet, les Figures 3.4b et 3.4c présentent deux prédictions pour la même +image, en haut, et leur superposition, en bas, avec l’image de vérité terrain présentée sur +la Figure 3.4a. La première prédiction montre des lignes divisées et fusionnées, une ligne +manquante (en rouge) et quelques faux positifs (en cyan). Au contraire, la seconde prédiction +montre des lignes plus épaisses mais pas de ligne manquante ni de faux positifs. Ainsi, la +seconde prédiction semble meilleure. Cependant, les valeurs d’IoU et de score F1 sont égales + +3.3 M É T R I Q U E S D’ É VA L U AT I O N +53 +(a) Image de label. +(b) Première prédiction : +IoU = 0,72 +F1 = 0,84 +P += 0,81 +R = 0,87 +AP@.5 = 0,68 +(c) Seconde prédiction : +IoU = 0,72 +F1 = 0,84 +P += 0,75 +R = 0,95 +AP@.5 = 0,94 +Figure 3.4 – Deux détections de lignes différentes obtenues pour une même image et obtenant les +mêmes scores d’IoU et de F1. Les superpositions sont générées avec l’outil DIVA +(Alberti et al. (2017)). Le vert et le noir correspondent respectivement aux pixels +d’arrière-plan et d’avant-plan correctement prédits. Le cyan représente les pixels fausse- +ment positifs et le rouge les pixels faussement négatifs. Ici, seul le score AP au niveau +objet (avec un seuil IoU de 50 %) permet d’évaluer et de comparer les prédictions avec +précision. +pour les deux prédictions. Cela illustre le fait que ces métriques ne sont pas les plus adaptées +pour évaluer les systèmes de détection d’objets, d’où le développement de nouvelles métriques +basées sur les objets. Cependant, le calcul de ces valeurs de précision et de rappel au niveau +de la ligne ou de l’objet n’est pas directement applicable car la décision qu’un objet soit bien +ou mal détecté est plus complexe. +Un autre aspect des faiblesses des métriques pixel a été étudié par Cheng et al. (2021). +Ils mettent en évidence le fait que la métrique IoU ne se concentre pas suffisamment sur les +contours des objets qui sont les positions les plus importantes à détecter dans des tâches +de détection d’objets. Ils présentent Boundary IoU, une nouvelle mesure d’évaluation de la +segmentation axée sur la qualité des frontières. Ils analysent différents types d’erreurs sur +différentes tailles d’objets et montrent que leur métrique est significativement plus sensible +aux erreurs de frontière pour les objets de grande taille, sans pénaliser les erreurs sur les +petits objets. De la même manière, ils proposent une adaptation de la précision moyenne se +basant sur le Boundary IoU. +De plus, lorsqu’un objet prédit ne possède pas d’intersection avec un objet réel, leur IoU +est égale à zéro. Cependant, la prédiction peut-être plus ou moins proche de la vérité, or l’IoU +ne peut pas refléter ce phénomène. Ainsi, Rezatofighi et al. (2019) ont proposé la GIoU +(Generalized IoU) qui permet de considérer l’espace entre les deux objets dans l’évaluation. + +54 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N +Focus 3.1 – PRÉCISION ET RAPPEL +Définition +Dans le cadre de la détection d’objets, la précision est le nombre d’éléments (pixels +ou objets) pertinents détectés, d’une classe considérée, rapporté au nombre total +d’éléments détectés de cette classe. Elle tente de répondre à la question « Quelle +proportion d’identifications positives est correcte ? ». +Le rappel est défini par le nombre d’éléments pertinents détectés rapporté au +nombre total d’éléments annotés de la classe considérée. Il montre donc la +proportion de positifs réels qui ont été correctement identifiés. +Équations +P = +TP +TP + FP +R = +TP +TP + FN +(3.1) +avec : +— TP : nombre de pixels ou d’objets positifs correctement prédits ; +— FP : nombre de pixels ou d’objets négatifs prédits comme positifs ; +— FN : nombre de pixels ou d’objets positifs prédits comme négatifs. +Focus 3.2 – INTERSECTION-SUR-UNION +Définition +La métrique Intersection-sur-Union (IoU) évalue la division entre la zone de +chevauchement et la zone d’union entre deux régions. En d’autres termes, elle +évalue le degré de chevauchement entre la vérité terrain et les prédictions. Elle est +comprise entre 0 et 1, où 1 correspond à un chevauchement parfait entre la vérité +terrain et la prédiction. +Équation +IoU = +TP +TP + FP + FN +(3.2) +avec : +— TP : nombre de pixels positifs correctement prédits ; +— FP : nombre de pixels négatifs prédits comme positifs ; +— FN : nombre de pixels positifs prédits comme négatifs. +Focus 3.3 – F1-SCORE +Définition +La F-mesure, aussi appelé F1-score, est la moyenne harmonique entre la précision +et le rappel. + +3.3 M É T R I Q U E S D’ É VA L U AT I O N +55 +Équation +F1-score = +2 × TP +2 × TP + FP + FN += 2 × P × R +P + R +(3.3) +avec : +— TP : nombre de pixels ou d’objets positifs correctement prédits ; +— FP : nombre de pixels ou d’objets négatifs prédits comme positifs ; +— FN : nombre de pixels ou d’objets positifs prédits comme négatifs. +3.3.2 +métriques orientées objets +Bien que des évolutions des métriques usuelles au niveau pixel aient été proposées, elles +ne fournissent toujours pas d’évaluation au niveau objet et ne permettent pas d’indiquer +le nombre d’objets correctement détectés. Un des problèmes pour la mise en place d’une +métrique objet est la difficulté à déterminer si un objet est correctement détecté ou non. +Pour résoudre ce problème, des métriques conçues à l’origine dans la communauté de la +recherche d’information (Information Retrieval) ont été adaptées aux images, et utilisées lors +du PASCAL VOC Challenge 2012 pour calculer la précision au niveau des objets. Lors de +cette compétition, la tâche de détection a été évaluée sur la base de la courbe Précision- +Rappel au niveau objet, où les détections sont considérées comme de vrais ou de faux positifs +en fonction de leur zone de recouvrement avec les objets de vérité terrain. +Suivant cette approche, Tarride et al. (2019) associent d’abord les objets prédits à ceux +annotés et considèrent une prédiction comme un vrai positif si son IoU est supérieure à un +seuil choisi t = 0,65. Ainsi, ils peuvent calculer la précision (P@.65), le rappel (R@.65) et +la précision moyenne (mAP@.65) au niveau des objets. Les calculs de précision moyenne +sont détaillés dans le Focus 3.4. Soullard et al. (2020) utilisent la mean Average Precision +(mAP), c’est-à-dire la précision moyenne calculée pour différents seuils d’IoU, afin d’évaluer +leur modèle de détection. +Wolf et al. (2006) ont montré l’importance de la qualité de détection (précision des objets +détectés) et de la quantité de détections (nombre d’objets) lors de l’évaluation d’un système. +La mesure mAP, qui est l’aire sous la courbe Précision-Rappel, permet d’évaluer la quantité +de détections en fonction d’un critère de qualité donné : le seuil d’IoU. Afin de pouvoir +mesurer autant la qualité que la quantité de détections, l’utilisation de la mAP moyennée +sur une gamme de seuils d’IoU a émergé pour la détection d’objets. Ainsi, Soullard et al. +(2020) ont utilisé cette moyenne mAP pour évaluer leur modèle de détection appliqué aux +journaux historiques. +La métrique ZoneMap proposée par Galibert et al. (2015) évalue également les systèmes +de détection au niveau objet et ne repose sur aucun seuil. Elle est basée sur les liens entre +les zones d’hypothèse et de référence. Les forces des liens sont d’abord calculées : si une zone +prédite est correcte, alors la force avec une zone de référence sera élevée. Au contraire, toutes +les forces pour une zone faussement positive seront faibles. Ensuite, les zones sont regroupées + +56 +E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N +en fonction de ces liens et chaque groupe reçoit une erreur de segmentation et une erreur +de classification, calculées en fonction du type de groupe (match, miss, false alarm, merge +ou split). Ces deux erreurs sont ensuite combinées pour donner une seule valeur. Même si +cette métrique s’est avérée complémentaire de la métrique IoU dans l’évaluation du projet +Maurdor (Oparin et al., 2014), elle n’est pas réellement utilisée à l’heure actuelle en raison de +la complexité de ses calculs et de sa difficile applicabilité aux images comportant de nombreux +objets. +Focus 3.4 – PRÉCISION MOYENNE / AVERAGE PRECISION +Définition +Le concept de la métrique d’évaluation de la précision moyenne est principalement +lié aux compétitions PASCAL VOC. Basé sur un seuil défini d’IoU, elle considère +les objets prédits comme vrais ou faux positifs, et calcule la précision moyenne grâce +à l’aire sous la courbe de la précision par rapport au rappel. +La mean Average Precision (mAP) est définie de plusieurs manières selon la com- +pétition. Dans le cas d’un problème à plusieurs classes, la mAP est définie comme +étant la valeur moyenne des AP calculées pour chaque classe. +Elle peut aussi correspondre à la moyenne arithmétique réalisée sur plusieurs seuils +d’IoU. En effet, pour s’abstenir d’un seuil prédéfini, la mAP est la moyenne des +AP calculées pour plusieurs seuils. Dans le cas des compétitions PASCAL VOC, la +moyenne est calculée sur des valeurs de seuil allant de 0,5 à 0,95 avec un pas de +0,05 (mAP@[.5 :.05 :.95] ou mAP@[.5,.95]). +Mise en oeuvre +— Toutes les prédictions sont ordonnées par leur confiance moyenne décroissante ; +— Les prédictions dont l’IoU est supérieur ou égal à un seuil t sont considérées +comme vrais positifs ; +— La courbe Précision-Rappel est construite à partir des prédictions ordonnées. +Cette courbe Précision-Rappel permet d’évaluer les performances d’un détec- +teur d’objets en fonction d’un seuil sur la confiance associée à la prédiction. +Il existe une courbe pour chaque classe d’objets. Un détecteur d’objets d’une +classe particulière est considéré comme bon si sa précision reste élevée alors que +le rappel augmente, ce qui signifie que si le seuil de confiance varie, la préci- +sion et le rappel resteront élevés. Habituellement, la courbe Précision-Rappel +commence par des valeurs de précision élevées, qui diminuent à mesure que le +rappel augmente ; +— La courbe est interpolée de telle sorte que la précision p pour un rappel r prenne +la valeur de la précision maximale des rappels supérieurs à r : +pinterp(r) = max +˜r⩾r p(˜r) +(3.4) +— La précision moyenne (AP@t) est égale à l’aire sous la courbe Précision-Rappel +interpolée. + +3.3 M É T R I Q U E S D’ É VA L U AT I O N +57 +Équations +Pour une classe donnée c et un seuil d’IoU t, nous avons : +AP@tc = +� 1 +0 +pc +t(rc +t)drc +t +(3.5) +Pour un seuil d’IoU t, la AP moyennée sur toutes les classes est calculée comme +suit : +mAP@t = +�C +c=1 AP@tc +C +(3.6) +Pour une classe donnée c, la AP moyennée sur plusieurs seuils est calculée comme +suit : +mAP@[.5,.95]c = +�0.95 +t=0.5 AP@tc +10 +(3.7) +Enfin, la AP moyennée sur plusieurs seuils et l’ensemble des classes est calculée +comme suit : +mAP = +�C +c=1 mAP@[.5, .95]c +C +ou mAP = +�0.95 +t=0.5 mAP@t +10 +(3.8) +avec : +— C : le nombre de classes ; +— p : la précision ; +— r : le rappel. +3.3.3 +métriques orientées vers la tâche finale +Trier et al. (1995) ont montré l’importance d’une évaluation orientée vers la tâche finale +pour les méthodes de binarisation, puisque l’évaluation par un expert humain dépend de ses +critères visuels. Ils ont appliqué onze méthodes de binarisations adaptatives locales à des +images de test avant de transmettre les résultats à un module de reconnaissance OCR. Les +méthodes de binarisation ont ensuite été comparées avec les taux de reconnaissance, d’erreur +et de rejet. Ils ont montré que les classements des méthodes en fonction de la qualité de +binarisation et des taux d’erreurs obtenus après le module OCR étaient différents, insistant +sur l’importance d’utiliser des métriques orientées vers la tâche finale. De même, Wolf et al. +(2006) ont montré l’importance de ces mesures pour la détection d’objets. Dans un cadre de +prédiction pixel, les objets sont reconstruits en regroupant les pixels connectés. Or, les objets +extraits sont similaires même si quelques pixels ont été mal prédits. Dans ce sens, une de +nos propositions vise à utiliser des métriques de reconnaissance de texte (HTR) pour évaluer +les systèmes de détection de lignes de texte. A notre connaissance, il n’existe pas de travaux +antérieurs dans la littérature à ce sujet. + + +4 +D É T E C T I O N D ’ O B J E T S D A N S D E S I M A G E S D E +D O C U M E N T S +La compréhension automatique de documents, et plus particulièrement l’analyse de la mise +en page de documents historiques, est toujours un domaine de recherche actif. Cette tâche +consiste à diviser un document en différentes régions en fonction de leur contenu. La grande +variété des documents existants rend cette tâche très complexe. Pour détecter des objets +dans des images de documents, de nombreux systèmes ont été proposés, la plupart assignant +une classe à chaque pixel de l’image donnée en entrée. Bien que ces systèmes aient montré +des performances intéressantes, ils nécessitent un grand nombre de données d’apprentissage +annotées et présentent des temps d’inférence longs. Dans ce chapitre, nous présentons deux +architectures à l’état de l’art et comparons leurs avantages et inconvénients. Par la suite, +nous proposons, en section 4.3, un système appelé Doc-UFCN mis au point afin de dépasser +les limitations mises en évidence par la comparaison des systèmes à l’état de l’art. Nous +présentons enfin une comparaison expérimentale des systèmes pour la détection des lignes de +texte, en section 4.4, et la détection d’actes, en section 4.5. +4.1 +P R É S E N TAT I O N D U P R O B L È M E +L’entraînement de modèles de détection d’objets dans les images de documents requiert +un grand nombre de données annotées. Cependant, dans le cas de documents historiques, ces +données annotées sont rarement disponibles. Pour pallier ce problème, des systèmes utilisant +des poids pré-entraînés tels que dhSegment (Ares Oliveira et al., 2018) ont été proposés. +Cela permet d’utiliser des réseaux avec plus de paramètres sur des jeux de données de tailles +réduites. De plus, l’utilisation du pré-entraînement a montré de nombreux avantages tels que +la diminution du temps d’apprentissage et l’amélioration de la précision du modèle. Cepen- +dant, ces poids sont souvent appris sur des images de scènes naturelles (ImageNet (Deng et +al., 2009)), puis appliqués à des images de documents, ce qui pose un problème d’adaptation +des modèles à un nouveau domaine. De plus, bien qu’ils obtiennent des performances élevées, +les systèmes actuellement à l’état de l’art peuvent présenter des temps d’inférence longs qui +peuvent avoir de grands impacts financiers et écologiques. Dans un cadre industriel, l’utilisa- +tion de tels systèmes ne semble pas appropriée. C’est pourquoi, nous proposons un nouveau +modèle, appelé Doc-UFCN, dans l’optique de répondre à ces problématiques de temps de +traitement. Les contraintes étant que ce système possède un nombre réduit de paramètres et +présente un temps d’inférence réduit par rapport aux systèmes existants, tout en obtenant +des performances à l’état de l’art. +59 + +60 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +4.2 +S Y S T È M E S À L’ É TAT D E L’ A RT +De nombreux modèles (Barakat et al., 2018 ; Mechi et al., 2019 ; Renton et al., 2018) +ont été proposés pour la détection d’objets dans les images de documents, notamment pour la +détection de lignes de texte. Ces modèles ont des architectures similaires, suivant une architec- +ture U-Net (Ronneberger et al., 2015). Le modèle dhSegment suit également l’architecture +U-Net, mais, contrairement aux autres systèmes, il intègre une partie pré-entraînée et a été +testé sur de nombreuses tâches telles que la détection de pages, de décorations ou encore de +photographies. C’est pourquoi, nous avons choisi de nous comparer à ce système. Il en est de +même pour le modèle de Yang et al. (2017) qui intègre l’information textuelle afin d’aider +la détection des objets et qui a obtenu de bonnes performances sur des images de documents +modernes. Ces deux systèmes sont détaillés dans les Focus 2.9 et 2.10. +4.3 +A R C H I T E C T U R E D U S Y S T È M E P R O P O S É : D O C- U F C N +Nous présentons, dans cette section, l’architecture du modèle que nous proposons +Doc-UFCN. Notre objectif est de concevoir un modèle comportant peu de paramètres afin +d’être entraîné sur des jeux de données restreints. De plus, celui-ci devra montrer des temps +d’inférence réduits par rapport aux modèles à l’état de l’art. Le développement de ce modèle +étant réalisé dans un cadre industriel, il est nécessaire d’avoir un modèle rapide en inférence, +capable de traiter des millions d’images de documents dans des délais raisonnables. En- +fin, ce modèle doit répondre à ces différents points tout en obtenant des performances élevées. +Pour concevoir notre système, Doc-UFCN, nous avons choisi d’utiliser le cœur du réseau +de Yang et al. (2017) car il possède un nombre réduit de paramètres et ne contient pas +de parties pré-entraînées. Pour réduire davantage le nombre de paramètres et être capable +d’entraîner notre modèle sur peu de données, seul le contenu visuel est utilisé. Par conséquent, +la carte d’incorporation de texte, le pont et le second décodeur pour la tâche de reconstruction +ne sont pas intégrés. Notre architecture est donc un réseau entièrement convolutif (FCN) en +forme de U composé d’un encodeur (blocs rouges sur la Figure 4.1) suivi d’un décodeur (blocs +bleus) et d’une couche de convolution finale. L’encodeur de Doc-UFCN diffère de celui de +dhSegment puisqu’il ne suit pas l’architecture ResNet-50. Notre encodeur possède beaucoup +moins de paramètres et est entièrement entraîné sur des images de documents. Cependant, +les deux systèmes ont des décodeurs similaires avec des connexions résiduelles où les cartes +de caractéristiques calculées pendant l’encodage, à une échelle donnée, sont utilisées pendant +l’étape de décodage de cette même échelle. +L’utilisation d’un FCN sans couche dense présente de nombreux avantages. Tout d’abord, +il réduit fortement le nombre de paramètres puisqu’il n’y a aucune connexion dense. De plus, +cela permet au réseau de traiter des images de taille variable et de conserver les informations +spatiales telles quelles. + +4.3 A R C H I T E C T U R E D U S Y S T È M E P R O P O S É : D O C- U F C N +61 +f +W +H +Dilated Block 1 +2f +W +2 +H +2 +Dilated Block 2 +4f +W +4 +H +4 +Dilated Block 3 +8f +Dilated Block 4 +4f +4f +W +4 +H +4 +Conv. +Block 1 +|| +2f +2f +W +2 +H +2 +Conv. +Block 2 +|| +f +f +W +H +Conv. +Block 3 +|| +c c +W +Softmax +Dilated convolution +Convolution +Max pooling +Upscaling +Softmax +|| +Concatenation +c +Number of classes +Figure 4.1 – Schéma de l’architecture du modèle Doc-UFCN. L’encodeur est représenté en rouge et le +décodeur en bleu avec respectivement H et W les hauteur et largeur de l’image d’entrée +et f le nombre de cartes de caractéristiques. +4.3.1 +encodeur +L’encodeur vise à extraire les caractéristiques importantes de l’image d’entrée. Il consiste +en quatre blocs dilatés. Ceux-ci sont légèrement différents de ceux présentés par Yang et al. +(2017) puisqu’ils consistent en cinq convolutions dilatées consécutives. L’utilisation de convo- +lutions dilatées au lieu de convolutions standards permet au champ réceptif d’être plus grand +et au réseau d’avoir plus d’informations contextuelles. De plus, exécuter ces convolutions +successivement plutôt qu’en parallèle permet d’agrandir le champ réceptif. Chaque bloc est +suivi d’une couche de max-pooling, sauf le dernier. +4.3.2 +décodeur +L’objectif du décodeur est de reconstruire l’image d’entrée avec un étiquetage pixel par pixel +à la résolution de l’image d’entrée originale. Cette déconvolution est généralement effectuée +à l’aide de convolutions transposées ou d’une mise à l’échelle. Comme suggéré par Mechi +et al. (2019), nous avons décidé de remplacer les couches de déconvolution du système de +Yang par des convolutions transposées afin de conserver la même résolution en entrée et en +sortie. Par conséquent, le chemin de décodage est composé de trois blocs convolutifs, chacun +consistant en une convolution standard suivie d’une convolution transposée. De plus, les +caractéristiques calculées lors de l’étape d’encodage sont concaténées avec celles de l’étape +de décodage (flèches violettes sur la Figure 4.1). + +62 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +La dernière couche convolutive produit des cartes de caractéristiques en pleine résolution. +Elle renvoie C cartes de caractéristiques de la même taille que l’image d’entrée, C étant le +nombre de classes concernées dans l’expérience. Une couche softmax est ensuite appliquée +pour transformer ces cartes de caractéristiques en cartes de probabilités. +4.3.3 +détails d’implémentation +Nous donnons, dans cette section, des détails techniques sur l’implémentation utilisée lors +de nos expériences. +taille des images en entrée +Nous avons décidé d’utiliser la même taille d’image d’entrée que celle de Yang et al. (2017). +Les images d’entrée ainsi que leurs vérités terrain sont donc redimensionnées en images plus +petites telles que la plus grande dimension de l’image soit égale à 384 pixels tout en conservant +le ratio de l’image originale. Cela permet de réduire le temps d’apprentissage et de traitement +sans perdre trop d’informations. Nous avons également entraîné le modèle sur une taille +d’entrée plus grande, de 768 pixels, pour voir l’impact de ce choix (voir en section 4.4.4). +blocs dilatés +Comme indiqué précédemment, tous les blocs dilatés sont composés de cinq convolutions +dilatées consécutives avec des taux de dilatation de 1, 2, 4, 8 et 16. Les blocs comportent +respectivement 32, 64, 128 et 256 filtres. Chaque convolution a un noyau de taille 3×3, un +stride de 1 et un padding adapté pour garder la même taille de tenseur dans tout le bloc. +Toutes les convolutions des blocs sont suivies d’une couche de batch normalization, d’une +activation ReLU et d’une couche de dropout. Le modèle comportant peu de paramètres, les +couches de dropout permettent d’éviter le sur-apprentissage. +blocs convolutifs +Les blocs convolutifs sont utilisés pendant l’étape de décodage. Le décodeur est composé +de trois blocs convolutifs et chaque bloc est composé d’une convolution standard suivie d’une +convolution transposée. Les blocs ont respectivement 128, 64 et 32 filtres. Chaque convolu- +tion standard a un noyau de taille 3×3, un stride et un padding de 1. Chaque convolution +transposée a un noyau de taille 2×2 et un stride de 2. Comme pour les blocs dilatés, toutes +les convolutions standards et transposées sont suivies d’une couche de normalisation, d’une +activation ReLU et d’une couche de dropout. +La dernière couche de convolution est paramétrée comme suit : C (nombre de classes) +filtres, noyau 3×3, stride et padding de 1. Elle est suivie d’une couche softmax qui calcule les +probabilités de chaque pixel d’appartenir aux C classes. + +4.4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E +63 +Table 4.1 – Statistiques des jeux de données utilisés pour la détection de lignes de texte. +Jeu de données +Images +Lignes +train +valid +test +train +valid +test +Balsac +Vézina et al. (2020) +730 +92 +91 +36 941 +4 592 +4 323 +DIVA-HisDB +Simistira et al. (2016) +60 +30 +30 +6 037 +2 999 +2 897 +Horae +Boillet et al. (2019) +522 +20 +30 +12 568 +270 +958 +READ-BAD +Grüning et al. (2017) +388 +49 +49 +22 885 +2 699 +2 481 +post-traitement +Comme étape de post-traitement, nous appliquons les mêmes opérations que celles appli- +quées par dhSegment : les pixels ayant une probabilité supérieure à un seuil t sont conservés +comme appartenant à la classe correspondante, les autres étant assignés au fond. Les compo- +santes connexes composées de moins de min_cc pixels sont supprimées. +4.4 +E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E +Dans cette section, nous comparons Doc-UFCN et dhSegment sur une tâche de détection +de lignes de texte. Nous montrons que Doc-UFCN obtient de meilleures performances tout en +ayant moins de paramètres et un temps de prédiction réduit. Nous présentons tout d’abord +les données utilisées, puis nous discutons des entraînements et des résultats obtenus. +4.4.1 +jeux de données +Les systèmes sont comparés sur quatre jeux de données annotés pour la détection de +lignes de texte : Balsac (Vézina et al., 2020), DIVA-HisDB (Simistira et al., 2016), Ho- +rae (Boillet et al., 2019) et READ-BAD (Grüning et al., 2017). La Table 3.1 présente les +détails de ces bases et la Table 4.1 en présente les statistiques. +Ces jeux de données sont très différents, ce qui permet de tester les systèmes sur des tâches +à complexité variable. En effet, la base Balsac contient des pages avec uniquement du texte +réparti en actes. Il s’agit de documents structurés semblables les uns aux autres. Les pages de +la base DIVA-HisDB ne contiennent également que du texte, mais les mises en page sont plus +complexes avec des commentaires dans les marges et entre les lignes. Les images de la base +Horae présentent des pages hétérogènes qui peuvent contenir des illustrations, une quantité +variable de lignes de texte et d’initiales. Enfin, READ-BAD comporte deux sous-ensembles, +l’un dit "simple" et l’autre "complexe", qui permettent d’évaluer et de comparer les systèmes +sur une grande diversité de données. + +64 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +4.4.2 +résultats et discussion +Nous avons entraîné Doc-UFCN et dhSegment dans les mêmes conditions sur les quatre +jeux de données. Cette section détaille les entraînements et présente les résultats obtenus. +détails des entraînements +Doc-UFCN est implémenté à l’aide du framework PyTorch. Nous l’avons entraîné avec +un learning rate initial de 5e − 3, l’optimiseur Adam (Kingma et al., 2015) et la fonction +de coût d’entropie croisée. Les poids sont initialisés grâce à l’initialisation Glorot (Glorot +et al., 2010). De plus, nous avons utilisé des mini-batchs de taille 4 pour réduire le temps +d’apprentissage. Nous avons testé différentes probabilités de dropout et décidé de conserver +une probabilité de 0,4, car elle correspond au modèle ayant donné les meilleures performances, +en moyenne, sur l’ensemble de validation. Le modèle est entraîné sur un maximum de 200 +époques et nous gardons le meilleur modèle en validation. +Nous avons également entraîné dhSegment sur ces mêmes données pour un maximum de +60 époques puisque le modèle est pré-entraîné et converge plus rapidement que le nôtre. Nous +avons utilisé des mini-batchs de taille 4 et des patchs de taille 400×400 pixels. Le learning +rate initial est de 5e − 5 et nous avons choisi d’utiliser un ResNet-50 (He et al., 2016) comme +encodeur pré-entraîné puisque les bonnes performances présentées dans Ares Oliveira et al. +(2018) ont été obtenues avec ResNet. Comme pour Doc-UFCN, le meilleur modèle obtenu +pendant l’apprentissage sur l’ensemble de validation est sélectionné. +Les deux modèles ont la même étape de post-traitement avec les mêmes hyper-paramètres. +Après avoir comparé des valeurs de seuil allant de 0,5 à 0,9, nous avons conservé le seuil t = +0,7 qui permet d’obtenir les meilleurs résultats sur l’ensemble de validation, avec une bonne +capacité d’acceptation des pixels attendus comme des lignes de texte et de rejet des pixels +appartenant au fond. Enfin, les composantes connexes de moins de min_cc = 50 pixels sont +écartées. Plusieurs valeurs ont également été comparées pour ce paramètre, cependant, il n’a +que peu d’impact sur les résultats. +évaluation des modèles +La plupart des méthodes existantes sont évaluées avec la métrique IoU. Cette métrique +mesure la similarité moyenne entre les pixels prédits et les pixels de la vérité terrain. Alberti +et al. (2017) ont conçu un outil pour évaluer la performance d’un modèle en calculant l’IoU, +la précision, le rappel et la F-mesure. Nous avons utilisé cet outil pour évaluer les modèles +car il permet d’obtenir plus d’informations concernant les performances du modèle au niveau +du pixel que l’IoU seule. +Par conséquent, pour évaluer les modèles, nous avons calculé différentes métriques au niveau +du pixel. Nous rapportons l’IoU ainsi que la précision (P), le rappel (R) et le score F1 dans +la Table 4.2. Pour être comparables, les images prédites par dhSegment sont redimensionnées +de sorte que leur plus grande dimension soit égale à 384 pixels avant de calculer les métriques. +De plus, les valeurs ne sont présentées que pour la classe des lignes de texte (le fond n’est +pas considéré ici). + +4.4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E +65 +Table 4.2 – Résultats obtenus par Doc-UFCN et dhSegment au niveau pixel. Résultats donnés sur les +ensembles de test pour la tâche de détection de lignes de texte. +Jeu de données +Système +IoU +P +R +F1-score +Doc-UFCN +0,84 +0,95 +0,88 +0,91 +Balsac +dhSegment +0,74 +0,92 +0,79 +0,85 +Doc-UFCN +0,76 +0,92 +0,81 +0,86 +DIVA-HisDB +dhSegment +0,74 +0,92 +0,79 +0,85 +Doc-UFCN +0,64 +0,78 +0,80 +0,85 +Horae +dhSegment +0,65 +0,72 +0,89 +0,82 +Doc-UFCN +0,64 +0,82 +0,76 +0,77 +READ-Simple +dhSegment +0,65 +0,85 +0,72 +0,77 +Doc-UFCN +0,54 +0,84 +0,62 +0,73 +READ-Complex +dhSegment +0,53 +0,79 +0,59 +0,69 +Figure 4.2 – Détections de lignes produites par dhSegment, au centre, et notre modèle Doc-UFCN, à +droite, sur l’image de la page 5 verso du Livre d’heures Horae 1. +Les résultats obtenus par notre méthode sont en moyenne supérieurs à ceux obtenus par +dhSegment. Sur le jeu de données Balsac, notre modèle surpasse dhSegment jusqu’à 6 points +pour la métrique F1-score. Nous observons également des gains respectifs de 3 et 4 points +de F1-score pour les bases Horae et READ-Complex. Ces résultats s’expliquent par une +meilleure séparation des lignes de texte proches qui sont souvent fusionnées par dhSegment +(Figure 4.2). Notre modèle aide à séparer ces lignes là où dhSegment échoue, mais permet +également d’obtenir des contours plus lisses et plus précis. Enfin, dhSegment semble avoir +plus de difficultés à détecter les lignes verticales et prédit parfois de très petites lignes dans +les miniatures, contrairement à Doc-UFCN qui semble plus robuste sur ce type d’images. +1. https://www.digitale-sammlungen.de/en/view/bsb00070331 + +nichil.DuoofactumetinipfoBitaetatierBita +etatfup fominumset up intenebrielucetatef +neBieeam noncomprefenjerit.fuicfomomi +ueaSeotcuinomenetatiofannes.DicBenitin +tefimoniumBtteftimonii petBieretSelumi +ne:Btomneecreberentperilfum.Donetatifle +faitefitage,fepfue +fup fesBtteftimoniumpetfibetetBelumine. +EratfupBeraqueifluminatomnem fomine +quifutoncaperceu +BenienteminfuncmunJum.nmunSoetata +munJueperipfumfactueeft:etmunoueeum +non cognouit.Fn piopia Benitetfuieum non +teceperunt.Duotquotautem tecepetunteum +Bediteie potefatem filios Jeifieri fiequtcte +ountinnomineeiue.Duinonepfanguinifue +neg3epBoluntatecatnie:neqsepBoluntateBi +tifeo epDeonatifuntEtBetbumcatofactu +efetfabBitauitinnobie,EtBiimuegforiam +eius gloziam quafi Bnigenitia pafre plenum +gtatieet Beritatie.Deo gtafiae.Detreuange +ceft par pofle Sun +ficaSicta Defeantut noftra Belicta:lmen. +ange.fufue bune +Ceinuocamue.teadoramue.telaubamug +DBeataetgfortofafanctatrinitab.bfub.Sit +nomenSomini BeneSictum.Rn.Eofoc nic +etBfg3infecuum. +Dratio. +Rofector in tefpetantium Beue finequo +nicfileftBalidum/nicilfanctum.mul +tipficafupetnoemifericoriamtuam:Bttete +Dteniceu.alectopone +ScBetitequetufue +rumpre Bitginite. +Birgeconceu,feft +Brap noue lauone66 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Table 4.3 – Temps d’inférence, en secondes par image, rapportés pour Doc-UFCN et dhSegment, et +calculés sur les ensembles de test. La colonne Ratio contient les ratios d’amélioration +(dhSegment/Doc-UFCN). +Jeu de données +Temps d’inférence† +Ratio +Doc-UFCN +dhSegment +Balsac +0,41 +2,95 +7,20 +DIVA-HisDB +0,80 +12,90 +16,13 +Horae +0,97 +7,87 +8,11 +READ-Simple +0,45 +3,73 +8,29 +READ-Complex +0,59 +4,70 +7,97 +† Prédictions faites sur une carte graphique GeForce RTX 2070 8G. +comparaison des modèles +Jusqu’à présent, notre modèle a obtenu, en moyenne, de meilleures performances que +dhSegment bien qu’il ne bénéficie pas d’un encodeur pré-entraîné. Un autre point intéressant +est que notre modèle comporte moins de paramètres à apprendre que dhSegment : 4,1 +millions pour Doc-UFCN contre 32,8 millions pour dhSegment, dont 9,36 millions, correspon- +dant au décodeur non pré-entraîné, qui doivent être entièrement entraînés. Cette diminution +du nombre de paramètres conduit à une réduction significative du temps de prédiction : +Doc-UFCN est jusqu’à 16 fois plus rapide que dhSegment, comme illustré dans la Table 4.3. +Cette réduction du temps d’inférence peut également s’expliquer par le fait que dhSegment +utilise des patchs de taille 400×400 pixels. Ainsi, pour la base DIVA-HisDB, il devra prédire +en moyenne 117 patchs de cette taille, là où notre modèle ne fera qu’une seule prédiction de +taille moyenne 768×512 pixels. +Grâce à ces premières expériences, nous avons montré que notre modèle Doc-UFCN obtient +de meilleures performances que dhSegment tout en étant plus rapide en inférence. Nous allons +maintenant étudier l’impact du pré-entraînement sur les résultats finaux. +4.4.3 +impact du pré-entraînement +Nous analysons maintenant l’impact du pré-entraînement sur des images de documents. +dhSegment est pré-entraîné sur des images de scènes naturelles (Deng et al., 2009), ce qui +lui permet d’obtenir des résultats satisfaisants, même avec peu de données annotées. Nous +nous questionnons donc sur l’intérêt que pourrait avoir un pré-entraînement de Doc-UFCN sur +des images de documents. Pour cela, nous avons entraîné Doc-UFCN ainsi que dhSegment +sur l’ensemble des quatre jeux de données présentés en section 4.4.1. Ces modèles ont été +entraînés avec 1700 images d’entraînement et 191 de validation, dans les mêmes conditions +que les expériences précédentes, puis évalués sur chaque base indépendamment. Les deux +modèles ainsi obtenus sont dits "génériques" dans la suite de cette section. Les résultats +obtenus sont résumés dans la Table 4.4. Pour plus de lisibilité, seules les valeurs d’IoU et de +F1 sont présentées dans la Table. En plus des résultats des modèles pré-entraînés (génériques), + +4.4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E +67 +Table 4.4 – Résultats obtenus par Doc-UFCN et dhSegment au niveau pixel pour la tâche de détection +de lignes de texte. Les résultats montrent les performances des modèles génériques sur les +ensembles de test avec et sans adaptation. +Jeu de données +Système +IoU +F1-score +Générique +Adapté +Générique +Adapté +Doc-UFCN +0,85 +0,86 +0,92 +0,92 +Balsac +dhSegment +0,74 +0,75 +0,85 +0,85 +Doc-UFCN +0,75 +0,75 +0,85 +0,85 +DIVA-HisDB +dhSegment +0,73 +0,74 +0,84 +0,85 +Doc-UFCN +0,69 +0,68 +0,89 +0,88 +Horae +dhSegment +0,61 +0,63 +0,82 +0,80 +Doc-UFCN +0,68 +0,68 +0,79 +0,79 +READ-Simple +dhSegment +0,65 +0,64 +0,81 +0,77 +Doc-UFCN +0,60 +0,60 +0,78 +0,78 +READ-Complex +dhSegment +0,53 +0,53 +0,68 +0,69 +nous montrons, dans cette table, les résultats après adaptation (fine tuning) des modèles +génériques sur chaque jeu de données. +comparaison des systèmes +Les résultats obtenus par les modèles Doc-UFCN et dhSegment génériques confirment +ceux obtenus par les modèles spécifiques présentés en section 4.4.2. En effet, avec et sans +adaptation, Doc-UFCN obtient quasiment toujours de meilleurs résultats que dhSegment +puisqu’il obtient des valeurs d’IoU et de F1 plus élevées. +pré-entraînement +La Figure 4.3 compare les résultats obtenus par le modèle Doc-UFCN générique par rapport +aux modèles spécifiques présentés précédemment. Nous pouvons noter que, pour les deux +métriques, le modèle générique obtient de meilleures performances sur toutes les bases sauf +sur DIVA-HisDB où il perd un point d’IoU et de F1. Sur les autres jeux de données, nous +observons des gains allant jusqu’à 6 points d’IoU, ce qui indique que le modèle a réussi à +apprendre des caractéristiques suffisamment génériques pour représenter des données très +variées, et qu’il a donc réussi à tirer profit de chaque jeu de données, malgré des données de +pré-entraînement non équilibrées. +adaptation +Nous observons également sur la Figure 4.3 qu’adapter le modèle générique à chaque +jeu de données n’apporte que très peu voire aucun gain de performance. Notre hypothèse +est que le modèle générique avait déjà appris au mieux sur ces données, l’adaptation +n’apportant pas de nouvelles données. Ce comportement peut être expliqué par le faible +nombre de paramètres que comporte le modèle et la taille réduite des jeux de données annotés. + +68 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Figure 4.3 – Impact du pré-entraînement de Doc-UFCN, évalué sur les ensembles de test. Les résultats +montrent les performances des modèles génériques avec et sans adaptation. +Pour conclure, nous avons montré qu’utiliser un modèle générique permet d’améliorer la +qualité des détections par rapport à un modèle spécifique, même avec une quantité limitée +de données. Dans le chapitre 5, nous cherchons à aller plus loin dans cette idée en analysant +et levant les défis liés à l’obtention d’un modèle encore plus générique et robuste, obtenant +de bonnes performances sur de nouvelles données sans aucune adaptation. +4.4.4 +étude ablative +Après avoir démontré l’intérêt de notre modèle Doc-UFCN pour la détection de lignes de +texte, nous synthétisons ici les expérimentations réalisées afin de valider nos choix d’architec- +ture et d’évaluer l’impact de certains composants et hyper-paramètres sur les performances +finales. Les paramètres étudiés dans les paragraphes suivants sont l’utilisation de la normali- +sation des batchs, l’utilisation du dropout, les taux de dilatation dans les blocs dilatés et la +taille des images en entrée. Les Tables 4.5, 4.6 et 4.7 regroupent les résultats de cette étude. + +4.4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E +69 +Table 4.5 – Étude ablative de Doc-UFCN sur la détection de lignes de texte. Résultats donnés pour +les ensembles de test. "BN" indique l’utilisation de la batch normalization durant l’entraî- +nement. +Jeu de données +Version +IoU +F1-score +∅ +0,79 +0,88 +BN +0,81 +0,89 +Balsac +BN + dropout +0,84 +0,91 +∅ +0,41 +0,56 +BN +0,74 +0,85 +DIVA-HisDB +BN + dropout +0,76 +0,86 +∅ +0,56 +0,78 +BN +0,64 +0,81 +Horae +BN + dropout +0,64 +0,85 +∅ +0,59 +0,73 +BN +0,58 +0,72 +READ-Simple +BN + dropout +0,64 +0,77 +∅ +0,39 +0,56 +BN +0,50 +0,69 +READ-Complex +BN + dropout +0,54 +0,73 +batch normalization +La normalisation des batchs appliquée durant l’entraînement de modèles neuronaux est +souvent utilisée, car elle a un impact positif sur la vitesse de convergence, mais parfois aussi sur +les performances (Ioffe et al., 2015). L’entraînement de Doc-UFCN avec cette normalisation +confirme ces résultats puisque le modèle a convergé plus de deux fois plus rapidement par +rapport au modèle sans normalisation. D’après la Table 4.5, la normalisation a également un +réel impact sur les valeurs de F1, en particulier pour Diva-HisDB, Horae et READ-Complex. +En plus de ces valeurs quantitatives, nous avons noté que les résultats visuels sont meilleurs +avec normalisation. Elle aide à séparer les objets proches mais aussi à joindre les parties +d’objets qui, sans normalisation, étaient sur-segmentés. En outre, les contours des objets +prédits sont souvent plus précis et plus lisses. +dropout +Le dropout (Srivastava et al., 2014) est également souvent utilisé dans les réseaux de +neurones profonds, car il permet notamment de limiter le sur-apprentissage. Nos expériences, +présentées en Table 4.5, montrent qu’entraîner avec dropout permet, en effet, d’obtenir de +meilleures performances sur toutes les bases, d’autant plus que notre modèle comporte assez +peu de paramètres. +taux de dilatation +Lors de la conception de notre modèle, nous avons choisi d’utiliser une version modifiée du +bloc dilaté proposé par Yang et al. (2017) dans le but de prendre en compte davantage de + +70 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Table 4.6 – Impact du taux de dilatation dans les blocs d’encodeur de Doc-UFCN sur la détection de +lignes de texte. Résultats donnés pour l’ensemble de test du jeu de données Balsac. +Dilatation +IoU +F1-score +[1] +0,76 +0,86 +[1, 1, 1, 1, 1] +0,80 +0,89 +[16] +0,77 +0,87 +[1, 2, 4, 8, 16] +0,84 +0,91 +Table 4.7 – Impact de la taille des images en entrée de Doc-UFCN sur la détection des lignes de texte. +Résultats donnés pour les ensembles de test. +Jeu de données +Taille +IoU +F1-score +384 +0,84 +0,91 +Balsac +768 +0,87 +0,93 +384 +0,76 +0,86 +DIVA-HisDB +768 +0,77 +0,87 +contexte pour prédire les lignes de texte. Pour justifier nos choix de taux de dilatation, nous +avons testé quatre configurations sur le jeu de données Balsac. Ainsi, nous avons entraîné des +modèles avec des blocs dilatés configurés comme suit : +— 1 convolution et un taux de dilatation de 1 : [1] ; +— 1 convolution et un taux de dilatation de 16 : [16] ; +— 5 convolutions et des taux de dilatation de 1 : [1, 1, 1, 1, 1] ; +— 5 convolutions et des taux de dilatation de 1, 2, 4, 8 et 16 : [1, 2, 4, 8, 16]. +Les résultats obtenus sont présentés dans la Table 4.6. Les résultats de la dernière configu- +ration sont meilleurs que ceux des autres puisque le champ réceptif est beaucoup plus grand +et que le modèle considère davantage de contexte pour prédire les lignes de texte. Le fait +d’avoir des convolutions dilatées au lieu de convolutions standards a un réel impact sur la +taille du champ réceptif, ce qui permet d’utiliser plus de contexte pour prédire les lignes de +texte et d’obtenir de meilleures performances. +taille des images d’entrée +Comme indiqué précédemment, nous avons choisi, pour Doc-UFCN, d’utiliser la même +taille d’image en entrée que celle utilisée par Yang et al. (2017). C’est pourquoi, pour tous +les résultats présentés jusqu’à maintenant, les images étaient redimensionnées de sorte que +leur plus grande dimension soit égale à 384 pixels, tout en conservant l’aspect de l’image. +Cependant, il est important de connaître l’impact de ce choix sur les résultats du modèle. +Dans cette optique, nous avons entraîné un modèle sur les données du jeu Balsac et un autre +sur celles de DIVA-HisDB sur des images redimensionnées à 768 pixels. +La Table 4.7 montre que l’entraînement sur des images plus grandes améliore les résultats. +Les lignes étant souvent de petite hauteur sur ces bases, agrandir les images d’entrée permet +au modèle de mieux séparer les lignes proches et de les prédire avec une plus grande précision. + +4.5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S +71 +Grâce à ces premières expériences, nous avons montré que notre modèle Doc-UFCN obtient +de meilleures performances sur la tâche de détection de lignes de texte dans des images de +document que dhSegment, tout en comportant moins de paramètres et en étant plus rapide +en inférence. De plus, nous avons montré qu’entraîner un modèle sur plusieurs bases permet +d’obtenir un modèle plus générique et d’améliorer la qualité des prédictions par rapport à +des modèles spécifiques. Nous évaluons, dans la section suivante, le modèle Doc-UFCN sur +une tâche plus complexe de détection et de classification d’actes. +4.5 +E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S +Les registres sont des types très courants de documents historiques qui contiennent des +listes d’enregistrements, appelés "actes", se rapportant à des personnes, des objets ou des +événements. Ils peuvent être présentés sous forme de tableaux ou de séquences de textes. +Dans le cas des registres royaux, des cartulaires médiévaux ou des registres paroissiaux et +civils, les actes sont des segments textuels composés d’un ou plusieurs paragraphes. Pour +traiter le problème de la détection d’actes, la plupart des méthodes existantes utilisent soit +le contenu textuel des documents, soit le contenu visuel. Les systèmes récents basés sur des +règles heuristiques ou des réseaux neuronaux se basent uniquement sur les caractéristiques +visuelles des images pour détecter les actes, en ignorant le texte des documents. En effet, +Tarride et al. (2019) combinent des règles et un réseau neuronal pour segmenter les +registres paroissiaux français en actes. Ils détectent d’abord les signatures des prêtres situées +à la fin de chaque acte à l’aide d’un réseau neuronal (dhSegment (Ares Oliveira et al., +2018) ou ARU-Net (Grüning et al., 2019)) avant d’utiliser un système à base de règles +pour générer les actes. Même s’ils ont obtenu un système avec 80 % de rappel au niveau des +actes, leur méthode repose principalement sur l’hypothèse que chaque acte se termine par +une signature, ce qui n’est pas toujours le cas, et si elle est effectivement présente, celle-ci +n’est pas toujours détectée par le système automatique. La méthode que nous proposons +comprend également des caractéristiques basées sur des règles, mais combinées avec l’image +originale. +De plus, les documents historiques peuvent avoir un contenu textuel riche qui peut per- +mettre un meilleur processus de détection. Ainsi, Prieto et al. (2020) ont étudié le cas où +l’aspect graphique des images n’est pas suffisant pour segmenter les chartes médiévales en +actes. Ils ne visent pas seulement à détecter les actes mais cherchent également à les classer +comme début, milieu, fin d’acte ou acte complet. Pour cela, ils utilisent une carte d’indexation +probabiliste pour construire des caractéristiques supplémentaires fondées sur le contenu tex- +tuel, puis les caractéristiques graphiques et textuelles sont fusionnées afin d’obtenir une seule +entrée pour le système de détection. Ils montrent que l’ajout de contenu textuel peut faciliter +la détection des actes et que l’ajout de connaissances a priori permet d’améliorer encore les +performances (73 % à 88 % de la F-mesure). Inspiré par cette idée et celle proposée par +Yang et al. (2017), notre travail se concentre sur l’utilisation des deux modalités en entrée +d’un système de détection par apprentissage profond pour améliorer la détection des actes + +72 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Table 4.8 – Statistiques des jeux de données utilisés pour la détection d’actes : nombre de pages, lignes +transcrites et actes par type. +Jeu de données +Images +Lignes +Actes +complet +début +milieu +fin +Balsac +train +730 +36 941 +1 474 +503 +2 +487 +Vézina et al. (2020) +valid +92 +4 592 +181 +66 +1 +58 +test +91 +4 323 +173 +62 +0 +52 +train +132 +– +144 +46 +21 +40 +Himanis-Act +valid +19 +– +29 +3 +3 +2 +Bluche et al. (2017) +test +411 +– +172 +203 +115 +196 +Himanis-GMV +train +– +18 504 +– +– +– +– +valid +– +2 367 +– +– +– +– +test +– +2 241 +– +– +– +– +Figure 4.4 – Annotations manuelles pour la détection et la classification d’actes sur une image du jeu +de données Balsac, à gauche, et Himanis-Act, à droite. +présents dans des documents historiques. Dans cette section, nous présentons tout d’abord +les données utilisées, notre approche pour résoudre la tâche de détection d’actes, puis nous +discutons des entraînements et des résultats obtenus. +4.5.1 +jeux de données +Pour nos expériences, nous avons utilisé deux jeux de données, Balsac (Vézina et al., 2020) +(présenté en section 3.1) et Himanis-Act (Prieto et al., 2020). Pour traiter les actes répartis +sur plusieurs pages, les actes sont annotés avec quatre classes : acte complet, début d’acte, +milieu d’acte et fin d’acte. Les statistiques de ces ensembles de données sont présentées dans +la Table 4.8. La Figure 4.4 présente un exemple d’annotation pour chaque jeu de données. + +Fin d'acte +091 +et +Mr +lglCom +uJiehulaluta +Acte complet +Nhin ehh +mari J +hleak +Acte complet +as +thut +92 +Début d'acteFin d'acte +HAMD +Acte complet +nammeppeneplamcoep +gobonu3gennmg +Débutd'acte4.5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S +73 +D´etection +lignes +HTR +"Premi`ere ligne" +"Autre ligne" +Classification +Classification +Incorporation +des lignes +Incorporation +des classes +D´etection et +classification +d’actes +Figure 4.5 – Chaîne de traitement proposée pour la détection et la classification d’actes avec l’utili- +sation du contenu textuel. +Le jeu de données Himanis est extrait du corpus Chancery, une grande collection de registres +produits par la chancellerie royale française. Il est composé de 80 000 images contenant des +chartes promulguées par les rois de France aux 14e et 15e siècles. Ces documents consignent +les décisions royales comme les donations ou les grâces, et sont organisés en actes. +Le jeu de données Himanis-Act consiste en un échantillon de 739 images extraites des +données Himanis et est annoté au niveau des actes. Pour réaliser nos expériences, nous avons +utilisé la répartition proposée par Prieto et al. (2020), obtenue après avoir éliminé les +pages ne contenant aucune information telles que les pages blanches. Ce jeu de données +est uniquement utilisé pour entraîner et évaluer le système de détection d’actes puisque les +annotations de lignes de texte et de transcription ne sont pas disponibles. La répartition +finale est présentée dans la Table 4.8. +Le jeu de données annoté Himanis-GMV est composé de 1 435 images extraites des données +Himanis, alignées automatiquement au niveau des lignes, des éditions imprimées (Guérin +et al., 1881-1958 ; +Guyotjeannin et al., 2005 ; +Viard, Jules, 1899) avec Transkribus +(Kahle et al., 2017). Après le processus d’alignement, 23 112 lignes de texte sont disponibles +pour l’entraînement et l’évaluation des systèmes de reconnaissance de l’écriture manuscrite, +dont la répartition est présentée dans la Table 4.8. Ce jeu de données est uniquement utilisé +pour entraîner un système de reconnaissance HTR puisque la segmentation en actes n’est pas +disponible. +4.5.2 +approche proposée +La Figure 4.5 détaille l’approche proposée pour résoudre la tâche de détection et de classi- +fication d’actes. L’approche consiste en plusieurs traitements successifs de l’image d’entrée : +1. L’image d’entrée est d’abord traitée par un détecteur de lignes de texte ; +2. Les lignes prédites sont extraites et traitées par un reconnaisseur de texte manuscrit ; + +Thauuc +tth +giniLn +B贝人 +Bleo +Bge +cloghLen +dmuhLe neuf fevrier mil neuf centun,nouspretresoussigneavons74 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Table 4.9 – Résultats du modèle générique de détection de lignes de texte sur l’ensemble de test du +jeu de données Balsac. +Jeu de données +Système +IoU +AP@.5 +AP@.75 +mAP +Balsac +Doc-UFCN +0,87 +0,98 +0,91 +0,76 +dhSegment +0,74 +0,94 +0,54 +0,51 +3. Chaque ligne est classifiée selon le texte qu’elle contient en "première ligne" (s’il est +probable que la ligne soit la première de l’acte) et "autre ligne" ; +4. L’image d’entrée est enrichie par ces lignes classifiées en dessinant les lignes de texte de +couleurs différentes selon leurs classes ; +5. Enfin, les actes sont détectés et classifiés à partir de cette image enrichie. +Les paragraphes suivants présentent et discutent chacune de ces étapes en détail. +détection des lignes de texte +La première étape du traitement des images consiste à détecter les lignes de texte sur les +images. Pour cette tâche, nous avons utilisé Doc-UFCN. Afin de créer un modèle générique +pouvant être utilisé sur différents jeux de données, le modèle a été entraîné sur neuf jeux +de données historiques dont Balsac et à l’exception d’Himanis, puisqu’il ne contient aucune +annotation au niveau des lignes. Les images ont été redimensionnées de manière à ce que leur +plus grande dimension soit égale à 768 pixels, tout en conservant le rapport d’aspect original. +Les annotations originales ont également été uniformisées grâce au processus détaillé dans +la section 5.1.1. dhSegment (Ares Oliveira et al., 2018) a également été entraîné afin de +fournir une comparaison de référence avec Doc-UFCN. +Les deux modèles de détection de lignes de texte ont été évalués à l’aide de l’IoU. Les +modèles ont également été évalués à l’aide de l’AP, qui quantifie le nombre d’objets correcte- +ment détectés, alors que la mesure IoU considère uniquement le nombre de pixels correctement +prédits. Plus de détails sur le calcul de la mAP sont donnés dans le Focus 3.4. +Les résultats sont présentés dans la Table 4.9, pour le jeu de données Balsac uniquement, +car aucune annotation manuelle n’est disponible pour les jeux de données Himanis. Les résul- +tats montrent que Doc-UFCN surpasse dhSegment pour toutes les métriques et obtient des +performances très satisfaisantes. +reconnaissance de texte +La reconnaissance de texte manuscrit (HTR) est appliquée aux lignes de texte détectées +et produit le texte correspondant. Le modèle de reconnaissance est construit avec la librairie +Kaldi 2, basée sur un modèle DNN-HMM (Deep Neural Network - Hidden Markov Model). +Notre modèle est comparable à celui décrit dans Arora et al. (2019). +2. https://kaldi-asr.org/ + +4.5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S +75 +Table 4.10 – Résultats de reconnaissance de textes manuscrits sur les jeux de données Balsac et +Himanis-GMV. Les résultats d’un modèle HTR+ entraîné avec Transkribus sont donnés +à titre de référence mais ne sont pas directement comparables à notre système puisque +les séparations train/valid ne sont pas identiques. +Jeu de données +Système +CER (%) +WER (%) +train +valid +test +train +valid +test +Balsac +Kaldi +4,1 +6,2 +6,4 +12,4 +17,1 +17,4 +Transkribus +12,2 +9,5 +– +– +– +– +Kaldi +5,4 +9,4 +8,0 +11,9 +19,3 +18,1 +Himanis-GMV +Transkribus +9,5 +5,3 +– +– +– +– +Nous avons entraîné un modèle Kaldi sur le jeu de données Balsac en suivant la répartition +présentée dans la Table 4.1. Aucune donnée supplémentaire n’a été utilisée pour le modèle +linguistique ni pour le modèle optique. Pour Himanis, nous avons entraîné un modèle sur le jeu +de données annoté Himanis-GMV présenté ci-dessus. Pour les deux modèles, les lignes d’entrée +ont été redimensionnées à une hauteur de 40 pixels tout en conservant le rapport d’aspect. +De plus, les lignes ayant des largeurs similaires ont été regroupées pour un entraînement plus +efficace. À titre de comparaison, un modèle HTR+ a été entraîné à l’aide de la plateforme +Transkribus (Kahle et al., 2017). +Les performances des modèles HTR sont décrites dans la Table 4.10. Concernant notre +système HTR (Kaldi), les résultats obtenus pour les deux jeux de données sont similaires. +Même si les modèles présentent des taux d’erreurs de mots (Word Error Rate (WER)) rela- +tivement élevés, nous pensons qu’ils sont capables de prédire une transcription suffisamment +correcte pour la détection de mots-clés et la détection des actes. L’interface d’entraînement +de Transkribus ne fournit qu’une évaluation pour le Character Error Rate (CER) et ne peut +pas être directement comparée à la nôtre puisque les répartitions train/valid/test ne sont pas +identiques. Cependant, les résultats sont du même ordre de grandeur. Le modèle de Trans- +kribus présente un CER plus élevé sur l’ensemble d’entraînement en raison de son processus +d’augmentation des données, processus non utilisé dans notre système Kaldi. +classification des lignes de texte +La classification des lignes de texte est effectuée à l’aide de règles. Le modèle utilise les +transcriptions prédites en entrée et prédit si une ligne de texte est la première ligne d’un +acte en se basant sur la présence de phrases clés définies a priori. +Pour le jeu de données Balsac, la plupart des actes commencent par une date telle que "Le +trente un janvier, mil neuf", et il n’y a souvent pas d’autres dates dans la suite des +actes. Ainsi, la règle est donc de compter le nombre de mots qui sont des chiffres ou des mois +pour que la ligne soit considérée comme une date. Expérimentalement, trois mots semblent +être suffisants pour que la ligne soit considérée comme contenant une date. +Pour les actes de Himanis-Act, la tâche est plus complexe car ils ne commencent pas +toujours par les mêmes mots. Nous avons donc analysé les premières lignes des actes + +76 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Table 4.11 – Résultats de classification des lignes de texte en première ligne d’un acte / autre ligne +sur les jeux de données Balsac et Himanis-Act. +Jeu de données +Précision +Rappel +F1-score +Balsac +train +0,69 +0,87 +0,77 +valid +0,72 +0,87 +0,79 +test +0,68 +0,86 +0,76 +train +0,79 +0,65 +0,71 +valid +0,81 +0,53 +0,64 +Himanis-Act +test +0,68 +0,86 +0,76 +d’entraînement et avons conservé les phrases clés les plus fréquentes (par exemple "dei +gratia francorum rex" ou "par la grâce de dieu roys de france"). Si une phrase clé +est incluse dans une ligne, elle est considérée comme se trouvant au début d’un acte. +La Table 4.11 montre la précision, le rappel et le F1-score de la classe "première ligne". +Seule la classe "première ligne" est donnée car c’est la seule classe apportant des informations +à la détection des actes, d’autant plus que la distribution des classes est très déséquilibrée. +Pour le jeu de données Balsac, les résultats sont stables entre les trois ensembles et le rappel +est élevé, ce qui est favorable à l’inclusion de cette information dans le système visuel. Pour +le jeu de données Himanis-Act, le rappel est plus faible et les résultats varient entre les +ensembles, ce qui montre bien que la tâche est plus complexe et que la détection des phrases +clés est plus difficile, et donc moins fiable. +4.5.3 +résultats et discussion +La Table 4.12 présente les résultats des différents modèles de détection des actes : Doc- +UFCN entraîné sur des images brutes (visuel) et Doc-UFCN entraîné sur les images brutes +avec les polygones des lignes de texte dessinés de deux couleurs dépendant de la classe de la +ligne (visuel + textuel). +Pour le jeu de données Balsac, nous ne présentons pas les résultats de la classe milieu d’acte +car il n’y en a pas dans le jeu de test. D’après la Table 4.12, les résultats sont en moyenne +meilleurs pour le modèle utilisant les contenus visuels et textuels. Pour les classes de début +et de fin d’acte, les deux systèmes sont presque équivalents pour toutes les métriques. En +revanche, nous constatons que l’ajout de la date directement à l’image d’entrée améliore les +performances de la classe d’actes complets de 36 points de pourcentage de mAP. Cela conduit +à une meilleure séparation des actes complets consécutifs dans les prédictions, ce qui était +l’un de nos principaux objectifs. +Pour le jeu de données Himanis-Act, les résultats sans contenu textuel sont significative- +ment meilleurs que ceux l’incorporant. Nous pensons que ces résultats sont dûs aux raisons +suivantes. Tout d’abord, le contenu textuel des documents de la base Himanis-Act est plus +complexe et diversifié que celui de la base de données Balsac, qui est très standardisé. De +plus, nous avons pu trouver de nombreux actes imbriqués dans le jeu de données (Vidimus), + +4.5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S +77 +Table 4.12 – Résultats de détection d’actes sur les ensemble de test des jeux de données Balsac et +Himanis-Act. +Jeu de données +Système +Actes +IoU +AP@.5 +AP@.75 +mAP +Visuel +complet +0,84 +0,57 +0,37 +0,38 +début +0,58 +0,86 +0,85 +0,76 +fin +0,58 +0,85 +0,64 +0,59 +complet +0,82 +0,89 +0,81 +0,74 +Visuel + +début +0,58 +0,90 +0,87 +0,78 +Balsac +textuel +fin +0,54 +0,86 +0,73 +0,63 +complet +0,61 +0,75 +0,73 +0,70 +début +0,76 +0,84 +0,82 +0,77 +milieu +0,88 +0,84 +0,83 +0,83 +Visuel +fin +0,73 +0,73 +0,65 +0,62 +complet +0,64 +0,54 +0,51 +0,49 +Visuel + +début +0,68 +0,69 +0,64 +0,60 +textuel +milieu +0,84 +0,80 +0,80 +0,80 +Himanis-Act +fin +0,70 +0,64 +0,63 +0,58 +ce qui peut ajouter de la confusion au système. La définition des phrases clés s’est avérée +plus complexe et la Table 4.11 montre que le rappel est faible, même pour l’ensemble +d’entraînement, ce qui conduit à des caractéristiques textuelles peu fiables pour entraîner le +modèle de détection d’actes. De plus, les modèles de détection de lignes et d’HTR n’ont pas +été entraînés sur l’ensemble de données Himanis-Act. Le modèle de détection de lignes de +texte a été entraîné sur des données similaires mais sans images venant du jeu de données +Himanis-Act. Il en est de même pour le système HTR qui n’a pas été entraîné directement +sur les mêmes volumes, ce qui peut créer un décalage entre les conditions d’entraînement et +de test. +En plus de ces expériences, nous avons comparé nos résultats avec ceux de l’état de l’art de +Prieto et al. (2020). Ils ont testé différentes configurations avec et sans le contenu textuel +pour détecter où se terminent les actes. Pour obtenir une comparaison juste, nous avons +utilisé le même protocole d’évaluation. L’évaluation est effectuée à l’aide du Transkribus +Baseline Evaluation Scheme (TBES) (Diem et al., 2017). Cet outil a été conçu pour évaluer +la détection de la ligne de base. Ainsi, pour l’utiliser, la ligne de base des actes est définie +comme étant la ligne droite horizontale à la fin d’un acte complet ou d’une fin d’acte. Pour +être en accord avec leurs résultats, nous avons utilisé la même valeur de tolérance de 128 +pixels. +D’après la Table 4.13, nous pouvons voir que Doc-UFCN utilisant uniquement l’image +améliore les résultats de l’état de l’art. En effet, en comparaison au système visuel de +Prieto et al. (2020), notre méthode obtient des performances supérieures de 10 points de +pourcentage. En outre, nous constatons que les deux modèles utilisant le contenu textuel +se comportent de la même manière et sont moins performants que notre système basé + +78 +D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Table 4.13 – Résultats obtenus par Doc-UFCN et Prieto et al. (2020) sur le jeu de données Himanis- +Act avec et sans l’information textuelle. +Système +Visuel +Visuel + textuel +Doc-UFCN +train +0,96 +0,96 +valid +0,96 +0,91 +test +0,90 +0,88 +Prieto et al. (2020) +test +0,80 +0,88 +uniquement sur les informations visuelles. Pour ce jeu de données, il semble préférable de se +concentrer sur les caractéristiques visuelles avec un système d’apprentissage profond robuste, +plutôt que d’ajouter un contenu textuel trop peu fiable. +Dans cette partie, nous avons présenté une chaîne de traitement simple permettant d’en- +richir des images d’entrée avec le contenu textuel de documents. Ces images enrichies per- +mettent d’effectuer une tâche de détection d’actes en utilisant simultanément le contenu visuel +et la position des lignes de texte contenant des phrases clés définies manuellement. Nous avons +montré que l’utilisation de ces images peut améliorer la détection des actes, en particulier des +actes consécutifs. Sur le jeu de données Balsac, pour lequel des règles de détection de phrases +clés ont pu être définies de manière fiable, l’utilisation de ces images augmente la détection +d’actes de 38 % à 74 % de mAP par rapport à un modèle standard se basant uniquement sur +le contenu visuel. +4.6 +C O N C L U S I O N +Dans ce chapitre, nous avons présenté Doc-UFCN, un nouveau système de détection d’ob- +jets dans les images de documents. Nous avons montré que ce système permet d’entraîner +des modèles plus performants, plus rapides et comportant moins de paramètres que ceux +de l’état de l’art pour la détection de lignes de texte. Le code de ce système est disponible +publiquement 3. De plus, les expérimentations décrites dans ce chapitre ont permis d’amorcer +une analyse sur l’intérêt des modèles génériques, qui seront l’objet du prochain chapitre. +Enfin, nous nous sommes intéressés aux méthodes combinant l’image et le texte pour la +détection et la classification d’actes. Dans un cadre dans lequel la séparation visuelle des +actes suit la séparation logique du texte, nous avons montré que l’incorporation du contenu +textuel dans l’image d’entrée permet de réellement améliorer la détection des actes. +3. https://pypi.org/project/doc-ufcn/ + +5 +E N T R A Î N E M E N T E T É VA L U AT I O N D ’ U N M O D È L E +R O B U S T E D E D É T E C T I O N D ’ O B J E T S +La littérature montre que des systèmes compétitifs et robustes ont été développés +pour résoudre le problème de la détection des lignes de texte, obtenant des performances +satisfaisantes sur des jeux de données individuels. Cependant, leurs performances sont +souvent insuffisantes sur d’autres documents hors échantillon, car ils manquent de ca- +pacités de généralisation. Or, dans un cadre industriel avec de nombreux projets et des +données très différentes, il est nécessaire de développer des modèles plus génériques, ob- +tenant des performances élevées sur des documents très variés provenant de différents projets. +L’entraînement de systèmes sur des données très diverses permettrait d’obtenir de tels +modèles, montrant de meilleures capacités de généralisation. Pour cela, il est nécessaire de +combiner plusieurs ensembles de données. Cependant, les schémas d’annotation ne sont pas +toujours compatibles entre les jeux de données publics (comme décrit en section 3.2), ce qui +rend difficile leur combinaison dans un seul ensemble d’entraînement unifié. Ces différents +schémas ne permettent pas une comparaison équitable des approches de détection d’un jeu +de données à l’autre. Par conséquent, dans la littérature, aucune comparaison de systèmes n’a +été effectuée sur un jeu de données large et diversifié, tant en entraînement qu’en évaluation. +De plus, dans la littérature, les études sur la détection des lignes de texte manuscrites se +concentrent généralement sur le développement d’une architecture de réseau neuronal spéci- +fique, ainsi que sur une bonne stratégie d’entraînement. Cependant, elles omettent souvent +d’analyser les annotations utilisées lors de l’entraînement et de l’évaluation, alors qu’elles +sont aussi importantes que l’architecture du réseau elle-même, puisqu’elles guident la phase +d’entraînement et les résultats finaux. En effet, le biais d’annotation est particulièrement +important lorsque nous voulons analyser l’impact de l’étape de détection sur l’étape de recon- +naissance. Cependant, il est rarement étudié dans les études se concentrant sur la détection de +lignes de texte. Il n’est pas non plus considéré dans les études portant sur la reconnaissance de +l’écriture manuscrite, car le processus de détection n’entre pas dans leur champ d’application. +Par exemple, la première lettre dans les documents historiques est parfois ornée, l’ajouter +ou non dans les lignes de texte pendant le processus d’annotation peut avoir un réel impact +sur les résultats de la reconnaissance finale, d’où l’importance de créer et d’analyser soigneu- +sement les annotations. Un autre problème se pose lors de la détection des lignes de texte +lorsque deux boîtes de délimitation annotées se touchent. Dans une telle situation, le réseau +fournit généralement des lignes fusionnées qui ne seront pas reconnues par le système de re- +connaissance HTR, alors que généralement les métriques d’évaluation ne tiennent pas compte +79 + +80 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +de ces problèmes. En effet, la métrique IoU, très souvent utilisée, est incapable de détecter +ces situations et de compter correctement les séparations de lignes correctes et incorrectes. +Dans ce chapitre, nous abordons ce problème, en section 5.1.1, en introduisant une +stratégie d’uniformisation d’étiquetage des jeux de données, qui réduit les chevauchements +des polygones englobants afin d’obtenir un résultat cohérent avec l’entrée requise des +systèmes de reconnaissance (HTR ou OCR). +Toujours dans un cadre industriel où l’objectif est la reconnaissance du texte des documents, +et pas uniquement la détection des lignes, il est nécessaire d’avoir une bonne évaluation des +modèles de détection. Cependant, l’évaluation de tels modèles est complexe. En effet, les +métriques d’évaluation au niveau pixel ne sont pas toujours représentatives de l’impact réel +de l’étape de détection de lignes de texte sur l’étape de reconnaissance de texte. De plus, +les comparaisons des systèmes de détection de lignes de texte en termes de taux d’erreur de +reconnaissance sont rarement rapportées en raison de la complexité de cette évaluation. +La plupart des méthodes de détection de lignes de texte existantes sont évaluées et +comparées à l’aide de métriques au niveau pixel telles que l’IoU, la précision, le rappel et +le score F1. Même si ces mesures indiquent les performances du modèle, elles ne donnent +aucune information réelle sur la quantité d’informations détectées, comme le nombre de +lignes. Certaines mesures au niveau de l’objet, telles que la précision moyenne (AP), ont été +proposées pour surmonter ce problème, mais elles reposent toujours sur un seuil d’IoU fixe. +Dans la section 5.3.2, nous analysons donc ces limitations et introduisons la métrique mAP +déjà utilisée dans les défis de détection COCO en vision, qui ne nécessite pas de seuil de +détection pour être mise en œuvre. De plus, comme la détection des lignes manuscrites est +la première étape de l’ensemble du processus de reconnaissance, il devrait être plus réaliste +d’évaluer son véritable impact sur les résultats de reconnaissance finaux (taux d’erreur +caractères et mots), en effectuant une évaluation orientée vers la tâche finale. À cet égard, +nous proposons, en section 5.4.1, une stratégie d’évaluation fondée sur les résultats de +reconnaissance obtenus après un système de reconnaissance de texte (HTR). +Dans cette partie, nous fournissons une évaluation juste et approfondie de trois approches +pour détecter les lignes de texte, Doc-UFCN (Boillet et al., 2021a), dhSegment (Ares +Oliveira et al., 2018) et ARU-Net (Grüning et al., 2019), sur une large collection de jeux +de données historiques et avec plusieurs métriques, y compris une métrique orientée recon- +naissance de texte (HTR). Nous analysons les métriques de détection de lignes de texte de +la littérature par rapport à de multiples jeux de données publiquement disponibles et mon- +trons certaines incohérences entre les jeux de données. Nous proposons, en section 5.1.1, une +stratégie d’uniformisation des annotations des jeux de données qui évite le biais d’étiquetage +pour la tâche de détection de lignes de texte. Cet étiquetage modifié permet de considérer +la variabilité des annotations et d’entraîner des modèles avec une plus grande capacité de +généralisation. Dans un second temps, nous effectuons une évaluation de l’état de l’art grâce +à différentes métriques et protocoles d’entraînement. Ces protocoles permettent de construire +des modèles de détection plus génériques qui considèrent les limitations mentionnées ci-dessus + +5.1 U N I F O R M I S AT I O N D E S A N N O TAT I O N S +81 +Table 5.1 – Statistiques des jeux de données utilisés pour la détection de lignes de texte. +Jeu de données +Images +Lignes +train +valid +test +train +valid +test +AN-Index† +19 +3 +12 +433 +62 +171 +Balsac +Vézina et al. (2020) +730 +92 +91 +36 941 +4 592 +4 323 +BNPP† +7 +2 +3 +705 +218 +358 +Bozen +Sánchez et al. (2016) +350 +50 +50 +8 366 +1 040 +1 138 +cBAD2019 +Diem et al. (2019) +720 +716 +45 266 +42 672 +DIVA-HisDB +Simistira et al. (2016) +60 +30 +30 +6 037 +2 999 +2 897 +HOME-NACR +Boros et al. (2020) +338 +40 +42 +6 590 +602 +909 +Horae +Boillet et al. (2019) +522 +20 +30 +12 568 +270 +958 +READ-Complex +Grüning et al. (2017) +216 +27 +27 +17 768 +2 160 +1 758 +READ-Simple +Grüning et al. (2017) +172 +22 +22 +5 117 +539 +723 +HOME-Alcar +Stutzmann et al. (2021) +– +– +55 +– +– +2 727 +ScribbleLens +Dolfing et al. (2020) +– +– +21 +– +– +563 +† Jeux de données privés utilisés durant la thèse. +et obtiennent des résultats similaires, voire meilleurs, que les modèles entraînés sur des en- +sembles de données uniques. +5.1 +U N I F O R M I S AT I O N D E S A N N O TAT I O N S +L’un de nos objectifs étant de développer un détecteur générique de lignes de texte sur +des documents historiques, nous avons collecté neuf jeux de données historiques dont sept +jeux publics pour mener les expérimentations. Ces jeux de données sont présentés en section +3.2 et une description est donnée dans la Table 5.1. De plus, comme nous souhaitons évaluer +la capacité de généralisation des modèles obtenus, nous avons collecté deux jeux de données +supplémentaires : ScribbleLens (Dolfing et al., 2020) et HOME-Alcar (Stutzmann et al., +2021) utilisés uniquement pendant l’étape d’évaluation. + +82 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +Tous ces ensembles de données ont été choisis pour leur diversité en termes de tailles, +d’écritures et de mises en page. La Figure 3.2 présente la variété des documents en montrant +un exemple d’image de chaque ensemble de données avec ses annotations lignes. La réparti- +tion utilisée pour entraîner les modèles a été obtenue en regroupant simplement les données +d’entraînement et de validation respectives des sous-ensembles. De plus, puisque le jeu de +données HOME-Alcar ne contenait pas d’ensembles d’entraînement, de validation et de test +officiels au moment des expériences présentées ci-après nous avons rassemblé, pour générer un +ensemble de test, 55 pages annotées au niveau ligne avec leurs transcriptions correspondantes +parmi tous les manuscrits. +5.1.1 +analyse des annotations +Tous les ensembles de données cités ci-dessus ont été utilisés pour entraîner des modèles +génériques de détection de lignes de texte avec les modèles Doc-UFCN (Boillet et al., 2021a), +dhSegment (Ares Oliveira et al., 2018) et ARU-Net (Grüning et al., 2019). Ces modèles +nécessitent des images annotées au niveau pixel pour l’entraînement. Nous présentons les +défis rencontrés pour générer un ensemble d’entraînement annoté unifié. +diversité dans les annotations +Pour générer les images d’annotations au niveau pixel, les polygones englobants sont ex- +traits de la vérité terrain et dessinés sur une image de fond noir de même taille que l’image +originale. Comme nous pouvons le voir sur la Figure 3.2, les annotations sont très variées +parmi les ensembles de données : +— Dans la plupart des jeux de données (AN-Index, Balsac, Bozen, BNPP, HOME et +Horae), les images sont annotées à l’aide de simples polygones incluant les ascendants +et descendants ; +— Dans les bases cBAD2019, READ-Simple et READ-Complex, les ascendants et descen- +dants ne sont généralement pas inclus dans les polygones, qui sont très fins par rapport +au premier cas ; +— Dans les images de la base ScribbleLens, les annotations sont de larges rectangles qui +incluent de nombreux pixels appartenant au fond ; +— Dans la base DIVA-HisDB, les lignes de texte sont annotées à l’aide de polygones plus +complexes qui suivent précisément le contour de chaque lettre. Les polygones des lignes +de texte du jeu HOME-Alcar sont également précis mais légèrement moins que ceux de +DIVA-HisDB. +Cette diversité dans les annotations nous empêche d’entraîner directement un modèle gé- +nérique qui pourrait être appliqué à de nouveaux jeux de données, car l’annotation serait +parfois incohérente entre deux exemples provenant de deux jeux de données différents. De +telles incohérences dégraderaient considérablement les performances du système. Corriger +les incohérences d’annotation entre les ensembles de données est donc une nécessité pour +permettre l’unification des ensembles d’entraînement. + +5.1 U N I F O R M I S AT I O N D E S A N N O TAT I O N S +83 +Figure 5.1 – Processus de génération d’annotations pour une image du jeu de données de Bozen. +À gauche : génération d’annotations à la taille originale de l’image. Au centre : redi- +mensionnement de l’image à la taille de l’entrée du réseau (768 pixels). À droite : redi- +mensionnement des polygones englobants à la taille d’entrée du réseau, atténuation des +chevauchements et génération de l’image d’annotations à la taille d’entrée du réseau (768 +pixels). +chevauchement des polygones +Un autre problème qui accentue les incohérences des annotations est le chevauchement des +polygones englobants annotés. Même si certains ensembles de données ont été annotés de +telle sorte que jamais les polygones ne se touchent ni ne se chevauchent, d’autres ont été +annotés moins précisément, ce qui entraîne des polygones qui se touchent et se superposent +(page ScribbleLens sur la Figure 3.2), comme le montre l’image de gauche de la Figure 5.1. +Évidemment, une telle vérité terrain ne peut pas servir à une évaluation précise de la capacité +d’un système à détecter chaque ligne de texte. De plus, Doc-UFCN et ARU-Net utilisent des +sous-résolutions des images d’entrée, ce qui peut entraîner des fusions supplémentaires dans +les images d’annotations. La Figure 5.1 présente cet effet indésirable en montrant l’image +d’annotation originale, à gauche, et sa version redimensionnée à 768 pixels, au centre. +Dans la littérature, la plupart des études utilisent la vérité terrain telle quelle, sans prêter +beaucoup d’attention à ce problème de fusion. La raison principale à cela est probablement que +les mesures d’évaluation utilisées comptent uniquement la précision des pixels, sans évaluer +plus précisément le processus de détection des lignes. Cependant, un nombre important de +fusions dans l’ensemble de données d’entraînement va certainement faire dévier le réseau vers +la fusion de plus de lignes que souhaité, avec un effet induit négatif sur le système HTR +incapable de reconnaître les lignes fusionnées verticalement. +Lors de nos expériences, nous avons atténué ce problème en supprimant, autant que pos- +sible, les chevauchements entre les lignes, tout en perdant le moins d’informations possible, +comme on peut le voir sur l’image de droite de la Figure 5.1. + +84 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +stratégie d’uniformisation +Pour unifier les annotations, nous avons choisi d’utiliser uniquement des polygones simples +pendant l’étape d’entraînement. Par conséquent, les rectangles englobant les polygones de +DIVA-HisDB ont été utilisés pendant l’apprentissage au lieu des polygones complexes origi- +naux. De plus, pour résoudre le problème du chevauchement des polygones englobants de +certains jeux de données, nous avons utilisé la stratégie suivante. Pour une image donnée, +nous recherchons les paires de lignes qui se touchent et se chevauchent. Ensuite, trois cas ont +été identifiés pour chaque paire : +— Les polygones se touchent : nous les érodons de 1 pixel ; +— Les polygones se chevauchent de moins de 20 % chacun : nous appliquons le processus +de scission des polygones décrit ci-dessous ; +— Les polygones se chevauchent de plus de 20 % : nous les gardons tels quels car la scission +peut entraîner la perte de trop d’informations (perte d’un polygone ou séparation en +deux polygones). +Dans le cas d’un petit chevauchement (moins de 20 % de l’aire des polygones), le processus +suivant est appliqué : la ligne ayant le plus petit ratio de surface de chevauchement par +rapport à sa surface totale est détectée, et son intersection avec l’autre ligne est supprimée, +l’autre ligne étant conservée telle quelle. Enfin, tous les polygones sont dessinés sur une image +de fond noir. Ce seuil de 20 % a été choisi car il correspond, à peu près, à la hauteur des +ascendants et descendants des lignes de texte. +Comme le redimensionnement de l’image d’annotation peut provoquer des fusions indési- +rables, les polygones englobants sont d’abord redimensionnés à la taille de l’image d’entrée +du réseau, puis la scission est appliquée à cette échelle. Ainsi, l’image d’annotation est di- +rectement générée à la résolution souhaitée de l’entrée du réseau, ce qui empêche la fusion +de certaines lignes. L’image de droite de la Figure 5.1 présente le résultat de ce processus. +Comme nous pouvons le voir, l’image d’annotation produite contient des polygones mieux +séparés. Même si certaines lignes se chevauchent encore sur certaines pages, nous espérons +avoir généré une vérité terrain plus appropriée qui aidera à entraîner le modèle de détection et +à améliorer sa capacité à prédire des lignes de texte séparées. Le code permettant de générer +ces annotations modifiées et les images de labels utilisées dans les expériences sont accessibles +publiquement 1. +5.2 +C O M PA R A I S O N D E S A P P R O C H E S D E D É T E C T I O N +Pour nos expériences, nous avons choisi d’étudier trois systèmes à l’état de l’art : Doc-UFCN +(Boillet et al., 2021a), dhSegment (Ares Oliveira et al., 2018) et ARU-Net (Grüning +et al., 2019). Doc-UFCN a été choisi pour ses bonnes performances sur des jeux de données +historiques et son nombre réduit de paramètres. De plus, nous avons sélectionné les systèmes +dhSegment et ARU-Net car ils sont open-source, faciles à entraîner et ont obtenu des perfor- +mances satisfaisantes sur des tâches de détection sur des documents historiques. ARU-Net est +1. https://gitlab.com/teklia/dla/arkindex_document_layout_training_label_normalization + +5.2 C O M PA R A I S O N D E S A P P R O C H E S D E D É T E C T I O N +85 +également le modèle de détection de lignes de texte utilisé dans Transkribus (Kahle et al., +2017). Nous présentons maintenant les systèmes et les détails d’entraînements puisque nous +les avons tous entraînés afin de pouvoir comparer équitablement leurs performances. +5.2.1 +doc-ufcn +Le système Doc-UFCN est le même que celui présenté en section 4.3. La seule différence +est la taille d’entrée du réseau. En effet, nous avons montré dans le chapitre précédent, que +de meilleures performances sont obtenues avec des images d’entrée plus grandes. Ainsi, les +images sont redimensionnées telles que leur plus grande dimension soit de 768 pixels tout +en conservant leur rapport d’aspect original. Par conséquent, pour entraîner Doc-UFCN, les +annotations sont directement générées à 768 pixels grâce au processus présenté dans la section +5.1.1. Pour les expériences suivantes, Doc-UFCN est entraîné avec un taux d’apprentissage +initial de 5e − 3, des mini-batchs de taille 2, l’optimiseur Adam, la fonction de coût Dice et +l’arrêt anticipé (early stopping). +5.2.2 +dhsegment +L’encodeur pré-entraîné présent dans le système dhSegment nécessite que les images d’en- +trée soient de taille fixe de 400×400 pixels. Ainsi, contrairement aux deux autres systèmes, +dhSegment est entraîné sur des patchs de 400×400 pixels d’images en taille réelle. Le proces- +sus de scission est donc appliqué sur les polygones de taille originale. Le modèle est entraîné +avec un arrêt anticipé et des mini-batchs de taille 4. De plus, nous avons conservé le post- +traitement proposé dans Ares Oliveira et al. (2018) en seuillant les probabilités avec t += 0,7. Différentes valeurs ont été testées pour ce paramètre et le seuil de 0,7 a donné les +meilleurs résultats sur l’ensemble de validation. +5.2.3 +aru-net +Le système ARU-Net (Grüning et al., 2019) est une version étendue du modèle U-Net +(Ronneberger et al., 2015) standard. Deux concepts ont été ajoutés : une attention spatiale +et une structure résiduelle. L’attention spatiale (A-Net) est un CNN multicouche et est utilisée +pour gérer différentes tailles de police sur une même page. Les blocs résiduels sont introduits +pour permettre la rétro propagation des erreurs sur les couches basses du réseau. +Pour les annotations ARU-Net, nous utilisons le même processus que pour Doc-UFCN mais +sur des polygones redimensionnés à 33 % de leur taille originale. Le modèle est entraîné en +utilisant l’arrêt anticipé. Nous avons utilisé la fonction de coût d’entropie croisée et un taux +d’apprentissage initial de 1e − 3. Comme pour dhSegment, il faut seuiller les probabilités +pour obtenir les prédictions finales. Cependant, le choix du seuil pour ARU-Net n’a pas été +une tâche facile car il a un impact réel sur les résultats. Finalement, nous avons choisi un +seuil bas de t = 0,3 car une valeur plus élevée éliminerait une quantité importante de pixels +de lignes de texte. + +86 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +Table 5.2 – Comparaison des systèmes Doc-UFCN, dhSegment et ARU-Net : nombre de paramètres, +en millions, et temps d’inférence moyen, en secondes par image, mesuré sur l’ensemble +de test du jeu de données Balsac. dhSegment possède 32,8 millions de paramètres mais +comme l’encodeur est pré-entraîné, seulement 9,36 millions doivent être entraînés. +Système +Temps d’inférence† +Paramètres +Doc-UFCN +0,41 +4,09 +dhSegment +2,95 +32,8 (9,36) +ARU-Net +1,39 +4,14 +† Prédictions faites sur une carte graphique GeForce RTX +2070 8G. +Pour toutes les architectures, nous conservons les meilleurs modèles en validation. En outre, +les éléments détectés de taille inférieur à 50 pixels sont supprimés. Cette paramétrisation est +optimisée sur l’ensemble de validation. +La Table 5.2 montre le nombre de paramètres et les temps d’inférence des trois systèmes. +Doc-UFCN et ARU-Net ont des poids similaires en nombre de paramètres tout en étant +beaucoup plus légers que dhSegment. Pour les temps d’inférence, dhSegment et ARU-Net +sont peu compétitifs, étant bien plus lents que Doc-UFCN. +5.3 +É VA L U AT I O N D E S D É T E C T I O N S +Les trois systèmes ont été entraînés sur toutes les images d’entraînement afin de dispo- +ser de modèles génériques. Dans cette section, nous présentons les résultats au niveau des +pixels et des objets. Pour une comparaison équitable, toutes les prédictions sont d’abord +redimensionnées à la taille de l’image originale avant de procéder à l’évaluation. En outre, +pour être comparables à d’autres résultats publiés dans la littérature, les modèles sont éva- +lués avec les lignes de vérité terrain originales. Il est intéressant de noter que, malgré des +résultats visuels solides, cette évaluation basée sur les annotations originales est en défaveur +des systèmes testés puisque les polygones d’entraînement sont beaucoup plus fins que ceux +de la vérité terrain. Puisque notre objectif est de développer un modèle historique générique +obtenant des performances satisfaisantes sur des jeux de données hors échantillon, nous rap- +portons également les résultats sur les jeux de données ScribbleLens (Dolfing et al., 2020) +et HOME-Alcar (Stutzmann et al., 2021), deux jeux de données qui n’ont pas été utilisés +pour l’entraînement. Enfin, nous montrons l’impact de l’unification des annotations sur les +résultats de détection lors de l’entraînement du système Doc-UFCN. +5.3.1 +métriques niveau pixel +Dans cette section, nous présentons les résultats d’évaluation des systèmes par les mé- +triques pixel IoU et F1-score, souvent utilisées dans la littérature pour évaluer les systèmes +de détection d’objets. + +5.3 É VA L U AT I O N D E S D É T E C T I O N S +87 +Table 5.3 – Résultats au niveau pixel obtenus par les systèmes Doc-UFCN, dhSegment et ARU-Net +sur les ensembles de test. Les résultats présentent les performances des modèles génériques +sans adaptation. ScribbleLens* rapporte les résultats des modèles spécifiques. +Jeu de données +IoU +F1-score +Doc-UFCN +dhSegment +ARU-Net +Doc-UFCN +dhSegment +ARU-Net +AN-Index +0,69 +0,68 +0,68 +0,82 +0,81 +0,74 +Balsac +0,87 +0,74 +0,98 +0,93 +0,85 +0,84 +BNPP +0,65 +0,60 +0,90 +0,78 +0,75 +0,75 +Bozen +0,82 +0,70 +0,99 +0,90 +0,82 +0,74 +cBAD2019 +0,66 +0,62 +0,89 +0,79 +0,76 +0,74 +DIVA-HisDB +0,67 +0,46 +0,96 +0,80 +0,60 +0,60 +HOME-NACR +0,60 +0,55 +0,94 +0,77 +0,73 +0,67 +Horae +0,64 +0,63 +0,87 +0,79 +0,79 +0,75 +READ-Complex +0,49 +0,58 +0,81 +0,70 +0,73 +0,73 +READ-Simple +0,60 +0,57 +0,88 +0,73 +0,71 +0,71 +HOME-Alcar +0,35 +0,49 +0,58 +0,49 +0,60 +0,70 +ScribbleLens +0,35 +0,36 +0,41 +0,51 +0,51 +0,58 +ScribbleLens* +0,80 +0,95 +– +0,89 +0,97 +– +comparaison des systèmes sur les ensembles de test des jeux d’entraîne- +ment +Les résultats obtenus par les trois réseaux sur les ensembles de test des jeux d’entraînement +sont présentés en haut de la Table 5.3. Le réseau ARU-Net semble plus performant en termes +d’IoU, alors qu’il l’est moins que les autres systèmes si nous considérons le score F1. En +effet, le score F1 repose réellement sur les mesures de précision et de rappel (non présentées +ici), en les résumant de manière précise. Ceci peut expliquer les faibles résultats obtenus par +ARU-Net puisque ses scores en précision ne sont jamais supérieurs à 60 % (sauf pour Balsac). +Au contraire, le score IoU étant moins focalisé sur les pixels correctement prédits (TP est +considéré deux fois pour le score F1 et seulement une fois pour IoU), les scores IoU sont plus +élevés, ce qui conduit à un meilleur classement dans le tableau des résultats. +Ces valeurs de précision faibles mais de rappel élevées obtenues par ARU-Net suggèrent +que le modèle a correctement prédit la majorité des pixels de lignes de texte alors que, par +ailleurs, beaucoup de pixels d’arrière-plan ont été classés comme lignes de texte. Cela reflète +la présence de fusions dans les lignes détectées. La Figure 5.2 montre les prédictions obtenues +par les modèles sur une image tirée au hasard dans l’ensemble de test Horae (Boillet et +al., 2019). Elle confirme notre hypothèse selon laquelle ARU-Net a fusionné certaines lignes. +Cependant, comme indiqué précédemment, l’utilisation d’un seuil plus élevé aurait conduit +à manquer un grand nombre de pixels de lignes de texte. Nous pensons qu’ARU-Net n’est +peut-être pas le système le plus approprié pour détecter des objets proches. En effet, il a +souvent obtenu de très bonnes performances lorsqu’il était entraîné avec des lignes de base, +où les objets sont plus espacés et plus fins que les polygones englobants des lignes de texte. + +88 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +Figure 5.2 – Détections de lignes produites sur une image du jeu de données Horae : Doc-UFCN à +gauche, dhSegment au centre et ARU-Net à droite. Doc-UFCN et dhSegment produisent +des résultats similaires, tandis que ARU-Net surestime l’épaisseur des lignes et fusionne +plusieurs lignes (l’une d’elles est mise en évidence en vert foncé). +Comparer Doc-UFCN et dhSegment est un peu plus facile car ils se comportent de la +même manière pour les scores IoU et F1. Doc-UFCN surpasse dhSegment sur la majorité +des jeux de données pour les deux mesures. Il est cependant moins bon sur le jeu de données +READ-Complex. Nous supposons que cela est dû au nombre élevé de petits objets dans les +images des documents de ce jeu qui peuvent avoir été manqués par Doc-UFCN puisqu’il +travaille à une faible résolution, contrairement à dhSegment et ARU-Net. +L’évaluation et la comparaison des trois modèles sur la base de l’IoU uniquement condui- +raient à choisir ARU-Net comme étant le meilleur modèle. Or, nous avons montré que ses +faibles précisions peuvent conduire à une faible capacité à distinguer des lignes proches. Les +mesures au niveau objet, qui peuvent rendre compte des lignes fusionnées, devraient être +utilisées en complément de ces valeurs de pixels. +évaluation hors échantillon +La Table 5.3 présente également les résultats des modèles génériques appliqués aux jeux de +données ScribbleLens et HOME-Alcar. Nous avons également entraîné Doc-UFCN et dhSeg- +ment sur ScribbleLens afin de disposer de modèles spécifiques pour la comparaison. Pour le jeu +de données HOME-Alcar, nous ne disposons pas d’images d’entraînement pour la détection +de lignes, donc seuls les résultats génériques sont présentés. +Les performances obtenues par les trois systèmes sur les jeux de données ScribbleLens et +HOME-Alcar sont bien inférieures à celles obtenues sur les jeux de données d’entraînement, +et également à celles obtenues par les modèles spécifiques de ScribbleLens*. Pour le jeu +de test ScribbleLens, la précision est égale à 97 % pour les modèles génériques dhSegment +et Doc-UFCN alors qu’elle se situe entre 82 et 85 % pour HOME-Alcar. Cela suggère que + +htretrneiru +ermtrateaste +moeumtr +防· +HHUS +e +KOJIRIOhmraememmm +toot:iairemtre ars-ne +iomttmrmf·oremm +tortmmm:mohtm +mmtotottmmtirgtrrg +toot +metm +m2 +1smm·oeumtn +mmon +mmmy +temrrommt +H +XU +cuott5.3 É VA L U AT I O N D E S D É T E C T I O N S +89 +Table 5.4 – Résultats au niveau pixel obtenus par Doc-UFCN avec et sans uniformisation des labels. +Les résultats montrent les performances des modèles génériques sans adaptation. +Jeu de données +IoU +F1-score +Originaux +Uniformes +Originaux +Uniformes +AN-Index +0,67 +0,69 +0,80 +0,82 +Balsac +0,71 +0,87 +0,83 +0,93 +BNPP +0,63 +0,65 +0,77 +0,78 +Bozen +0,67 +0,82 +0,80 +0,90 +cBAD2019 +0,61 +0,66 +0,75 +0,79 +DIVA-HisDB +0,65 +0,67 +0,78 +0,80 +HOME-NACR +0,56 +0,60 +0,74 +0,77 +Horae +0,64 +0,64 +0,79 +0,79 +READ-Complex +0,53 +0,49 +0,74 +0,70 +READ-Simple +0,58 +0,60 +0,72 +0,73 +HOME-Alcar +0,51 +0,35 +0,63 +0,49 +ScribbleLens +0,42 +0,35 +0,59 +0,51 +presque tous les pixels prédits étaient corrects, alors qu’un grand nombre de pixels de la +vérité terrain n’ont pas été détectés. Notre hypothèse est que les modèles ont prédit de +bons polygones mais très fins par rapport aux polygones annotés très larges des pages +ScribbleLens, ce qui a conduit à des valeurs d’IoU dégradées. Il en est de même pour les +images HOME-Alcar, où des polygones fins rectangulaires comprenant uniquement quelques +pixels d’arrière-plan ont probablement été prédits. +Sur la base de ces métriques, nous ne pouvons pas être certains que les systèmes ne par- +viennent pas à généraliser sur les deux nouveaux ensembles de données. D’autres mesures +pourraient donner un meilleur aperçu des capacités de généralisation réelles des modèles. +impact de l’unification des annotations +Pour évaluer l’impact de l’unification des annotations sur les résultats, nous avons entraîné +Doc-UFCN sur tous les jeux de données avec des annotations non uniformisées. Selon la Table +5.4, l’entraînement avec les annotations uniformisées améliore les performances au niveau +pixel jusqu’à +16 points de pourcentage d’IoU sur Balsac. Cependant, comme expliqué pour +ARU-Net dans la section 5.3.1, les mesures de rappel sont plus élevées sans le processus +d’unification. En effet, les pixels entre les lignes consécutives et ceux le long des bords des +lignes sont plus souvent prédits comme des lignes de texte, ce qui augmente les valeurs de +rappel. Cependant, certains de ces pixels ne sont pas censés faire partie des lignes de texte +(puisqu’ils créent des fusions), ce qui diminue les valeurs de précision. Sur la base de ces +métriques, la scission des lignes proches semble être nécessaire pour aider le modèle à les +distinguer. + +90 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +limitation des métriques pixel +Même si ces mesures au niveau pixel peuvent donner une première idée de la performance +d’un modèle, nous présentons, sur la Figure 3.4, deux exemples prouvant qu’elles peuvent +ne pas être suffisantes. En effet, deux prédictions différentes peuvent être qualifiées par les +mêmes valeurs d’IoU et de score F1 malgré une différence importante de qualité. Or, dans +la littérature, les systèmes sont souvent comparés par leurs valeurs d’IoU et de F1-score, +nous montrons donc ici que ces métriques ne sont pas appropriées pour choisir le meilleur +modèle car elles ne prennent pas en compte le nombre d’objets détectés. En conclusion, ces +métriques ne nous permettent pas de déterminer la capacité de généralisation des modèles +entraînés. Pour surmonter ces problèmes, la section suivante présente et analyse les résultats +des métriques au niveau objet. +5.3.2 +métriques niveau objet +Nous avons montré, dans la section précédente, que les métriques au niveau pixel +peuvent ne pas être suffisantes pour une évaluation et une comparaison approfondies des +modèles. Nous présentons maintenant les métriques au niveau objet et montrons qu’elles +sont complémentaires aux métriques précédentes, et peuvent donner des informations plus +parlantes sur la qualité d’un résultat de détection. +Comme indiqué précédemment, déterminer si un objet doit être considéré comme positif +ou négatif est complexe. En se basant sur l’idée proposée dans les compétitions PASCAL +VOC, il est possible de calculer la précision, le rappel et la précision moyenne (AP) au niveau +de l’objet. Pour ce faire, les objets prédits et les objets annotés sont d’abord appariés en +fonction de leurs scores IoU, de sorte qu’un seul objet prédit soit apparié à un objet annoté +et inversement. +Ensuite, les objets appariés sont classés par score de confiance décroissant. Pour chaque +objet prédit i, les mesures de précision Pi et de rappel Ri sont calculées en considérant uni- +quement les objets ayant des scores de confiance supérieurs ou égaux à celui de l’objet courant +i. Ces mesures sont calculées en fonction d’un seuil d’IoU choisi t, à l’aide des équations 5.1 +suivantes. +Pi = +TPi +Totali +Ri = +TPi +TotalGT +(5.1) +Ces équations s’appliquent avec : +— TPi : nombre d’objets positifs correctement prédits ayant une confiance supérieure ou +égale à celle de l’objet i ; +— Totali : nombre d’objets prédits ayant une confiance supérieure ou égale à celle de +l’objet i ; +— TotalGT : nombre d’objets annotés à retrouver ; +où un objet est considéré comme positif si son IoU est supérieur au seuil choisi t. + +5.3 É VA L U AT I O N D E S D É T E C T I O N S +91 +Table 5.5 – Résultats au niveau ligne obtenus par les systèmes Doc-UFCN, dhSegment et ARU-Net +sur les ensembles de test. Les résultats présentent les performances des modèles génériques +sans adaptation. ScribbleLens* rapporte les résultats des modèles spécifiques. +Jeu de données +AP@.5 +AP@[.5, .95] +Doc-UFCN +dhSegment +ARU-Net +Doc-UFCN +dhSegment +ARU-Net +AN-Index +0,75 +0,76 +0,51 +0,34 +0,35 +0,17 +Balsac +0,98 +0,94 +0,76 +0,76 +0,51 +0,34 +BNPP +0,83 +0,78 +0,50 +0,31 +0,27 +0,13 +Bozen +0,99 +0,74 +0,01 +0,69 +0,35 +0,0 +cBAD2019 +0,86 +0,71 +0,29 +0,48 +0,24 +0,07 +DIVA-HisDB +0,77 +0,39 +0,10 +0,36 +0,17 +0,04 +HOME-NACR +0,85 +0,78 +0,19 +0,46 +0,28 +0,04 +Horae +0,83 +0,85 +0,56 +0,38 +0,34 +0,17 +READ-Complex +0,60 +0,62 +0,22 +0,23 +0,24 +0,08 +READ-Simple +0,69 +0,58 +0,21 +0,28 +0,21 +0,05 +HOME-Alcar +0,16 +0,76 +0,0 +0,03 +0,26 +0,0 +ScribbleLens +0,06 +0,02 +0,0 +0,02 +0,02 +0,0 +ScribbleLens* +0,94 +0,0 +– +0,61 +0,0 +– +La courbe Précision-Rappel est ensuite calculée et interpolée et la précision moyenne (AP) +est définie comme l’aire sous cette courbe. Cette AP est calculée pour toutes les classes d’une +expérience, puis la moyenne est calculée pour toutes les classes, ce qui donne la précision +moyenne (mAP). Pour la détection des lignes de texte, nous n’avons qu’une seule classe +d’objets, la mAP est donc égale à l’AP et est notée AP@t dans la suite, t étant toujours le +seuil IoU. +comparaison des systèmes sur les ensembles de test des jeux d’entraîne- +ment +La Table 5.5 présente les résultats d’AP obtenus sur les ensembles de test pour un seuil +d’IoU de 50 % (AP@.5). En outre, et afin de s’affranchir de tout seuil, la moyenne des AP +sur une plage de valeurs d’IoU (50 % – 95 %) est également calculée et présentée comme +AP@[.5,.95]. +Les résultats présentés ici renforcent notre hypothèse précédente selon laquelle ARU-Net +ne parvient pas à séparer les objets proches. En effet, tous les résultats de ARU-Net sont très +inférieurs à ceux des deux autres systèmes, sauf pour le jeu de données Balsac où les polygones +des lignes de texte sont vraiment espacés dans les annotations. De plus, nous observons que, +pour un seuil bas de 50 %, Doc-UFCN surpasse légèrement dhSegment. En passant de 50 % +à la moyenne des AP, nous constatons que les résultats des deux modèles se dégradent, ce qui +signifie que, avec des seuils plus élevés, certaines lignes deviennent considérées comme fausses +positives car leur localisation n’est pas assez précise. Cependant, cette dégradation est plus +faible pour Doc-UFCN que pour dhSegment, ce qui signifie que la localisation des objets par +dhSegment est moins précise que celle de Doc-UFCN. La Figure 5.3 présente les résultats des + +92 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +Figure 5.3 – Détections de lignes produites sur une image du jeu de données Bozen : Doc-UFCN à +gauche, dhSegment au centre et ARU-Net à droite. Doc-UFCN prédit des lignes bien +séparées alors que ARU-Net prédit des lignes fusionnées. dhSegment ne produit pas de +lignes fusionnées mais elles sont plus proches que celles produites par Doc-UFCN. De +plus, les polygones de dhSegment incluent plus d’espace en haut des lignes, ce qui peut +avoir un impact négatif sur la reconnaissance du texte. +trois modèles sur une image tirée au hasard dans le jeu de données Bozen. Ces prédictions +confirment l’intérêt des mesures AP pour évaluer les prédictions de détection puisqu’elles +mettent en évidence les mauvais comportements, comme ceux montrés par ARU-Net. +évaluation hors échantillon +La Table 5.5 présente également les résultats des modèles génériques appliqués à Scribble- +Lens et HOME-Alcar et les modèles spécifiques de ScribbleLens. Les résultats obtenus par les +modèles spécifiques au niveau objet sont totalement opposés à ceux obtenus au niveau pixel. +Les résultats obtenus par Doc-UFCN confirment que le modèle fonctionne bien lorsqu’il est +entraîné directement sur ScribbleLens, sauf sur quelques images comme montré sur la Figure +5.4. Au contraire, alors que dhSegment a obtenu de bonnes mesures au niveau des pixels, ses +valeurs d’objet sont toutes à 0. En effet, comme pour les résultats précédents obtenus par +ARU-Net, nous observons de nombreuses lignes fusionnées prédites par le modèle spécifique +dhSegment, ce qui signifie que ce dernier n’a pas réussi à apprendre directement à partir des +images ScribbleLens. +Les faibles scores AP des modèles génériques peuvent être expliqués par la façon dont +le jeu de données ScribbleLens a été annoté : des rectangles englobants très larges. Les +modèles ayant été entraînés sur des polygones bien divisés et beaucoup plus fins, seuls quelques +polygones réels ont été appariés aux polygones prédits lors du calcul de l’AP. La même +observation s’applique aux résultats du jeu HOME-Alcar. Les Figures 5.4 et 5.5 présentent +une visualisation des résultats obtenus sur les images des bases ScribbleLens et HOME-Alcar. +Malgré des valeurs de métriques peu élevées, les modèles génériques semblent nettement +surpasser les modèles spécifiques, d’où l’importance de développer des modèles génériques. De +plus, il est nécessaire de les évaluer sur des annotations cohérentes avec celles de l’ensemble +d’entraînement des modèles. + +州5.3 É VA L U AT I O N D E S D É T E C T I O N S +93 +Figure 5.4 – Détections de lignes produites par les modèles génériques, en haut, et spécifiques, en +bas, sur une image du jeu de données ScribbleLens. Les images de gauche montrent les +résultats produits par Doc-UFCN et celles de droite par dhSegment. +impact de l’unification des annotations +Sans surprise, selon la Table 5.6, presque toutes les valeurs sont meilleures lorsque nous +utilisons le modèle entraîné sur les annotations unifiées, parfois avec une marge assez impor- +tante (+33 points de pourcentage pour Balsac et +37 pour Bozen). Pour le jeu de données +DIVA-HisDB, les résultats sont mitigés. Nous supposons que cela est dû au processus d’unifi- +cation qui peut considérablement modifier les annotations en réduisant la hauteur de la ligne. +Ces métriques au niveau objet ont souligné la nécessité de les utiliser avec celles au ni- +veau du pixel pour évaluer et comparer les modèles. Cependant, il est encore difficile de +voir l’avantage d’utiliser des modèles génériques sur des documents hors échantillon. Les mé- +triques orientées vers les objectifs, décrites dans la section suivante, permettront une meilleure +comparaison des objets prédits et des objets réels. + +89A +13080 +.AA +la9e8f8. +enetalrmycawl +moeneeim +4se3 +2-0002 +9020004 +080 +00n489840 +1380.A147 +enxelaermiek +Ae +XA& +aAg +9020894 +80058. +mrceaLon48p8f +ongop +yhyerayo +Ramse +Lem +G- +Coesusi94 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +Figure 5.5 – Détections de lignes produites par les modèles génériques Doc-UFCN, à gauche, et dh- +Segment, à droite, sur une image du jeu de données HOME-Alcar. +Table 5.6 – Résultats au niveau ligne obtenus par Doc-UFCN avec et sans uniformisation des labels. +Les résultats montrent les performances des modèles génériques sans adaptation. +Jeu de données +AP@.5 +AP@[.5,.95] +Originaux +Uniformes +Originaux +Uniformes +AN-Index +0,69 +0,75 +0,28 +0,34 +Balsac +0,95 +0,98 +0,44 +0,76 +BNPP +0,81 +0,83 +0,30 +0,31 +Bozen +0,77 +0,99 +0,31 +0,69 +cBAD2019 +0,71 +0,86 +0,25 +0,48 +DIVA-HisDB +0,86 +0,77 +0,40 +0,36 +HOME-NACR +0,82 +0,85 +0,33 +0,46 +Horae +0,84 +0,83 +0,34 +0,38 +READ-Complex +0,61 +0,60 +0,24 +0,23 +READ-Simple +0,60 +0,69 +0,19 +0,28 +HOME-Alcar +0,86 +0,16 +0,27 +0,03 +ScribbleLens +0,41 +0,06 +0,08 +0,02 +5.4 +É VA L U AT I O N O R I E N T É E V E R S L A TÂ C H E D E R E C O N N A I S S A N C E +Dans les sections précédentes, nous avons discuté des résultats aux niveaux pixel et objet +pour les trois modèles. Nous avons montré que les mesures d’objets donnent plus d’informa- +tions sur la qualité et les performances d’un modèle de détection de lignes de texte que les +mesures au niveau du pixel. Cependant, elles sont toujours limitées lorsque les objets prédits +sont trop fins par rapport aux objets réels. Les mesures orientées vers la tâche finale peuvent +aider à déterminer les capacités réelles des modèles dans ce cas. + +mx +373 +hacebetehicatoteqmalo +feDmoarentoacuntioo +gottonatzatn.qofutzpuaguntt +fhaebnptiontanta non ondta-cictanmo +eltotmoarpen.tumoeticcncomcem +Dhitdo-ceefabotetotutpibadta +tomoqo.ftugao.cttceetupgbomall +oquaumttamncobemtocet +mmametumcomotceaeto +cenunetambmmaenutafinmum +mafplenteuctamtuno +atpentumeotmtovnqtetutnauo +axmomcecnammonotaltttem +oocftbttoarentaqueiaobitemote +Focumtacmutqotnntaptentacomih +ptomoatnettattewmamac +otica attecutet enanqoeleanteo +tascetobeittoetagucetunztemmo +Stuteteneteceotintebtwomtlttuen +btenrabb.cocotenttntbiopongtp +Lemofimamcacliebtoponigcumtampat +tamincremeneicentaucooinielno(rut +temutataenncetiteapoream +neacgsafinnttutfonbecemttceteib +helamanevoa btopontg ncenttuta +qtaoobsbeicotenuecetiqinammatt +tbpomigcetelhotttcutemtnneeabno +tmqtinctenauoto0cebomaalletp +neiopctaintaqungubpaiamm +cetin-KonotcizanccerolegcTo c +tolutnccamuobontemttumvtcno +Otilerotatiottineapaa ctactinnocoao +tucutigtbtoponzgpomcnponem citm +ouoaxumcenatr copitalfcenttutcotoy +posegimesometttauuoaeuinda +ctlrobtettcobomersomttecalttet +quehemonnetacpsv-mb +iong cmeima tabutinitecagnegface +patfibrtouttntcomnittuccomnob +atmqpoamuenottontemcont-otguit +Go.coeiieeeuocabbycrconmentn +tazabmcooecetiecr crcont aancttenot +teteneanoimpeuumcmtucremen +tonem.pteatutbatnonueteonnt +uuereetpicstcedne cacoy +uumAeuo cmelne ecegnetqutan +enoenotp2omtcetneantottotxgczct +rune-quttmonotatnealemetonet +atuavaomowcetinpeoohac +nem cecoencrntonetnoateteueier +qutatonemfactctane-tjonegfuia-incn +atosuonuemencmtumeqotet +nemoam.enztemomtpfenteut +Soatabipttuenotcafmenozattauu +secattzgntemmtmnetecioo +gmentngaeuuabmncaouttco +tal.ad.amnoom.cy.coxoqrw.tmente +futentotneg fianc cone othiccaattejolm +ato.c +quteu fiecnca frontaneanon cacta +qumjubunteaareibneotarng +citreiytonccoraitetyteaitoquccumop +itieptenrefxtteartmfichutto +momabbducepieacmecfc +Metafcurpatinonotauuemfld +TobntdamtenbteonemtotztteLaoatut +tamfaemaoantaotetzeconfitutot +tcontetamnobtheonaeg +oybeptieenmtetseyomettumeane +nout tebnbufieaaitaotecrconen +tateeczqosptimbetamctmmcanonem +qtmnggtnra foitiog may cacem qutinan +onetactenadceammo.ontt.mcxevt +tuauoacltctemtptnenfmm +menteuitoac.cmpngugmd +Dtunonmatepoemamictmttet +pteroc +iolicecfnznumerpentu.tnceqo +igptitoneorttna aug caie +mkaruebionol.gofuootutbupbzc +mmuenoigmutbnnpienn373 +hacuebeethiatoequomalto +Leomayeubtslncuintcono +pmenttbieaoctanquttam +qouoatumetanqofutpagunct +facacbabfjontaneanoncacta-@tammo +clomoarpen-neeCencocemtey +omoqotuegumocectpqoamalt +oqtreutumtimeocemtoocot +muowtda +Cefinefanbapungenatainternmn +uadpenetucatmfiumt +aruntumotmovmyqtitecutenatton +ancbropacccnemnonotaittmem +Co ciwooenentaquejacobsbetoce +ocuimtacmuiqomntapteuaconth +uotooarenrtmtawmaad +Cut jolkytetmtaoo emeineeutmm +ewhicaaerueocnanqo eieante-o +abtetobutoeragucfettutenomo +Gruteompteceotntebungomtitretuen +ucetpteaonafienpueamacueruame +onceramrabbyevcoxuentbropongto +temotmamcctebroponig qutimtam pat +tumnereneneiceufuCgobeumopmt +temtuutaetineetieapoream +neargsetimttucfoubecemmnceb +fidanapeptiotabropomgucentiua +gjaobybacoceutecetqhammam +Btoboniscrtefiotntcniceuttneecotho +Eamqfactonattotodcel)omtaafteeti +naiop.ttnnqumngubttamztor +ce tia-Romnontctaut-oetjoxg cjobn c +tolnuturcamobconfeftftmvtenot +xozqotptnenotcmtptcitabbrceco +Duteotaruottinea paaatactmmocoao +ttezubropongpoczcpoizez centun +outoccumCenatrcpitaintcenfiuteoncg +cepfoagtimeronienfaiuoachds +tuttoorettcnobbuettapzomteieautel +queenemumeitafoatmsiban +ioheg emeitma taouifntcvagnegfi +Dattibrtotunucconufemtunccomnob +atmqooemiteottotemconofitgare +Ce.conoeieectoemabbrctconucnn +tazabncpoeccueecgrcontanctienot +ucteneanomauuudintucemeum +tonempteautpealtrnonentertunfit +Denttcne qtuenepicanice Gnemactd +nuum ceuo cmema cragnetquitaio +ondonbtpzomtcttncautattotkgceo +Tunequtcoontainmea teimerone +xn:fitesthcuroalgocontitenotm +atlfacaomocftmgomc +nem oecotcetitonenpoafsertetetpe +qutatomenfacctantefjonnia-incti +aluoguonuemencinuumcecoet +retuemnoamctehmommpietett +pocatabiptnenotmfmemozattabbrce +oienntgaeruzabntaoutcoo +tat.ac.amo.ot.dy.c.wodtw.nce +futeuotnes ftaneoneomtScaauejoh +matose +qutentficeanta twontaneanonCaacta +qmuottetaeueibreocarnp +itebetonecotaaetaltoquctmo +mttfptenrertttafmmeut +mocoapbdcoceptiecmletfoo +obutdamttenottonemwolutteLataatut +tamfaemqomnaptetzaonintuut +concelliero2atobtiteCitaceteao +oix.oepntiecuterseotnettmealle +noutttebnbtutieaccitelbtecconttent +titeeoqototmteumetmmconem +quinggiea toitoog tura caoxm.qttran +onetactenn.aceannoonit.oscoxav +llg pmitoneotttma atyg cohie +nnheinozamtbnmat5.4 É VA L U AT I O N O R I E N T É E V E R S L A TÂ C H E D E R E C O N N A I S S A N C E +95 +Nous avons effectué une évaluation orientée vers la reconnaissance de texte sur les cinq +ensembles de données pour lesquels la transcription des lignes de texte est disponible, en +calculant le taux d’erreur de caractère (CER) et le taux d’erreur de mot (WER). Par souci de +clarté, dans les tables suivantes, seuls les CER sont présentés car les WER y sont fortement +corrélés. +Pour réaliser cette évaluation axée sur la reconnaissance du texte, nous avons utilisé un +reconnaisseur de texte manuscrit (Boros et al., 2020) basé sur la bibliothèque Kaldi (Arora +et al., 2019). Le modèle est composé de deux éléments principaux : un modèle optique utilisant +un modèle hybride Deep Neural Network-Hidden Markov Model et un modèle de langue fondé +sur un modèle n-gram entraîné sur des sous-mots générés par la méthode Byte Pair Encoding +(BPE). Contrairement au modèle de détection de lignes de texte entraîné sur tous les jeux +de données, nous avons entraîné un modèle de reconnaissance spécifique pour chaque jeu de +données et utilisé ces modèles pour l’évaluation. +Les paragraphes suivants présentent et analysent les résultats de la détection à l’aide de +deux métriques basées sur le CER au niveau des pages et des lignes. +5.4.1 +cer niveau page +Pour commencer l’évaluation, nous avons d’abord choisi de calculer le CER au niveau de +la page. Les calculs sont détaillés dans l’Algorithme 5.1. Tout d’abord, tous les polygones de +lignes prédits et annotés d’une image sont triés de haut en bas et de gauche à droite de l’image. +En suivant cet ordre, toutes les transcriptions sont concaténées en une seule ligne de texte +et le CER@page est calculé. La Table 5.7 présente les CER@page obtenus par les systèmes. +En outre, nous avons calculé le CER obtenu par le système HTR lors de la transcription des +polygones annotés manuellement. Par conséquent, la colonne "Manuel" des tables suivantes +correspond au meilleur CER réalisable avec le système de détection idéal. Il s’agit du CER +que nous aurions si nous avions 100 % pour toutes les métriques pixel et objet. +Algorithme 5.1 Calcul du CER@page +Entrée: HTR ← modèle de reconnaissance entraîné +Entrée: DLA ← modèle de détection de lignes de texte entraîné +Entrée: image ← image à évaluer +Entrée: transcription ← transcription manuelle de l’image, texte ordonné de +haut en bas, gauche à droite +1: lignes ← DLA(image) +2: ord(lignes) {ordonne les lignes de haut en bas, gauche à droite} +3: prediction ← ”” +4: pour chaque ligne ∈ lignes faire +5: +prediction ← concat(prediction, HTR(ligne)) +6: fin pour +7: cer ← CER(prediction, transcription) +Sortie: cer + +96 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +Table 5.7 – Résultats de reconnaissance niveau page obtenus par les systèmes Doc-UFCN, dhSeg- +ment et ARU-Net sur les ensembles de test. Les résultats présentent les performances +des modèles génériques sans adaptation. ScribbleLens* rapporte les résultats des modèles +spécifiques. +Jeu de données +CER@page (%) +Manuel +Doc-UFCN +dhSegment +ARU-Net +Balsac +4,3 +14,9 +15,8 +31,5 +BNPP +15,5 +37,2 +38,2 +46,5 +Bozen +5,8 +11,7 +13,2 +74,9 +HOME-NACR +11,9 +38,6 +22,3 +75,2 +Horae +10,3 +14,8 +12,1 +31,5 +HOME-Alcar +12,5 +37,4 +43,5 +43,3 +ScribbleLens +4,4 +9,5 +21,9 +15,4 +ScribbleLens* +4,4 +25,2 +92,6 +– +comparaison des systèmes sur les ensembles de test des jeux d’entraîne- +ment +Les résultats de la Table 5.7 montrent la faible performance de la reconnaissance de texte +sur les lignes détectées par ARU-Net. Ce taux d’erreur élevé est la conséquence de la fusion de +nombreuses lignes de texte détectées qui ne peuvent pas être correctement reconnues, comme +cela a déjà été mis en évidence avec l’évaluation au niveau de l’objet. +Doc-UFCN est plus performant que dhSegment sur trois des cinq jeux de données et l’est +légèrement moins que dhSegment pour le jeu de données Horae. Ces résultats confirment +ceux obtenus avec les évaluations au niveau du pixel et de l’objet. Doc-UFCN est cependant +loin derrière dhSegment sur le jeu de données HOME-NACR, contrairement aux résultats +obtenus avec les métriques pixel et objet. +Si nous analysons davantage les résultats de détection obtenus par Doc-UFCN sur le jeu de +données HOME-NACR, nous constatons qu’environ la moitié des pages ont été parfaitement +segmentées sans aucune fusion, ce qui augmente considérablement les scores AP. Cependant, +les autres pages contiennent des lignes prédites qui sont des fusions de deux, trois lignes, ou +même des fusions de lignes de paragraphes entiers. Cela conduit à une légère diminution des +scores AP mais à une dégradation drastique des performances en termes de CER. En effet, si +une seule fusion n’a qu’un faible impact sur le score AP, elle a un impact direct sur le CER +par deux types d’erreurs : +— Le CER entre la prédiction et sa ligne annotée correspondante (qui est souvent élevé +dans le cas d’une fusion) ; +— Le CER des lignes annotées non appariées, égal à la longueur de chaque ligne. +Cette seconde erreur n’est pas significative lorsque seules quelques lignes annotées ne sont +pas appariées. C’est le cas pour les quatre premiers ensembles de données où le nombre de +fusions est négligeable. Elle est encore moins significative lorsque les lignes non appariées ont +un petit nombre de caractères. Cependant, HOME-NACR est le jeu de données avec la plus +grande densité de caractères par ligne (jusqu’à six fois plus que les autres jeux de données). + +5.4 É VA L U AT I O N O R I E N T É E V E R S L A TÂ C H E D E R E C O N N A I S S A N C E +97 +Figure 5.6 – Simulation des scores lorsque deux lignes sont bien séparées, à gauche, et fusionnées, à +droite, sur une image du jeu de données HOME-NACR. À gauche, AP[.5,.95]=60 % et +CER@page=7.3 % ; à droite, AP[.5,.95]=51.3 % et CER@page=20.4 %. +C’est pourquoi, cette seconde erreur a un réel impact sur le CER final de l’ensemble de +données HOME-NACR. C’est également la raison pour laquelle les scores AP ne révèlent pas +le phénomène. +La Figure 5.6 illustre ce point : l’image de gauche est correctement segmentée alors que +sur celle de droite, deux lignes sont fusionnées. Dans ce cas, l’introduction d’une fusion dans +les prédictions entraîne une diminution relative de la moyenne AP@[.5,.95] de 15 % (60 % à +51,3 %) tandis que le CER@page se dégrade de 179 % (7,3 % à 20,4 %). Cela prouve qu’une +fusion n’affecte pas les différentes métriques de la même manière. +Toujours sur le jeu de données HOME-NACR, dhSegment montre une localisation moins +précise des lignes de texte (scores AP plus faibles) par rapport à Doc-UFCN mais très +peu de fusions, ce qui conduit à de meilleures performances de reconnaissance. En effet, +HOME-NACR est le jeu de données où l’écriture est la plus dense et où les lignes sont les plus +proches les unes des autres parmi les dix jeux de données. En raison de la mince hauteur des +lignes de texte et du redimensionnement à 768 pixels, nous pensons que Doc-UFCN n’est pas +l’architecture la plus adaptée pour travailler avec ces pages, contrairement à dhSegment qui +présente une meilleure détection puisqu’il est appliqué sur les images dans leur taille originale. +Cette métrique supplémentaire donne de nouveau un aperçu des performances des modèles, +en étant complémentaire aux métriques vues précédemment. Elle peut, en effet, détecter des +comportements qui ne sont pas mis en évidence par les mesures de pixels ou d’objets. +évaluation hors échantillon +Les résultats de la généralisation sont également présentés dans la Table 5.7. Pour le jeu +ScribbleLens, nous constatons l’avantage d’utiliser un modèle générique : les résultats des +modèles spécifiques (Doc-UFCN 25,2 % de CER, dhSegment 92,5 de % CER) sont nettement +moins bons que ceux des modèles génériques (Doc-UFCN 9,5 % de CER, dhSegment 21,9 % +de CER). Les valeurs de CER du jeu HOME-Alcar sont élevées pour tous les systèmes, ce +qui peut être dû à la complexité de certaines images de documents : mauvaise qualité de la +numérisation, mauvaises conditions de conservation (certaines pages ont été déchirées, par +exemple). Cependant, ces résultats mettent en évidence les capacités de généralisation du +modèle générique Doc-UFCN, donnant de meilleurs résultats sur ScribbleLens que le modèle +spécifique. + +ACK +124ACK +12498 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +Table 5.8 – Résultats de reconnaissance niveau page obtenus par Doc-UFCN avec et sans uniformi- +sation des labels. Les résultats présentent les performances des modèles génériques sans +adaptation. +Jeu de données +CER@page (%) +Originaux +Uniformes +Balsac +14,4 +14,9 +BNPP +34,4 +37,2 +Bozen +27,6 +11,7 +HOME-NACR +33,5 +38,6 +Horae +15,1 +14,8 +HOME-Alcar +43,2 +37,4 +ScribbleLens +12,9 +9,5 +impact de l’unification des annotations +Comme présenté dans la Table 5.8, les deux Doc-UFCN avec et sans le processus d’unifi- +cation ont des résultats assez similaires sur quatre jeux de données sans aucune dégradation +significative. Cependant, l’impact de l’uniformisation des annotations est plus important sur +la base de données Bozen. Pour la même raison que ARU-Net, le modèle entraîné avec les +annotations originales prédit un grand nombre de lignes fusionnées, ce qui conduit à un taux +d’erreur caractères très élevé par rapport au modèle entraîné avec les annotations uniformes. +Le processus d’unification n’a pas amélioré les résultats sur Balsac et BNPP, car les anno- +tations originales étaient déjà fines et constituaient une entrée correcte pour le système de +reconnaissance HTR. +Même si l’entraînement avec les annotations uniformisées n’a pas montré d’amélioration +significative des valeurs de CER pour quatre ensembles de données, il a eu un réel impact +sur les prédictions de Bozen. Concernant les jeux de données hors échantillon, l’unification +des annotations a également un impact positif. Le modèle entraîné avec les annotations +uniformisées donne un CER de 9,5 % pour ScribbleLens et 37,4 % pour HOME-Alcar, ce qui +correspond respectivement à 26 % et à 13 % de diminution relative de l’erreur caractère. +5.4.2 +cer niveau ligne +Cette dernière mesure est étroitement liée au CER au niveau de la page. Ici, le CER n’est +pas calculé sur le texte complet de la page, mais sur chaque ligne de texte prédite. À cet égard, +les lignes prédites et les lignes annotées doivent d’abord être appariées. Dans la littérature, +elles sont souvent appariées sur la base d’un seuil IoU de t = 50 %. Comme pour l’AP, nous +avons calculé le CER pour ce seuil d’IoU de 50 % (CER@.5) ainsi qu’une moyenne sur la plage +50 % – 95 % d’IoU (CER@[.5,.95]). Les lignes prédites sont appariées avec celles annotées +manuellement qui ont l’IoU la plus élevée de sorte qu’une seule prédiction soit appariée avec +une annotation et inversement. Une fois les lignes appariées, nous calculons le CER pour tous + +5.4 É VA L U AT I O N O R I E N T É E V E R S L A TÂ C H E D E R E C O N N A I S S A N C E +99 +Table 5.9 – Résultats de reconnaissance niveau ligne obtenus par les systèmes Doc-UFCN, dhSeg- +ment et ARU-Net sur les ensembles de test. Les résultats présentent les performances +des modèles génériques sans adaptation. ScribbleLens* rapporte les résultats des modèles +spécifiques. +Jeu de données +CER@.5† (%) +CER@[.5, .95] (%) +Doc-UFCN +dhSegment +ARU-Net +Doc-UFCN +dhSegment +ARU-Net +Balsac +7,2/0,95 +8,2/0,95 +29,7/0,64 +14,2 +14,9 +52,0 +BNPP +22,4/0,93 +21,5/0,83 +32,1/0,73 +44,2 +42,8 +53,2 +Bozen +8,8/0,94 +10,0/0,93 +86,5/0,12 +20,7 +18,2 +93,3 +HOME-NACR +36,1/0,61 +23,3/0,79 +80,1/0,18 +61,8 +42,0 +91,1 +Horae +15,2/0,98 +12,0/0,97 +30,3/0,90 +22,6 +29,0 +58,0 +HOME-Alcar +22,9/0,92 +27,6/0,73 +30,0/0,72 +46,6 +49,3 +54,0 +ScribbleLens +9,8/0,80 +18,2/0,37 +15,8/0,81 +40,3 +60,0 +45,5 +ScribbleLens* +24,3/0,76 +90,9/0,10 +– / – +32,6 +93,3 +– +† CER@.5 / Proportion de caractères des lignes annotées appariés à une ligne de prédiction. +1 signifie que 100 % des caractères de l’annotation ont été appariés à une ligne de prédiction. +Figure 5.7 – Résultats de reconnaissance niveau ligne obtenus, sur les ensembles de test, par les +modèles génériques Doc-UFCN, dhSegment et ARU-Net sans adaptation. +les couples dont l’IoU est supérieur au seuil. En outre, le CER final est pénalisé par toutes +les lignes qui ne sont pas appariées. +La Table 5.9 et la Figure 5.7 présentent les résultats obtenus après le système HTR au +niveau ligne. Pour les résultats de CER@.5, la Table montre la proportion de caractères des +lignes annotées appariés à une ligne prédite pour calculer les valeurs de CER. Il aurait été +possible de faire cela au niveau de la ligne (proportion de lignes appariées) pour voir sur +quelle quantité de lignes les CER ont été calculés. Cependant, comme les lignes peuvent +contenir un nombre variable de caractères, cela ne refléterait pas précisément le nombre réel +de correspondances. + +100 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +Table 5.10 – Résultats de reconnaissance niveau ligne obtenus par Doc-UFCN avec et sans uniformi- +sation des labels. Les résultats montrent les performances des modèles génériques sans +adaptation. +Jeu de données +CER@.5 (%) +CER@[.5,.95] (%) +Originaux +Uniformes +Originaux +Uniformes +Balsac +7,2/0,97 +7,2/0,95 +17,1 +14,2 +BNPP +18,8/0,87 +22,4/0,93 +36,0 +44,2 +Bozen +27,3/0,67 +8,8/0,94 +47,3 +20,7 +HOME-NACR +29,4/0,73 +36,1/0,61 +46,6 +61,8 +Horae +15,7/0,98 +15,2/0,98 +20,6 +22,6 +HOME-Alcar +31,7/0,79 +22,9/0,92 +61,3 +46,6 +ScribbleLens +14,3/0,71 +9,8/0,80 +54,2 +40,3 +Comme pour les métriques précédentes, ARU-Net n’est pas compétitif : pas assez de carac- +tères appariés et des valeurs de CER très élevées. Pour comparer Doc-UFCN et dhSegment, +il est nécessaire d’étudier le CER et la proportion d’appariement dans leur ensemble. En effet, +avoir un CER très bas lorsqu’il n’est calculé que sur une petite partie des lignes prédites n’est +pas significatif puisque certaines lignes peuvent être plus faciles à reconnaître. Il est préférable +d’avoir un bon compromis entre le nombre de caractères appariés et le taux d’erreur. +comparaison des systèmes sur les ensembles de test des jeux d’entraîne- +ment +Les résultats obtenus au niveau des lignes reflètent réellement ceux obtenus avec les scores +AP et CER au niveau des pages. À 50 % d’IoU, Doc-UFCN semble meilleur pour les jeux +de données Balsac, BNPP et Bozen. En outre, si nous considérons les résultats moyens +CER@[.5,.95], dhSegment et Doc-UFCN ont tous deux des performances similaires sur les +jeux de données Balsac, BNPP et Bozen. Dans le but d’avoir un modèle historique générique, +les deux architectures semblent appropriées, obtenant de bons résultats au niveau des pixels +et des objets, et des taux d’erreurs caractères acceptables au niveau des pages et des lignes. +évaluation hors échantillon +Les résultats de l’évaluation hors échantillon sont également présentés dans la Table 5.9. Les +résultats pour ScribbleLens confirment ceux obtenus au niveau page. Les modèles génériques +sont, en effet, meilleurs que les modèles spécifiques puisqu’ils montrent des scores de CER +plus bas (Doc-UFCN 9,8 % de CER par rapport à 24,3 %, dhSegment 18,2 % de CER +par rapport à 90,9 %) et des proportions d’appariement supérieures. Les performances sur +les données HOME-Alcar sont comparables à celles obtenues au niveau page avec des taux +d’erreurs élevés, malgré de hautes proportions d’appariement. + +5.5 C O N C L U S I O N +101 +impact de l’unification des annotations +La Table 5.10 présente les résultats, au niveau ligne, de reconnaissance obtenus par Doc- +UFCN avec et sans uniformisation des annotations. Les résultats présentés dans cette table +sont semblables à ceux présentés au niveau page, à savoir des résultats assez similaires sur +quatre jeux de données, avec et sans uniformisation, sans aucune dégradation significative. +L’impact est cependant très important sur le jeu de données Bozen puisqu’il y a un gain de +18,5 points de pourcentage de CER@.5 en uniformisant les labels, impact expliqué par les +mêmes raisons que celles exposées dans la section 5.4.1. +Concernant les jeux de données hors échantillon, l’unification des annotations a également +un impact très positif avec des diminutions de CER@.5 de 8,8 et 4,5 points de pourcentages +respectivement sur les bases HOME-Alcar et ScribbleLens, par rapport aux labels originaux. +5.5 +C O N C L U S I O N +Dans ce chapitre, nous avons montré qu’il est possible d’entraîner un modèle générique pour +détecter les lignes de texte dans les documents historiques. Nous avons entraîné trois systèmes +à l’état de l’art qui ont obtenu de bonnes performances sur différents ensembles de données. +Ceci a été rendu possible par la création d’un large jeu de données d’entraînement, qui est, à +notre connaissance, le plus grand et le plus diversifié des jeux de données historiques utilisés +pour comparer les systèmes de segmentation de documents. Nous avons également montré +que, lors de l’agrégation de différents ensembles de données, l’uniformisation des polygones +englobants annotés réduit les incohérences d’annotation entre les jeux annotés et permet +d’entraîner de meilleurs modèles. En outre, les modèles génériques entraînés sur plusieurs +ensembles de données peuvent être meilleurs, non seulement sur les ensembles de données +individuels, mais également sur les documents hors échantillon, ce qui prouve leurs capacités +de généralisation. +Pour une évaluation pertinente des performances des trois systèmes, ce chapitre compare +et analyse également plusieurs métriques de détection d’objets. Nous avons montré que les +métriques standards au niveau pixel ne sont pas suffisantes car elles ne tiennent pas compte +de la qualité des objets prédits. Pour pallier cet inconvénient, des métriques au niveau des +lignes ont été introduites. Celles-ci ont montré que le système ARU-Net n’est pas approprié +pour la tâche de détection de lignes de texte lorsqu’il est entraîné avec de telles annotations, +le nombre de lignes fusionnées étant important par rapport aux deux autres approches. Ce +système est, en effet, souvent utilisé pour détecter les lignes de base des documents, qui sont +plus fines et plus espacées que les polygones englobants des lignes. Ces mesures ont également +confirmé les bonnes performances de Doc-UFCN et dhSegment sur la plupart des jeux de +données, fournissant une détection précise et exacte des objets. Ces résultats n’auraient pas +été possibles en utilisant uniquement des mesures au niveau pixel. Nous sommes convaincus +que l’utilisation des scores de précision moyenne est nécessaire pour évaluer correctement + +102 +E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S +les modèles de détection de lignes de texte. Notre bibliothèque d’évaluation a été rendue +publique 2, elle peut être utilisée sur n’importe quel jeu de données. +Enfin, ce chapitre fournit une évaluation orientée vers la tâche de reconnaissance de texte +qui, à notre connaissance, n’a jamais été réalisée jusqu’à présent. Les métriques d’évaluation +de reconnaissance HTR donnent encore davantage d’informations sur les objets prédits, étant +complémentaires aux métriques au niveau des objets. De plus, elles permettent d’explorer +l’impact de la qualité des lignes détectées sur les résultats finaux de reconnaissance. +2. https://gitlab.com/teklia/dla/document_image_segmentation_scoring + +6 +E S T I M AT I O N D E L A C O N F I A N C E D E S +P R É D I C T I O N S +Malgré les performances remarquables des réseaux de neurones profonds obtenus dans les +travaux scientifiques, leur utilisation dans des applications réelles exige qu’ils soient, non +seulement performants, mais aussi capables d’évaluer la confiance de leurs décisions. Ceci +est particulièrement important pour les applications liées aux images médicales ou à la +conduite autonome, par exemple. Le problème se pose également dans le cas de l’adaptation +d’un modèle à un nouveau domaine, où nous souhaitons fournir au système le minimum de +nouveaux exemples étiquetés pour réaliser l’adaptation. Le choix des exemples pertinents +à soumettre à un annotateur humain est crucial pour optimiser le processus d’adaptation. +Ce cadre, connu sous le nom d’apprentissage actif (active learning), exige qu’un premier +système effectue la tâche finale tout en évaluant automatiquement sa confiance sur de +nouvelles données non vues, de sorte que les décisions moins confiantes puissent être +soumises à un opérateur humain pour une annotation manuelle, tandis que les décisions plus +confiantes prises par le système seraient conservées telles quelles pour fournir un étiquetage +automatique. Dans ce chapitre, nous visons à développer des mesures de confiance pour +l’adaptation d’un modèle de détection d’objets dans un cadre d’apprentissage actif, afin de +réduire au minimum l’effort d’annotation humaine. +Pour cela, notre objectif est de construire un estimateur de confiance pour la détection +d’objets dans des images de documents dans un scénario d’apprentissage actif. Dans ce but, +nous étudions trois approches afin d’estimer la confiance. La première consiste à utiliser +les probabilités de classe a posteriori du modèle de détection pour estimer la confiance. La +seconde approche proposée est inspirée de la méthode de Monte Carlo (Gal et al., 2016) +et consiste à construire des estimations de confiance en utilisant la méthode de dropout +au moment du test. Le principal avantage de cette approche est qu’aucun entraînement +supplémentaire n’est nécessaire pourvu que le modèle ait été entraîné avec des couches de +dropout. Elle peut être appliquée à des modèles déjà entraînés sans aucune modification. +Cette approche est cependant coûteuse en calculs, c’est pourquoi notre dernière proposition +consiste à construire un système dédié qui peut prédire une estimation de confiance avec +une seule prédiction pendant l’inférence. Indépendant du système de prédiction, ce système +nécessite cependant une phase d’entraînement spécifique. +Ce chapitre présente tout d’abord, en section 6.1, les estimateurs de confiance que nous +proposons. La configuration utilisée pour les expériences (données, détails de l’entraînement +103 + +104 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S +des modèles de détection et ceux des estimateurs de confiance) est ensuite détaillée dans la +section 6.2. Enfin, dans la section 6.3, nous présentons et discutons les résultats obtenus. +6.1 +M É T H O D E S D’ E S T I M AT I O N D E L A C O N F I A N C E +Comme énoncé dans la section 2.2, très peu de travaux ont été proposés afin d’estimer la +confiance des objets prédits par un modèle de détection d’objets dans les images. Certains +travaux utilisent le dropout de Monte Carlo (Gal et al. (2016)) et analysent la distribution des +prédictions afin d’estimer la confiance de la prédiction sans dropout. Dans d’autres travaux, +un réseau adverse est entraîné pour estimer la proximité des prédictions avec la vérité terrain. +Dans ce qui suit, nous proposons quatre estimateurs de confiance. Le premier se base sur +les probabilités a posteriori des classes données par le modèle de détection. Les deux suivants +s’inspirent des travaux réalisés sur le dropout de Monte Carlo et sont déduits de la variance des +prédictions calculées avec dropout. Enfin, le dernier se base sur des statistiques descriptives +des objets attendus et prédits. +Dans la suite de ce chapitre, les objets prédits font référence aux composantes obtenues +après l’application d’un modèle de détection au niveau pixel suivi d’un seuillage. Le seuillage +assigne à chaque pixel la classe (ou fond) de plus grande probabilité. +6.1.1 +estimateur basé sur les probabilités a posteriori +Les réseaux neuronaux de détection d’objets produisent des probabilités au niveau pixel +qui sont ensuite seuillées afin de créer des objets. Le premier estimateur que nous proposons, +dénoté Posterior probability-based Confidence Estimator (PCE), utilise directement ces pro- +babilités a posteriori afin d’estimer la confiance des prédictions. +Tout d’abord, le modèle de détection est appliqué à une image d’entrée, les probabilités pj +obtenues pour chaque pixel sont ensuite seuillées afin d’en extraire les objets. Pour chaque +objet prédit sur une image, nous calculons d’abord la moyenne des probabilités des pixels +prédits par le modèle de détection. Ensuite, le score PCE de l’image est déduit en calculant +la moyenne de toutes les probabilités des objets. Le calcul du score PCE est détaillé dans +l’équation 6.1. Les valeurs calculées par cet estimateur sont comprises entre 0 et 1, une valeur +de 1 étant interprétée comme un indicateur d’une détection correcte. +PCE = 1 +N × +N +� +i=1 + + 1 +Ni +× +Ni +� +j=1 +pj + + +(6.1) +avec : +— N : le nombre d’objets prédits sur l’image d’entrée ; +— Ni : le nombre de pixels composant l’objet i ; +— pj : la probabilité du pixel j d’appartenir à la classe d’objet. + +6.1 M É T H O D E S D’ E S T I M AT I O N D E L A C O N F I A N C E +105 +Cet estimateur présente les avantages d’être simple et rapide à calculer. De plus, il ne +nécessite aucun entraînement supplémentaire autre que le modèle de détection, et peut ainsi +être utilisé pour n’importe quel modèle de détection produisant des probabilités en sortie. +6.1.2 +estimateurs basés sur le dropout de monte carlo +L’estimation de la confiance d’une prédiction avec le dropout de Monte Carlo consiste à +calculer N prédictions de la même observation et à analyser la distribution des prédictions. +La variance entre les N prédictions est un indicateur de l’incertitude du modèle et peut donc +être considérée comme une estimation de la confiance. Dans cette partie, nous proposons +deux scores résumant la variance des prédictions : la précision moyenne (Dropout Average +Precision (DAP)) et la variance du nombre d’objets (Dropout Object Variance (DOV)). +dropout average precision +Comme démontré dans le chapitre 5, la précision moyenne (mAP) utilisée dans les défis +PASCAL VOC et décrite dans Boillet et al. (2022b) permet d’évaluer une prédiction au +niveau objet par rapport à une annotation manuelle. L’avantage de cette métrique est qu’elle +considère la taille et la position des objets prédits puisqu’elle s’appuie sur une correspondance +des objets basée sur l’IoU. Inspirés de cette métrique, nous dérivons l’estimateur DAP qui est +calculé en considérant chaque paire de prédictions ((pi, pj) où pi et pj sont deux prédictions +distinctes de la même image avec i, j ∈ [1, N] et i ̸= j) et en calculant la mAP (voir +Focus 3.4) pour chaque paire, une des deux prédictions étant considérée comme vérité terrain +arbitrairement. Enfin, le DAP est la moyenne de tous les scores mAP (voir l’équation 6.2). +Les valeurs calculées par cet estimateur sont comprises entre 0 et 1, un score DAP élevé +indique que les N prédictions sont très similaires et est interprété comme un indicateur d’une +détection correcte. +DAP = +1 +N2 − N × +N +� +i=1,j=1,i̸=j +mAP(pi, pj) +(6.2) +dropout object variance +Le second estimateur que nous proposons est basé uniquement sur la variance du nombre +d’objets prédits parmi les N prédictions avec dropout. Lorsque le modèle est peu confiant, nous +avons observé qu’un nombre très variable d’objets est prédit avec de nombreux petits objets +autour de l’objet principal (comme le montre l’image de droite de la Figure 6.1). Pour obtenir +une valeur unique, nous calculons la variance du nombre d’objets dans les prédictions avec +dropout comme indiqué dans l’équation 6.3, où ni est le nombre d’objets dans la prédiction +pi. Les valeurs calculées par cet estimateur sont comprises entre 0 et 1, un score DOV de 0 +indique que toutes les prédictions ont le même nombre d’objets et est interprété comme un +indicateur d’une détection correcte. + +106 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S +Figure 6.1 – Deux images issues du jeu de données Horae, à gauche, avec leurs prédictions, au centre +et la variance pour N =10 prédictions avec dropout, à droite. Une variance élevée est +représentée en jaune alors que les zones sans variance sont en noir. L’image de gauche +a des estimations de confiance de DOV=0,0, DAP=1,0 et mAP-RFR=1,0 et celle de +droite DOV=17,36, DAP=0,0993 et mAP-RFR=0,5553. +DOV = +1 +N − 1 × +N +� +i=1 +(ni − n)2 +avec n = 1 +N × +N +� +i=1 +ni +(6.3) +Comme l’estimateur PCE, le calcul des scores DAP et DOV ne nécessitent pas d’autre +entraînement que celui du modèle de détection. En effet, ces estimateurs peuvent être utilisés +pour tout modèle de détection possédant des couches de dropout. +6.1.3 +estimateur basé sur les statistiques d’objets +Pour ce dernier estimateur, nous adoptons une approche basée sur une extraction de ca- +ractéristiques pour estimer la confiance. Nous concevons un système qui analyse les caracté- +ristiques des objets détectés et estime la mAP, car aucune vérité terrain n’est disponible au +moment du test. Contrairement à nos premières propositions, le système étant indépendant +du détecteur, cette approche peut être appliquée à tout type de détecteur. +statistiques descriptives d’objets +Les modèles de détection développés dans nos travaux fournissent, pour chaque pixel, les +probabilités d’appartenir à une classe d’objet ou d’arrière-plan. Les pixels sont d’abord affec- +tés à la classe ayant la plus forte probabilité, puis nous détectons les éléments constitués des +pixels connexes, ce qui conduit à plusieurs objets prédits pour une image donnée. Ensuite, +nous extrayons les polygones englobants des éléments détectés ainsi que leurs rectangles englo- +bants. À partir de ces informations, nous calculons les huit caractéristiques d’objets suivantes. +Pour chaque image et chaque objet, nous calculons : +1. Ratio entre la hauteur du rectangle englobant et la hauteur de l’image ; +2. Ratio entre la largeur du rectangle englobant et la largeur de l’image ; +3. Ratio entre la hauteur et la largeur du rectangle englobant ; +4. Ratio entre l’aire du polygone et l’aire de l’image ; +5. Ratio entre l’aire du polygone et l’aire du rectangle englobant ; +6. Ratio entre l’aire du rectangle englobant et l’aire de l’image ; + +ELE +s.oao.6.2 C A D R E E X P É R I M E N TA L +107 +et pour chaque image : +7. Distances en y (hauteur) entre les centroïdes de tous les rectangles englobants, norma- +lisées par la hauteur de l’image ; +8. Distances en x (largeur) entre les centroïdes de tous les rectangles englobants, norma- +lisées par la largeur de l’image. +Les distances sont calculées en considérant chaque paire de rectangles englobants. Les +caractéristiques permettent de décrire les tailles, les formes et les positions des objets détectés +dans les images de documents. Pour une image donnée et chacune des huit caractéristiques, +les ratios sont calculés pour chaque objet détecté dont les valeurs résultantes sont regroupées +en B intervalles pour fournir un histogramme. Les histogrammes de caractéristiques sont +ensuite concaténés pour constituer un vecteur de statistiques d’objets de taille 8×B. Ces +statistiques sont ensuite utilisées pour entraîner un modèle de régression. +mean average precision - random forest regressor +Pour construire l’estimateur de confiance, nous avons choisi d’estimer la mAP des pré- +dictions, car nous avons montré, en section 5.3.2, qu’elle est plus significative que l’IoU +(Boillet et al., 2022b). Pour estimer la mAP d’une prédiction, plusieurs méthodes de ré- +gression peuvent être utilisées telles que la régression par vecteur de support (SVR) ou le +régresseur Random Forest (RFR). Dans nos expériences, nous avons utilisé RFR, car il a +obtenu les meilleurs résultats dans nos travaux préliminaires. Après l’application du modèle +de régression, aucun traitement supplémentaire n’est nécessaire puisqu’il fournit directement +un score unique considéré comme l’estimation de confiance. Dans ce qui suit, cet estimateur +est appelé mean Average Precision - Random Forest Regressor (mAP-RFR). +6.2 +C A D R E E X P É R I M E N TA L +Nous avons évalué et comparé les estimateurs présentés en 6.1 sur deux tâches de difficultés +différentes : la détection de pages et la détection de lignes de texte manuscrites. La détection +de pages correspond au détourage des pages dans des prises de vues de doubles ou de simples +pages dont les dimensions ne correspondent pas exactement aux dimensions des images pro- +duites par l’imageur (scanner ou caméra). Il s’agit d’une tâche assez simple puisqu’il y a +souvent un ou deux objets sur une image. La détection de lignes de texte manuscrites est une +tâche plus complexe car les pages de documents peuvent contenir un nombre variable, parfois +important, de lignes de texte qui ont des formes et des positions très différentes. +6.2.1 +jeux de données +Pour les expériences de détection de pages, nous avons utilisé les jeux de données READ- +BAD (Grüning et al., 2017) et Horae (Boillet et al., 2019). Notre objectif est d’adapter +le modèle de détection pré-entraîné sur les données READ-BAD aux images de documents +de la base Horae en annotant le moins de données possible. Pour la tâche de détection de + +108 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S +Table 6.1 – Statistiques des jeux de données utilisés pour la détection de pages. +Jeu de données +Images +Pages +simple +double +anormal +READ-BAD +train +1 635 +1 459 +171 +5 +Grüning et al. (2017) +valid +200 +179 +21 +– +test +200 +179 +20 +1 +train +1 630 +1 801 +– +– +READ-BAD* +valid +200 +221 +– +– +Grüning et al. (2017) +test +199 +219 +– +– +train +522 +789 +– +– +Horae +valid +20 +27 +– +– +Boillet et al. (2019) +test +30 +51 +– +– +test-300 +300 +364 +– +– +lignes de texte, notre objectif est d’adapter un modèle générique pré-entraîné à un nouvel +ensemble de documents hors échantillon d’apprentissage, à savoir le jeu de données Hugin- +Munin (Maarand et al., 2022), détaillé en section 3.1. Les statistiques de ces jeux de données +sont présentés dans la Table 6.1. +jeu de données read-bad +Le jeu de données READ-BAD (Grüning et al., 2017), présenté en section 3.1, contient +2 035 images de documents manuscrits utilisées lors des compétitions READ-BAD pour la +détection des lignes de base. Le jeu de données a été annoté aux niveaux simple et double +pages 1. Dans nos expériences, nous prédisons au niveau simple page, ce qui conduit à détecter +deux objets sur les images qui présentent un document en double-page. De plus, les images +ayant été annotées comme "anormales" dans la base ont été supprimées car leurs annotations +n’étaient pas assez précises. Dans ce qui suit, cette version du jeu de données est appelée +READ-BAD* et comprend 1 630 images d’entraînement, 200 images de validation et 199 +images de test avec respectivement 1 801, 221 et 219 pages simples. +jeu de données horae +Le jeu de données Horae (Boillet et al., 2019) est semblable à celui présenté en section +3.1. Afin d’avoir des résultats plus significatifs, nous avons étendu l’ensemble de test original +qui ne contenait que 30 images en annotant 300 images supplémentaires choisies au hasard +parmi les 1 158 livres d’heures, ce qui représente 364 pages simples. Cet ensemble de test est +dénommé Horae-test-300 dans la suite. +Le corpus complet Horae est composé de 1 158 livres d’heures présentant une grande +diversité d’images de documents non annotés en termes de types de numérisations, de fonds +et de formes. Ce corpus est utilisé pour comparer les différents estimateurs lorsqu’ils sont +utilisés dans un cadre d’apprentissage actif. +1. https://github.com/ctensmeyer/pagenet + +6.2 C A D R E E X P É R I M E N TA L +109 +6.2.2 +entraînement des systèmes de détection +Pour nos expériences, nous avons utilisé le système Doc-UFCN comme détecteur d’objets, +car, comme détaillé dans les deux chapitres précédents 4 et 5, il a montré de bonnes +performances pour la détection d’objets sur des documents historiques tout en ayant un +temps d’inférence réduit par rapport aux autres systèmes. +Pour les deux tâches, les modèles Doc-UFCN pré-entraînés (désignés par "référence" dans +la suite) sont entraînés avec les images redimensionnées de telle sorte que leur plus grande +dimension soit égale à 768 pixels, en conservant leur rapport d’aspect. Un prétraitement est +appliqué aux labels d’entraînement afin d’éviter que les zones annotées ne se touchent lors +du redimensionnement des images (prétraitement détaillé dans la section 5.1.1). Les modèles +sont entraînés pendant 150 époques avec un taux d’apprentissage de 5e − 3 et l’optimiseur +Adam. La configuration (poids) qui minimise la fonction de perte sur l’ensemble de validation +est conservée à l’issue de l’apprentissage. +métriques d’évaluation +Outre les métriques de détection standards, les modèles de lignes de texte sont également +évalués à l’aide de métriques orientées vers la tâche finale, notamment le CER et le WER au +niveau page. À cette fin, un reconnaisseur de texte manuscrit (HTR) basé sur Kaldi (Arora +et al., 2019) a été entraîné sur les lignes transcrites Hugin-Munin. Nous avons choisi cet HTR +parce qu’il s’agit d’un outil prêt à l’emploi qui fonctionne généralement assez bien dans la +plupart des cas d’utilisation et qui a obtenu des performances compétitives sur les documents +Hugin-Munin (Maarand et al., 2022). +Le modèle de reconnaissance entraîné est appliqué à toutes les lignes prédites par Doc- +UFCN, ordonnées par leur centroïde du coin supérieur gauche de la page au coin inférieur +droit. Les textes prédits sont concaténés dans ce même ordre pour fournir une transcription +unique au niveau de la page. Les transcriptions manuelles sont ordonnées de la même manière +et les CER et WER au niveau de la page sont calculés. Le modèle de détection de référence +obtient environ 24 % de CER sur les images Hugin-Munin. En outre, nous calculons la +WordCountFMeasure (WCFM) (Pletschacher et al., 2015) qui évalue les modèles HTR +sur la base du nombre de mots correctement prédits, indépendamment de leur position. Nous +avons utilisé le PRIMA Text Evaluation Toolkit 2 pour calculer les scores WCFM. Kaldi +obtient un WCFM de 59 % par rapport aux transcriptions manuelles. Ces valeurs de CER +relativement élevées et de WCFM faibles indiquent que les lignes détectées par le modèle de +référence ne sont pas de très bonne qualité pour le modèle de reconnaissance. Elles peuvent +refléter des lignes détectées mal placées (pas de texte), des lignes trop fines (texte coupé) ou +des lignes manquées. + +110 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S +Table 6.2 – Résultats de détection de pages obtenus par le modèle de référence entraîné sur le jeu de +données READ-BAD* et évalué sur les jeux de données READ-BAD* et Horae-test-300. +Jeu de données +IoU +F1-score +mAP +READ-BAD* +train +0,97 +0,98 +0,92 +valid +0,97 +0,98 +0,91 +test +0,97 +0,98 +0,94 +Horae +test-300 +0,90 +0,94 +0,60 +Table 6.3 – Résultats de détection de lignes de texte obtenus par le modèle de référence entraîné sur +19 jeux de données et évalué l’ensemble de test du jeu de données Hugin-Munin. +Jeu de données +IoU +F1-score +mAP +CER (%) +WCFM +Hugin-Munin +test +0,48 +0,63 +0,21 +24,37 +0,59 +résultats des systèmes de détection +Pour la tâche de détection de pages, le modèle de référence est entraîné sur des images +READ-BAD* dont les résultats sont présentés dans la Table 6.2. Il obtient une IoU de 97 % +et une mAP de 94 % sur READ-BAD*. Cependant, la mAP sur les images de l’ensemble +Horae-test-300 est d’environ 60 %, ce qui laisse une marge d’amélioration importante. Dans +ce qui suit, les images ayant les plus faibles scores de confiance estimés dans le corpus Horae +sont annotées afin d’améliorer la détection sur Horae-test-300. +Pour la détection des lignes de texte, nous avons entraîné un modèle générique de détection +des lignes de texte, différent de celui présenté dans le chapitre 5, ainsi que les estimateurs de +confiance sur de nombreux jeux de données. À cet égard, nous avons rassemblé 19 bases de +données principalement publiques, comprenant des documents historiques et modernes. Au +total, ce jeu de données contient 9 432 images d’entraînement, 1 907 images de validation +et 6 669 images de test, ce qui correspond à 374 316 lignes annotées d’entraînement, 85 208 +lignes de validation et 190 502 lignes de test. Ce modèle générique appliqué à l’ensemble de +test Hugin-Munin a été évalué à 48 % d’IoU et 21 % de mAP (Table 6.3). Ces résultats +assez faibles étaient attendus puisque les documents sont beaucoup plus complexes que ceux +utilisés lors du pré-entraînement. +6.2.3 +entraînement des estimateurs de confiance +Aucun apprentissage supplémentaire n’est requis pour les estimateurs basés sur le dropout +de Monte Carlo, puisque seuls les modèles de détection d’objets sont utilisés pour estimer +la confiance. En revanche, les régresseurs doivent être entraînés sur les statistiques d’objets +décrites en section 6.1.3. Tout d’abord, le modèle de détection d’objets est appliqué à toutes les +images (READ-BAD* pour la détection de pages et les 19 jeux de données pour la détection +2. https://www.primaresearch.org/tools/PerformanceEvaluation + +6.3 R É S U LTAT S E T D I S C U S S I O N +111 +Figure 6.2 – Courbes de rejet présentant l’évolution des performances du modèle de détection de +pages de référence sur l’ensemble de test Horae-test-300. Courbes présentées pour les +estimateurs DAP, à gauche, et DOV, à droite, en fonction du nombre de prédictions +avec dropout N. +de lignes de texte), ce qui permet de calculer les statistiques. Comme les jeux de données +sont annotés, le modèle de détection est ensuite évalué sur chaque image séparément, ce qui +fournit un IoU et une mAP pour chaque image. Ces valeurs de mAP sont utilisées comme +cible pour l’entraînement des régresseurs. +Pour entraîner les modèles de régression, nous avons utilisé le RandomForestRegressor de +scikit-learn avec les paramètres par défaut. Les modèles de régression présentent de faibles +erreurs quadratiques moyennes (MSE) sur les ensembles de données d’entraînement (0,0164 +MSE sur l’ensemble de test de READ-BAD*). +6.3 +R É S U LTAT S E T D I S C U S S I O N +Dans cette section, nous évaluons et comparons les estimateurs de confiance à l’aide de +courbes de rejet, puis nous comparons leurs performances lorsqu’ils sont intégrés dans un +cadre d’apprentissage actif. +6.3.1 +nombre de prédictions avec dropout +Pour les expérimentations avec le dropout de Monte Carlo (DAP et DOV), nous devons +définir le nombre de prédictions N à calculer pour estimer la qualité des prédictions. La +Figure 6.2 montre la mAP en fonction du taux de rejet pour les estimateurs DAP et DOV +calculés pour différentes valeurs de N (2, 5, 10, 25 et 50). Nous avons choisi ces valeurs +car nous recherchons un ordre de grandeur de N plutôt qu’une valeur précise. L’idée est de +savoir si nous avons besoin d’un nombre important de prédictions pour obtenir une variance +suffisamment fiable, ou si quelques prédictions suffisent. De plus, nous ne sommes pas allés +au-delà de 50 prédictions car nous voulons garder un temps de calcul raisonnable. + +112 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S +Figure 6.3 – Courbes de rejet présentant l’évolution du score mAP en fonction du taux de rejet. +Les courbes présentent les résultats du modèle de détection de pages de référence sur +l’ensemble de test Horae-test-300. +Les résultats sont présentés sur Horae-test-300 pour la détection de pages. Les courbes +de rejet sont construites en ordonnant les images en fonction de leur confiance estimée, +les exemples ayant une valeur DAP inférieure (ou une valeur DOV supérieure) à un seuil +prédéfini sont retirés de l’ensemble d’évaluation et la mAP est calculée sur les exemples +restants. Pour DAP, le seuil varie de 0 à 1 avec un pas de 0,05. Pour DOV, les valeurs ne +sont pas bornées, le seuil varie donc de 10 à 0 avec un pas de -1. +Ces graphiques montrent que l’utilisation de N =10 prédictions pour l’estimation avec dro- +pout est suffisante et qu’aucune amélioration n’est observée avec N =25 ou N =50. De plus, +le coût de calcul est réduit avec seulement 10 prédictions. Sur la base de cette observation, +nous avons utilisé N =10 prédictions avec dropout pour estimer les scores de confiance dans +le reste des expériences. +6.3.2 +performances des estimateurs en rejet +Dans une première expérience, nous évaluons la capacité des estimateurs de confiance à +détecter les exemples mal prédits. Pour ce faire, nous évaluons les performances du modèle +de détection lorsque les images ayant le score de confiance estimé le plus faible sont retirées +de l’ensemble d’évaluation. Cette évaluation est réalisée grâce à des courbes de rejet. Sur les +courbes de rejet, chaque point correspond à un seuil pour lequel les images dont le score +estimé est inférieur à ce seuil sont retirées de l’évaluation. Les courbes n’atteignent pas 100 +% car, au-dessus d’un seuil donné, il reste uniquement des images ayant le même score, de +sorte qu’elles ne peuvent plus être retirées sans que l’ensemble d’évaluation soit vide. Par +souci de clarté, nous montrons seulement l’évolution de la mAP, les résultats d’IoU suivant +la même tendance. + +6.3 R É S U LTAT S E T D I S C U S S I O N +113 +La Figure 6.3 montre l’évolution des performances du modèle de référence sur Horae- +test-300 pour la tâche de détection de pages par rapport au taux de rejet pour différents +estimateurs de confiance. Nous montrons les courbes médianes ainsi que les intervalles de +confiance (10eet 90e percentiles) obtenus en calculant 100 courbes de rejet générées par 100 +ré-échantillonnages avec remplacement à partir de l’ensemble de test original. La courbe +aléatoire montre les résultats obtenus pour 100 échantillonnages aléatoires. +Notre objectif est d’avoir un modèle avec une mAP élevée et un faible taux de rejet. Nous +pouvons constater que les estimateurs basés sur le dropout ne sont pas compétitifs par rapport +au régresseur basé sur les statistiques. De plus, comme mAP-RFR ne nécessite qu’une seule +prédiction en inférence, ce premier résultat montre que l’utilisation de mAP-RFR au lieu du +dropout de Monte Carlo est plus intéressante. Les résultats de PCE étant comparables à ceux +de DAP et DOV, il semble que les estimateurs dropout de Monte Carlo ne fassent pas de +meilleurs indicateurs que les probabilités a posteriori des détecteurs. Cela peut s’expliquer +par le fait qu’aucune information supplémentaire à part les prédictions du réseau neuronal ne +soit fournie à ces trois estimateurs. Cette première expérience montre que notre proposition +mAP-RFR a une grande capacité à estimer la confiance des pages prédites. Il surpasse DAP +et DOV qui sont eux-mêmes à peine meilleurs que PCE. +Sur la Figure 6.1, nous montrons deux prédictions obtenues par le modèle de référence pour +la tâche de détection de pages. À gauche, nous montrons une bonne prédiction où la variance +est faible, sauf sur les bords des objets. Les estimations de confiance DOV=0,0, DAP=1,0 et +mAP-RFR=1,0 reflètent bien la bonne qualité de la détection de l’image de gauche tandis +que les estimations de confiance DOV=17,36, DAP=0,0993 et mAP-RFR=0,5553 de l’image +de droite reflètent également la très mauvaise qualité de la détection, qui contient un nombre +élevé de petits objets prédits autour du principal. +6.3.3 +apprentissage actif +Dans un cadre d’apprentissage actif, l’objectif est d’entraîner un bon détecteur d’objets +tout en minimisant la quantité d’exemples à annoter manuellement. Pour y parvenir, il est +crucial de bien choisir les données à annoter. +Dans nos expériences, nous suivons une configuration standard d’apprentissage actif (Cohn +et al., 1996). Tout d’abord, un modèle Doc-UFCN de référence est entraîné, puis appliqué +à des documents non vus et non annotés provenant d’un nouveau jeu de données. Ensuite, +ces images sont classées en fonction de leur confiance estimée, celles dont la confiance est +la plus faible sont sélectionnées pour une annotation manuelle et utilisées pour entraîner +un nouveau modèle. Bien que de nombreuses stratégies de sélection des données à annoter +aient été proposées pour améliorer au mieux les modèles (Settles et al., 2008), nous nous +concentrons, dans cette section, sur la sélection des images ayant la plus faible confiance. +Ainsi, les images ayant une confiance inférieure à un seuil prédéfini sont sélectionnées. Le +seuil varie d’une itération à l’autre en fonction de la distribution des confiances estimées. +Nous présentons une analyse de deux stratégies de sélection dans la section 6.4. + +114 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S +Table 6.4 – Résultats des modèles de détection de pages sur l’ensemble de test Horae-test-300 après +apprentissage actif. La colonne Itération indique le nombre d’itérations réalisées afin +d’obtenir le meilleur modèle. Le nombre d’images annotées est indiqué dans la colonne +Images. +Estimateur +Itération +Images +IoU +mAP +Référence +– +0 +0,90 +0,60 +Aléatoire +5 +300 +0,93 +0,86 +PCE +9 +90 +0,93 +0,86 +mAP-RFR +8 +107 +0,94 +0,89 +DAP +9 +129 +0,94 +0,91 +DOV +9 +168 +0,95 +0,92 +Figure 6.4 – Évolution des performances de détection de pages (mAP) sur l’ensemble de test Horae- +test-300 pendant les itérations d’apprentissage actif. +Chaque modèle de détection est entraîné dans la même configuration que les modèles de +référence décrits dans la section 6.2.2. Pendant les itérations d’apprentissage actif, plusieurs +stratégies d’initialisation des poids des modèles peuvent être envisagées. Ils peuvent être +initialisés avec les poids des derniers modèles entraînés, ceux du modèle de référence (à chaque +itération) ou encore ceux du meilleur modèle entraîné durant les itérations précédentes. Pour +nos expériences, nous initialisons les poids avec ceux des derniers modèles entraînés. +Enfin, pour les expériences suivantes, nous avons calculé un intervalle de confiance sur +les modèles de la dernière itération. Pour cela, nous avons utilisé le bootstrapping empirique +(Wasserman, 2004) avec 100 ré-échantillonnages avec remplacement. En outre, les expé- +riences avec la sélection aléatoire sont répétées cinq fois et les valeurs moyennes et les écarts +types sont présentés. +détection de pages +La Figure 6.4 et la Table 6.4 présentent les résultats obtenus pour la tâche de détection de +pages. À chaque itération, le modèle courant est appliqué aux images du corpus Horae. Les +images dont le score de confiance estimé est inférieur à un seuil sont annotées manuellement + +6.3 R É S U LTAT S E T D I S C U S S I O N +115 +et ajoutées à l’ensemble d’entraînement de l’itération précédente afin d’entraîner un nou- +veau modèle. Comme pour les courbes de rejet, ces graphiques montrent que les estimateurs +sont capables de détecter les mauvaises prédictions afin d’entraîner des modèles plus perfor- +mants, avec seulement une petite quantité de données annotées. En effet, les estimateurs sont +meilleurs qu’une sélection aléatoire puisqu’avec deux fois moins de données, les modèles pré- +sentent des augmentations relatives de 6 % de mAP (+5 points de pourcentage) pour DAP, +7 % (+6 points de pourcentage) pour DOV et presque 3,5 % (+3 points de pourcentage) +pour mAP-RFR. Sur la Figure 6.4, nous observons également que la courbe correspondant à +mAP-RFR est presque toujours supérieure à celles des autres estimateurs, ce qui indique des +modèles plus performants avec moins de données annotées. +Ces résultats montrent que l’estimateur mAP-RFR est plus performant que les estimateurs +basés sur le dropout de Monte Carlo puisqu’il présente une mAP plus élevée tout en ne +nécessitant qu’une seule prédiction pendant l’inférence et moins de données annotées. Une +explication possible à ces résultats, que nous avons déjà formulée précédemment, est que les +estimateurs DAP et DOV sont non supervisés : ils n’ont aucune connaissance préalable de +ce qu’est une prédiction correcte. Au contraire, mAP-RFR est entraîné avec les mAPs réelles +calculées sur les données annotées. +détection de lignes de texte +La Figure 6.5 et la Table 6.5 montrent les résultats obtenus avec les estimateurs mAP-RFR, +DAP et PCE pour l’apprentissage actif. Nous ne montrons pas les résultats de DOV car ils +sont équivalents à ceux de DAP. De plus, le WER n’est pas rapporté ici puisqu’il est fortement +corrélé au CER. +D’après la Figure 6.5, il apparaît que la sélection aléatoire donne de bons résultats avec +seulement 50 images. Cependant, ces résultats dépendent fortement des données choisies, ce +qui conduit à des performances très variables d’une sélection à l’autre. Par conséquent, au +vu de cette grande variabilité des résultats, nous pensons qu’il est préférable de se concentrer +sur un estimateur plus robuste et moins aléatoire qui peut obtenir des résultats tout aussi +satisfaisants. +D’après la Table 6.5, l’estimateur DAP se distingue des autres en obtenant des valeurs d’IoU +et de mAP plus faibles que les autres estimateurs mais un CER bien moins élevé. mAP-RFR +ne semble pas ici faire un meilleur estimateur que PCE ou que la sélection aléatoire. Malgré des +résultats bien moins bons en termes d’IoU et de mAP, DAP présente de meilleures valeurs de +CER et de WCFM par rapport à mAP-RFR. En effet, mAP-RFR a été conçu pour estimer +la mAP de chaque prédiction et ainsi maximiser la mAP des modèles. Cependant, nous +avons montré, en section 5.4.1, que la maximisation de la mAP ne signifie pas nécessairement +l’amélioration de l’entrée pour le reconnaisseur. +Pour cette tâche, il serait intéressant de sélectionner les images en fonction d’un score de +confiance lié à la reconnaissance de texte. Le modèle de détection s’adapterait pour améliorer +directement la reconnaissance du texte. + +116 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S +Table 6.5 – Résultats des modèles de détection de lignes de texte sur l’ensemble de test du jeu de +données Hugin-Munin après apprentissage actif. La colonne Itération indique le nombre +d’itérations réalisées afin d’obtenir le meilleur modèle. Le nombre d’images annotées est +indiqué dans la colonne Images. +Estimateur +Itération +Images +IoU +mAP +CER (%) +WCFM +Référence +– +0 +0,48 +0,21 +24,37 +0,59 +Aléatoire +1 +50 +0,63 +0,45 +22,18 +0,64 +PCE +6 +83 +0,67 +0,46 +22,79 +0,66 +mAP-RFR +9 +139 +0,64 +0,44 +22,50 +0,66 +DAP +6 +110 +0,63 +0,40 +20,23 +0,68 +Figure 6.5 – Évolution des performances de détection de lignes de texte (mAP) sur l’ensemble de test +du jeu de données Hugin-Munin pendant les itérations d’apprentissage actif. +6.4 +S T R AT É G I E D’ E N T R A Î N E M E N T : S É L E C T I O N E T A N N O TAT I O N D E S D O N N É E S +Dans le cadre de l’apprentissage actif, de nombreuses stratégies d’entraînement peuvent +être exploitées. Bien que de nombreuses méthodes aient été proposées dans la littérature, +aucune ne semble réellement surpasser les autres. C’est pourquoi, dans cette section, nous +étudions deux stratégies d’entraînement qui concernent la sélection des données ainsi que leur +annotation. +La première est la même que celle utilisée dans les expériences précédentes : les exemples +avec les confiances estimées les plus faibles sont annotés manuellement puis ajoutés à l’en- +semble d’entraînement. La seconde stratégie sélectionne les exemples avec les confiances les +plus élevées et utilise les prédictions du modèle de détection comme labels pour les entraî- +nements suivants. Cette stratégie de sélection permet de réduire le coût d’annotation ma- +nuelle au minimum puisqu’aucune donnée n’est annotée manuellement. Nous présentons tout +d’abord les résultats pour la détection de pages, en section 6.4.1, puis pour la détection de +lignes de texte, en section 6.4.2. Pour l’ensemble des résultats présentés dans ce qui suit, les +modèles sont entraînés dans les mêmes conditions que dans les expériences précédentes. + +6.4 S T R AT É G I E D’ E N T R A Î N E M E N T : S É L E C T I O N E T A N N O TAT I O N D E S D O N N É E S +117 +Table 6.6 – Résultats des modèles de détection de pages sur l’ensemble de test Horae-test-300 après +apprentissage actif et pour différentes stratégies de sélection de données. La colonne Ité- +ration indique le nombre d’itérations réalisées afin d’obtenir le meilleur modèle. La sélec- +tion "Faible" correspond à la sélection des images avec les confiances les plus faibles. La +sélection "Élevée" correspond à la sélection des images avec les confiances les plus élevées +où leurs prédictions sont directement utilisées comme labels d’entraînement. Les colonnes +"Manuelle" et "Auto." indiquent respectivement les nombres d’images d’entraînement +avec annotations manuelles et automatiques permettant d’obtenir le meilleur modèle. +Estimateur +Sélection +Itération +Images +IoU +mAP +Manuelle +Auto. +Référence +– +– +– +– +0,90 +0,60 +Faible +8 +107 +– +0,94 +0,89 +mAP-RFR +Élevée +9 +– +444 +0,90 +0,84 +Faible +9 +129 +– +0,94 +0,91 +DAP +Élevée +8 +– +475 +0,90 +0,72 +Faible +9 +168 +– +0,95 +0,92 +DOV +Élevée +3 +– +163 +0,90 +0,64 +Figure 6.6 – Évolution des performances de détection de pages (mAP) sur l’ensemble de test Horae- +test-300 pendant les itérations d’apprentissage actif pour différentes stratégies de sélec- +tion de données. Les courbes "Manuelle" correspondent à la sélection des exemples avec +les confiances les plus faibles et une annotation manuelle de ces exemples. Les courbes +"Automatique" correspondent à la sélection des exemples avec les confiances les plus +élevées et l’utilisation des prédictions comme labels d’entraînement. +6.4.1 +détection de pages +Les Figure 6.6 et Table 6.6 présentent les résultats obtenus pour les deux stratégies de +sélection pour la tâche de détection de pages. Les courbes et valeurs correspondant à la +sélection basée sur les faibles confiances sont les mêmes que celles présentées en section +6.3.3. D’après la Figure 6.6, l’utilisation des prédictions comme labels d’entraînement pour +l’estimateur DOV ne permet pas réellement d’améliorer le modèle de détection par rapport +au modèle de référence. Au contraire, pour les estimateurs DAP et mAP-RFR, l’utilisation +des prédictions permet une importante amélioration des performances par rapport au modèle +de référence. En effet, le modèle mAP-RFR permet une amélioration de 40 % de mAP (+24 + +118 +E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S +Table 6.7 – Résultats des modèles de détection de lignes de texte sur l’ensemble de test du jeu de +données Hugin-Munin après apprentissage actif et pour différentes stratégies de sélection +de données. La colonne Iteration indique le nombre d’itérations réalisées afin d’obtenir +le meilleur modèle. La sélection "Faible" correspond à la sélection des images avec les +confiances les plus faibles. La sélection "Élevée" correspond à la sélection des images +avec les confiances les plus élevées où leurs prédictions sont directement utilisées comme +labels d’entraînement. Les colonnes "Manuelle" et "Auto." indiquent respectivement les +nombres d’images d’entraînement avec annotations manuelles et automatiques permettant +d’obtenir le meilleur modèle. +Estimateur +Sélection +Iteration +Images +IoU +mAP +CER (%) +WCFM +Manuelle +Auto. +Référence +– +– +– +– +0,48 +0,21 +24,37 +0,59 +Faible +9 +139 +– +0,64 +0,44 +22,50 +0,66 +mAP-RFR +Élevée +4 +– +54 +0,53 +0,28 +22,89 +0,62 +Faible +6 +110 +– +0,63 +0,40 +20,23 +0,68 +DAP +Élevée +1 +– +38 +0,51 +0,26 +21,98 +0,63 +Figure 6.7 – Évolution des performances de détection de lignes de texte (mAP) sur l’ensemble de +test du jeu de données Hugin-Munin pendant les itérations d’apprentissage actif pour +différentes stratégies de sélection de données. Les courbes "Manuelle" correspondent à la +sélection des exemples avec les confiances les plus faibles et une annotation manuelle de +ces exemples. Les courbes "Automatique" correspondent à la sélection des exemples avec +les confiances les plus élevées et l’utilisation des prédictions comme labels d’entraînement. +points de pourcentage) et le modèle DAP de 20 % (+12 points de pourcentage) par rapport +au modèle de référence, sans aucune donnée annotée manuellement. +Le modèle obtenu avec l’estimateur DAP et les données annotées automatiquement est +tout de même bien moins performant que celui obtenu avec les données annotées manuelle- +ment (-21 % de mAP). Pour l’estimateur mAP-RFR, l’écart est moins important puisque les +performances sont dégradées de seulement 5,5 % de mAP en passant des données annotées +manuellement aux labels automatiques. Ainsi, nous constatons que l’estimateur mAP-RFR +est le plus robuste pour cette tâche puisqu’il permet de détecter de manière fiable les bonnes +ainsi que les mauvaises prédictions. Cet estimateur permet d’obtenir l’amélioration de perfor- +mance la plus intéressante sans annotation manuelle. + +6.5 C O N C L U S I O N +119 +6.4.2 +détection de lignes de texte +Les Figure 6.7 et Table 6.7 présentent les résultats obtenus pour les deux stratégies de +sélection pour la tâche de détection de lignes de texte. Les courbes et valeurs correspondant +à la sélection basée sur les faibles confiances sont les mêmes que celles présentées en section +6.3.3. D’après la Table 6.7, l’utilisation des prédictions comme labels d’entraînement mène à +une légère amélioration des performances par rapport au modèle de référence (-6 % de CER +pour mAP-RFR et -11 % de CER pour DAP). Ceci permet de valider l’utilité des estimateurs +mAP-RFR et DAP dans la sélection des mauvais autant que des bons exemples. +Comme pour la détection de pages, les modèles obtenus avec les données annotées automa- +tiquement sont moins performants que ceux obtenus avec les données annotées manuellement +(+1,7 % de CER pour mAP-RFR et +8,7 % de CER pour DAP). +6.5 +C O N C L U S I O N +Dans ce chapitre, nous avons comparé quatre estimateurs de confiance pour les modèles +de détection d’objets. Nous avons montré que, dans un contexte d’apprentissage actif, ces +estimateurs peuvent être utilisés pour entraîner des modèles atteignant des performances +élevées pour la détection d’objets en termes d’IoU et de mAP tout en ne nécessitant qu’un +faible effort d’annotation manuelle. Lorsque les métriques optimisées sont étroitement liées à +l’objectif, comme pour la mAP et la détection de pages, nous avons montré que l’estimateur +mAP-RFR permet d’obtenir de meilleures performances de détection que celles basées sur le +dropout de Monte Carlo, tout en ayant un coût de calcul réduit. Cependant, cet estimateur +est supervisé et doit être entraîné, ce qui n’est pas le cas pour DAP, DOV et PCE. Dans +le cas d’une adaptation à de nouvelles données, il est donc avantageux, dans un premier +temps, d’utiliser l’estimateur DAP basé sur le dropout. Si les résultats n’atteignent pas les +performances attendues, il semble alors plus intéressant d’utiliser un estimateur entraîné tel +que mAP-RFR. D’autre part, lorsque les métriques sont moins étroitement liées à l’objectif, +comme pour la détection des lignes de texte, les méthodes basées sur le dropout sont plus +compétitives. +À l’avenir, nous envisageons d’adapter l’estimateur mAP-RFR afin qu’il estime la confiance +au niveau de l’objet directement de façon à ne plus rejeter les images mais les objets. Cela +permettrait de savoir exactement quels objets posent un problème et de les corriger. De plus, +il serait intéressant de créer automatiquement des vecteurs de description d’objets à travers +des représentations apprises. Enfin, nous avons montré que l’utilisation de métriques orientées +vers la tâche finale permet d’évaluer l’impact des modèles de détection sur les résultats finaux. +Il semblerait donc intéressant de sélectionner les images ou les objets en se basant sur les +résultats de reconnaissance de texte. Dans cette optique, nous prévoyons de mettre en place +un nouvel estimateur qui reflète les résultats de la reconnaissance de texte. + + +7 +D É T E C T I O N S É Q U E N T I E L L E D ’ O B J E T S D A N S D E S +I M A G E S D E D O C U M E N T S +Les systèmes à base de Transformers proposés récemment, et détaillés dans le Focus 2.11, +obtiennent désormais les meilleures performances de l’état de l’art tant sur des tâches de +traitement de la langue que des tâches de classification d’images. Un de leurs avantages réside +dans leur capacité à modéliser et à générer des séquences et même des objets structurés. Il +semble désormais possible de prédire automatiquement la structure complète d’une image +de document, avec l’ensemble de ses éléments organisés de manière hiérarchique. De plus, +ces systèmes sont capables de prédire séquentiellement les coordonnées des objets à détecter +(Chen et al., 2022), sans avoir à passer par une prédiction pixel à pixel. Bien qu’il n’y ait +pas, à notre connaissance, de travaux proposés dans la littérature afin de réaliser une telle +tâche, une prédiction directe de coordonnées présente de nombreux avantages comparée à +une prédiction standard niveau pixel. C’est pourquoi, dans ce chapitre, nous avons choisi +d’explorer les modèles Transformers pour construire un nouveau modèle de détection +séquentielle d’objets dans les images de documents. +Un premier point ayant motivé nos travaux dans ce sens est lié à la capacité d’un tel +système à passer outre les problèmes liés aux boîtes englobantes qui se touchent et se +superposent. En effet, le modèle n’est pas appris avec des images de labels mais directement +avec les coordonnées des éléments à détecter, telles que les boîtes englobantes ou les lignes de +base. Similairement aux approches par régression de boîtes englobantes, il devient possible +de détecter plusieurs objets d’une même classe au même endroit sur l’image. Un autre +avantage de cette approche tient au fait qu’elle permet d’apprendre un ordre de lecture +implicitement représenté par la séquentialité du processus de détection des éléments. Le +modèle est, en effet, entraîné à détecter les éléments dans l’ordre imposé par la séquence des +objets représentés dans la vérité terrain. Cette séquentialité de la vérité terrain définit donc +un ordre de détection, et donc de lecture des objets présents dans l’image. Resitué dans +le contexte de la reconnaissance de documents, il devient possible d’apprendre à prédire +les lignes de texte dans l’ordre de lecture du texte. Enfin, comme énoncé plus tôt, cette +approche permet d’avoir une sortie structurée des résultats. Si nous imaginons un problème +à deux classes telles que les paragraphes et les lignes de texte, il est possible d’apprendre +un modèle qui détecte le début d’un paragraphe, toutes les lignes qu’il contient puis la fin +de ce paragraphe avant de passer au suivant. Nous pouvons donc obtenir directement une +détection hiérarchique des éléments sur une image. +121 + +122 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +La plupart des systèmes proposés traitant la détection d’objets prédisent un masque de +probabilités à la résolution de l’image d’entrée. Bien que cette tâche de détection puisse +être traitée à l’aide de Vision Transformers (voir le Focus 2.14), qui utilisent un encodeur +Transformer suivi d’un décodeur CNN, elles ne bénéficient pas de l’avantage principal des +Transformers qui réside en leur capacité à prédire des éléments de manière séquentielle, capa- +cité induite par le décodeur. De même, dans le domaine de la vision, la plupart des travaux +proposés dans la littérature ont exploré les architectures Transformers pour constituer de nou- +veaux extracteurs de caractéristiques, des encodeurs Transformers, et ainsi tenter d’améliorer +les architectures convolutives. D’autres travaux prédisent un nombre fixe d’objets (Carion +et al., 2020) ou utilisent la sortie du décodeur comme entrée d’un CNN afin d’avoir une +prédiction dense (Zheng et al., 2020). Dans ce domaine, la séquentialité du processus de +décision n’est pas non plus exploitée. +Les architectures Transformers, et plus particulièrement les décodeurs Transformers, fonc- +tionnent selon un nouveau paradigme qui traite un élément en entrée séquentiellement, au +rythme d’une séquence d’attention visuelle. Cela nécessite donc de repenser le type de sorties +attendues qui doivent nécessairement être structurées sous forme d’une séquence d’objets. Il +semble complexe de réaliser une prédiction pixel à pixel de manière séquentielle puisqu’elle +induirait des temps et coût de traitement très élevés. Cependant, l’application de ce para- +digme pour résoudre un problème d’analyse de document est relativement directe puisque, +dans la plupart des applications, les sorties du modèle de détection et de reconnaissance +ont besoin d’être organisées dans l’ordre naturel de lecture. La tâche de détection doit donc +être reformulée afin de profiter pleinement de la capacité de prédiction séquentielle de ces +nouvelles architectures. C’est pourquoi, nous présentons, dans la section 7.1, une étude et +comparaison de différentes modélisations du problème de détection d’objets permettant une +prédiction séquentielle. +Comme nous venons de l’évoquer, très peu de systèmes ont été proposés dans la littérature +permettant de prédire séquentiellement les objets présents dans des images. Le seul modèle +réalisant une telle tâche est Pix2Seq (Chen et al., 2022), détaillé dans le Focus 2.15, appliqué +aux images de scènes naturelles. Ce système possédant un grand nombre de paramètres et +nécessitant un pré-entraînement sur des milliers d’images, il n’est pas directement applicable +à nos jeux de données réduits d’images de documents. Ainsi, inspiré par Pix2Seq, nous nous +sommes intéressés à la mise en place d’un système permettant de prédire séquentiellement +les objets tout en possédant un nombre réduit de paramètres afin d’être entraîné sur des jeux +de données réduits. Ce système est détaillé dans la section 7.2. +7.1 +M O D É L I S AT I O N D E L A TÂ C H E D E D É T E C T I O N +Dans cette section, nous présentons et comparons différentes modélisations possibles de la +tâche de détection d’objets. Dans un premier temps, nous comparons plusieurs modélisations +de la position et de la forme des objets. Nous comparons ensuite plusieurs stratégies de +prédiction des coordonnées. Enfin, nous présentons la modélisation des classes des objets que +nous avons retenue. + +7.1 M O D É L I S AT I O N D E L A TÂ C H E D E D É T E C T I O N +123 +(x1, y1) +(x2, y2) +Rectangle +englobant +(x1, y1) +Point +d’origine +(x1, y1) +h +Position d´ebut ++ hauteur +(x1, y1) (x2, y2) +(xN, yN) +Ligne +de base +(x1, y1) (x2, y2) +(xN, yN) +Polygone +englobant +(x1, y1) (x2, y2) +(xN, yN) +h +Ligne de base ++ hauteur +Figure 7.1 – Représentation de différentes modélisations de la position et de la forme des objets à +détecter. Exemple pour la détection d’une ligne de texte. +7.1.1 +modélisation de la position et de la forme des objets +Afin de réaliser une prédiction séquentielle des éléments présents sur une image de +document, il est nécessaire de passer d’une prédiction pixel à une prédiction de plus haut +niveau, au niveau de l’objet. Pour cela, différentes formulations du problème de détection +d’objets ont été proposées dans la littérature. Nous les présentons, dans le cadre d’une +détection de lignes de texte, sur la Figure 7.1 et détaillons leurs caractéristiques dans la +Table 7.1. +Dans le domaine de la vision, la grande majorité des systèmes tels que les modèles R- +CNN (Girshick, 2015 ; +Girshick et al., 2014 ; +Ren et al., 2015) et YOLO (Redmon +et al., 2016 ; 2017 ; 2018) définissent les objets à détecter par leur rectangle englobant. C’est +également le cas de Pix2Seq (Chen et al., 2022) qui prédit la séquence suivante : ordonnée du +point supérieur gauche, abscisse du point supérieur gauche, ordonnée du point inférieur droit, +abscisse du point inférieur droit et classe de l’objet. Cette détection permet une extraction +directe de l’objet mais n’est pas correctement applicable à des éléments non rectangulaires. +Lors de la compétition ANDAR-TL de détection de lignes de texte (Murdock et al., +2015), la tâche de détection correspond à l’identification des points d’origine des lignes de +texte, à savoir la ligne de base du premier caractère d’une ligne. D’un autre côté, dans +Moysset et al. (2017), les auteurs proposent une localisation des lignes de texte basée +sur des régressions dans lesquelles seules les positions du début des lignes de texte et leurs +hauteurs sont prédites. Le reconnaisseur de texte est alors chargé de reconnaître le texte de + +124 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Table 7.1 – Tableau récapitulatif de différentes modélisations de la position et forme des objets à +détecter. La colonne Prédiction indique les valeurs à prédire pour un objet. La colonne +"Extraction directe" indique si l’objet peut directement être extrait en sortie du détecteur +ou si des traitements supplémentaires sont nécessaires tels que l’estimation de la largeur +et/ou de la hauteur de l’objet. La colonne "Optimisation mémoire" indique si la quantité +de mémoire nécessaire pour prédire un objet est importante ou non, cette quantité étant +directement corrélée au nombre de valeurs à prédire. La dernière colonne indique si la +détection est applicable à des objets non rectangulaires ainsi qu’à des lignes inclinées ou +incurvées. +Modélisation +Prédiction +Extraction +Optimisation +Objets non- +directe +mémoire +rectangulaires +Rectangle englobant +(x1, y1, x2, y2) +Pix2Seq (Chen et al. (2022)) +→ 4 valeurs +✓ +✓ +✗ +Point d’origine +(x1, y1) +(Murdock et al., 2015) +→ 2 valeurs +✗ +✓ +✗ +Position du début + hauteur +(x1, y1, h) +(Moysset et al., 2017) +→ 3 valeurs +✗ +✓ +✗ +Ligne de base +(x1, y1, ...xN, yN) +(Diem et al., 2019) +→ N valeurs +✗ +✗ +✓ +(x1, y1, ...xN, yN) +Polygone englobant +→ N valeurs +✓ +✗ +✓ +(x1, y1, ...xN, yN, h) +Ligne de base + hauteur +→ N+1 valeurs +✗ +✗ +✓ +la ligne et de s’arrêter lorsqu’il n’y a plus de texte à reconnaître. Ces propositions permettent +d’envisager une détection optimisée des éléments puisque, pour chaque objet, seules deux +ou trois valeurs sont à prédire. Cependant, elles ne permettent pas une détection complète +de l’objet puisque la largeur est inconnue. Il serait donc nécessaire d’avoir des traitements +supplémentaires afin d’extraire les objets de l’image. Sans cela, il serait impossible d’appliquer +un reconnaisseur de texte standard sur les lignes de texte par exemple. +Enfin, la détection basée sur les lignes de base (Diem et al., 2017 ; Diem et al., 2019) ou +les polygones englobants présente l’avantage de localiser précisément les contours d’objets +rectangulaires ou non, tels que des lignes inclinées et incurvées. Cependant, ces propositions +sont très coûteuses en mémoire puisque le système doit prédire un nombre de points inconnu +à l’avance, et qui peut-être très variable d’un objet à l’autre, en fonction de la taille et de la +forme des objets à localiser. +Cette représentation pose également un problème de paramétrage du Transformer. En +effet, dans un Transformer, la taille maximale de la séquence pouvant être générée pour une +image est fixée durant la phase d’entraînement afin de réduire la mémoire utilisée. Durant +l’inférence, il est donc impossible de prédire plus de valeurs que cette limite. Bien que +celle-ci puisse être fixée à plusieurs milliers de valeurs, il est toujours possible de rencontrer +un document avec un très grand nombre d’objets, menant à une séquence plus longue. +Dans le cas d’une détection de rectangles englobants, cette limite est facile à fixer puisque +seules quatre valeurs sont à prédire pour chaque objet. Ainsi, le problème est réduit à + +7.1 M O D É L I S AT I O N D E L A TÂ C H E D E D É T E C T I O N +125 +Table 7.2 – Stratégies de prédiction séquentielle des rectangles englobants. Pour un rectangle donné +i de coordonnées (xi +0, yi +0, xi +1, yi +1), à chaque pas t, une coordonnée unique, un point ou le +rectangle complet peut être prédit. +Séquence +t = 0 +t = 1 +t = 2 +t = 3 +t = 4 +t = 5 +t = 6 +... +Coordonnée +x0 +0 +y0 +0 +x0 +1 +y0 +1 +x1 +0 +y1 +0 +x1 +1 +... +Point +(x0 +0, y0 +0) +(x0 +1, y0 +1) +(x1 +0, y1 +0) +(x1 +1, y1 +1) +... +Rectangle +(x0 +0, y0 +0, x0 +1, y0 +1) +(x1 +0, y1 +0, x1 +1, y1 +1) +... +quelques dizaines de valeurs à prédire par image. Le problème se complexifie lorsque nous +souhaitons prédire des polygones plus précis puisque nous ignorons à l’avance combien +de valeurs sont nécessaires pour prédire chaque objet. Il serait possible de simplifier les +polygones englobants afin de fixer le nombre de coordonnées les définissant, cependant, cela +mènerait à un traitement supplémentaire et à une perte de précision. De plus, de nombreuses +questions en découlent telles que le nombre de points à utiliser pour décrire un polygone an- +noté, leurs espacements, l’évaluation des points obtenus pour le calcul de la fonction de perte. +Pour toutes ces raisons, nous pensons que la détection des rectangles englobants par la +prédiction du point supérieur gauche et du point inférieur droit, semblable à Pix2Seq, repré- +sente un bon compromis entre performance et précision de la localisation. Cette formulation +est assez simple et rapide à appliquer, et peut permettre à un système d’apprendre malgré +la quantité relativement faible de données annotées. En effet, plus le système doit prédire de +points, plus il sera en difficultés et nécessitera un grand nombre de données d’apprentissage. +7.1.2 +stratégie de prédiction des coordonnées : singleton vs n-uplet +La modélisation de la tâche de détection explicitement définie, il est maintenant nécessaire +de choisir la stratégie de prédiction des rectangles englobants. En effet, le système devra être +capable de prédire séquentiellement les coordonnées des rectangles englobants. Cependant, +la Table 7.2 présente trois stratégies différentes afin de réaliser cette tâche. Ainsi, pour un +rectangle donné i de coordonnées (xi +0, yi +0, xi +1, yi +1) avec (xi +0, yi +0) les coordonnées du point +supérieur gauche et (xi +1, yi +1) les coordonnées du point inférieur droit, il est possible de prédire, +à chaque pas de la séquence : +— Un singleton correspondant à une coordonnée d’un des deux points du rectangle ; +— Un couple de valeurs correspondant à un des deux points du rectangle ; +— Un quadruplet correspondant aux coordonnées du rectangle complet. +La première stratégie, qui consiste à prédire un singleton à chaque pas dans la séquence, +permet d’avoir un modèle possédant quelques paramètres en moins par rapport à la prédic- +tion de couples qui elle-même nécessite moins de paramètres que la prédiction de quadruplets. +En effet, la dernière couche du modèle produisant les coordonnées finales sera différente d’une +stratégie à l’autre. Cependant, la prédiction de singletons requiert davantage d’itérations puis- + +126 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +qu’elle nécessite deux fois plus d’itérations que la prédiction de couples, elle-même nécessitant +deux fois plus d’itérations que la prédiction du quadruplet. +Dans Pix2Seq, les auteurs ont choisi de traiter cette tâche de telle sorte qu’à chaque pas, +une seule valeur de coordonnée est prédite. Ainsi, quatre prédictions sont nécessaires afin de +prédire un objet. Dans la suite de ce chapitre, nous décidons d’adopter la même stratégie. +7.1.3 +stratégie de prédiction des coordonnées : classification vs régres- +sion +Une autre stratégie à étudier dans la conception du modèle concerne le type de prédiction +souhaité. En effet, la prédiction de coordonnées peut être réalisée de deux manières. La +première consiste à réaliser une régression où le but est de prédire une coordonnée de la +boîte sur l’image. La seconde consiste à considérer chaque pixel de l’image d’entrée comme +étant une classe distincte et à réaliser une classification. Le but est alors de maximiser les +probabilités de la classe correspondant à la coordonnée de la boîte dans l’image. +L’avantage de la régression est qu’elle nécessite légèrement moins de paramètres que la clas- +sification puisqu’une seule valeur est produite par le modèle, directement considérée comme +la coordonnée finale. Ainsi, la dernière couche du modèle ne produira qu’une seule valeur. +Cependant, les valeurs prédites ne sont pas bornées, il est donc possible que le modèle pré- +dise des valeurs en dehors de l’image. Pour pallier ce problème, les coordonnées peuvent être +normalisées, ce qui permet également d’obtenir une cohérence des labels entre les images +pouvant être de tailles variables. +La classification est quant à elle plus simple à mettre en œuvre. Dans Pix2Seq, les auteurs +choisissent de traiter les images ainsi, en redimensionnant les images dans une taille fixe et +en considérant une classe pour chaque valeur possible en abscisse et en ordonnée. Dans un +premier temps, nous avons choisi d’utiliser cette même stratégie. +Dans Chen et al. (2022), les auteurs considèrent les classes permettant de représenter les +positions des objets en abscisse et en ordonnée comme appartenant à un "vocabulaire". Cela +leur permet de distinguer les positions des objets des "classes", utilisées pour représenter les +classes des objets à détecter telles que, dans leur application, les classes "chaise" ou "voiture". +Nous utilisons ces mêmes termes dans la suite de ce chapitre. Dans Pix2Seq, les auteurs +utilisent un vocabulaire partagé pour les deux axes et pour toutes les classes de position, la +taille du vocabulaire est donc égale au nombre maximal de pixels sur les deux axes. Pour une +image de taille 600×600 pixels, le vocabulaire a donc une taille de 600. De la même manière, +nous disposons, dans nos expériences, d’un vocabulaire V de taille TV = max(H, W) avec +H et W respectivement les hauteur et largeur de l’image d’entrée. +7.1.4 +stratégie de prédiction de la classe des objets +En plus des coordonnées des objets présents dans les images, il est nécessaire de prédire leurs +classes. En effet, dans les chapitres précédents, nous avons principalement abordé la tâche +de détection de lignes de texte uniquement, ainsi une seule classe est à prédire. Cependant, + +7.1 M O D É L I S AT I O N D E L A TÂ C H E D E D É T E C T I O N +127 +Ordre de pr´ediction : +y0 x0 cp +0 +y0 x0 cl +0 +y1 x1 cl +1 +y0 x0 cl +0 +y1 x1 cl +1 +y0 x0 cl +0 +y1 x1 cl +1 +y1 x1 cp +1 +eos +Figure 7.2 – Exemple de séquence à deux classes : paragraphe et ligne de texte. L’ordre de prédiction +préserve la hiérarchie des objets : point supérieur gauche du paragraphe, point supérieur +gauche de la première ligne de texte, point inférieur droit de la première ligne de texte, +..., point inférieur droit du paragraphe, fin de séquence (eos). +certaines tâches plus complexes considèrent davantage de classes d’objets qui doivent être +prédites par le modèle. +Ces classes peuvent être représentées de différentes manières. Ainsi, dans Pix2Seq, le +modèle prédit la classe de l’objet après les quatre coordonnées du rectangle englobant. +Chaque objet est donc défini par cinq prédictions successives. Cette représentation ne permet +cependant pas de représenter la hiérarchie des objets présents sur une image. +Afin de représenter au mieux ces informations, nous proposons de représenter un objet par +ses deux points supérieur gauche et inférieur droit avec, pour chacun de ces points, une classe +indiquant s’il s’agit du premier ou du second point de l’objet. Cette représentation permet +au modèle d’apprendre, en plus de l’ordre de lecture, la hiérarchie des objets. La Figure +7.2 présente un exemple de séquence construite pour deux classes d’objets : paragraphe et +ligne de texte. Chaque ligne représente un point avec son ordonnée, son abscisse et la classe +correspondante. Pour chaque classe d’objet, deux classes sont définies : une classe indiquant +le début de l’objet et une classe indiquant la fin. Ainsi, un objet paragraphe est défini par +deux points, le premier avec la classe cp +0 et le second avec la classe cp +1. De la même manière, +les lignes de texte sont définies par les classes cl +0 et cl +1. +Cette représentation permet également de reconstruire les boîtes englobantes de manière +plus fiable, même dans le cas d’une prédiction manquante ou supplémentaire. En effet, si le +modèle prédit deux points de début de ligne à la suite, lors de la reconstruction des objets, +un des deux points devra être mis de côté. Sans ces indicateurs de début et de fin d’objet, +il serait impossible de détecter ce phénomène. Le processus étant séquentiel, l’ensemble des +boîtes reconstruites après le point erroné seraient fausses. + +0 +Sentence Database +M04-251 +100 +ThmauSem,cndRe.iunds +200 - +locofal +300 +400 - +009 +600 + 002 +0 +100 +200 +300 +400 +500128 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +En conclusion, nous avons choisi de représenter un objet par les deux séquences "y0, x0, +c0" et "y1, x1, c1" avec : +— y0 et y1, respectivement les ordonnées des points supérieur gauche et inférieur droit du +rectangle englobant de l’objet ; +— x0 et x1, respectivement les abscisses des points supérieur gauche et inférieur droit du +rectangle englobant de l’objet ; +— c0 et c1, les jetons de début et de fin de la classe de l’objet. +7.2 +A R C H I T E C T U R E D U S Y S T È M E P R O P O S É : D O C2 S E Q +Dans cette section, nous présentons le modèle que nous avons développé, appelé Doc2Seq. +Nous détaillons l’architecture ainsi que les choix que nous avons faits lors de sa conception. +Très peu de modèles ont été proposés pour la détection séquentielle d’objets dans les images. +Seul Pix2Seq (Chen et al., 2022) a été proposé, appliqué aux images de scènes naturelles. +Il s’agit d’un modèle comportant un très grand nombre de paramètres (341 millions pour +le meilleur modèle) qui montre des résultats satisfaisants lorsqu’il est pré-entraîné sur des +milliers d’images. Or, nous ne disposons pas d’une telle quantité d’images annotées. C’est +pourquoi, il est nécessaire que notre système comporte moins de paramètres afin de pouvoir +être entraîné sur les jeux de données d’images de documents. C’est dans cet objectif que +nous avons conçu Doc2Seq, dont l’architecture est présentée en Figure 7.3. Il s’agit d’un +modèle hybride encodeur-décodeur où l’encodeur extrait les caractéristiques importantes de +l’image d’entrée et le décodeur prédit séquentiellement les éléments à partir de l’image encodée +et des prédictions précédentes. L’encodeur génère une matrice de caractéristiques 2D de +l’image d’entrée. Un encodage positionnel 2D est additionné à cette matrice afin de conserver +l’information spatiale, avant d’être aplani en une séquence 1D de caractéristiques. Comme +pour un FCN, cette représentation est calculée une seule fois et sert d’entrée au décodeur. Le +décodeur suit un processus récurrent : étant donné l’image encodée et les éléments prédits +précédemment ( ˆy0, ˆy1, ..., ˆyt−1), il produit les caractéristiques de l’élément suivant. Enfin, la +branche de classification produit des probabilités à partir de la sortie du décodeur et l’élément +prédit ˆyt est celui qui a la plus forte probabilité. Chacun de ces composants est détaillé dans +les paragraphes suivants. +7.2.1 +encodeur doc-ufcn +L’encodeur de Doc2Seq est identique à l’encodeur de Doc-UFCN présenté dans le chapitre +4. Il est donc composé de quatre blocs dilatés comportant chacun cinq convolutions dilatées +consécutives. Chaque bloc est suivi d’une couche de max-pooling, sauf le dernier. +Nous avons choisi d’utiliser l’encodeur de Doc-UFCN car, d’après les expériences présentées +précédemment, il a démontré de bonnes capacités d’extraction de caractéristiques sur les +images tout en possédant un nombre réduit de paramètres, ce qui nous permet d’entraîner le + +7.2 A R C H I T E C T U R E D U S Y S T È M E P R O P O S É : D O C2 S E Q +129 +W +H +Doc-UFCN +Encoder +8f +W +8 +H +8 ++ +8f +W +8 +H +8 +2D positional +encoding +H +8 × W +8 +8f +Image +features +Flatten ++ transpose +ˆy0 ˆy1 +... +ˆyt−1 +t − 1 +8f ++ +t − 1 +8f +1D positional +encoding +t − 1 +8f +Embedding +Masked +Self-Attention +Add & Norm +Multi-Head +Attention +Add & Norm +Feed Forward +Add & Norm +4× +8f +C +ˆyt +Linear +Argmax +Figure 7.3 – Schéma de l’architecture du modèle Doc2Seq avec respectivement H et W les hauteur et +largeur de l’image d’entrée, f le nombre de cartes de caractéristiques et ˆyi les prédictions. +système complet sur des jeux de données restreints. De plus, nous avons opté pour un FCN +comme encodeur car ces modèles peuvent traiter des entrées de tailles variables. +Certains systèmes tels que les Vision Transformers remplacent les encodeurs convolutifs +par des encodeurs Transformer. Ces systèmes sont appliqués sur des patchs d’image à la +résolution originale projetés dans la dimension dmodel. Bien que cet encodeur ait montré +de légèrement meilleures performances dans Chen et al. (2022) par rapport à un encodeur +CNN, il augmente significativement le nombre de paramètres et nécessite donc une plus +grande quantité de données d’entraînement. +Dans nos expériences, l’encodeur prend en entrée une image de document de taille (H × +W × 3) avec H la hauteur, W la largeur et 3 le nombre de canaux de l’image. Il produit une +matrice de caractéristiques de taille ( H +8 × W +8 × 8f) avec f = 32 comme pour Doc-UFCN. + +130 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +7.2.2 +encodage positionnel 2d +Une fois l’image encodée, sa matrice de caractéristiques est additionnée à une matrice de +codage positionnel 2D afin de garder en mémoire à quelle partie de l’image chaque pixel +correspond. Le Transformer original a été conçu pour traiter des séquences en entrée en +1D. Pour lui donner des représentations 2D en entrée, il suffit de transformer les cartes de +caractéristiques en les sérialisant ligne par ligne. Cependant, il est important d’associer à ces +représentations un encodage positionnel cohérent avec l’information originale, c’est-à-dire un +encodage 2D. C’est ce que nous avons réalisé comme proposé par Singh et al. (2021). Ainsi, il +s’agit toujours d’un codage fixe basé sur les fonctions cosinus et sinus, mais, au lieu de coder +une position 1D sur tous les canaux, la première moitié est dédiée à l’encodage positionnel +vertical et la seconde à l’encodage positionnel horizontal (voir équations 7.1). +PE(posx, posy, 2i) = sin(wi · posy) ∀i ∈ +� +0, dmodel +4 +� +PE(posx, posy, 2i + 1) = cos(wi · posy) ∀i ∈ +� +0, dmodel +4 +� +PE(posx, posy, dmodel +2 ++ 2i) = sin(wi · posx) ∀i ∈ +� +0, dmodel +4 +� +PE(posx, posy, dmodel +2 ++ 2i + 1) = cos(wi · posx) ∀i ∈ +� +0, dmodel +4 +� +(7.1) +avec : +wi = +1 +10000 +2i +dmodel +La position de l’élément dans la séquence 2D est donnée par posx et posy. dmodel correspond +à la dimension d’encodage de l’image d’entrée et des éléments au sein du Transformer. Dans +notre cas, dmodel = 8f = 256. +La matrice de caractéristiques ainsi enrichie de la position des éléments est ensuite aplanie +afin de pouvoir être utilisée lors du décodage. +7.2.3 +décodeur transformer +Pour le décodeur, nous utilisons un Transformer standard puisqu’il permet la prédiction +de séquences de longueurs variables. Celui-ci est constitué d’un empilement de quatre +couches de décodeur Transformer. Chaque couche suit une architecture standard avec un +mécanisme d’auto-attention et un mécanisme d’attention croisée. L’auto-attention modélise +les dépendances entre les éléments de la séquence prédite, contrairement à l’attention croisée, +utilisée pour extraire des informations visuelles de l’encodeur, sur la base des prédictions +précédentes. En d’autres termes, étant donné les prédictions précédentes, elle indique où le +modèle doit regarder pour prédire le prochain élément. + +7.3 D É TA I L S D’ I M P L É M E N TAT I O N E T S T R AT É G I E S D’ E N T R A Î N E M E N T +131 +Le décodeur suit donc un processus récurrent où à chaque itération, il prend en entrée les +caractéristiques visuelles aplanies et les éléments prédits précédemment ( ˆy0, ˆy1, ..., ˆyt−1) et +produit un vecteur de caractéristiques pour la prédiction de l’élément au pas de temps t. +Chaque prédiction précédente est encodée dans un vecteur de taille 8f grâce à une couche +d’embedding apprise, puis les vecteurs sont concaténés afin de former une matrice de taille +(t − 1 × 8f). Ainsi, les caractéristiques visuelles et les vecteurs des prédictions précédentes +ont la même dimension dmodel = 8f = 256. +Comme pour un Transformer standard, la matrice des embeddings est additionnée à une +matrice de codage positionnel 1D de même taille qui permet de savoir où se trouve cette +prédiction dans la séquence. Cet encodage positionnel 1D est détaillé dans le Focus 2.13. +7.2.4 +branche de classification +La branche de classification génère des probabilités à partir de la sortie du décodeur. Elle est +composée d’une couche linéaire qui permet de modifier la dimension de la sortie de dmodel = +256 à TC = TV + 2 × C + 2, C étant le nombre de classes considérées dans l’expérience. +Deux sorties supplémentaires sont ajoutées afin de représenter les jetons de fin de séquence +(EOS) et de padding. Le jeton de padding est utilisé afin de permettre un entraînement par +batchs dans lesquels les séquences doivent être de même longueur. Cette couche linéaire est +suivie d’une fonction argmax qui assigne à ˆyt la position ou la classe d’objet pour laquelle la +probabilité est maximale. +7.3 +D É TA I L S D’ I M P L É M E N TAT I O N E T S T R AT É G I E S D’ E N T R A Î N E M E N T +Dans cette section, nous donnons des détails techniques sur l’implémentation utilisée lors +de nos expériences. +Doc2Seq est implémenté à l’aide du framework PyTorch. Il est entraîné avec un learning +rate initial de 5e − 5, l’optimiseur Adam et la fonction de coût d’entropie croisée. Les poids +sont initialisés grâce à l’initialisation Glorot. Nous utilisons le teacher forcing, qui est une +stratégie d’apprentissage de modèle qui utilise la vérité terrain comme entrée au lieu de +la sortie du modèle de l’itération précédente. Cela permet de paralléliser les calculs en +prédisant en parallèle tous les éléments de la séquence de sortie, et donc de réduire le temps +d’entraînement. +Au total, le modèle comporte 6,6 millions de paramètres répartis comme suit : +— 3,5 millions venant de l’encodeur Doc-UFCN ; +— 0,2 million pour la couche d’embedding ; +— 2,6 millions venant du décodeur Transformer ; +— 0,2 million pour la branche de classification. +Ces valeurs sont données dans un contexte dans lequel une seule classe d’objets est considérée +(C = 1). Les nombres de paramètres venant de la couche d’embedding et de la branche de +classification varient légèrement en fonction de ce nombre de classes C. + +132 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +7.3.1 +taille des images en entrée +Puisque nous utilisons l’encodeur du modèle Doc-UFCN, nous avons décidé d’utiliser la +taille d’image en entrée ayant obtenu les meilleurs résultats dans les expériences présentées +dans les chapitres précédents. Ainsi, les images d’entrée sont redimensionnées en images plus +petites telles que la plus grande dimension de l’image soit égale à 768 pixels tout en conservant +le ratio de l’image originale. Les coordonnées des objets présents sur les images sont également +mises à l’échelle. Ainsi, nous disposons d’un vocabulaire de taille TV = 768. +7.3.2 +augmentation de données +Durant l’entraînement des modèles, nous utilisons une stratégie d’augmentation des don- +nées. Tout d’abord, des augmentations sont appliquées à l’image d’entrée telles qu’un ajout +de bruit et de flou gaussien, un changement de luminosité, une inversion des canaux de cou- +leur ou une mise en niveaux de gris. De plus, des transformations linéaires, telles que des +translations et des rotations, sont appliquées. +Enfin, nous augmentons les séquences d’entrée pendant l’apprentissage pour inclure des +jetons bruités. Cela améliore la robustesse du modèle contre les prédictions bruitées et dupli- +quées. Les séquences sont augmentées de trois manières : +— Ajout de bruit sur les coordonnées des boîtes englobantes (translation et redimension- +nement aléatoires avec une probabilité de 0,3) ; +— Suppression aléatoire de 20 % des boîtes ; +— Inversion des jetons de classes de début et de fin avec une probabilité de 0,1. +7.3.3 +décodeur transformer +Nous utilisons quatre couches de décodage avec la dimension dmodel = 8f = 256. Chaque +couche de décodage possède quatre têtes d’attention et utilise une activation ReLU. +Comme les différentes images comportent souvent un nombre différent d’objets, les sé- +quences générées auront des longueurs différentes. Pour indiquer la fin d’une séquence, nous +incorporons donc un jeton de fin de séquence (EOS). Ainsi, le processus de prédiction se +termine lorsque le jeton EOS est prédit ou après un nombre prédéfini de valeurs prédites. +7.3.4 +choix du meilleur modèle +Dans le cadre d’une application de reconnaissance de document, l’objectif principal est +d’obtenir le texte contenu dans celui-ci ainsi que sa position sur l’image. Comme montré +dans le chapitre 5, il est nécessaire d’évaluer les modèles de détection de lignes de texte +grâce aux métriques de reconnaissance. Ainsi, nous pouvons évaluer l’impact des résultats de +détection sur les résultats finaux. +Dans nos expériences, nous avons souhaité poursuivre dans cette direction en intégrant un +reconnaisseur pré-entraîné dans les processus de sélection des meilleurs modèles et d’évalua- + +7.4 E X P É R I E N C E S E T R É S U LTAT S +133 +Table 7.3 – Statistiques du jeu de données IAM utilisé pour la détection de lignes de texte. +Jeu de données +Images +Lignes +train +valid +test +train +valid +test +IAM +Marti et al. (2002) +747 +220 +232 +6 482 +1 926 +1 965 +tion. Ainsi, chaque expérience impliquant une détection de lignes de texte intègre un modèle +de reconnaissance de texte niveau ligne. Pour cela, un modèle de reconnaissance est tout +d’abord entraîné sur les lignes transcrites provenant du même jeu de données que celui consi- +déré dans l’expérience. Ensuite, à partir de la 500e époque et toutes les cinq époques, le +modèle de reconnaissance est appliqué à l’ensemble des lignes prédites sur l’ensemble de vali- +dation et un CER@page est calculé (voir algorithme 5.1). Le modèle final est celui obtenant le +CER le plus bas. Cette stratégie de sélection du meilleur modèle est comparé à une sélection +standard basée sur la fonction de coût dans la section 7.4.3. +Enfin, ce même modèle de reconnaissance est appliqué durant la phase d’évaluation afin +d’obtenir le CER@page sur l’ensemble de test. +7.4 +E X P É R I E N C E S E T R É S U LTAT S +Nous décrivons, dans cette section, les résultats préliminaires obtenus avec Doc2Seq pour +la détection de lignes de texte sur le jeu de données IAM (Marti et al., 2002). +7.4.1 +jeu de données +La Table 7.3 présente les statistiques du jeu de données issu de la base IAM utilisé pour +l’entraînement et l’évaluation du modèle Doc2Seq. Nous avons choisi ce jeu car il est annoté +au niveau ligne et nous disposons des transcriptions pour chaque ligne de texte. De plus, il +s’agit d’un jeu de données assez simple, annoté en rectangles englobants. +Durant l’entraînement, les lignes de texte sont ordonnées de haut en bas afin que le modèle +apprenne cet ordre de lecture. +7.4.2 +entraînement des modèles de détection +Le modèle est entraîné avec des mini-batchs de taille 12 pour réduire le temps d’apprentis- +sage sur un maximum de 1500 époques. De plus, dans un processus d’entraînement standard, +le meilleur modèle est choisi comme étant celui obtenant les meilleures performances sur l’en- +semble de validation. Il est choisi selon les valeurs de la fonction de coût ou d’une métrique +directement liée à la tâche. Dans nos expériences, nous choisissons le meilleur modèle comme +étant celui obtenant le plus faible CER. Nous comparons l’impact de ce choix par rapport +à une sélection standard basée sur la valeur de la fonction de coût dans les paragraphes +suivants. + +134 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Table 7.4 – Résultats de reconnaissance de textes manuscrits sur le jeu de données IAM. Nous pré- +sentons également les résultats du modèle de Moysset et al. (2019), modèle obtenant les +résultats à l’état de l’art sans modèle de langue. +Système +CER (%) +WER (%) +train +valid +test +train +valid +test +PyLaia +0,32 +6,50 +7,68 +1,26 +19,12 +19,82 +Moysset et al. (2019) +– +4,62 +7,73 +– +17,31 +25,22 +modèle de reconnaissance pylaia +Un modèle de reconnaissance PyLaia (Puigcerver, 2017) est entraîné au préalable sur +les mêmes données et en suivant la même répartition dans les ensembles d’entraînement, de +validation et de test. Il est ensuite intégré à l’entraînement de Doc2Seq et appliqué aux boîtes +prédites sur l’ensemble de validation. Le modèle PyLaia a été choisi car il obtient des résultats +satisfaisants sur les textes manuscrits tout en étant assez rapide ce qui permet de l’intégrer +à l’entraînement. De plus, le système est facilement interfaçable avec le code PyTorch de +Doc2Seq. +Les résultats du modèle de reconnaissance niveau ligne sont présentés dans la Table 7.4. +Cette table montre des performances assez satisfaisantes que nous considérons suffisantes afin +d’évaluer et de comparer les modèles de détection. +7.4.3 +résultats et discussion +Dans cette section, nous présentons, tout d’abord, les résultats quantitatifs du modèle +Doc2Seq. Nous visualisons ensuite les prédictions et analysons les erreurs obtenues. +Durant l’inférence, les valeurs prédites sont regroupées par six (les quatre coordonnées +et les deux classes de points) afin de créer les rectangles englobants. Le modèle a très bien +appris sur le jeu de données IAM puisque qu’aucune valeur n’est prédite en plus et que les +jetons de classes ont tous été correctement prédits. +Pour comparaison, nous avons entraîné un modèle Doc-UFCN sur les mêmes données IAM. +Le modèle a été entraîné durant 150 époques dans les mêmes conditions que celles décrites +dans les chapitres précédents : +— Images redimensionnées telles que la plus grande dimension de l’image soit égale à 768 +pixels ; +— Taux d’apprentissage initial de 5e − 3, mini-batchs de taille 4, optimiseur Adam, fonc- +tion de coût Dice et arrêt anticipé (early stopping). +De plus, afin d’avoir des résultats comparables, les rectangles englobants des composantes +connexes prédites par Doc-UFCN sont extraites et le même schéma d’évaluation que Doc2Seq +est appliqué. + +7.4 E X P É R I E N C E S E T R É S U LTAT S +135 +Table 7.5 – Résultats des modèles de détection de lignes sur le jeu de données IAM, donnés en fonction +du critère de sélection. Les colonnes "Manuel" présentent les résultats du reconnaisseur +sur les lignes annotées manuellement contrairement aux colonnes "Prédit" qui présentent +les résultats du reconnaisseur sur les lignes prédites automatiquement. +Système +Critère de +Set +AP@.5 +AP@.75 +mAP +CER (%) +WER (%) +sélection +Manuel +Prédit +Manuel +Prédit +train +0,98 +0,83 +0,68 +0,31 +1,21 +1,43 +3,82 +Doc2Seq +CER +valid +0,98 +0,68 +0,60 +5,72 +6,98 +20,11 +22,67 +test +0,98 +0,71 +0,62 +6,65 +7,58 +20,97 +22,62 +train +0,99 +0,82 +0,69 +0,31 +1,17 +1,43 +3,86 +Doc2Seq +Entropie +valid +0,97 +0,70 +0,61 +5,72 +6,98 +20,11 +22,49 +croisée +test +0,98 +0,70 +0,62 +6,65 +7,66 +20,97 +22,68 +train +0,98 +0,61 +0,57 +0,31 +3,37 +1,43 +9,13 +Doc-UFCN +CER +valid +0,97 +0,59 +0,57 +5,72 +8,47 +20,11 +25,48 +test +0,95 +0,61 +0,55 +6,65 +10,64 +20,97 +27,34 +train +0,92 +0,89 +0,71 +0,31 +6,83 +1,43 +9,69 +Doc-UFCN +DICE +valid +0,92 +0,87 +0,70 +5,72 +12,68 +20,11 +28,24 +test +0,88 +0,83 +0,68 +6,65 +16,59 +20,97 +30,68 +résultats quantitatifs +La Table 7.5 présente les performances des modèles obtenus pour le jeu de données IAM. +Dans cette Table, nous montrons les résultats de deux modèles issus du même entraînement +mais sélectionnés selon un critère différent. Les modèles sont évalués par différentes métriques : +— Les métriques objet fournies par COCO 1, notamment la précision moyenne (AP) pour +différentes valeurs de seuil : AP@.5, AP@.75 et AP@[.5,.95] (mAP) ; +— Les métriques de reconnaissance niveau page : CER et WER. +Comme dans les chapitres précédents, les colonnes "Manuel" présentent les résultats du +reconnaisseur sur les lignes annotées manuellement. Elles correspondent donc au CER entre +les transcriptions et les résultats d’HTR appliqué sur les mêmes lignes de texte. Ainsi, ces +valeurs représentent les meilleures atteignables dans le cas d’un détecteur de lignes idéal. +Les deux modèles Doc2Seq présentent des résultats très satisfaisants. En effet, les valeurs +d’AP sont relativement élevées pour de la détection de lignes de texte dans les images de +documents. De plus, les valeurs de CER sur les ensembles de validation et de test sont très +proches des valeurs obtenues sur les lignes annotées manuellement. Cela signifie que les lignes +obtenues sont localisées avec précision sur les images. En effet, pour le modèle ayant obtenu +le CER le plus faible, nous perdons moins d’un point de pourcentage de CER (de 6,65 % à +7,58 %) en passant des lignes manuelles aux lignes prédites sur l’ensemble de test. +De plus, les deux modèles Doc2Seq obtiennent des résultats similaires. En effet, pour les +métriques AP, les deux modèles obtiennent, en moyenne, un point de pourcentage d’écart. +Le modèle sélectionné sur la base du CER présente un faible gain de performances en CER +1. https://github.com/cocodataset/cocoapi + +136 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +et WER sur l’ensemble de test, ce qui était attendu puisqu’il a été choisi afin de minimiser le +CER. Cependant, le modèle sélectionné sur la base de l’entropie croisée présente des résultats +très satisfaisants, ce qui valide l’utilisation de la classification comme type de prédiction +couplé à la fonction de perte d’entropie croisée. Le calcul du CER durant l’entraînement, qui +nécessite le texte des documents transcrits ainsi qu’un modèle de reconnaissance entraîné, +ne semble donc pas nécessaire pour obtenir un modèle très performant. Bien que cela +permette d’optimiser les résultats de reconnaissance, il est envisageable d’entraîner un +modèle performant sans utilisation de reconnaisseur de texte. +Les résultats obtenus par le modèle Doc-UFCN sont légèrement meilleurs en termes de +précision moyenne par rapport aux modèles Doc2Seq, notamment sur l’ensemble de validation. +Cependant, les résultats finaux de reconnaissance niveau page sont bien moins bons que ceux +des modèles Doc2Seq. En effet, pour le critère de sélection basé sur la fonction de coût, +nous notons une augmentation de +5,70 points de pourcentage de CER sur l’ensemble de +validation et de +5,75 points de WER. Ces écarts sont plus faibles lorsque nous comparons +les modèles sélectionnés sur la base du CER, cependant, le modèle Doc-UFCN reste bien +moins bon que le modèle Doc2Seq. +Ainsi, pour des résultats niveau objet équivalents, le modèle Doc2Seq obtient de bien +meilleures performances en reconnaissance de texte. Cela s’explique, entre autres, par sa +capacité à apprendre l’ordre de lecture, ordre non disponible dans les prédictions de Doc- +UFCN pour lequel nous avons dû ordonner les boîtes de haut en bas. Ces résultats montrent +une nouvelle fois l’intérêt des métriques orientées vers la tâche finale dans l’évaluation et la +comparaison de modèles de détection. +Le modèle Doc2Seq présente un temps de prédiction moyen de 284 ms par image sur +une carte graphique Tesla V100-SXM2-16GB pour le jeu de données IAM. Dans les mêmes +conditions, Doc-UFCN est lui deux fois plus rapide, avec un temps moyen de 134 ms par +image. Le temps d’inférence présenté par Doc2Seq est très raisonnable sachant que le modèle +permet l’extraction directe des objets dans l’ordre de lecture demandé. +visualisation des prédictions +La Figure 7.4 montre les résultats visuels sur quatre images de l’ensemble de test du jeu +de données IAM. Les boîtes annotées manuellement sont représentées en bleu et les boîtes +prédites par le modèle sont en rouge. Visuellement, les prédictions sont très proches des +boîtes annotées pour les trois images en haut et en bas à gauche. Nous remarquons très +peu de problèmes liés à la largeur des boîtes et à leur position selon l’axe des abscisses. De +plus, nous notons que les principales erreurs viennent de la hauteur des boîtes ainsi que +leur position selon l’axe des ordonnées. En effet, sur les images en haut à droite et en bas à +gauche, nous voyons que certaines boîtes sont trop hautes par rapport aux boîtes annotées +correspondantes. L’image en bas à gauche montre des boîtes mal localisées, trop basses selon +l’axe des ordonnées. + +7.4 E X P É R I E N C E S E T R É S U LTAT S +137 +Figure 7.4 – Détections de lignes produites par le modèle Doc2Seq, sélectionné sur les valeurs du +CER, sur quatre images de l’ensemble de test du jeu de données IAM. Les rectangles +englobants annotés manuellement sont représentés en bleu et les boîtes prédites en rouge. + +P03-185 +Sentence Database +Thus had they parted the previous evening and now Diana was trailing up the grav- +elled drive to the hospital alone. Of course one couldn't say for certain when a doctor +would be free during the day; tea was served from four until five-thirty in the residents' +common-room, which proved the elasticity of medical commitments. Something had +cropped up which required Nigel's attention, she was convinced, or he would have +granted her small request to be met at the gates. +s +hadfheypavledthe prevlous evenins aud nou +Diany was railing up Hhe graveled drve fo He osofta +alone.OfCouvse +eowe couldu'f say fo cerhuin whe +neduri +lochov +r)nom +beee +duvn +fea +WaS +gevee +4m +ve-trhnenesdenhs +may +Ymm +proved He elasdahy ofwedical commitreruhs +1m00 +Somethly laedCropped +upwmuvegnineeNil's aHenHor +SMME +convinced +or would have +gfaedlersalle +b meFat He +geules +Name:SentenceDatabase +M01-131 +When he finally beckoned to them to enter, the action gave the impression of having +the floor. He seemed completely unaware of their presence. They just stared at him, +turning their heads like tennis spectators as he walked up and down, up and down. +When he Bualy beduoned to Hhin to enles, thu acha +gaveHheimpresscmap +benhougktoufand +deaded upan.lnside they sat down un bidd,whie +Dan paced Hhefor.lle Seemed Complelely unawae +mun +Name:Sentence Database +N04-048 +So he put up for the night at The Admiral's Head, that famous Portsmouth hostelry, +during the last war. Having deposited his baggage and unpacked his overnight-bag he +went in search of a drink. The lower bar was empty, save for the lady known by all +habitue*?2s as 'Seaweed', and a youngish, sharp-eyed man who was staring moodily +into a gin and tonic. +So kepuf up Perthemightat he Aimiral HeadtMe +lamirrotmeihhoytelny,Olisndonlymhimeri +mitiretoThebeepee,lnhappiluydeokruyeel NyCemar +lemlr dunne Nu lafar. lhanng depaotd lu +lriggaeg and cinpaihed his oemglt-lug +etnsearR aladmnk.DheXoerlarws +emphy,Mve for thu lady hnom luyall haluilue +X?ls a Seumeed , anila youngsh, yhanp-eyel +manAho ansiorng moodilys inkea gin +inelenie +Name:Sentence Database +P02-115 +Doc gave her hand a shake. "Wake up Gay, and don't even contemplate throwing +yourself away on a chap like that. You're a fine girl, intelligent, and pretty, and I +had thought you were sensible too. Don't make a fool of yourself over someone who +doesn't care two jots for your feelings. If he behaves like this now what is your married +lifegoing tobelike? +hana +gae +May +qshahe. +throwing +pup +een +don +yoursel +that +You're +qway +on +Chao +Tike +Pine +inteligent +gir/ +prel +anc +bpy +thoughl +you +were +sensibk +too +Don't +make +100/ +0f +yourselp +over +who +doesnt +two +2015 +your +he +behaves +this +whal +IS +youl +lihe +buob +fo +be +Name:138 +D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S +Le modèle prédit, tout d’abord, l’ordonnée puis l’abscisse de chaque point. Ainsi, notre +hypothèse est que, pour prédire l’abscisse d’un point, le modèle a déjà connaissance de son +ordonnée, il sait donc exactement où regarder sur l’image (sur quelle ligne d’ordonnée) pour +se positionner. Les coordonnées prédites selon l’axe des abscisses sont donc plus précises. +Au contraire, afin de prédire la première coordonnée, le modèle peut regarder partout sur +l’image, ou du moins en dessous des lignes précédemment prédites, ce qui peut mener à des +localisations moins précises. Cette hypothèse s’applique également au second point à prédire. +Afin de vérifier cette hypothèse, il serait nécessaire d’entraîner un second modèle à prédire +d’abord l’abscisse puis l’ordonnée de chaque point. Ainsi, nous pourrions voir si les problèmes +seraient désormais liés à la largeur des boîtes et leur localisation selon l’axe des abscisses. Ces +expérimentations sont actuellement en cours. +7.5 +C O N C L U S I O N +Dans ce chapitre, nous avons présenté un nouveau système de détection d’objets dans les +images de documents. Celui-ci se base sur les Transformers, algorithmes les plus performants +actuellement dans de nombreux domaines. Nous avons montré que ce système pouvait être +entraîné sur un jeu de données restreint d’images de documents. Il permet d’obtenir des +premiers résultats prometteurs. En effet, il montre des temps d’inférence raisonnables tout +en obtenant des performances à l’état de l’art et en permettant une prédiction séquentielle. +Nous avons également proposé une modélisation complète du problème de détection d’ob- +jets. Cette modélisation permet de représenter les objets de manière simple et une détection +efficace de ceux-ci. Cette modélisation a le potentiel à se généraliser à d’autres tâches plus +complexes, qui feront l’objet de nos futures recherches. + +8 +C O N C L U S I O N S E T P E R S P E C T I V E S +8.1 +C O N C L U S I O N S +Dans cette thèse, nous avons proposé deux systèmes à base de réseaux neuronaux afin +de résoudre la tâche de détection d’objets dans les images de documents. Le premier mo- +dèle, Doc-UFCN, a présenté de grandes capacités de détection sur de nombreux jeux de +données manuscrits hétérogènes. Il a également montré une grande robustesse en obtenant +des résultats très compétitifs sur de nouveaux documents hors échantillon. Le second modèle, +Doc2Seq, a également obtenu des premiers résultats encourageants qui permettent de traiter +la détection d’objets à un plus haut niveau. +Le développement et l’application de ces modèles ont mené à des études plus globales +concentrées sur les données et leurs annotations, les métriques d’évaluation et les scores de +confiance. +Nous avons répondu aux principales problématiques liées à la détection d’objets dans les +images de document dans un cadre industriel. La première problématique concerne le déve- +loppement de modèles avec de grandes capacités de généralisation. En effet, dans ce cadre +dans lequel de nouveaux projets sont régulièrement mis en place, il n’est pas envisageable +d’annoter de nombreuses données pour chacun de ces projets, d’où la nécessité de dévelop- +per des modèles plus génériques, montrant des performances élevées sur des documents très +hétérogènes. +Dans cette optique, nous avons entraîné plusieurs systèmes sur un grand volume de +données très différentes. Ces entraînements ont mené à des modèles plus robustes, obtenant +de meilleures performances que des modèles spécifiques entraînés sur un jeu de données +unique. Comme la plupart des modèles de type FCN, ces modèles ont cependant montré +des difficultés à prédire des éléments qui se touchent ou se chevauchent. Pour cette raison, +nous avons proposé une uniformisation des annotations, ainsi qu’une scission des boîtes +afin de réduire ces chevauchements dans les annotations, utilisées durant la phase d’en- +traînement. Ces traitements permettent la prédiction de boîtes plus précises et non fusionnées. +Une seconde problématique induite par le cadre de production dans lequel se situe cette +thèse concerne l’efficacité des modèles de détection : ceux-ci doivent fournir des détections +de grande précision tout en montrant des temps de traitement réduits et en pouvant être +entraînés sur des jeux de données restreints. Dans la littérature, de nombreux modèles ont +été proposés pour la détection d’objets dans les images de documents, cependant, la plupart +requièrent un grand nombre de données annotées. Pour pallier ce problème, des systèmes +139 + +140 +C O N C L U S I O N S E T P E R S P E C T I V E S +utilisant des poids pré-entraînés sur des images de scènes naturelles, tels que dhSegment, ont +été proposés, mais ces systèmes montrent des temps d’inférence encore trop élevés pour une +utilisation à l’échelle industrielle. +Pour répondre à ces problématiques, nous avons proposé le système Doc-UFCN. Ce +système a montré des temps d’inférence réduits et obtenu des performances à l’état de l’art. +Celui-ci peut être entraîné sur peu de données en comparaison avec les systèmes dédiés à la +détection d’objets dans les images de scènes naturelles. +Dans de nombreux projets, il y a peu, voire aucune donnée annotée. Comme énoncé plus +tôt, il est nécessaire d’avoir un détecteur assez générique afin de traiter ces documents plus +facilement. Cependant, les modèles génériques peuvent parfois ne pas être suffisamment per- +formants sur ces nouveaux documents, qui peuvent avoir une mise en page très différente +de celles des documents que le modèle a rencontrés durant sa phase d’entraînement. Pour +cela, l’apprentissage actif a été introduit, permettant d’entraîner itérativement des modèles +en ajoutant, à chaque itération, de nouvelles données annotées sélectionnées dans le but +d’améliorer les résultats du modèle de détection tout en réduisant le coût d’annotation ma- +nuelle. Dans ce cadre, il est nécessaire que le modèle fournisse les détections tout en estimant +automatiquement leur qualité. +Nous avons proposé et évalué quatre estimateurs de confiance. Ceux-ci ont permis +d’entraîner des modèles atteignant des performances élevées pour la détection d’objets +tout en ne nécessitant qu’un faible nombre d’images annotées manuellement. Nous avons +également démontré que deux de ces estimateurs permettent de sélectionner les détections +les plus précises afin d’être utilisées dans un entraînement autosupervisé. Les modèles +ainsi entraînés ont permis d’obtenir des gains significatifs de performances par rapport aux +modèles génériques tout en ne nécessitant aucune donnée annotée. +La reconnaissance de documents consiste généralement en l’application successive de diffé- +rents modèles. Dans ce cadre, les lignes de texte produites par un modèle de détection sont +généralement fournies à un modèle de reconnaissance de texte. Ainsi, l’amélioration de la +détection des lignes de texte doit permettre d’améliorer les résultats de reconnaissance, or les +deux tâches ne sont pas étroitement liées. Il est donc nécessaire d’évaluer, à chaque étape du +traitement, son impact sur les résultats de l’étape suivante ou finale. +Cette problématique a été très peu étudiée dans la littérature. Afin d’avoir une évaluation +de la détection de lignes de texte davantage cohérente avec la tâche finale, nous avons donc +proposé des métriques liées à la reconnaissance de texte. Ces métriques permettent de voir +directement l’impact de la tâche de détection sur les résultats finaux. De plus, nous avons +constaté que l’utilisation de ces métriques durant l’entraînement de modèles de détection +permet d’optimiser les résultats du reconnaisseur de texte. +Enfin, les modèles à base de réseaux de neurones profonds de type Transformers ont récem- +ment été proposés. Ceux-ci ont été introduits pour des tâches de traitement de la langue et +notamment la tâche de traduction de texte. Ils ont été initialement établis afin de pallier le + +8.2 P E R S P E C T I V E S +141 +trop faible contexte disponible pour traiter les longues séquences de texte par les réseaux ré- +currents, systèmes largement utilisés jusqu’alors pour traiter ces tâches. Par la suite, certains +travaux ont cherché à adapter ces modèles aux tâches de vision, motivés par leur capacité +de modélisation des dépendances des éléments en entrée réalisée à l’aide du mécanisme d’at- +tention. Bien que ces travaux aient montré des avancées significatives pour les tâches de +classification d’images, très peu se sont intéressés à leur application aux tâches de détection +d’objets. +Dans cette thèse, nous nous sommes intéressés à adapter ces modèles à base de Transfor- +mers à la tâche de détection d’objets dans les images de documents. Nous avons donc proposé +Doc2Seq, un modèle hybride combinant un encodeur convolutif et un décodeur Transfor- +mer. Ce modèle a permis d’obtenir des premiers résultats encourageants, tout en respectant +l’ensemble des contraintes évoquées précédemment. Il permet de modéliser correctement les +dépendances entre les différentes parties de l’image d’entrée mais aussi celles entre l’image +d’entrée et les coordonnées prédites. Ce système apporte également d’autres avantages tels +que sa capacité à produire des résultats séquentiels et structurés. +8.2 +P E R S P E C T I V E S +Dans de nombreux domaines d’application des réseaux de neurones profonds, les modèles +sont entraînés sur des milliers, voire des millions d’exemples. En effet, le premier Vision Trans- +former proposé, pour la tâche de classification d’images, a nécessité un pré-entraînement sur +303 millions d’images. De la même manière, Pix2Seq a été entraîné sur le jeu de données de +référence MS-COCO 2017 comportant 118 000 images d’entraînement. De plus, leur meilleur +modèle a été obtenu grâce à un pré-entraînement sur les données Objects365, qui représentent +600 000 images d’entraînement. Ces modèles ont montré des performances très élevées, mon- +trant l’intérêt d’utiliser de telles quantités de données. +Dans cette optique, une perspective de cette thèse est de collecter encore davantage de jeux +de données, toujours plus divers, et d’étudier la capacité du modèle à apprendre à partir de +ces données. La plupart de ceux utilisés jusqu’ici étaient principalement historiques, il serait +également envisageable de collecter des documents modernes afin d’obtenir un détecteur de +lignes de texte très générique. +Nous souhaiterions également comparer cette approche à une stratégie de collecte dans +laquelle un nombre plus restreint d’exemples serait utilisé, mais qui chercherait à maximiser +la diversité des mises en page et des contenus. Durant cette étude, il serait également +intéressant d’évaluer l’impact du balancement des différentes données dans l’ensemble +d’entraînement. +De plus, nous avons proposé, durant cette thèse, quatre estimateurs de confiance permet- +tant de sélectionner les images à annoter. Pour le moment, ceux-ci sont utilisés dans le cadre +d’une adaptation d’un modèle générique à un nouveau domaine, afin de sélectionner les +images à annoter pour mettre en place un nouveau système. Une perspective serait d’utiliser +les confiances estimées afin de suivre la qualité des résultats en production. De plus, dans un + +142 +C O N C L U S I O N S E T P E R S P E C T I V E S +cadre non supervisé, les confiances estimées par le modèle de détection peuvent permettre de +vérifier que celui-ci s’améliore, durant les différentes itérations, grâce à sa confiance moyenne. +De nombreux modèles de détection d’objets traitent les images de documents à partir +de sous-résolutions. C’est également le cas des deux modèles que nous avons proposés, +Doc-UFCN et Doc2Seq. Bien que, dans la plupart des cas, cette sous-résolution soit +suffisante pour obtenir des résultats satisfaisants, dans le cas où les objets sont très petits +et très proches, la détection est impossible puisque de nombreuses fusions sont produites. +Il serait intéressant de comparer différentes sous-résolutions mais aussi une approche par +patchs, bien que celle-ci soit beaucoup plus coûteuse en ressources et en temps d’inférence. +De même, nous pourrions estimer la sous-résolution optimale pour un jeu de données ou +pour une image de manière automatique. +Nous souhaitons également évaluer le modèle Doc2Seq sur d’autres jeux de données et +d’autres tâches, et notamment des problèmes plus complexes avec des objets imbriqués tels +que des paragraphes et lignes de texte. Il serait intéressant de le tester sur d’autres tâches +telles que l’analyse de mise en page de tableaux avec la détection des éléments structurels +tels que les lignes et colonnes de titre et les pieds de tableaux 1. +De plus, l’ensemble des modèles de détection traitent les images de manière isolée. Ils ne +possèdent aucune mémoire quant aux prédictions réalisées sur les images précédentes. Or, +dans le cadre du traitement de séries (ouvrages ou collections), il pourrait être bénéfique +de considérer, lors du traitement d’une nouvelle image, des propriétés établies sur d’autres +images ou au niveau de la série. Le système Doc2seq permet d’envisager un tel traitement. En +effet, l’utilisation des prédictions précédentes dans le décodeur Transformer permet d’imagi- +ner des tâches de plus haut niveau. 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In : IEEE Conference on Computer Vision and Pattern +Recognition (CVPR), p. 625-632. + diff --git a/ztFKT4oBgHgl3EQfNC2a/content/tmp_files/load_file.txt b/ztFKT4oBgHgl3EQfNC2a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3149a506da67da11cb48438465b52ccc6df64f3b --- /dev/null +++ b/ztFKT4oBgHgl3EQfNC2a/content/tmp_files/load_file.txt @@ -0,0 +1,10252 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf,len=10251 +page_content="THÈSE Pour obtenir le diplôme de doctorat Spécialité INFORMATIQUE Préparée au sein de l'Université de Rouen Normandie Détectiοn d'οbjets dans les dοcuments numérisés par réseaux de neurοnes prοfοnds Présentée et soutenue par MELODIE BOILLET Thèse soutenue le 10/01/2023 devant le jury composé de M." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" ANDREAS FISCHER PROFESSEUR DES UNIVERSITES, Haute école d'ingéniérie et d'archi." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Rapporteur du jury M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' HAROLD MOUCHERE PROFESSEUR DES UNIVERSITES, UNIVERSITE NANTES Rapporteur du jury M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' CHRISTOPHER KERMORVANT , Membre du jury MME LAURENCE LIKFORMAN-SULEM PROFESSEUR ASSOCIE, TELECOM PARISTECH Membre du jury MME CAROLINE PETITJEAN PROFESSEUR DES UNIVERSITES, Université de Rouen Normandie Membre du jury M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" THIERRY PAQUET PROFESSEUR DES UNIVERSITES, Université de Rouen Normandie Directeur de thèse Thèse dirigée par THIERRY PAQUET (Laboratoire d'Informatique, du Traitement de l'Information et des Systèmes) UNIVERSITE DE ROUEN MANMIIE R E M E R C I E M E N T S Je tiens, en premier lieu, à remercier Thierry Paquet, qui a accepté d’encadrer et de superviser ma thèse, pour son expérience et ses précieux conseils." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Je tiens également à remercier mon co-encadrant de thèse, Christopher Kermorvant, pour l’opportunité qu’il m’a offert en me proposant ce sujet, et pour m’avoir encadrée et soutenue durant ces trois années.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Leur patience mais aussi leurs expériences ont permis des discussions enrichissantes et très intéressantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Je remercie les membres de mon jury d’avoir accepté d’évaluer mes travaux de thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, mes deux rapporteurs Harold Mouchère et Andreas Fischer, pour le temps consacré à la lecture de ce manuscrit et leurs commentaires avisés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Merci également à Laurence Likforman et Caroline Petitjean d’avoir accepté de faire partie du jury de thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Merci à Teklia de m’avoir permis de réaliser ma thèse dans les meilleures conditions, en m’encourageant à publier et en valorisant mon travail en l’intégrant dans de nombreux projets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' J’aimerais également remercier toutes les personnes de l’entreprise avec qui j’ai eu la chance de collaborer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Je tiens tout particulièrement à remercier l’équipe de recherche : Martin, Marie-Laurence, Blanche, Chaza et Solène pour nos rencontres et discussions très inspirantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Merci à toute l’équipe de Grenoble pour leur gentillesse et leur soutien, et plus spécialement à Bastien, avec qui j’ai pu échanger sur de nombreux points techniques, pour son aide et sa patience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Je me dois aussi d’être très reconnaissante envers mon laboratoire, le LITIS, pour m’avoir offert tout le confort et le soutien matériel nécessaire au bon déroulement de ma thèse et plus particulièrement les membres de l’équipe Apprentissage pour leur accueil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un chaleureux merci à Denis avec qui j’ai eu la chance de partager mon bureau pendant ces trois dernières années mais également de longues et captivantes discussions toujours remplies de joie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur un plan plus personnel, je tiens à remercier toutes les personnes présentes durant ces trois années.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, mes parents et mes frères pour les nombreuses distractions qu’ils m’ont apportées mais aussi leur soutien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Merci également à ma belle famille pour leur présence et leurs relectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, un grand merci à Quentin pour m’avoir soutenue et écoutée jour après jour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' iii R É S U M É Qu’ils soient historiques ou modernes, imprimés ou manuscrits, les documents constituent un ensemble précieux d’informations souvent difficilement accessible dans leur forme originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La transformation de ces documents en documents digitaux est désormais possible grâce à leur numérisation et à l’extraction automatique de leurs contenus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette extraction nécessite la détection de différents éléments tels que les lignes de texte, éléments cruciaux afin d’obtenir la transcription du texte présent dans les images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que de nombreuses méthodes aient été proposées pour détecter ces éléments, l’analyse de la structure des documents reste un problème difficile : les modèles proposés souffrent de difficultés à généraliser à de nouvelles données et à des structures plus complexes, et ils nécessitent de nombreux exemples d’apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette thèse, nous étudions différentes tâches liées à l’analyse de la mise en page de do- cuments telles que la détection de lignes de texte, la séparation en actes ou encore la détection du support d’écriture (page).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, nous proposons deux modèles fondés sur des réseaux de neurones profonds suivant deux approches différentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les réseaux neuronaux ont démontré de bonnes capacités d’apprentissage dans de nombreux domaines d’application et notamment dans la détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Récemment, de nouveaux types de réseaux neuronaux ont vu le jour, les réseaux à base de Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceux-ci permettent de traiter plus efficacement les tâches de prédiction séquence-à-séquence telles que la traduction de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Leur adaptation aux tâches de vision a rapidement suscité l’engouement grâce à leurs performances élevées et leur capacité à produire des résultats séquentiels et structurés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre objectif est de proposer un modèle permettant de détecter les objets en tenant compte des difficultés liées au traitement de documents, notamment le nombre restreint de données d’entraînement disponibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les systèmes existants peuvent présenter des temps de traitement longs qui peuvent entraîner des coûts financiers importants et des impacts écologiques négatifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un cadre industriel, l’utilisation de tels systèmes ne semble pas appropriée, il est donc nécessaire de proposer des modèles plus parcimonieux en termes de nombre de paramètres afin d’obtenir des temps d’entraînement et d’inférence plus réduits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette optique, nous proposons un modèle de détection niveau pixel et un second modèle de détection niveau objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous commençons par proposer un modèle de détection comportant peu de paramètres, rapide en prédiction, et qui permet d’obtenir des masques de prédiction précis à partir d’un nombre réduit de données d’apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le pré-entraînement de ce modèle sur différents jeux de données annotés a permis d’obtenir des gains significatifs de performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces résultats nous ont donc conduits à mettre en place une stratégie de collecte et d’uniformisation de nombreux jeux de données, utilisés afin d’entraîner un modèle unique de détection de lignes démontrant de grandes capacités de généralisation à des documents hors échantillon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous proposons également un modèle de détection à base de Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La conception d’un tel modèle a nécessité de redéfinir la tâche de détection d’objets dans les images de documents et à en étudier différentes modélisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Suite à cette étude, nous proposons une stratégie de détection d’objets consistant à prédire séquentiellement les coordonnées des rectangles englobant les objets grâce à une classification pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette stratégie permet d’obtenir un modèle comportant peu de paramètres et rapide en inférence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les expériences v préliminaires de détection de lignes de texte montrent des bonnes performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, dans un cadre industriel, de nouvelles données non annotées sont souvent disponibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, dans le cas de l’adaptation d’un modèle à ces nouvelles données, on s’attend à fournir au système le minimum de nouveaux exemples annotés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le choix des exemples pertinents pour l’annotation manuelle est donc crucial pour permettre une adaptation réussie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est donc nécessaire que les systèmes effectuent la tâche finale tout en évaluant automatiquement leur confiance quant à leurs décisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, les décisions moins confiantes peuvent être soumises à un opérateur humain pour une annotation manuelle, tandis que les décisions plus confiantes sont conservées telles quelles pour fournir une annotation automatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À cet égard, nous proposons des estimateurs de confiance issus d’approches différentes pour la détection d’objets dans des images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La première approche proposée est inspirée de la méthode de Monte Carlo et consiste à construire des estimations de confiance en utilisant la méthode du dropout au moment du test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre seconde proposition consiste à construire un système dédié indépendant, entraîné à prédire une estimation de confiance de- puis une seule prédiction pendant l’inférence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous montrons que ces estimateurs permettent de réduire fortement la quantité de données annotées tout en optimisant les performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' vi A B S T R A C T Whether they are historical or modern, printed or handwritten, documents constitute a valuable collection of information that is usually difficult to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' The transformation of these documents into digital documents is now possible through their digitization and the automatic extraction of their contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' This extraction requires the detection of different elements such as text lines, which are essential to obtain the transcription of the image’s textual contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Although many methods have been proposed to detect these elements, the analysis of document structure remains a difficult problem : the proposed models suffer from difficulties in generalizing to new data and more complex structures, and they require many training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In this thesis, we study multiple tasks related to document layout analysis such as the detection of text lines, the splitting into acts or the detection of the writing support (page).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Thus, we propose two deep neural models following two different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Neural networks have shown good learning capabilities in many application domains, and in particular in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Recently, new types of neural networks have emerged, the Transformer- based networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' These systems allow processing more efficiently sequence-to-sequence tasks such as text translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Their adaptation to vision tasks has quickly become popular thanks to their high performance and their ability to produce sequential and structured outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' We aim at proposing a model for object detection that considers the difficulties associated with document processing, including the limited amount of training data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Moreover, existing systems can have long processing times that can result in significant financial costs and negative ecological impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In an industrial setting, the use of such systems does not seem appropriate, so it is necessary to propose more parsimonious models in terms of number of parameters to obtain reduced training and inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In this respect, we propose a pixel-level detection model and a second object-level detec- tion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' We first propose a detection model with few parameters, fast in prediction, and which can obtain accurate prediction masks from a reduced number of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' The pre-training of this model on different annotated datasets allowed us to obtain significant per- formance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' These results led us to implement a strategy of collection and uniformization of many datasets, which are used to train a single line detection model that demonstrates high generalization capabilities to out-of-sample documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' We also propose a Transformer-based detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' The design of such a model required redefining the task of object detection in document images and to study different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Following this study, we propose an object detection strategy consisting in sequentially predicting the coordinates of the objects enclosing rectangles through a pixel classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' This strategy allows obtaining a fast model with only few parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Preliminary experiments on text line detection show good performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Finally, in an industrial setting, new non-annotated data are often available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Thus, in the case of a model adaptation to this new data, it is expected to provide the system as few new annotated samples as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' The selection of relevant samples for manual annotation is therefore crucial to enable successful adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Thus, it is necessary for the systems to perform the final task while automatically assessing their confidence about their own vii decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' This way, less confident decisions can be submitted to a human operator for manual annotation, while more confident decisions are kept as is to provide an automatic annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' For this purpose, we propose confidence estimators from different approaches for object detection in document images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' The first proposed approach is inspired by the Monte Carlo method and consists in building confidence estimates using the dropout method at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Our second proposal consists in building an independent dedicated system, trained to predict a confidence estimate with a single prediction during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' We show that these estimators greatly reduce the amount of annotated data while optimizing the performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' viii TA B L E D E S M AT I È R E S Remerciements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' iii Résumé .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' v Liste des figures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' xii Liste des tableaux .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' xv Liste des focus .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 38 3 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Métriques d’évaluation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Métriques basées sur les pixels .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Métriques orientées objets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Métriques orientées vers la tâche finale .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 57 4 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Présentation du problème .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Expériences de détection de lignes de texte .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 63 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Jeux de données .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Étude ablative .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Expériences de détection d’actes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Résultats et discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 78 5 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C- T I O N D’ O B J E T S .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 101 6 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Méthodes d’estimation de la confiance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Estimateur basé sur les probabilités a posteriori .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Estimateurs basés sur le dropout de Monte Carlo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 105 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Estimateur basé sur les statistiques d’objets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 106 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Cadre expérimental .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 111 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Nombre de prédictions avec dropout .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 117 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Détection de lignes de texte .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 119 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 119 7 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 121 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Modélisation de la tâche de détection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 122 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Modélisation de la position et de la forme des objets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 123 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Stratégie de prédiction des coordonnées : singleton vs n-uplet .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 125 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Stratégie de prédiction des coordonnées : classification vs régression .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 126 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Stratégie de prédiction de la classe des objets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 133 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Résultats et discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 134 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 138 8 C O N C L U S I O N S E T P E R S P E C T I V E S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 139 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Perspectives .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 141 B I B L I O G R A P H I E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 143 xi L I S T E D E S F I G U R E S Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Pages présentant des difficultés de traitement : pages arrachées, dé- gradées et parties de pages manquantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Chaîne de traitement standard impliquant une détection de lignes de texte, une reconnaissance du texte manuscrit (HTR) suivi d’une détection des entités nommées (NER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Détection d’objets sur l’image de la page 17 recto du Livre d’heures Horae ad usum Romanum, Bibliothèque nationale de France, Dépar- tement des manuscrits, NAL 3111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Architecture du modèle LeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 15 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Architecture du modèle AlexNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 15 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Architecture du modèle VGG-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 16 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Architecture du modèle ResNet-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 16 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Schéma d’une convolution 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 17 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 Système R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 18 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 Système YOLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 20 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 Architecture du modèle U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 24 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 Schéma d’une convolution transposée 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 25 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 Schéma d’une convolution dilatée 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 25 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 Architecture du modèle dhSegment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 27 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12 Architecture du modèle de Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 28 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13 Architecture du modèle Transformer original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 33 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='14 Architecture du modèle Vision Transformer original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 36 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='15 Système Pix2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 37 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='16 Apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 41 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Représentation des modélisations d’une ligne de texte proposées dans la littérature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 47 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Visualisation des différents taux de relâchement détectés dans les jeux de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les taux de relâchement indiquent la quantité de fond présent autour des pixels de texte dans les annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 50 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Masques de segmentation comparés par Peskin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 51 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Deux détections de lignes différentes obtenues pour une même image et obtenant les mêmes scores d’IoU et de F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 53 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Architecture du modèle Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 61 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Détections de lignes obtenues par dhSegment et Doc-UFCN sur l’image de la page 5 verso du Livre d’heures Horae .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 65 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Impact du pré-entraînement de Doc-UFCN, évalué sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 68 xii Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Annotations manuelles pour la détection et la classification d’actes sur les jeux de données Balsac et Himanis-Act.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 72 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Chaîne de traitement proposée pour la détection et la classification d’actes avec l’utilisation du contenu textuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 73 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Processus de génération d’annotations pour une image du jeu de don- nées de Bozen.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 83 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Détections de lignes produites sur une image du jeu de données Horae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 88 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Détections de lignes produites sur une image du jeu de données Bozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 92 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Détections de lignes produites par les modèles génériques et spécifiques sur une image du jeu de données ScribbleLens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 93 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Détections de lignes produites par les modèles génériques Doc-UFCN et dhSegment sur une image du jeu de données HOME-Alcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 94 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 Simulation des scores, pour deux prédictions avec et sans fusion, sur une image du jeu de données HOME-NACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 97 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 Résultats de reconnaissance niveau ligne obtenus par les modèles gé- nériques Doc-UFCN, dhSegment et ARU-Net sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 99 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Deux images issues du jeu de données Horae avec leurs prédictions et la variance pour N =10 prédictions avec dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 106 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Courbes de rejet présentant l’évolution des performances du modèle de détection de pages de référence sur l’ensemble de test Horae-test-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Courbes présentées pour les estimateurs DAP et DOV en fonction du nombre de prédictions avec dropout N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 111 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Courbes de rejet présentant l’évolution du score mAP en fonction du taux de rejet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes présentent les résultats du modèle de détection de pages de référence sur l’ensemble de test Horae-test-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 112 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Évolution des performances de détection de pages sur l’ensemble de test Horae-test-300 pendant les itérations d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 114 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Évolution des performances de détection de lignes de texte sur l’en- semble de test du jeu de données Hugin-Munin pendant les itérations d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 116 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 Évolution des performances de détection de pages sur l’ensemble de test Horae-test-300 pendant les itérations d’apprentissage actif pour différentes stratégies de sélection de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 117 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 Évolution des performances de détection de lignes de texte sur l’en- semble de test du jeu de données Hugin-Munin pendant les itérations d’apprentissage actif pour différentes stratégies de sélection de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='118 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Représentation de différentes modélisations de la position et de la forme des objets à détecter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Exemple pour la détection d’une ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 123 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Exemple de séquence à deux classes : paragraphe et ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’ordre de prédiction préserve la hiérarchie des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 127 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Architecture du modèle Doc2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 129 xiii Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Détections de lignes produites par le modèle Doc2Seq, sélectionné sur les valeurs du CER, sur quatre images de l’ensemble de test du jeu de données IAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 137 xiv L I S T E D E S TA B L E A U X Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Tableau récapitulatif des différents jeux de données utilisés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 44 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Tableau récapitulatif du type d’annotation des différents jeux de don- nées utilisés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 49 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Métriques d’évaluation utilisées dans les récents travaux liés à la dé- tection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 52 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Statistiques des jeux de données utilisés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 63 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Résultats obtenus par Doc-UFCN et dhSegment au niveau pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ré- sultats donnés sur les ensembles de test pour la tâche de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 65 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Temps d’inférence rapportés pour Doc-UFCN et dhSegment calculés sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 66 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Résultats obtenus par Doc-UFCN et dhSegment au niveau pixel pour la tâche de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats montrent les performances des modèles génériques sur les ensembles de test avec et sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 67 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Étude ablative de Doc-UFCN sur la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 69 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 Impact du taux de dilatation dans les blocs d’encodeur de Doc-UFCN sur la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 70 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 Impact de la taille des images en entrée de Doc-UFCN sur la détection des lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 70 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 Statistiques des jeux de données utilisés pour la détection d’actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 72 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 Résultats du modèle générique de détection de lignes de texte sur l’ensemble de test du jeu de données Balsac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 74 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 Résultats de reconnaissance de textes manuscrits sur les jeux de don- nées Balsac et Himanis-GMV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 75 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 Résultats de classification des lignes de texte sur les jeux de données Balsac et Himanis-Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 76 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12 Résultats de détection d’actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 77 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13 Résultats obtenus par Doc-UFCN et le système de Prieto sur le jeu de données Himanis-Act avec et sans l’information textuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 78 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Statistiques des jeux de données utilisés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 81 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Comparaison des systèmes Doc-UFCN, dhSegment et ARU-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 86 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Résultats au niveau pixel obtenus par les systèmes Doc-UFCN, dh- Segment et ARU-Net sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 87 xv Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Résultats au niveau pixel obtenus par Doc-UFCN avec et sans unifor- misation des labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 89 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Résultats au niveau ligne obtenus par les systèmes Doc-UFCN, dh- Segment et ARU-Net sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 91 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 Résultats au niveau ligne obtenus par Doc-UFCN avec et sans unifor- misation des labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 94 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 Résultats de reconnaissance niveau page obtenus par les systèmes Doc- UFCN, dhSegment et ARU-Net sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 96 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 Résultats de reconnaissance niveau page obtenus par Doc-UFCN avec et sans uniformisation des labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 98 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 Résultats de reconnaissance niveau ligne obtenus par les systèmes Doc- UFCN, dhSegment et ARU-Net sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 99 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 Résultats de reconnaissance niveau ligne obtenus par Doc-UFCN avec et sans uniformisation des labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 100 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Statistiques des jeux de données utilisés pour la détection de pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 108 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Résultats de détection de pages obtenus par le modèle de référence entraîné sur le jeu de données READ-BAD* et évalué sur les jeux de données READ-BAD* et Horae-test-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 110 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Résultats de détection de lignes de texte obtenus par le modèle de référence entraîné sur 19 jeux de données et évalué l’ensemble de test du jeu de données Hugin-Munin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 110 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Résultats des modèles de détection de pages sur l’ensemble de test Horae-test-300 après apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 114 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Résultats des modèles de détection de lignes de texte sur l’ensemble de test du jeu de données Hugin-Munin après apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 116 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 Résultats des modèles de détection de pages sur l’ensemble de test Horae-test-300 après apprentissage actif et pour différentes stratégies de sélection de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 117 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 Résultats des modèles de détection de lignes de texte sur l’ensemble de test du jeu de données Hugin-Munin après apprentissage actif et pour différentes stratégies de sélection de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 118 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Tableau récapitulatif de différentes modélisations de la position et forme des objets à détecter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 124 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Stratégies de prédiction séquentielle des rectangles englobants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 125 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 Statistiques du jeu de données IAM utilisé pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 133 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 Résultats de reconnaissance de textes manuscrits sur le jeu de données IAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 134 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 Résultats des modèles de détection de lignes sur le jeu de données IAM, donnés en fonction du critère de sélection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 19 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 Architecture FCN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 23 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 Convolution transposée .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 24 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 Convolution dilatée .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 25 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 Système dhSegment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 26 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 Système de Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 27 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 Architecture Transformer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 33 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13 Encodage positionnel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 34 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='14 Architecture Vision Transformer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': 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+page_content=' Boillet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Kermorvant et D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Stutzmann (mai 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Books of Hours : the First Li- turgical Corpus for Text Segmentation ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : 12th Language Resources and Evaluation Conference (LREC), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 776-784 — M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Boillet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Kermorvant et T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Paquet (jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Multiple Document Datasets Pre-training Improves Text Line Detection With Deep Neural Networks ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : 25th International Conference on Pattern Recognition (ICPR), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2134-2141 — M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Boillet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Maarand, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Paquet et C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Kermorvant (sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Inclu- ding Keyword Position in Image-Based Models for Act Segmentation of Historical Registers ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : 6th International Workshop on Historical Document Imaging and Processing (HIP), 31–36 — J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Debezia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Boillet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Kermorvant et Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Barral (sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Drilling a Large Corpus of Document Images of Geological Information Extraction ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : Machine Learning and Principles and Practice of Knowledge Discovery in Database (ECML PKDD), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 525-530 — M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Boillet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Kermorvant et T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Paquet (mars 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Robust Text Line Detection in Historical Documents : Learning and Evaluation Methods ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : Interna- tional Journal on Document Analysis and Recognition (IJDAR), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1433-2825 — M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Boillet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Kermorvant et T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Paquet (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Confidence Estimation for Document Object Detection ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : Submitted to Pattern Recognition Letters (PRL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='xix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='A C R O N Y M E S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='AP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Average Precision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='CER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Character Error Rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Convolutional Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='DAP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Dropout Average Precision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='DLA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Document Layout Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='DOV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Dropout Object Variance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='FCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Fully Convolutional Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='HTR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Handwritten Text Recognition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='IoU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Intersection-over-Union ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='mAP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Mean Average Precision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='mAP-RFR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Mean Average Precision - Random Forest Regressor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Multi-Layer Perceptron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='NER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Named Entity Recognition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='OCR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Optical Character Recognition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='PCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Posterior Probability-based Confidence Estimator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='WER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Word Error Rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='xxi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='I N T R O D U C T I O N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 C O N T E X T E Les documents historiques constituent un patrimoine précieux que les archives, biblio- thèques et certaines entreprises cherchent à protéger, préserver et rendre accessible au plus grand nombre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Après de nombreuses années de numérisation, des millions d’images de do- cuments sont maintenant disponibles dans le monde entier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le contenu de ces images est cependant souvent compréhensible uniquement par des experts, qui travaillent à rendre ac- cessible cette grande quantité de contenus afin de permettre aux chercheurs et au grand public de travailler plus facilement et efficacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, un long et coûteux travail de transcription manuelle des documents est souvent nécessaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Afin de rendre cette tâche plus efficace, de nombreuses institutions cherchent à automatiser ce processus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Grâce aux nouvelles technologies, et notamment l’amélioration majeure des méthodes d’apprentissage profond, il devient désormais possible de transformer automatiquement les documents originaux en documents digitaux, qui peuvent facilement être lus, traduits ou encore dans lesquels il est possible de faire des recherches avancées, tout en nécessitant une quantité plus raisonnable de travail de transcription manuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le même temps, ces évolutions ouvrent de nouvelles perspectives de recherche à la communauté du traitement de document en mettant en évidence des documents complexes pour lesquels les avancées récentes restent encore insuffisantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les différentes tâches liées au traitement automatique de documents numérisés telles que l’analyse de la mise en page (Document Layout Analysis (DLA)) ou la reconnaissance de texte (Handwritten Text Recognition (HTR)) sont des problématiques étudiées depuis de nombreuses années.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Des solutions industrielles existent déjà mais sont souvent limitées à Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Pages présentant des difficultés de traitement : pages arrachées, dégradées et parties de pages manquantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À gauche et au centre, images 4 et 141 du Cartulaire de la famille de Boussac 1et, à droite, pages 14 verso et 19 recto du Livre d’heures du Vatican 09488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://bvmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='irht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='fr/resultRecherche/resultRecherche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='COMPOSITION_ID=28605 1 cimefcndpatewleSeae 135 peaaarayteudeyortapalarafujponarcnaerofa 1262 fugu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Ya anaCetamacutzCagbrenaggepaAdond anca astBinrglootnaerovnuederla nieeBeSonatCanaialey 6F116OC RIGH 14V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 19r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' CopyrightBiblioteca Apostolica Vaticana http://digi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='vatlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='it/view/MSSVat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9488/0016 poweredbyAMLAD·NTTDATA2 I N T R O D U C T I O N D´etection des lignes Extraction des lignes HTR NER Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Chaîne de traitement standard impliquant une détection de lignes de texte, une recon- naissance du texte manuscrit (HTR) suivi d’une détection des entités nommées (NER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' des documents modernes ou simples (mise en page simple, documents non dégradés).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les récentes avancées en Machine Learning et plus particulièrement en Deep Learning permettent désormais de lever ces limitations et d’améliorer la qualité des traitements automatiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces méthodes nécessitent cependant une quantité importante de données annotées manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les documents dont les mises en page sont simples et dont il est facile et rapide d’en annoter de grandes quantités, le traitement automatique a obtenu de très bons résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au contraire, les jeux de données disponibles pour le traitement de documents plus complexes, tels que des documents historiques, sont très réduits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela est principalement dû au fait que les documents sont très variés, et donc coûteux à annoter manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, comme montré sur la Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, les conditions de conservation et de numérisation peuvent mener à des manuscrits abîmés avec notamment des pages tâchées, arrachées ou dégradées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour toutes ces raisons, de nombreuses recherches s’intéressent à améliorer le traitement automatique de tels documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Avoir une version digitale d’un manuscrit historique permet, entre autres, de pouvoir faire de la recherche par mots-clés ou de retrouver des noms de personnes ou encore des dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour parvenir à cela, plusieurs étapes sont appliquées à chaque page numérisée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une chaîne de traitement utilisée dans de nombreux projets est présentée sur la Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 analyse de la mise en page En entrée de la chaîne, nous disposons d’une image d’une page ou d’une double-page d’un document numérisé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une première étape réalisée consiste à analyser la mise en page du document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’objectif de ce premier module est d’identifier les diverses régions physiques d’un document et leurs caractéristiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela revient donc à détecter différents éléments sur l’image tels que les blocs de texte, images, graphiques ou encore lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces régions ne s’excluent pas mutuellement et une région peut contenir d’autres types de régions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En plus de ces entités physiques, des étiquettes fonctionnelles ou logiques telles que des titres ou légendes peuvent être attribuées à certaines de ces régions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le processus d’analyse Thauuc tth giniLn B贝人 Bleo Bge cloghLen dmuhLe neuf fevrier mil neuf centLe date neuf fevrier mil neuf cent1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 C O N T E X T E 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Détection d’objets sur l’image de la page 17 recto du Livre d’heures Horae ad usum Romanum, Bibliothèque nationale de France, Département des manuscrits, NAL 3111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Source https://gallica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='bnf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='fr/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' de la structure et de la mise en page d’un document tente donc de décomposer l’image d’un document donné en ces régions et de comprendre leurs rôles fonctionnels et leurs relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans de nombreux cas d’usage, l’analyse de la mise en page d’un document revient à détecter les lignes de texte dans le but d’appliquer un reconnaisseur sur ces lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, certaines études s’intéressent également à d’autres éléments tels que les miniatures et initiales dans les livres d’heures (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) (exemple Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3), les actes (Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) ou encore les tableaux de recensement (Constum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces tâches nécessitent des traitements plus spécifiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, la détection d’actes est souvent accompagnée d’une classification selon le type d’acte présent (baptême, mariage, décès).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il en est de même pour les tableaux, qui peuvent être traités de différentes manières : détection des lignes uniquement ou conjointement avec les colonnes ou encore détection des cellules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 reconnaissance de texte Une fois les lignes obtenues, elles subissent chacune un traitement menant à une version digitale du texte manuscrit (HTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, ce texte peut être conservé tel quel, traduit dans une autre langue, ou encore traité afin d’obtenir les entités présentes dans le document (Named Entity Recognition (NER)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Des recherches récentes commencent à proposer des systèmes qui s’affranchissent de la détection des lignes de texte et permettent de transcrire le texte de l’image complète (Bluche, 2016 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Coquenet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Yousef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Malgré des premiers résultats prometteurs, ces systèmes sont encore limités à des documents simples ou avec une grande régularité de mise en page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' tnma lotme ntabusndmoiti hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content="amauutulhsnrtus eterpobantnarouh omtouuaammMiniature L'igne Text eterpobatntmaouh Marge ornée4 I N T R O D U C T I O N Dans cette thèse, nous étudions les tâches liées à l’analyse de la mise en page telles que la détection de lignes de texte, d’actes ou encore de pages." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous nous concentrons sur l’applica- tion de méthodes basées sur les réseaux de neurones profonds pour la détection d’objets dans les images de documents, principalement historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De nombreux systèmes permettant de résoudre ces différentes tâches ont été proposés dans la littérature (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Mechi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021), cependant, ils sont souvent évalués uni- quement sur la tâche de détection de lignes de texte, et sont difficilement généralisables à des documents aux structures plus diverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, dans cette thèse, nous cherchons à développer des modèles plus génériques et à réaliser des évaluations plus complètes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les systèmes actuellement utilisés pour la détection d’objets dans les images de scènes naturelles, tels que les modèles YOLO (Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2017 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2018) et R-CNN (Girshick, 2015 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2014 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015), sont difficilement applicables aux documents historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une des raisons à cela est l’importante quantité de données annotées qu’ils nécessitent pour être entraînés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, il devient nécessaire de développer des systèmes moins complexes en termes de nombre de paramètres, de combiner plusieurs bases, de recourir au Transfer learning (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) ou d’augmenter la quantité de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces différents points ont été étudiés durant la thèse et les conclusions seront présentées dans la suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, des systèmes à base de Transformers (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017) ont commencé à être proposés afin de résoudre plus efficacement les tâches liées aux problèmes séquence-à-séquence telles que la traduction de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À la suite de cela, certains travaux ont adapté ces systèmes aux tâches de vision et ont montré qu’ils permettent d’obtenir de très bonnes performances pour la classification d’images (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021) ou la détection d’objets (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous nous sommes également intéressés à cette catégorie de systèmes, qui permettent d’avoir des sorties structurées des objets prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 C A D R E D E L A T H È S E Cette thèse a été réalisée au sein de l’entreprise Teklia 2 dans le cadre d’une collaboration avec le Laboratoire d’Informatique, de Traitement de l’Information et des Systèmes (LITIS) 3 à l’Université de Rouen Normandie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Teklia a été fondée en 2014 et est spécialisée dans la compréhension automatique de docu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’entreprise travaille sur diverses applications telles que le traitement automatique de documents historiques (livres d’heures, chartes), mais également le traitement de documents plus récents comme des tableaux de recensement de la population française.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les activités de recherche de l’entreprise s’inscrivent dans des projets de recherche français mais aussi internationaux comme les projets HOME 4, HuginMunin 5 et Balsac 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://teklia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='com 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='litislab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='fr 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='history-of-medieval-europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='eu 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://hugin-munin-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='io/ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://balsac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='uqac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='ca 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 O B J E C T I F S E T C O N T R I B U T I O N S 5 L’entreprise possède également une équipe spécialisée dans le développement, qui réalise à la fois des projets pour des clients, mais intègre également les résultats de l’équipe de recherche dans des applications 7 et facilite le travail de recherche en développant, notamment une plateforme d’annotation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’entreprise travaille sur de nombreux projets et les produits des travaux de recherche y sont directement intégrés, et donc appliqués dans de réelles conditions industrielles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour répondre aux demandes des projets, il est nécessaire d’avoir un détecteur d’objets robuste, performant et rapide pour traiter de grandes quantités de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, il n’est pas rare que dans un projet il y ait peu, voire aucune donnée annotée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est donc également nécessaire d’avoir un détecteur assez générique afin de traiter ces documents plus facilement et d’estimer automatiquement la qualité des résultats fournis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 O B J E C T I F S E T C O N T R I B U T I O N S Les défis liés à la tâche de détection d’objets dans des images de documents sont nombreux, d’autant plus dans un cadre dans lequel de nouvelles données sont souvent disponibles, tou- jours plus variées et complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les problématiques auxquelles nous cherchons à répondre sont les suivantes : — Comment détecter efficacement les objets présents dans des images de documents variés, et à partir de peu d’exemples annotés manuellement ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Comment évaluer les modèles de détection pour représenter correctement la qualité des objets prédits ainsi que leurs impacts sur les tâches suivantes ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Comment estimer la confiance d’un modèle de détection quant à la qualité de ses prédictions ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour répondre à toutes ces problématiques, plusieurs contributions ont été proposées durant cette thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles nous permettent de proposer une étude complète de détection d’objets allant de l’annotation manuelle à l’évaluation finale : — Certains réseaux de neurones utilisés pour la détection d’objets fournissent un masque de prédiction où chaque pixel appartient à une classe d’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous proposons un mo- dèle de détection possédant peu de paramètres et rapide en inférence, produisant des masques de prédiction très précis tout en nécessitant un nombre réduit de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — D’autres systèmes plus récents permettent de générer une sortie structurée des objets détectés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Suivant cette idée, nous proposons un second modèle de détection qui montre des performances encourageantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Nous montrons que malgré une grande hétérogénéité entre les documents mais aussi entre leurs annotations manuelles, l’entraînement de réseaux de neurones génériques permet d’obtenir des modèles encore plus performants et applicables à de nouvelles 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://arkindex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='teklia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='com 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://callico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='teklia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='com 6 I N T R O D U C T I O N données sans ré-entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, l’uniformisation des annotations entre les diffé- rents jeux de données permet d’entraîner des modèles de meilleure qualité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Nous proposons d’utiliser des métriques d’évaluation qui sont davantage en accord avec la tâche finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En particulier, nous proposons des métriques liées à la reconnaissance de texte afin d’évaluer les modèles de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Les données annotées sont souvent disponibles en faible quantité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, nous proposons différents estimateurs de confiance et montrons, dans un cadre d’active learning, qu’ils permettent d’obtenir des modèles de détection d’objets plus performants avec moins d’exemples annotés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 O R G A N I S AT I O N D U M A N U S C R I T Cette thèse est composée, outre cette introduction, de six chapitres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chapitre 2 : État de l’art Le chapitre 2 présente un aperçu de l’état de l’art dans plusieurs domaines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une revue des différentes approches de détection de lignes de texte et d’objets dans des images est présentée, allant des premières méthodes de traitements d’images aux plus récents systèmes établis à partir de réseaux neuronaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les récentes méthodes combinant l’utilisation de l’image et du texte pour la détection d’objets sont décrites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, nous y présentons les techniques permettant d’estimer une confiance reflétant la qualité d’une pré- diction, élément crucial lorsque les systèmes de détection sont utilisés en phase de production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chapitre 3 : Entraînement et évaluation des systèmes de détection Nous présentons, dans le chapitre 3, une revue des différents jeux de données utilisés pour la détection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par la suite, nous proposons une étude des stratégies d’entraînement et d’évaluation utilisées par les systèmes récents avec, notamment, le détail des métriques d’évaluation basées sur les pixels et sur les objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chapitre 4 : Détection d’objets dans des images de documents Dans le chapitre 4, nous introduisons une architecture simple, rapide et efficace, mise au point afin de détecter des objets dans les images de documents au niveau pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La détection est réalisée grâce à un réseau de neurones entièrement convolutif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce chapitre décrit les détails d’architecture ainsi que les avantages de celle-ci par rapport aux systèmes existants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, les résultats de différentes expérimentations sur les tâches de détection de lignes de texte et d’actes y sont présentés et discutés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chapitre 5 : Entraînement et évaluation d’un modèle robuste de détection d’objets Le chapitre 5 propose une étude avancée des techniques d’entraînement et d’évaluation des systèmes de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il expose la grande hétérogénéité et les incohérences des an- notations des différents jeux de données actuellement disponibles, et présente une technique 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 O R G A N I S AT I O N D U M A N U S C R I T 7 d’uniformisation des annotations mise au point durant la thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, ce chapitre met en lumière les limitations des métriques d’évaluation actuellement utilisées et détaille plusieurs métriques que nous proposons afin de lever ces limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats d’entraînements de modèles de détection de lignes de texte à grande capacité de généralisation sont enfin présentés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chapitre 6 : Estimation de la confiance des prédictions Nous proposons, dans le chapitre 6, différents estimateurs de confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un premier temps, des estimateurs basés sur le modèle de détection d’objets entraîné sont présentés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Des estimateurs basés sur un apprentissage externe au détecteur sont ensuite détaillés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une étude comparative des différentes approches est menée sur deux tâches de détection de pages et de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chapitre 7 : Détection séquentielle d’objets dans des images de documents Le chapitre 7 présente une seconde architecture de détection d’objets, celle-ci étant établie à partir de Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les détails de l’architecture sont présentés ainsi que la justification des choix de conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Des premiers résultats d’expérimentations sont également présentés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chapitre 8 : Conclusions et perspectives Dans le chapitre 8, nous concluons sur l’ensemble des travaux proposés et énonçons des pistes de recherche complémentaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2 É TAT D E L’ A RT Les recherches axées autour de la mise en place et de l’amélioration de modèles de détection d’objets sont toujours très actives, et ont motivé un nombre croissant de travaux ces dernières années du fait d’importantes avancées dans le domaine de l’apprentissage automatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce chapitre, nous présentons une étude des travaux liés à la détection d’objets en évoquant les premiers systèmes permettant de séparer les blocs de texte du fond des images ainsi que les méthodes les plus récentes basées sur des réseaux de neurones profonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous passons en revue différents systèmes d’estimation de la qualité des prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, nous décrivons les méthodes ad hoc de détection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons ensuite les méthodes proposées à base d’apprentissage profond, avec notamment les approches pixel dans la section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 et celles à base de Transformers en section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, la section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 présente les travaux permettant d’estimer la qualité des prédictions, peu de travaux ayant été publiés pour la tâche de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S La mise en page d’un document fait référence à la position physique et aux limites des différentes régions dans l’image du document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le processus d’analyse de la mise en page d’un document vise à décomposer une image de document en une hiérarchie de régions, telles que les figures, l’arrière-plan, les blocs de texte, les lignes de texte, les mots, les caractères, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Depuis plusieurs années, différentes méthodes permettant de détecter des objets dans des images de documents ont émergé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces différentes approches peuvent être divisées en deux groupes : les méthodes ad hoc et les méthodes par apprentissage automatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, dans chacun de ces deux groupes, il existe des algorithmes dits ascendants et descendants (Namboodiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2007 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les algorithmes ascendants partent des plus petits composants d’un document (pixels ou composantes connexes) et les regroupent de manière itérative pour former des régions plus grandes telles que les caractères, qui sont ensuite regroupés en mots, lignes ou blocs de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En revanche, les algorithmes descendants partent de l’image complète du document et la divisent itérativement en sous-images pour former des régions de plus en plus petites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La procédure de découpage s’arrête lorsqu’une certaine condition est vérifiée, les sous-images obtenues à ce stade constituent les résultats finaux de la segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, il existe également des approches hybrides qui utilisent une combinaison de stratégies ascendantes et descendantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 9 10 É TAT D E L’ A RT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 méthodes ad hoc Les approches ad hoc sont basées sur la combinaison de différentes techniques d’analyse d’image telles que le regroupement, les profils de projection ou encore le filtrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles sont établies pour un type d’images donné et sont peu généralisables à un grand nombre et une grande variété d’images de documents mais sont encore aujourd’hui utilisées (Eskenazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les premières méthodes ayant vu le jour permettaient de séparer les contenus textuels des contenus graphiques d’une image sans nécessiter d’annotations manuelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Parmi les algo- rithmes descendants, le Run-Length Smoothing Algorithm (RLSA) (Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 1982) a été proposé pour segmenter les pages de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet algorithme fonctionne sur des images bi- naires dans lesquelles deux pixels noirs voisins éloignés d’une distance maximale donnée sont fusionnés en une séquence continue de pixels noirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le RLSA est d’abord appliqué ligne par ligne, puis colonne par colonne, et les deux bitmaps résultants sont combinés en appliquant une opération logique "ET" à chaque position de pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’inconvénient de cet algorithme est qu’il ne peut être utilisé que pour extraire de petits blocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par la suite, la méthode du XY- Cut (Nagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 1984) a été proposée afin de détecter les blocs de texte dans des images en niveaux de gris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode consiste à utiliser une projection horizontale et verticale des valeurs des niveaux de gris des pixels afin de trouver les espaces interlignes et intercolonnes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les projections sont faites de manière itérative menant à des objets homogènes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette tech- nique permet d’obtenir une détection de grande qualité mais est limitée à des documents dont la mise en page est simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, elle est incapable de prédire des objets corrects sur des images dans lesquelles les lignes sont mal alignées, ou si le document est légèrement incliné.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Akindele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (1993) ont proposé une amélioration de cette méthode afin de corriger l’in- clinaison des lignes, cependant, d’autres problèmes persistent tels que la difficulté du système à traiter des documents comportant des illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces méthodes semblent difficilement applicables à des documents historiques qui ne sont pas que textuels et qui ont des mises en page complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour résoudre le problème posé par les images de pages obliques, Pavlidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (1992) ont proposé une méthode basée sur les « flux blancs ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils émettent l’hypo- thèse que les colonnes de texte d’une page contiennent un type unique de données (texte ou illustration) et qu’elles sont suffisamment espacées pour être distinguées des autres espaces tels que l’espacement entre les mots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La méthode identifie donc les larges espaces blancs afin d’estimer l’angle d’inclinaison de la page puis de localiser les objets comme étant les régions entre ces espaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode permet également de traiter des documents plus complexes contenant, entre autres, des illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les systèmes présentés ci-dessus permettent de traiter des documents ayant des mises en page de type Manhattan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit de pages ayant des composants de formes arbitraires (rectangulaires) où les segments des blocs sont parallèles ou perpendiculaires les uns par rapport aux autres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette thèse, nous nous intéressons principalement aux documents historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces méthodes semblent donc difficilement applicables à de tels documents dont les colonnes contiennent rarement des types uniques de données, comme montré sur les Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 et 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3, et dont les mises en page sont non-Manhattan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 11 Les méthodes ascendantes permettent de traiter des documents beaucoup plus variés aux mises en page complexes mais sont en général plus lentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une des premières méthodes, présentée par Kise et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (1998), se base sur le diagramme de Voronoi pour la segmentation d’images de pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les auteurs détectent tout d’abord les points des bords des composantes connexes et construisent un diagramme de Voronoi à partir de ces points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les arêtes de Voronoi détectées entre des caractères, mots et lignes de texte d’un même bloc sont ensuite filtrées pour garder uniquement celles qui séparent les blocs du document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un désavantage à cette méthode est qu’elle segmente parfois les illustrations et les titres ayant des polices d’écriture larges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' O’Gorman (1993) a présenté DocStrum, qui repose sur un regroupement des plus proches voisins appliqué aux composantes connexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par la suite, ces deux propo- sitions ont été conjointement utilisées et améliorées par Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2009) avec leur système appelé Voronoi++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a été mis au point pour répondre au manque d’adaptation des systèmes existants aux variations de taille, d’orientation et de distance des composants d’une page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au lieu d’utiliser des relations linéaires entre la distance et le rapport de surfaces des composantes connexes, les auteurs montrent que la détermination dynamique de ces relations et la combinaison des caractéristiques angulaires et des caractéristiques de voisinage, ces dernières venant de l’approche de DocStrum, améliorent la précision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour lever les limitations liées aux systèmes présentés ci-dessus telles que le temps de traitement, d’autres méthodes ont émergées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celles-ci sont basées sur des algorithmes plus robustes face aux images de documents couleurs et en niveaux de gris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, elles ne sont plus limitées aux documents possédant des structures et contenus simples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette optique, Coüasnon (2006) a conçu et publié un langage de grammaire de mise en page appelé DMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il permet de décrire une grande variété de mises en page et l’analyseur syn- taxique associé reconnaît cette disposition dans une image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La grammaire permet également d’associer une étiquette à chaque région.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par la suite, la méthode a été améliorée (Lemaitre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2008) en intégrant une approche multirésolution lui permettant de segmenter des lettres manuscrites et d’identifier les lignes de texte dans des documents administratifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la même idée, Shafait et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2008) ont proposé un autre algorithme de grammaire basé sur une formulation probabiliste de la mise en page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’utilisateur définit un ensemble de coupes horizontales et verticales dont la position est définie de manière approximative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, pour chaque image, un ajustement probabiliste est effectué pour obtenir les régions finales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet algorithme est capable de segmenter des mises en page serrées avec de faibles marges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que cette méthode ainsi que DMOS aient obtenu de très bonnes performances, les systèmes reposent sur l’hypothèse que les documents à traiter ont une mise en page homogène puisqu’ils nécessitent que l’utilisateur définisse des règles de mise en page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’autres systèmes ont ensuite été proposés afin de traiter des documents complexes et ne nécessitant pas de modèle de mise en page prédéfini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par exemple, Louloudis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2007) ont utilisé la transformée de Hough sur un ensemble de composantes connexes sélectionnées pour extraire les lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette approche, basée sur la transformée de Hough, n’est adaptée qu’aux images de documents où les lignes ne sont pas incurvées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Journet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 12 É TAT D E L’ A RT (2008) utilisent une approche ascendante basée sur les textures des images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils extraient cinq caractéristiques liées aux fréquences et orientations calculées à quatre résolu- tions, ainsi chaque pixel de l’image possède 20 valeurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils utilisent ensuite un algorithme de groupement afin de regrouper les pixels correspondant à des zones homogènes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont testé leur méthode sur des documents modernes et historiques, et ont souligné l’importance d’une approche multirésolution pour réduire le bruit dans les techniques ascendantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2009), les auteurs proposent une technique appelée ALCM (Adaptive Local Connectivity Map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils utilisent des filtres directionnels orientables pour détecter les lignes de texte et appliquent des post-traitements heuristiques pour séparer les lignes connectées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode descendante a obtenu des résultats intéressants sur la détection de lignes de texte, l’algorithme ayant été conçu pour résoudre les problèmes particulièrement complexes observés dans les documents manuscrits, notamment les lignes de texte qui fluctuent, se touchent ou se superposent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par la suite, Erkilinc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2012) ont proposé une méthode de segmentation robuste face aux fonds et aux structures complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette approche permet de résoudre un problème de détection à trois classes : texte, photographie et ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, l’image subit une étape de prétraitement qui consiste à réaliser un filtrage, une conversion de l’espace couleur et une correction gamma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les éléments sont ensuite détectés grâce à plusieurs techniques telles que la transformée en ondelettes et le codage par plages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les objets détectés sont enfin combinés par un algorithme de K-moyennes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode de classification en blocs et en pixels a montré de bons résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, comme la plupart des méthodes présentées ici, elle consiste en plusieurs opérations successives et est coûteuse en temps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, cette méthode ne permet de résoudre qu’un problème spécifique avec trois classes très distinctes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, une autre méthode ascendante qui a obtenu de bons résultats pour détecter les lignes de texte est décrite dans Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’approche est basée sur les super-pixels pour obtenir des composantes connexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les auteurs définissent une fonction de coût pour agréger les super-pixels en une ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode a gagné la compétition de l’International Conference on Document Analysis and Recognition (ICDAR) sur la détection des lignes de texte (Murdock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Concernant les méthodes hybrides, un travail récent proposé par Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015) utilise la méthode Multilevel Homogeneous Structure (MHS), et a remporté la compé- tition de segmentation de documents complexes en 2015 (Antonacopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La méthode implique à la fois l’analyse en composantes connexes et l’analyse des espaces blancs (arrière-plan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, l’image est binarisée puis les composantes connexes sont détectées et celles considérées de manière fiable comme étant du bruit ou des régions sans texte sont filtrées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une classification multiniveaux est effectuée, basée sur l’analyse des régions homogènes multiniveaux et des espaces blancs, pour identifier toutes les composantes textuelles et non textuelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode a montré de bonnes performances sur la compétition, notamment pour sa capacité à manquer très peu de régions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même si la plupart de ces méthodes ont obtenu de bons résultats sur un jeu de données spécifique, elles doivent être affinées manuellement, ce qui est une tâche fastidieuse et dépend 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 13 généralement de l’ensemble de données considéré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, une fois mises en place, ces mé- thodes sont souvent difficiles à maintenir et à améliorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, la plupart des algorithmes mentionnés ci-dessus ne créent pas de descriptions hiérarchiques ou ne permettent pas aux utilisateurs de préciser des informations sur la structure du document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, à part pour les modèles à base de grammaire, ils ne fournissent pas de méthodes d’estimation des pa- ramètres de l’algorithme à partir de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En d’autres termes, ils ne sont pas dotés de capacités d’apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 méthodes par apprentissage profond Pour répondre à ces difficultés, des méthodes basées sur un apprentissage ont été proposées afin d’apprendre automatiquement la variabilité des documents à partir de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nos travaux se positionnent dans ce cadre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les méthodes par apprentissage automatique sont actuellement principalement constituées d’algorithmes de réseaux de neurones profonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces algorithmes permettent à la fois d’ap- prendre automatiquement les caractéristiques importantes des images et d’effectuer la tâche requise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont tendance à être une « boîte noire » dont le fonctionnement est difficile à expliquer, cependant ils permettent de traiter des documents complexes que les méthodes ad hoc sont incapables de traiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les approches par apprentissage profond ont obtenu de bons résultats dans de nombreux domaines d’application (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015), ainsi, de nombreux travaux ont étudié leur utili- sation pour la détection d’objets dans les images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Puisque de multiples recherches s’orientent autour de la détection d’objets dans des images en général et non spécifiquement dans des images de documents, la section suivante passe en revue quelques travaux dans ces domaines connexes de détection d’objets et de textes dans des images de scènes naturelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le domaine de la vision par ordinateur, la littérature sur la détection d’objets peut être divisée en trois catégories principales : les systèmes basés sur la proposition de régions, l’estimation de la position des boîtes englobantes par régression et la détection au niveau du pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' proposition de régions Pour la tâche de détection de texte dans des images de scènes naturelles, les premiers travaux basés sur des approches par apprentissage profond utilisent une méthode de fenêtre glissante (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Des parties d’images sont d’abord extraites à l’aide d’une fenêtre glissante, puis elles sont étiquetées grâce à un réseau de neurones profond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’utilisation d’une fenêtre glissante induit un temps de traitement élevé et limite le contexte qui peut être utilisé pour prendre une décision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour limiter le temps de traitement, une solution consiste à utiliser un prétraitement pour extraire les candidats et ensuite prendre une décision pour chacun de ces candidats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est la méthode utilisée par Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2014), qui extrait les candidats grâce aux Maximally Stable Extremal Regions (MSER) et les classe à l’aide d’un réseau de neurones convolutif (Convolutional Neural Network (CNN)) (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’architecture CNN est décrite à la fin de cette section, dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 14 É TAT D E L’ A RT De la même manière, l’idée d’extraire les candidats avant de les classer a été utilisée pour la détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces systèmes, basés sur la proposition de régions, consistent en trois étapes consécutives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, un ensemble de propositions de régions indépendantes de la catégorie est généré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, un CNN est appliqué sur ces régions pour extraire les informations significatives, et un classificateur prédit la classe de chaque proposition de région.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette stratégie a été proposée pour la première fois par Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2014) avec leur système R-CNN, détaillé dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4, et appliquée aux images de scènes naturelles des jeux de données VOC 2010-2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Malgré le développement de systèmes plus avancés (Fast R-CNN (Girshick, 2015), Faster R-CNN (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) et Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017)), cette méthode a été peu adoptée par la communauté du traitement d’images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, ces systèmes sont bien adaptés aux images de scènes naturelles où seuls quelques objets sont présents sur les images, contrairement aux images de documents qui contiennent de nombreux objets de toutes tailles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, malgré différentes améliorations qui ont permis d’accélérer ces systèmes, ils restent complexes et peu efficients, d’où l’introduction des méthodes dites « one stage » où l’on s’abstient de l’étape de proposition de régions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certains de ces systèmes sont présentés dans les paragraphes suivants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – ARCHITECTURE CNN Définition Un réseau de neurones profond est une succession de plusieurs couches où chaque couche est généralement composée d’une fonction paramétrée suivie d’une fonction de non-linéarité (fonction d’activation), chaque couche calculant une nouvelle repré- sentation de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas d’un réseau neuronal convolutif (CNN), les fonctions paramétrées sont des opérations de convolution, détaillées dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La partie convolutive d’un CNN permet d’extraire et de compresser les caractéristiques de l’image d’entrée grâce à des couches de regroupement (pooling, expliqué dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Avantages — Un CNN est capable de capturer avec succès les dépendances spatiales d’une image par l’application de filtres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’architecture s’adapte au mieux à l’ensemble des données grâce à la réduction du nombre de paramètres impliqués et à la réutilisation des poids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — L’architecture CNN rend possible l’apprentissage de modèles profonds avec re- lativement peu de paramètres grâce au partage des poids entre les couches convolutives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Chaque filtre d’une couche de convolution est appliqué à l’ensemble de l’image d’entrée, ainsi le traitement d’une image par un CNN est invariant par transla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Comparés à d’autres algorithmes de traitement d’image, les CNN utilisent rela- tivement peu de prétraitement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 15 Inconvénients — Pour entraîner un CNN, il est souvent nécessaire d’avoir de nombreuses données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Comme pour la plupart des systèmes à base de réseaux neuronaux, il peut être coûteux en mémoire et en temps d’entraîner un CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Exemples de systèmes de type CNN — LeNet (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (1998)) : LeNet est la première architecture CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a été développé en 1998 et a été appliqué avec succès à la tâche de reconnaissance de chiffres manuscrits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’architecture LeNet se compose de plusieurs couches de convolution et de regroupement (pooling), suivies d’une partie entièrement connectée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle comporte cinq couches de convolution suivies de deux couches entièrement connectées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Schéma de l’architecture du modèle LeNet, issu de LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (1998), pour une image d’entrée de taille 32×32 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — AlexNet (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2012)) : AlexNet est l’architecture d’apprentis- sage profond qui a popularisé le CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le réseau AlexNet a une architecture très comparable à celle de LeNet, mais est plus profond, plus grand et comporte des couches convolutives empilées les unes sur les autres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' AlexNet a été utilisé pour remporter l’ImageNet Large Scale Visual Recognition Challenge (ILSVRC) en 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' AlexNet est composé de cinq couches convolutives avec une combinaison de couches de max-pooling, de trois couches entièrement connectées et de deux couches de dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le nombre total de paramètres dans cette architecture est d’environ 60 millions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Schéma de l’architecture du modèle AlexNet, issu de Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2012), pour une image d’entrée de taille 224×224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ici, deux cartes graphiques sont utilisées, une traite la partie haute de l’image et l’autre la partie basse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cs: ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' maps 16@10xi10 Ci: foature mapg INPUT S4: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' maps 16@5x5 6@28x28 32x82 s2: t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' mapg cs: layer t6: layer 120 OUIPUT 6@iNx 84 10 F ull conneean Ceuss en connectans Convoutong Subsamplng Convolutlang ubsamplng Bull connectbn16 É TAT D E L’ A RT — VGG (Simonyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015)) : VGGNet est un réseau CNN à 16 couches comptant jusqu’à 95 millions de paramètres et entraîné sur plus d’un milliard d’images (1000 classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il prend des images d’entrée de taille 224×224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il nécessite beaucoup de données d’entraînement, ce qui est la principale raison pour laquelle les architectures telles que AlexNet fonctionnent mieux pour la plupart des tâches de classification d’images où les images d’entrée ont une taille comprise entre 100×100 pixels et 350×350 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle VGG est efficace et sert de base solide pour de nombreuses applications en raison de son applicabilité à de nombreuses tâches, notamment la détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ses représentations profondes des caractéristiques sont utilisées dans de nombreuses architectures de réseaux neuronaux telles que YOLO (Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Schéma de l’architecture du modèle VGG-16 (Simonyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015)) pour une image d’entrée de taille 224×224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Schéma extrait de l’article de Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016)) : ResNet a été développé dans le cadre de la compétition pour la tâche de classification de l’ILSVRC 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le réseau contient des connexions résiduelles en plus des couches habituelles d’un CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Outre les tâches de classification d’images, ResNet a été utilisé avec succès pour résoudre des problèmes de traitement du langage naturel comme la complétion de phrases ou la compréhension automatique par l’équipe Microsoft Research Asia en 2016 et 2017 respectivement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Schéma de l’architecture du modèle ResNet-34, issu de He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' tr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 17 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – CONVOLUTION Définition La couche de convolution est le bloc de base utilisé dans les réseaux dits convolutifs (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une couche de convolution permet de générer une nouvelle représentation de l’image d’entrée ou intermédiaire (en sortie de la couche précédente).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, elle possède un ou plusieurs filtres de convolution qui traitent une portion limitée, le champ réceptif, de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque filtre est défini par un ensemble de poids appris durant l’entraînement du modèle et analyse une caractéristique de l’image d’entrée (caractéristique de couleur, de texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, chaque filtre est appliqué à chaque pixel de l’image, calculant une nouvelle représentation pour chacun de ces pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la plupart des cas, le filtre a une taille plus grande que 1 ce qui mène à utiliser du contexte, les pixels voisins, pour calculer la nouvelle représentation du pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Schéma d’une convolution 2D La Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 présente le schéma d’une convolution 2D avec X l’image d’entrée, W le filtre et Y la nouvelle représentation de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cet exemple, le filtre W a une taille 3×3, ce qui implique que, pour calculer la représentation d’un pixel, les valeurs de ses huit pixels voisins sont prises en compte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' x0,0 x1,0 x2,0 x3,0 x0,1 x1,1 x2,1 x3,1 x0,2 x1,2 x2,2 x3,2 x0,3 x1,3 x2,3 x3,3 X w0,0 w1,0 w2,0 w0,1 w1,1 w2,1 w0,2 w1,2 w2,2 W y1,1 y0,1 y1,0 y0,0 Y Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – Schéma d’une convolution 2D avec X l’image d’entrée, W le filtre et Y la nouvelle représentation de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – REGROUPEMENT / POOLING Définition Le pooling consiste à regrouper des représentations locales ou globales en résumant les valeurs de plusieurs pixels en une seule valeur unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les couches de regroupe- ment réduisent les dimensions des données en combinant plusieurs entrées, et ainsi extraient les caractéristiques dominantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’opérations simples, non para- métriques, telles qu’un min (min pooling), un max (max pooling), une somme ou encore une moyenne (average pooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un CNN, ces couches permettent à la fois de réduire la taille des images intermédiaires en résumant les caractéristiques qu’elles contiennent, mais aussi d’avoir davantage de contexte puisque les pixels voisins sont regroupés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 18 É TAT D E L’ A RT Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – SYSTÈME R-CNN Le système R-CNN a été proposé par Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2014) et permet de réaliser de la détection d’objets sur des images à partir de propositions de régions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un ensemble de régions (environ 2 000) est tout d’abord généré grâce à un algorithme de recherche sélective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les caractéristiques importantes de chacune de ces régions sont ensuite extraites par un CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, un SVM linéaire prédit la classe de chaque région.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que ce système ait été utilisé avec succès afin de détecter des objets dans les images de scènes naturelles, il reste très lent car prend en moyenne 47 secondes pour traiter une image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 – Schéma du système R-CNN, issu de Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' régression de boîtes englobantes La détection d’objets dans des images a également été réalisée à l’aide de modèles de prédiction des coordonnées des boîtes englobantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces systèmes, fondés sur un algorithme de régression, ont été introduits pour la première fois par Erhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2014) qui ont pro- posé la méthode MultiBox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celle-ci effectue une régression directe des positions des boîtes englobantes au lieu de s’appuyer sur des propositions d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils utilisent un CNN comme ré- gresseur pour directement prédire un nombre donné de coordonnées de boîtes et une confiance pour chaque boîte correspondant à sa probabilité de contenir un objet d’intérêt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il permet de détecter un nombre variable d’objets superposés de la même classe, la taille des objets n’étant pas limitée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Mais, lorsqu’il est nécessaire de détecter un grand nombre d’objets, le nombre de paramètres du modèle augmente et une grande quantité de données est nécessaire pour l’apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' YOLO et SSD peuvent être considérés comme des variantes de ce concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) ont proposé le modèle YOLO (You Only Look Once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’objectif de YOLO était de détecter et de classifier les objets en un seul traitement et d’être plus rapide que les méthodes R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’image est d’abord divisée en une grille régulière, puis chaque cellule de la grille prédit un nombre prédéfini de boîtes englobantes avec leurs confiances ainsi que les probabilités de classe grâce à un seul réseau neuronal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les détections finales sont les boîtes ayant le score de confiance le plus important et la probabilité de la classe la plus élevée dans cette boîte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système est présenté dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De multiples méthodes ont ensuite étendu l’idée originelle de YOLO (Bochkovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2018) mais très peu ont été appliquées aux images de documents, probablement pour la même raison que celle mentionnée ci-dessus : les images de documents contiennent trop d’objets à détecter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' waped rcgion aetopl ane?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' petson?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' CNN : tyontor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' mo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jput 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='. Hixtiract regjon 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Compute 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Classifty image proposals (-2k) CNN feauncs Tegjons2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 19 D’un autre côté, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) ont proposé SSD (Single Shot MultiBox Detector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le système discrétise l’espace de sortie des boîtes englobantes en un ensemble de boîtes prédé- finies par défaut avec différents rapports d’aspect et échelles par emplacement de carte de caractéristiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au moment de la prédiction, le réseau génère des scores reflétant la présence de chaque catégorie d’objets dans chaque boîte par défaut, et ajuste la boîte pour mieux cor- respondre à la forme de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, le réseau combine les prédictions de plusieurs cartes de caractéristiques de différentes résolutions pour traiter des objets de différentes tailles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' SSD est simple par rapport aux méthodes qui nécessitent des propositions d’objets, car il élimine la génération de propositions et les étapes ultérieures de ré-échantillonnage de pixels ou de caractéristiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode a montré de meilleurs résultats que Faster-RCNN et YOLO sur les données VOC 2017 tout en étant plus rapide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien qu’elles aient montré de très bonnes performances sur des images de scènes naturelles où peu d’objets sont à détecter, ces méthodes sont moins adaptées au traitement d’images de documents où il y a souvent un grand nombre d’éléments à localiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certains travaux ont tout de même adapté ces systèmes aux images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour la détection de lignes de texte, les premières contributions ont été présentées par Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016a) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016a), les auteurs proposent une approche basée sur MultiBox pour détecter les boîtes englobantes des lignes de texte en utilisant des poids partagés afin de permettre au système d’être entraîné sur une quantité de données annotées réduite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme les modèles YOLO et SSD, les sorties sont attribuées à des régions locales de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, le modèle est capable de prédire les objets dans sa région de support, ou en dehors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016b) proposent l’utilisation d’un réseau neuronal Multi Dimensional Long Short Term Memory (MDLSTM) combiné à des couches convolutives pour prédire une boîte englobante autour d’une ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils traitent la tâche de détection de lignes de texte comme étant un problème de régression, et prédisent les coordonnées des boîtes englobantes directe- ment à partir des valeurs des pixels des images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont comparé deux stratégies de régression : prédire directement les boîtes englobantes et prédire séparément les points inférieurs gauche et supérieurs droit avant de les coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La seconde stratégie a montré une réelle amélioration pour la tâche de détection sur les documents du jeu de données Maurdor (Oparin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2014) mais est limitée aux lignes horizontales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Malgré les améliorations apportées aux modèles de régression de boîtes, cette approche est toujours limitée aux éléments horizontaux et ne permet pas une détection précise des lignes de texte par exemple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pour cela que les méthodes niveau pixel ont été proposées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – SYSTÈME YOLO Le système YOLO (You Only Look Once) a été proposé par Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) et permet de réaliser de la régression de boîtes englobantes d’objets sur des images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' YOLO divise l’image d’entrée en une grille régulière.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque cellule de la grille prédit un nombre prédéfini de boîtes de délimitation et des scores de confiance pour 20 É TAT D E L’ A RT chacune de ces boîtes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, les boîtes ayant les scores de confiance les plus élevés et les probabilités de classe les plus élevées dans ces boîtes sont considérées comme détections finales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' YOLO est beaucoup plus rapide que R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il montre cependant plus de difficultés à détecter des objets proches et les petits objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 – Schéma du système YOLO, issu de Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' détection niveau pixel La détection d’objets au niveau pixel est actuellement l’approche la plus utilisée pour le traitement d’images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De nombreux systèmes ont été proposés et c’est également dans ce cadre que se positionnent nos principaux travaux de recherche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La grande majorité de ces systèmes se base sur l’architecture Fully Convolutional Network (FCN), expliquée en détail dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6, en fin de cette section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les premiers FCN proposés étaient composés d’une succession de convolutions et de couches de regroupement (pooling) permettant de résumer les caractéristiques importantes de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces systèmes, tels que le VGG (Simonyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) et le ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016), étaient principalement utilisés pour les tâches de classification avec une classe unique en sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, pour la segmentation sémantique ou la détection d’objets, il est nécessaire d’avoir également la position de la classe dans l’image, c’est-à-dire une classe pour chaque pixel de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Afin d’obtenir une telle sortie, Ciresan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2012) ont entraîné un réseau utilisant une fenêtre glissante et prédisant une classe pour chaque pixel grâce à une région locale autour du pixel (un patch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien qu’il ait montré de très bonnes performances en gagnant notamment la compétition sur la segmentation de structures neuronales (ISBI 2012), le principal inconvénient de ce système est qu’il est très lent à traiter une image, le modèle étant appliqué à chaque patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour pallier cela, Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015) ont proposé un FCN pixel-à-pixel pour la tâche de segmentation sémantique d’images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les auteurs ont proposé une modification de l’architecture FCN en ajoutant, après les couches standards de convolution et de regroupement (étape d’encodage), une étape de décodage constituée d’une succession de couches équivalentes à l’encodeur dans laquelle les opérations de regroupement sont remplacées par des opérations d’upsampling, augmentant la résolution de sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’upsampling étant réalisé à l’aide de convolutions transposées (Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, afin d’avoir une localisation plus précise, les auteurs proposent de combiner les caractéristiques calculées durant l’étape d’encodage à celles du décodage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Montrant de très bonnes performances et un temps d’inférence raisonnable, de nombreux 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 21 autres travaux similaires ont vu le jour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une modification de ce système a été proposée par Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015) avec leur architecture U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les auteurs se sont concentrés sur le décodeur, l’encodeur étant comparable aux FCN que nous avons présentés plus tôt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont proposé d’utiliser des matrices de caractéristiques avec de nombreux canaux durant l’étape de décodage afin de propager davantage de contexte aux couches finales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Appliquée sur différentes tâches de segmentation d’images médicales, cette architecture a montré des gains importants de performances par rapport aux méthodes existantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le domaine de la détection de texte dans des images de scènes naturelles, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016b) appliquent également un FCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, un FCN TextBlock est utilisé pour détecter les localisations approximatives des lignes de texte, qui sont ensuite extraites en tenant compte des informations locales des caractères.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, un autre FCN est appliqué pour rejeter les fausses lignes de texte détectées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le traitement de documents, les FCN ont également été largement utilisés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, l’intérêt porté à l’analyse des documents a été stimulé par les compétitions sur la détection des lignes de texte (Murdock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015), la détection des lignes de base (Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) ou l’analyse de la mise en page (Antonacopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La plu- part de ces tâches ont été abordées au niveau pixel, et donc de nombreux systèmes de type FCN ont été développés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) a été proposé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’un système complexe permettant de traiter des documents avec de nombreuses classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est un réseau avec une architecture proche du U-Net où l’encodeur est pré-entraîné sur des images de scènes naturelles (ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la suite de cette thèse, nous comparons certains de nos modèles à dhSegment, c’est pourquoi son architecture est détaillée dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans dhSegment, contrairement aux réseaux proposés par Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015) et Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015) où l’upsampling était réalisé à l’aide de convolutions transposées, la résolution de sortie est augmentée à l’aide d’interpolations bilinéaires, ce qui permet d’avoir moins de paramètres à apprendre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode a obtenu de bons résultats sur diverses tâches de traitement de documents historiques, telles que l’analyse de mise en page ou l’extraction de lignes de base, avec peu de données d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, malgré un grand nombre de paramètres, le temps d’entraînement est considérablement réduit grâce à l’encodeur pré-entraîné.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’autres systèmes similaires ont ensuite été proposés, la principale différence entre leurs architectures étant la manière dont la résolution est augmentée dans le décodeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Barakat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) ont proposé un réseau entièrement convolutif pour détecter les lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Leur proposition consiste à utiliser uniquement des cartes de caractéris- tiques de bas niveau pendant l’étape de décodage, en les sur-échantillonnant plusieurs fois, à l’aide de convolutions transposées, avant de les combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette architecture a donné de bons résultats sur des pages manuscrites arabes mais nécessite des images d’entrée binarisées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Mechi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) ont présenté une architecture U-Net adaptative pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Leur proposition est de réduire le nombre de filtres (deux fois moins) dans les convolutions de l’encodeur afin de diminuer la quantité de paramètres du modèle, et donc le temps d’inférence ainsi que le sur-apprentissage, leur quantité de données annotées étant faible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 22 É TAT D E L’ A RT Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) ont proposé un système plus complexe composé de deux étapes pour détecter les lignes de base dans les documents historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, un réseau de neurones hiérarchique (ARU-Net) est appliqué pour détecter les lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet ARU-Net est une version étendue de l’architecture U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) : d’une part, un réseau d’attention spatiale est incorporé pour traiter les différentes tailles de caractères dans les pages ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' d’autre part, des blocs résiduels sont ajoutés à l’architecture U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela permet d’entraîner des réseaux neuronaux plus profonds tout en obtenant de meilleurs résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, des traitements successifs sont appliqués pour regrouper les super-pixels afin de construire les lignes de base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les auteurs ont montré que leur méthode était capable d’extraire des lignes de texte courbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, de nombreuses étapes de post-traitement ont été introduites dans la seconde phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Mechi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021) ont également présenté une méthode en deux étapes pour segmenter les lignes de texte dans des images de documents historiques arabes ou latins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, un FCN est utilisé pour segmenter la zone centrale du texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La seconde étape affine les résultats du FCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle est basée sur une version modifiée du RLSA pour extraire les lignes complètes du texte (y compris les composantes ascendantes et descendantes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Des évaluations quantitatives et qualitatives sont rapportées sur un grand nombre d’images de documents arabes et latins collectés à partir des archives nationales tunisiennes ainsi que d’autres ensembles de données de référence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, ce système nécessite une binarisation de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Tensmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017), les auteurs présentent PageNet, un système mis au point pour identifier les pages dans des images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les pages détectées sont ensuite extraites, ce qui permet de supprimer le bruit induit par la numérisation des pages, et différents traitements d’analyse de la mise en page peuvent être appliqués.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans PageNet, un réseau entièrement convolutif obtient une segmen- tation par pixel post-traitée afin d’extraire une région quadrilatérale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celui-ci traite l’image d’entrée à quatre résolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le système est évalué sur différents jeux de données et les au- teurs montrent que PageNet peut segmenter des documents superposés à d’autres documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) ont conçu un réseau multimodal entièrement convolutif pour l’analyse de la mise en page de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils tirent parti du contenu textuel ainsi que de l’apparence visuelle pour extraire les structures sémantiques des images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette méthode a montré des scores élevés d’Intersection-over-Union (IoU) (voir le Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2) mais nécessite des annotations de données plus complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, pour chaque image de document, une image étiquetée pixel par pixel ainsi que son contenu textuel sont nécessaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils utilisent des convolutions dilatées dans l’encodeur afin d’avoir une information contextuelle plus large et des résultats plus précis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Puisque dans la suite de cette thèse nous comparons certains de nos modèles à ce système, nous détaillons son architecture dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans leurs travaux, Renton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) ont également démontré les avantages d’utiliser de telles convolutions, détaillées dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8, par rapport à des convolutions standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Leur réseau entièrement convolutif est composé de convolutions dilatées successives qui augmentent le champ réceptif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles sont suivies d’une dernière convolution standard qui produit les images étiquetées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans notre méthode, nous tirons également profit de ces convolutions dilatées afin d’avoir un champ réceptif assez grand pour détecter correctement les objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 23 Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 – ARCHITECTURE FCN Définition Un réseau entièrement convolutif (FCN) est une extension d’un réseau neuronal convolutif (CNN) qui ne contient aucune couche dense et accepte des entrées de tailles variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il permet de faire de la prédiction spatiale dense, au niveau pixel, de manière rapide et précise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour faire de la prédiction dense, il est souvent composé d’un encodeur, résumant les caractéristiques importantes de l’image d’entrée, et d’un décodeur, augmentant la résolution des cartes de caractéristiques et prédisant des probabilités de classe pour chaque pixel d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Avantages — La suppression des couches denses d’un CNN permet de travailler avec des tailles d’entrée variables car les couches convolutives ne nécessitent pas un nombre fixe d’entrées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Éviter les couches denses réduit fortement le nombre de paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Les FCN sont capables de conserver l’information spatiale et de produire une description spatiale de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Inconvénients — L’utilisation d’un réseau de neurones convolutif induit l’utilisation de couches de regroupement (pooling), qui réduisent la résolution d’entrée dans le but d’augmenter le champ réceptif sans augmenter le nombre de paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour avoir un étiquetage au niveau des pixels d’une image d’entrée, la résolution de sortie du réseau doit être augmentée soit à l’aide d’une interpolation, d’une convolution transposée (Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7), d’une opération d’unpooling ou encore d’une convolution dilatée (Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Système de type FCN Un des premiers systèmes de type FCN proposé pour la détection niveau pixel est le U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a montré de très bonnes performances sur différentes tâches de segmentation d’images biomédicales avec très peu de don- nées annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur la Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8, la partie gauche constitue l’encodeur, composé de couches de convolutions et de max pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La partie droite est le décodeur, constitué de convolutions standards et transposées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 24 É TAT D E L’ A RT Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 – Schéma de l’architecture du modèle U-Net, issu de Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015), pour une image d’entrée de taille 572×572 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 – CONVOLUTION TRANSPOSÉE Définition La couche de convolution transposée est utilisée pour inverser le sous- échantillonnage induit par les couches de convolution standard ou de regroupement utilisées dans les réseaux convolutifs (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le principe est d’avoir la couche inverse d’une couche de convolution standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette couche permet d’avoir une sortie de plus grande résolution en représentant la valeur d’un pixel d’entrée sur plusieurs pixels de sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette couche est souvent utilisée dans les réseaux suivant l’architecture encodeur-décodeur où la sortie du réseau a la même taille que l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Avantages — Les filtres de la couche de convolution transposée doivent être entraînés ce qui permet au réseau d’être plus expressif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Inconvénients — Le réseau est plus profond et comporte plus de paramètres qu’un réseau contenant uniquement des opérations d’upsampling sans paramètres entraînés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Schéma d’une convolution transposée 2D La Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 présente le schéma d’une convolution transposée 2D avec X l’image d’entrée, W le filtre et Y la nouvelle représentation de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' t conv 3x3, Relu + up-com 2x2 → co 1x12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 25 x1,1 x0,1 x1,0 x0,0 X w2,2 w1,2 w0,2 w2,1 w1,1 w0,1 w2,0 w1,0 w0,0 W y3,3 y2,3 y1,3 y0,3 y3,2 y2,2 y1,2 y0,2 y3,1 y2,1 y1,1 y0,1 y3,0 y2,0 y1,0 y0,0 Y Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 – Schéma d’une convolution transposée 2D avec X l’image d’entrée, W le filtre et Y la nouvelle représentation de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 – CONVOLUTION DILATÉE Définition Une convolution dilatée suit le principe de base d’une convolution standard, mais calcule les nouvelles représentations sur une plus grande fenêtre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les pixels considérés pour le calcul de la nouvelle représentation d’un pixel ne sont plus ses voisins directs mais des voisins plus éloignés, l’écart étant défini par le taux de dilatation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Avantages — L’utilisation de convolutions dilatées permet d’avoir un champ réceptif plus grand, la nouvelle représentation d’un pixel considérant davantage de contexte, sans augmenter le nombre de paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Elle est souvent utilisée à la place des couches de regroupement, ce qui permet de perdre moins d’informations qu’avec une fonction de min ou max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Schéma d’une convolution dilatée 2D La Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 présente le schéma d’une convolution dilatée 2D avec un taux de dilatation de 2 et X l’image d’entrée, W le filtre et Y la nouvelle représentation de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' x0,0 x1,0 x2,0 x3,0 x4,0 x5,0 x0,1 x1,1 x2,1 x3,1 x4,1 x5,1 x0,2 x1,2 x2,2 x3,2 x4,2 x5,2 x0,3 x1,3 x2,3 x3,3 x4,3 x5,3 x0,4 x1,4 x2,4 x3,4 x4,4 x5,4 x0,5 x1,5 x2,5 x3,5 x4,5 x5,5 X w0,0 w1,0 w2,0 w0,1 w1,1 w2,1 w0,2 w1,2 w2,2 W y1,1 y0,1 y1,0 y0,0 Y Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 – Schéma d’une convolution dilatée 2D avec un taux de dilatation de 2 et X l’image d’entrée, W le filtre et Y la nouvelle représentation de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 26 É TAT D E L’ A RT Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 – SYSTÈME DHSEGMENT dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) est un des systèmes de référence pour les tâches d’analyse d’images de documents historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il possède plusieurs avantages comme le fait de pouvoir être entraîné avec peu de données d’entraînement et un temps d’entraînement réduit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, le code permettant d’entraîner et de tester le modèle est open-source a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est un modèle profond puisqu’il possède jusqu’à 2048 cartes de caractéristiques et suit l’architecture encodeur-décodeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un premier temps, l’image d’entrée est traitée par l’encodeur qui va résumer les caractéristiques importantes de l’image dans une matrice de caractéristiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette matrice est ensuite trai- tée par le décodeur qui va générer une carte de probabilités de même taille que l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, une étape de post-traitement est réalisée afin notamment de seuiller les probabilités des pixels et de supprimer les petites composantes connexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Encodeur L’encodeur est principalement constitué d’un CNN pré-entraîné sur des images de scènes naturelles de la base ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2009) et représenté à gauche sur la Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce CNN pré-entraîné suit l’architecture du réseau ResNet-50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016) mais a été légèrement modifié afin de réduire le nombre de paramètres, et donc la mémoire requise pendant l’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est également possible de remplacer ce ResNet-50 par un VGG-16 (Simonyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) ou un U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette partie pré-entrainée présente l’avantage de réduire considérablement le nombre de paramètres à apprendre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, le réseau possède 32,8 millions de paramètres au total dont la plupart proviennent du CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, seuls 9,36 millions de paramètres restent à entraîner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela permet au réseau d’apprendre rapidement et correctement sur un nombre restreint de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Décodeur Le décodeur est standard et consiste en une succession de cinq blocs de déconvolu- tion composés d’une couche de convolution standard et d’une couche d’upscaling, et d’une couche finale de convolution afin de générer une carte de probabilités.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette partie est entièrement apprise sur les données d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Post-traitement En sortie du décodeur, nous disposons, pour chaque pixel, des probabilités d’appar- tenir aux différentes classes définies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Différentes techniques d’agrégation des résul- tats au niveau pixel sont possibles afin de détecter les objets pour la tâche considérée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Quatre principales techniques ont été implémentées et sont disponibles : — Seuillage : permet d’assigner une classe aux pixels ayant une probabilité supé- rieure à un seuil prédéfini ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Opérations de morphologie mathématique : opérations d’érosion, de dilatation, d’ouverture et de fermeture afin de créer des objets plus plausibles ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Analyse des composantes connexes : permet de filtrer les petites composantes connexes restantes après l’étape de seuillage ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Vectorisation des objets : transforme les régions détectées en un ensemble de coordonnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 27 Schéma de l’architecture de dhSegment S S 64 S 2 256 S 4 512 S 8 1024 512 S 16 2048 512 512 S 16 || 512 512 S 8 || 256 256 S 4 || 128 128 S 2 || 64 64 S || 32 c S Convolution Max pooling Bottleneck Bottleneck S/2 Upscaling || Concatenation c Number of classes Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 – Schéma de l’architecture du modèle dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='com/dhlab-epfl/dhSegment Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 – SYSTÈME DE YANG ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) ont proposé un réseau multimodal permettant de segmenter des documents en se basant sur le contenu visuel et textuel de ceux-ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’utilisation des textes permet d’assigner des classes spécifiques aux régions de texte en fonction de leur rôle dans le document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, dans l’article original, les classes considérées sont les suivantes : fond, image, tableau, paragraphe, titre, liste et légende.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le système a montré de bonnes performances sur des ensembles de données synthétiques et réelles d’images de documents modernes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, le code permettant d’entraîner un modèle est open-source a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle de Yang est un réseau multimodal entièrement convolutif (FCN) dont l’architecture est présentée sur la Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La base de ce modèle suit une archi- tecture encodeur-décodeur et est constituée de quatre modules : — Un encodeur ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Un décodeur ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Un décodeur auxiliaire ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Un pont (intégration du contenu textuel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Encodeur L’encodeur est constitué de quatre blocs dilatés, chaque bloc comportant cinq couches de convolutions dilatées, de taux de dilatation 1, 2, 4, 8 et 16, exécutées en parallèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’avantage d’utiliser de telles convolutions est que le champ réceptif est 28 É TAT D E L’ A RT plus grand, ce qui permet au modèle d’avoir davantage de contexte par rapport à une convolution standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Décodeurs Les deux décodeurs ont la même architecture avec trois blocs contenant une couche de convolution suivie par une couche d’unpooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le premier décodeur est stan- dard et vise à produire une carte de probabilités.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le décodeur auxiliaire est, quant à lui, utilisé uniquement durant l’entraînement et cherche à reconstruire l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a été montré qu’une branche auxiliaire de reconstruction aide à générer de meilleures représentations de l’image d’entrée, et donc améliore les performances de la tâche principale, ici, la tâche de segmentation (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Contenu textuel L’information textuelle est extraite grâce à un algorithme de reconnaissance (Optical Character Recognition (OCR) ou HTR) de textes de la manière suivante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’algorithme de reconnaissance est appliqué au document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour chaque phrase extraite du document, un embedding moyen est calculé à partir des embeddings des mots de cette phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, une carte de caractéristiques est construite à partir de ces embeddings : pour chaque phrase, les pixels du document initial lui appartenant prennent la valeur de cet embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les pixels n’appartenant à aucune phrase prennent la valeur 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, cette carte est concaténée à la carte de caractéristiques visuelles avant la dernière convolution du décodeur principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Schéma de l’architecture du système de Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' W H 32 W H 64 W 2 H 2 128 W 4 H 4 256 128 128 W 4 H 4 || 64 64 W 2 H 2 || 32 32 W H || c 64 64 W 2 H 2 32 32 W H c Reconstructed input Text Embedding Map Segmentation Convolution Dilated block Max pooling Upscaling Text embedding map || Concatenation c Number of classes Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12 – Schéma de l’architecture du modèle de Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' http://personal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='edu/xuy111/projects/cvpr2017_doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='html 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 29 modèles séquentiels à base de transformers Avant le développement des modèles à attention et des systèmes Transformers, les tâches de traitement du langage et notamment de traduction étaient réalisées à l’aide de réseaux encodeurs-décodeurs récurrents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’encodeur est utilisé pour traiter la phrase d’entrée entière et l’encoder dans un vecteur de contexte unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les couches du décodeur produisent ensuite, à partir du vecteur de contexte, les mots de la phrase les uns après les autres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le principal inconvénient de cette approche provient du traitement de la phrase d’entrée qui est résumée dans un unique vecteur de taille fixe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2014) ont démontré que la perfor- mance du modèle encodeur-décodeur se dégrade rapidement lorsque la longueur de la phrase d’entrée augmente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un autre problème est que le modèle n’a aucun moyen de donner plus d’importance à certains des mots en entrée par rapport à d’autres lors de la traduction de la phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pour résoudre ces problèmes que l’attention (voir le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12) a été introduite par Bahdanau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celle-ci permet de considérer tous les mots de la phrase d’en- trée dans le vecteur de contexte, mais également d’accorder une importance relative à chacun d’entre eux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, lorsque le modèle génère une phrase, il recherche un ensemble de positions dans les états cachés de l’encodeur, dans lesquels les informations les plus pertinentes sont disponibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette même optique, des systèmes à base de réseaux Transformers ont été proposés récemment afin de réaliser des tâches de détection en tenant compte de la séquentialité entre les éléments prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’architecture de ces systèmes est présentée dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit de modèles reposants sur le même mécanisme d’attention, qui sélectionne les caractéristiques pertinentes à chaque itération du processus de prédiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le système met en œuvre une seconde attention qui tient compte des éléments précédemment prédits en sortie, pour agir comme un modèle de langage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les premiers systèmes ont été principalement conçus pour le traitement automatique des langues sans utiliser ni récurrence ni convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le premier système à base de Transformers (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017) a été établi afin de résoudre plus efficacement la tâche de traduction de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les auteurs ont proposé une architecture composée d’un encodeur suivi d’un décodeur, qui génère une séquence de sortie, un élément à la fois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est auto-régressif : il intègre les éléments prédits précédemment comme entrée supplémentaire lors de la prédiction de l’élément suivant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’encodeur extrait les caractéristiques des données d’entrée grâce à un mécanisme d’attention qui permet de considérer le contexte, ici, l’ensemble des mots de la séquence d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet encodeur est constitué de six blocs successifs identiques, composés de deux principaux éléments : une couche d’auto-attention et un réseau dit entièrement connecté (feed-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’auto-attention permet de représenter l’interdépendance des mots de la séquence en entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le décodeur permet de modéliser le langage de sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est également composé de six blocs successifs identiques, chacun contenant une couche d’auto-attention, un réseau entièrement connecté et une couche d’attention dite d’attention croisée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette dernière permet au décodeur de réaliser l’attention entre la séquence d’entrée et celle de sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tous les réseaux entièrement connectés du modèle contiennent deux couches linéaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les séquences en entrée et en sortie sont additionnées à un encodage de position, 30 É TAT D E L’ A RT détaillé dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13, avant d’être respectivement traitées par les encodeur et décodeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces encodages de position permettent de garder l’ordre de la séquence durant l’ensemble des traitements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le Transformer a rapidement été largement utilisé car il a permis de remplacer les couches récurrentes, jusqu’alors utilisées, par des couches d’attention tout en conservant des performances similaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les couches récurrentes jusqu’ici utilisées empêchaient la parallélisation des calculs durant la phase d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette récurrence ayant été remplacée par ces fameuses couches Transformer non récurrentes, entraînées par une stratégie dite de teacher forcing, les calculs peuvent être parallélisés et le temps d’entraînement fortement réduit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette architecture est détaillée dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au vu des résultats obtenus par ces systèmes, certains travaux les ont adaptés à des tâches de vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, les Vision Transformers (ViT) ont été proposés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les premiers travaux intro- duisant les ViT ont été présentés par Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021) et sont détaillés dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils interprètent une image en entrée de l’encodeur comme étant une séquence de patchs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, la représentation vectorielle d’un caractère dans une tâche de traduction est ici remplacée par les valeurs des pixels d’un patch de l’image d’entrée mis à plat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette fois, l’encodage de position correspond à la position du patch dans l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Puisqu’il est appliqué à la tâche de classification d’images, qui ne nécessite pas de sortie séquentielle, seul l’encodeur Transformer est intégré au système.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, les auteurs utilisent un simple Multi-Layer Per- ceptron (MLP) chargé de prédire la classe de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système a obtenu des performances à l’état de l’art sur différents ensembles de classification d’images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, le système nécessite un pré-entraînement sur un nombre imposant de données, 303 millions d’images pour leur meilleur modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Plusieurs approches ont également été présentées afin d’appliquer les Transformers à la détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cadre d’images de scènes naturelles, DETR (DEtection TRansfor- mer) a été proposé par Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’un système hybride qui combine un encodeur CNN suivi d’un encodeur et d’un décodeur Transformer produisant un ensemble de boîtes englobantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est entrainé à prédire un nombre fixe de boîtes englobantes ainsi que leurs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Leur modèle obtient des résultats semblables à Faster R-CNN sur les images de scènes naturelles du jeu de données COCO 1, tout en obtenant de meilleurs résul- tats sur les grands objets grâce à l’auto-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il ne tire cependant pas profit de la capacité de prédiction séquentielle permise par les Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) ont ensuite pro- posé Pix2Seq, présenté dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='15, afin de traiter la détection de manière séquentielle en prédisant, pour chaque objet, une séquence de coordonnées suivie de la classe de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les auteurs comparent différents encodeurs à base de convolutions et de Transformers, suivis par un décodeur Transformer standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pix2Seq obtient des performances à l’état de l’art sur l’ensemble de données de référence COCO en obtenant des valeurs de précision moyenne (Average Precision (AP), voir le Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4) supérieures à celles obtenues par Faster-RCNN (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) et DETR (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) tout en nécessitant moins de paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, les résultats montrent que, pour tous les encodeurs, la détection est meilleure par rapport aux systèmes Faster-RCNN et DETR avec des encodeurs comparables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://cocodataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='org/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 31 montrent également qu’utiliser un encodeur Transformer est préférable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, davan- tage de données d’entraînement sont nécessaires puisque le modèle comporte beaucoup plus de paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Toujours dans le domaine de la détection dans des images naturelles, certains travaux ont été proposés afin de faire de la prédiction dense en augmentant la sortie du Transformer, et donc d’avoir une sortie de même taille que l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ont proposé un ViT (appelé SETR) où l’encodeur Transformer est suivi d’un décodeur composé de convolutions réalisant l’augmentation d’échelle (upsampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont obtenu les meilleurs résultats sur différentes bases de segmentation d’images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) ont proposé un modèle hybride CNN-Transformer, très comparable à SETR mais incluant un encodeur CNN avant l’encodeur Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que ces méthodes aient montré des gains de performances par rapport aux systèmes existants, elles ne tirent pas pleinement profit des Transformers qui permettent d’avoir des sorties séquentielles et structurées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, dans le domaine des Transformers, il n’y a pas, à notre connaissance, de travaux permettant de traiter la tâche de détection d’objets de manière séquentielle dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Quelques rares travaux ont appliqué les Transformers aux images de documents, principalement pour la re- connaissance de caractères niveau paragraphe ou page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, dans les travaux de Coquenet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) et Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021), les auteurs ont proposé des modèles hybrides combinant un encodeur CNN et un décodeur Transformer afin de prédire séquentiellement les caractères du texte d’un paragraphe ou document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le système proposé par Coquenet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) fournit également une structuration des résultats en générant des tags de mise en page dans la séquence des caractères reconnus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système est le premier à résoudre la tâche de recon- naissance de texte pleine page sans segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a obtenu des performances de même ordre que les systèmes à l’état de l’art travaillant au niveau ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De leur côté, Rouhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) utilisent un modèle hybride avec un encodeur CNN suivi d’un encodeur et déco- deur Transformer pour traiter la tâche de reconnaissance d’entités nommées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Leur approche consiste à créer une architecture qui reconnaît les textes et les entités nommées à partir d’images de paragraphes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils utilisent des labels dits "visuels" correspondant aux caractères du texte présent dans les images ainsi que des labels dits "contextuels" correspondant aux entités nommées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) ont proposé DONUT, un modèle de compréhension de documents sans OCR composé d’un encodeur et décodeur Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' DONUT obtient de très bons résultats en termes de temps d’exécution et de précision sur diverses tâches telles que la classification de documents, l’extraction d’informations et le Visual Question Answe- ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il nécessite cependant un important pré-entraînement sur des milliers de documents synthétiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 – ARCHITECTURE TRANSFORMER Définition Un modèle à base de Transformer est un modèle permettant de réaliser un trai- tement séquence-à-séquence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’un modèle auto-régressif prédisant séquen- tiellement les éléments et utilisant les éléments de la séquence d’entrée ainsi que 32 É TAT D E L’ A RT les éléments prédits précédemment en sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce modèle repose sur un mécanisme d’attention (présenté dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12), qui permet de représenter les données en utilisant le contexte et, notamment, les interdépendances entre les éléments des séquences d’entrée et de sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les modèles Transformers initialement proposés suivent une architecture encodeur- décodeur où l’encodeur génère une représentation de la séquence en entrée incluant l’interdépendance des éléments de cette séquence ainsi que leurs positions dans la séquence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le décodeur génère une séquence de sortie grâce à la séquence d’entrée encodée et les éléments précédemment prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Avantages — Dans un modèle à base de Transformer, les couches récurrentes d’un réseau à attention ont été remplacées par des couches non récurrentes pour réaliser cette attention, ce qui conduit à des temps d’entraînement réduits tout en conservant des performances similaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Par rapport à un réseau récurrent, ce modèle permet de mieux représenter les dépendances entre les éléments de la séquence d’entrée grâce au mécanisme d’attention, notamment pour des séquences longues, tout en conservant un temps de traitement raisonnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Système Transformer Le premier système à base de Transformer a été proposé pour la tâche de traduction de texte (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’un modèle encodeur-décodeur entière- ment basé sur l’attention dont l’architecture est présentée sur la Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a dépassé les résultats à l’état de l’art sur des tâches de traduction anglais-allemand et anglais-français.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Encodeur Sur la Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13, la partie de gauche compose l’encodeur qui traite la sé- quence d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans l’implémentation originale, l’encodeur comporte une couche d’embedding de la séquence puis six blocs d’encodage Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, un encodage de position est additionné à la représentation de la séquence d’entrée avant les couches d’encodage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celui-ci est réalisé à l’aide des fonctions cosinus et sinus comme détaillé dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les couches dites de Multi- Head Attention calculent les vecteurs d’auto-attention (voir Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12) et sont suivies d’un réseau entièrement connecté composé de deux couches linéaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Décodeur La partie de droite de la Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13 présente le décodeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il comporte une couche d’embedding de la séquence de sortie suivie de six blocs de décodage Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le même encodage de position utilisé dans l’encodeur est appli- qué sur la séquence partielle de sortie courante, avant les couches de Transfor- mer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, les couches Multi-Head Attention et le réseau entièrement connecté sont similaires à ceux de l’encodeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La seule différence concerne la seconde couche d’attention qui prend en entrée la séquence d’entrée encodée ainsi que la séquence de sortie encodée pour réaliser l’attention croisée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 33 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13 – Schéma de l’architecture du modèle Transformer original, issu de Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12 – ATTENTION Définition Le concept d’attention permet de considérer la corrélation entre les éléments de deux séquences grâce à des coefficients d’attention calculés entre chaque élément de chaque séquence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une fonction d’attention peut être décrite comme la mise en correspondance d’une requête (q) et d’un ensemble de paires clé-valeur (k-v) avec une sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La sortie est calculée comme une somme pondérée des valeurs, le poids attribué à chaque valeur étant calculé par une fonction de compatibilité de la requête avec la clé correspondante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Lorsque l’attention est réalisée sur une unique séquence, celle-ci est appelée auto- attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’attention croisée fait elle référence au mécanisme d’attention standard, appliqué sur deux séquences distinctes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas du traitement de la langue, le mécanisme d’attention permet de déterminer les mots sur lesquels le modèle doit porter le plus d’attention pour traiter la séquence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Auto-attention L’auto-attention, appelée self-attention, correspond au mécanisme d’attention ap- pliqué à une seule séquence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle détermine donc l’interdépendance (ou l’auto- corrélation) des éléments d’une même séquence entre-eux afin de lui associer une représentation pertinente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Output Probabilities Softmax Linear Add & Norm Feed Forward Add & Norm Add & Norm Multi-Head Feed Attention Forward Nx 分 Add & Norm Nx Add & Norm Masked Multi-Head Multi-Head Attention Attention Positional Positional Encoding Encoding Input Output Embedding Embedding 个 Inputs Outputs (shifted right)34 É TAT D E L’ A RT Attention multi-têtes Dans la multi-head attention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' le calcul d’attention est réalisé en parallèle par plu- sieurs blocs d’attention différents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela permet au modèle de considérer des informa- tions provenant de différents sous-espaces de représentations à différentes positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cadre d’un modèle de traitement du langage, cela permet de caractériser les mots vis-à-vis de différents points de vue ou rôles qu’ils occupent dans la phrase tels que sujet, verbe ou encore complément.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le vecteur de sortie correspond à la concaténation des vecteurs de sortie de chaque tête.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Mise en oeuvre — Pour calculer une sortie (vecteur d’attention), trois vecteurs pour chaque élé- ment de la séquence d’entrée sont considérés : — Vecteur requête q (query) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Vecteur clé k (key) de dimension dk ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Vecteur valeur v (value) de dimension dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les valeurs de chacun de ces vecteurs sont apprises pendant l’entraînement du modèle Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Pour chaque élément de la séquence d’entrée (requête q), les produits scalaires avec l’ensemble des éléments de la seconde séquence (clés k) sont calculés, puis divisés par la racine carrée de la dimension du vecteur k (dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette division assure la stabilité du gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Une opération softmax est ensuite appliquée à chaque sortie puis celle-ci est multipliée par le vecteur valeur v correspondant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Enfin, le vecteur d’attention d’une requête q correspond à la somme des vecteurs ainsi calculés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Équation En pratique, la fonction d’attention est calculée sur un ensemble de requêtes simultanément, regroupées dans la matrice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les clés et valeurs sont elles aussi regroupées respectivement dans des matrices K et V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les sorties sont calculées comme suit : Attention(Q, K, V ) = Softmax �QKT √dk � V (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1) avec : — dk : la dimension du vecteur clé k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13 – ENCODAGE POSITIONNEL Définition Dans un Transformer, chaque élément de la séquence d’entrée (ou de sortie) est traité simultanément dans la pile d’encodeurs (ou de décodeurs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, le modèle n’a pas connaissance de la position de chaque élément dans la séquence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi l’encodage positionnel est utilisé dans les réseaux à base de Transformers, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 35 afin de ne pas perdre l’ordre des éléments de la séquence d’entrée (ou de sortie) lors de la propagation des informations dans le modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Équation Le premier encodage de position a été proposé par Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’un encodage fixe qui se base sur les fonctions cosinus et sinus, et est calculé comme suit : PE(pos, 2i) = sin(wi · pos) ∀i ∈ � 0, dmodel 2 � PE(pos, 2i + 1) = cos(wi · pos) ∀i ∈ � 0, dmodel 2 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2) avec : wi = 1 10000 2i dmodel et : — pos : la position de l’élément dans la séquence ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — dmodel : la dimension d’encodage de l’élément.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='14 – ARCHITECTURE VISION TRANSFORMER Définition Un Vision Transformer (ViT) est une adaptation de l’architecture Transformer standard appliquée aux images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La séquence en entrée du système correspond à une séquence de patchs de taille fixe de l’image originale, où la couche d’embedding est remplacée par une projection linéaire des valeurs des patchs aplanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour la tâche de classification, l’encodeur est suivi d’un MLP standard produisant des probabilités pour chaque classe considérée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour la détection d’objets, il est suivi d’un décodeur convolutif semblable à ceux des FCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Avantages — Par rapport aux CNN, les performances sont au moins aussi bonnes tout en nécessitant moins de mémoire pour le traitement et en étant plus rapide en inférence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Inconvénients — Le modèle nécessite un très grand nombre de données d’apprentissage ou une étape de pré-entraînement afin d’obtenir des résultats satisfaisants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Système Vision Transformer Le premier Vision Transformer a été proposé pour la tâche de classification d’images (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle comporte un encodeur Transformer suivi d’un MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a obtenu des performances comparables aux systèmes CNN à l’état de l’art, tout en nécessitant beaucoup moins de ressources pour l’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 36 É TAT D E L’ A RT Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='14 – Schéma de l’architecture du modèle Vision Transformer original pour la classification d’images, issu de Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='15 – SYSTÈME PIX2SEQ Pix2Seq (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022) est un des premiers systèmes à base de Transformers proposé pour traiter la détection d’objets dans les images de scènes naturelles de manière séquentielle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle obtient des performances supérieures à celles obte- nues par les systèmes à l’état de l’art, tels que Faster-RCNN (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) et DETR (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020), tout en nécessitant moins de paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est composé d’un encodeur suivi d’un décodeur Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les auteurs ont comparé différents encodeurs à base de convolutions, de Transformers ou des encodeurs hybrides, leurs expériences montrant les meilleures performances avec un encodeur Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le décodeur est standard et comporte six couches de décodeur Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celui-ci produit une séquence de coordonnées et de classes représentant les objets détectés ainsi que leurs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Modélisation de la détection Le modèle Pix2Seq est entraîné à prédire séquentiellement chaque objet, une co- ordonnée à la fois, de la manière suivante : ordonnée du point supérieur gauche, abscisse du point supérieur gauche, ordonnée du point inférieur droit, abscisse du point inférieur droit et classe de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, un objet et sa classe sont détectés par cinq valeurs prédites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les auteurs considèrent la détection comme une tâche de classification en considérant une classe pour chaque valeur possible en ordonnée et en abscisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Class Bird MLP Ball Head Car IPatsh -- IPosd(dilm 161829st3*0 [cLe5 s] cmbedldig Linear Projection of Flattened Patches2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 37 Schéma de l’architecture de Pix2Seq Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='15 – Schéma du système Pix2Seq, issu de Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' approches combinant image et texte Les systèmes présentés en section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 sont actuellement les plus utilisés pour la détection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certaines recherches se sont également orientées vers des approches combinant l’image et le texte du document afin de réaliser la tâche de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas de documents complexes, l’ajout du texte dans le processus de détection peut aider à détecter et à classifier des éléments de plus haut niveau tels que des actes (Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la tâche de Visual Document Understanding (Delteil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022), les informations sont extraites à l’aide d’une combinaison des caractéristiques textuelles et visuelles de l’image d’un document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certaines propositions sont présentées dans cette section pour les tâches de détection, mais aussi de pré-entraînement pour différentes tâches de compréhension d’images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est important de noter que pour la plupart des systèmes présentés dans cette section, un reconnaisseur (HTR ou OCR) a été entraîné au préalable afin d’extraire le contenu textuel des images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) ont été parmi les premiers à proposer un réseau entièrement convolutif multimodal pour extraire des structures sémantiques de documents modernes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour aider à distinguer des classes similaires comme les paragraphes et les listes, ils incorporent des in- formations textuelles à l’aide d’une carte d’intégration de texte concaténée avant la dernière convolution du modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’ajout de cette carte n’a pas montré d’amélioration significative dans des conditions réelles d’utilisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Suivant cette idée, Barman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021) ont proposé un système capable de segmenter finement les journaux historiques et de gérer les variations de mise en page dans le temps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils utilisent la même représentation textuelle que Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) mais sur des jetons produits par un processus OCR au lieu de phrases, ce qui est plus réaliste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont montré que l’ajout des cartes d’intégration du texte au début du réseau donne de meilleures performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certains travaux ont également étudié la combinaison de carac- téristiques textuelles et visuelles pour classifier les pages des documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Wiedemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018), une combinaison de deux réseaux neuronaux convolutifs (CNN), l’un basé sur des données textuelles et l’autre sur des numérisations d’images, est utilisée pour classifier les pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les paramètres sont ensuite combinés et transmis à un perceptron multicouche pour la classification finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette combinaison a permis d’augmenter les performances par rapport à un seul CNN basé uniquement sur le texte ou l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Encoder38 É TAT D E L’ A RT Pour les documents modernes, LayoutLM (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) a été proposé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’une mé- thode de pré-entraînement simple pour les tâches de compréhension d’images de documents qui permet de modéliser conjointement les interactions entre le texte et les informations de mise en page dans les documents numérisés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, un processus de reconnaissance complet est appliqué à l’image d’entrée afin de détecter les objets textuels et de reconnaître l’ensemble des textes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, les auteurs utilisent une combinaison de BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019), où l’information textuelle d’entrée est principalement représentée par des plon- gements de mots, et des caractéristiques d’image données par Faster-RCNN (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' LayoutLM permet de modéliser conjointement les interactions entre le texte et les informations de mise en page dans les documents numérisés et est ensuite utilisé comme pré-entraînement pour un grand nombre de tâches de compréhension d’images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système a montré des performances à l’état de l’art sur des documents commerciaux numérisés, mais nécessite un nombre important de données d’apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021), les auteurs présentent VTLayout, un système qui fusionne les caractéristiques visuelles profondes, superficielles et textuelles des documents pour localiser et identifier les différents blocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la première étape, le modèle Cascade Mask R-CNN est appliqué directement sur l’image pour localiser tous les blocs du document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la seconde étape, les caractéristiques visuelles profondes, superficielles et textuelles sont extraites et fusionnées afin d’identifier les classes de chaque bloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les caractéristiques textuelles sont extraites par PaddleOCR (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) puis transformées par une application de TF-IDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce modèle a montré un gain de performances de détection par rapport aux systèmes standards manquant, notamment, de précision sur la classe de titre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ont également étudié le cas où l’aspect graphique des images n’est pas suffisant pour segmenter les chartes médiévales en actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ne visent pas seulement à détecter les actes mais cherchent également à les classifier comme début, milieu, fin d’acte ou acte complet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils utilisent une carte d’indexation probabiliste pour construire des caracté- ristiques supplémentaires basées sur le contenu textuel, puis les caractéristiques graphiques et textuelles sont fusionnées afin d’obtenir une seule entrée pour le système de segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils montrent que l’ajout de contenu textuel peut faciliter la segmentation des actes, et que l’ajout de connaissances préalables permet d’améliorer encore les performances, cependant, leur méthode reste complexe à mettre en place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 E S T I M AT I O N D E L A C O N F I A N C E D E S O B J E T S D É T E C T É S Les réseaux de neurones obtiennent désormais des performances remarquables dans de nom- breux domaines d’application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, leur utilisation pour des applications industrielles exige qu’ils soient à la fois capables de fournir le résultat attendu tout en évaluant leur propre certitude, ou incertitude, quant à cette décision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceci est particulièrement important pour les applications critiques telles que celles liées aux images médicales ou à la conduite autonome par exemple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’apprentissage actif (active learning, détaillé dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='16) (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 1995) est une méthode d’apprentissage automatique itératif dans lequel l’algorithme d’apprentissage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 E S T I M AT I O N D E L A C O N F I A N C E D E S O B J E T S D É T E C T É S 39 demande des données d’entraînement, celles jugées les plus pertinentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces données sont sélectionnées en fonction de la confiance de l’algorithme quant à ses propres décisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les premières propositions consistaient à utiliser directement les probabilités a posteriori du classifieur afin de sélectionner les exemples à annoter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, les exemples ayant une probabilité proche de 0,5 (uncertainty sampling) étaient sélectionnés pour l’itération suivante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les réseaux neuronaux de détection d’objets produisent également des probabilités qui pourraient directement être utilisées comme estimations de confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, il a été démontré que ces probabilités sont souvent des estimateurs trop confiants qui donnent une confiance élevée même sur des prédictions erronées (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour résoudre ce problème, plusieurs études ont été menées afin de concevoir de meilleurs estimateurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, toujours dans le cadre de l’apprentissage actif, l’une des premières approches proposées pour sélectionner les échantillons à annoter manuellement était basée sur les machines à vecteurs de support (SVM) linéaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette optique, Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2002) ont proposé SVM Min Margin qui consiste à entraîner un SVM linéaire et à choisir les échantillons étant les plus proches de la limite de décision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une autre approche populaire est l’échantillonnage d’incertitude (uncertainty sampling) (Settles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2008) où les échantillons menant à des prédictions avec une grande incertitude sont sélectionnés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour quantifier l’incertitude, plusieurs mesures basées sur les probabilités a posteriori ont été proposées, comme l’entropie ou le score de moindre confiance (Brust et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour modéliser l’incertitude des décisions des réseaux neuronaux, d’autres approches ont été proposées, comme le dropout de Monte Carlo (Gal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le dropout (Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2014) est une méthode de régularisation utilisée dans les réseaux neuronaux afin de lutter contre le manque de généralisation des modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il consiste à désactiver (mettre à 0) des valeurs, choisies aléatoirement, de l’image en entrée d’une couche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est appliqué uniquement durant la phase d’apprentissage et permet d’éviter le sur-apprentissage et la coadaptation, chaque neurone devant apprendre indépendamment des autres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le MC dropout, au lieu de calculer une seule prédiction au moment du test, il est demandé au réseau de fournir plusieurs prédictions avec dropout, dont la distribution est ensuite analysée pour dériver une estimation de la confiance de la prédiction sans dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette technique, qui se rapproche des modèles bayésiens par apprentissage profond, a été utilisée pour de nombreuses tâches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle s’est souvent révélée efficace pour la classification afin de choisir les données à étiqueter (Gal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Dechesne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021), le MC dropout est utilisé pour estimer l’incertitude de résultats de segmentation sémantique d’images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) utilisent le MC dropout comme technique de régularisation des probabilités de classe pour obtenir un meilleur classement ordinal des prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’autres travaux font appel à des modèles d’estimation de confiance profonds indépendants du modèle de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Granell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021), un réseau adversaire est entraîné en parallèle du modèle de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celui-ci est entraîné pour estimer la proximité des prédictions avec la vérité du terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 40 É TAT D E L’ A RT La plupart des travaux présentés ici se concentrent sur la tâche de classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, malgré les nombreux travaux présentant de nouveaux systèmes de détection d’objets, il y a très peu de travaux, dans la littérature, discutant l’estimation de la confiance pour cette tâche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='16 – APPRENTISSAGE ACTIF / ACTIVE LEARNING Définition L’Active learning (ou apprentissage actif) (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 1995) est une méthode d’apprentissage automatique qui permet à un algorithme d’interagir avec un oracle durant le processus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un cadre d’apprentissage classique, les données sont choi- sies au préalable et imposées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En apprentissage actif, c’est l’algorithme d’apprentis- sage qui demande les données jugées les plus pertinentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le processus est itératif et s’arrête lorsqu’un critère de performances ou un nombre défini de données annotées ou d’itérations est atteint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Avantages — L’utilisation de l’apprentissage actif permet de réduire fortement le coût d’an- notation manuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Les performances du modèle final sont améliorées en comparaison avec un entraî- nement classique puisque les données sont choisies afin d’optimiser les résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Inconvénients — Une fonction d’acquisition est nécessaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les fonctions d’acquisition permettent d’associer une donnée à une valeur qui encode soit l’incertitude du modèle sur cet exemple soit sa contribution dans l’ajustement du modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Plusieurs fonc- tions ont été proposées dans le domaine de l’uncertainty sampling telles que l’entropie, se basant directement sur les probabilités a posteriori, ou la dis- tance des exemples par rapport à la limite de décision dans les SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’autres approches se basent sur la différence entre les résultats donnés par plusieurs modèles (Query By Committee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Il est également nécessaire de définir une stratégie de sélection : quels exemples seront utilisés pour l’itération suivante ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certains travaux choisissent les exemples où les probabilités a posteriori sont les plus hautes ou les plus basses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas des SVM, certains choisissent les exemples les plus proches, d’autres les plus éloignés de la limite de décision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il n’y a, à notre connaissance, pas de consensus sur la stratégie à utiliser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Exemple d’apprentissage actif La Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='16 présente un exemple d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les paramètres du modèle sont initialisés ou pré-entraînés sur l’ensemble d’apprentissage annoté.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est appliqué aux exemples non annotés qui sont ensuite sélectionnés selon la stratégie choisie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les exemples sélectionnés sont annotés par un opérateur puis ajoutés à l’en- semble d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un nouveau modèle est entraîné sur ce nouvel ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le processus est répété jusqu’à ce que les conditions d’arrêt prédéfinies soient atteintes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 E S T I M AT I O N D E L A C O N F I A N C E D E S O B J E T S D É T E C T É S 41 Ensemble d’entraˆınement annot´e Mod`ele d’apprentissage profond Entraˆınement initial Corpus non annot´e Pr´ediction Oracle S´election Nouvel ensemble d’entraˆınement annot´e Annotation Entraˆınement Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='16 – Schéma présentant le processus d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N La mise en place et l’amélioration de modèles de détection d’objets conduisent à explo- rer différents axes de recherche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que la majorité des travaux dans la littérature se concentrent uniquement sur la proposition de nouvelles architectures, nous avons souhaité nous intéresser à des études et des solutions plus complètes, en évoquant notamment les problématiques liées aux annotations et évaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, l’évaluation de la qualité des algorithmes de détection ou de reconnaissance est cruciale dans la mise au point de systèmes et leurs comparaisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle nécessite donc l’utilisation de métriques appropriées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, il faut également étudier les données annotées utilisées pendant l’entraînement et l’évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Si les annotations des données ne sont pas cohérentes avec la métrique utilisée, la métrique ne peut pas refléter les performances réelles du modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce chapitre, nous mettons tout d’abord en avant les problèmes liés aux annotations des jeux de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans une première section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, nous présentons une étude des récents jeux de données utilisés dans les systèmes à l’état de l’art, principalement pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous mettons ensuite en évidence, en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2, les différentes règles d’annotation manuelle, ainsi que les défis liés et les solutions proposées dans la littérature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par la suite, nous discutons, en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3, des différentes métriques proposées et utilisées dans la littérature afin d’évaluer et de comparer les systèmes de détection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 J E U X D E D O N N É E S Dans cette partie, nous présentons les jeux de données utilisés dans les systèmes récemment proposés, notamment pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons également les jeux de données privés que nous avons utilisés durant la thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces jeux de données sont détaillés dans les paragraphes suivants, résumés dans la Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, et un exemple est montré sur la Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous nous focalisons sur la tâche de détection de lignes de texte car c’est une étape centrale de l’analyse de la mise en page des documents puisqu’elle est nécessaire à la reconnaissance de texte et qu’elle a un fort impact sur la qualité de la reconnaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, c’est une des tâches pour laquelle le type d’annotation et la définition même de la tâche peuvent être très variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une étude plus générale sur les jeux de données historiques est proposée par Nikolaidou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 43 44 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Tableau récapitulatif des différents jeux de données utilisés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le symbole "–" indique une résolution ou date non disponible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour chaque jeu de données, la colonne Taille indique la taille moyenne des images, calculée sur l’ensemble d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Date Images Lignes Langue(s) Résolution (dpi) Taille (pixels) AN-Index† – 34 666 Français – [1 949, 1 338] ± [796, 607] Balsac Anglais Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) 1850 – 1916 913 45 685 Français – [3 746, 2 671] ± [1 141, 627] BNPP† 19e – 20e siècle 12 1 281 Français – [3 710, 5 103] ± [21, 86] Bozen Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) 1470 – 1805 450 10 550 Allemand – [3 524, 2 398] ± [22, 62] cBAD2019 Diverses Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) – 3 021 193 858 européennes Variable [3 268, 2 751] ± [1 364, 1 504] DIVA-HisDB Italien, Latin Simistira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) 11e / 14e siècles 150 12 808 Allemand, Grec 600 [5 493, 3 843] ± [709, 728] HOME-Alcar Stutzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021) 12e – 14e siècle 1 845 136 206 Latin Variable [3 850, 4 506] ± [949, 1 820] Allemand HOME-NACR Latin Boros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) 1145 – 1491 496 7 614 Tchèque – [4 499, 6 206] ± [1 639, 2 292] Hugin-Munin Maarand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) 19e – 20e siècle 849 23 732 Norvégien – [3 998, 3 740] ± [1 405, 1 403] Horae Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) 14e – 15e siècle 573 13 796 Latin Variable [4 200, 4 648] ± [1 361, 2 112] IAM Marti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2002) 1999 1 539 13 353 Anglais 300 [3 542, 2 479] ± [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='33] RASM Clausner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) 10e – 19e siècle 120 2 619 Arabe 400 [7 674, 5 408] ± [1 384, 914] READ Diverses Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) 1470 – 1930 2 035 132 124 européennes Variable [3 966, 3 121] ± [1 350, 1 268] ScribbleLens Variable Dolfing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) 16e – 18e siècle 1 000 28 255 Néerlandais 150 – 300 [3 519, 2 375] ± [665, 459] † Jeux de données privés utilisés durant la thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 J E U X D E D O N N É E S 45 AN-Index – Ce premier jeu de données est composé de 34 images de documents des instruments de recherche numérisés des Archives nationales françaises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’une base privée dont les documents sont rédigés en français.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Balsac – Depuis 50 ans, le projet BALSAC 1 construit une importante base de données sur la population du Québec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour entraîner des modèles de traitement automatique et ainsi aider l’intégration de millions d’enregistrements, un échantillon du corpus contenant les actes de naissance, de mariage et de décès de la population québécoise de 1850 à 1916 a été annoté.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le jeu de données Balsac (Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) consiste donc en 913 images (pages simples ou doubles) extraites de 74 registres manuscrits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles ont été annotées au niveau des actes et des lignes avec leurs transcriptions et entités nommées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' BNPP – Ce jeu de données privé a été fourni par les Archives historiques de la banque BNP Paribas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il consiste en un échantillon de 12 images manuscrites extraites de cinq registres scannés de procès-verbaux du Comptoir National d’Escompte de Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles ont été sélectionnées parmi une centaine de registres rédigés en français entre le 19e et le 20e siècle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bozen – Ce jeu de données (Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016) fait partie du projet READ et consiste en 450 pages manuscrites annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les pages sont extraites de documents de la collection Ratsprotokolle écrits entre 1470 et 1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est annoté au niveau des lignes de texte avec leurs transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' cBAD2019 – Le jeu de données cBAD (Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) est constitué de 3 021 images de documents collectées dans sept archives européennes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a été utilisé lors de la compétition cBAD à ICDAR2019 pour la détection des lignes de base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' DIVA-HisDB – DIVA-HisDB (Simistira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016) est une base de données qui contient 150 images extraites de trois manuscrits médiévaux des 11e et 14e siècles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces manuscrits ont été choisis pour la complexité de leurs mises en page avec du texte principal, et des commentaires dans les marges et entre les lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour chaque manuscrit, 50 images ont été sélectionnées et réparties en 20 images d’entraînement, 10 images de validation, 10 images de test et 10 autres images de test « privées ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque image a été annotée manuellement au niveau du pixel pour les classes de corps de texte, décorations et commentaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' HOME-Alcar – Le jeu de données HOME-Alcar (Stutzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021) contient 17 cartulaires, recueils des chartes et des actes juridiques produits entre le 12e et le 14e siècle et écrits en latin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les images ont été annotées au niveau des lignes de texte avec leurs transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://balsac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='uqac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='ca/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='bnpparibas/ 46 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N HOME-NACR – Le jeu de données HOME-NACR (Boros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) est composé de 496 chartes médiévales sélectionnées parmi 43 000 chartes numérisées provenant des archives de la Couronne de Bohême et des archives des monastères.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles ont été rédigées de 1145 à 1491 en allemand, latin et tchèque du début de l’ère moderne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les chartes ont été annotées au niveau des lignes avec leurs transcriptions et entités nommées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Hugin-Munin – La base de données Hugin-Munin (Maarand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022) est constituée de pages provenant de correspondances et de journaux intimes de 12 artistes norvégiens écrits de 1820 à 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les documents ont été annotés au niveau des lignes avec leurs transcriptions correspondantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La base comporte 691 images d’entraînement, 85 de validation et 73 de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Horae – Durant le projet de recherche Hours : Recognition, Analysis, Edition (HORAE) (Stutzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019), un jeu de données a été créé (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) et consiste en 573 images annotées de livres d’heures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles ont été sélectionnées parmi 500 manuscrits car elles représentent la variété des mises en page et des contenus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les images ont été annotées à différents niveaux et avec différentes classes : page, paragraphe, ligne, miniature, initiale (simple, ornée ou illustrée), marge (ornée ou illustrée), ornementations et rubriques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' IAM – Le jeu de données IAM (Marti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2002) a été créé en 1999 et contient 1 539 images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque image comporte une page avec un texte imprimé extrait du corpus Lancaster - Oslo/Bergen corpus (LOB), puis ce même texte écrit à la main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les textes datent de 1961 et sont très divers : fictions, écrits scientifiques ou encore textes traitant de religion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' RASM – La compétition RASM 2018 (Clausner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) visait la reconnaissance de manuscrits historiques en arabe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un ensemble de 15 images de pages à une colonne a été utilisé pour l’entraînement et 85 pour évaluer les tâches de détection de lignes de texte, segmentation de pages et reconnaissance d’écriture manuscrite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au total, le jeu contient 120 images extraites parmi une collection de manuscrits scientifiques arabes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' READ-BAD – Ce jeu de données (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017) contient de 2 035 images de documents écrits entre 1470 et 1930 et extraits de neuf archives européennes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les données sont très variées avec des registres paroissiaux, des procès-verbaux ou encore des tables de recensement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce jeu a été utilisé lors de la compétition sur la détection de lignes de base cBAD : ICDAR2017 (Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le jeu de données est divisé en sous-ensembles simples et complexes dépendant de la complexité de mise en page des documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 A N N O TAT I O N D E S D O N N É E S 47 ScribbleLens – Le jeu de données ScribbleLens (Dolfing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) contient 1 000 images de pages de documents néerlandais datant du début de l’ère moderne, tels que des journaux de bord de navires et des journaux de bord quotidiens produits entre le 16e et le 18e siècle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les manuscrits consistent en des voyages écrits par des capitaines et des commerçants de la Vereenigde Oost-indische Company (VOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’ensemble de test est composé de 21 images annotées et transcrites au niveau de la ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous observons que de nombreux jeux de données ont été présentés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est important de noter que ces jeux de données ont été annotés à l’aide de différents outils et pour différentes tâches : détection de lignes de texte ou de lignes de base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous détaillons les différents types d’annotations dans la section suivante 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 A N N O TAT I O N D E S D O N N É E S Bien que la tâche de détection de lignes de texte soit assez triviale dans le cas de documents imprimés, dans le cas de documents manuscrits de nombreux aspects peuvent venir perturber la bonne détection des lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, il n’est pas rare que des lignes se chevauchent ou que des initiales soient de taille très différente comparé au corps de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, la qualité de numérisation et les possibles dégradations liées à la conservation peuvent rendre cette tâche d’autant plus complexe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un autre défi avec la détection de lignes de texte concerne la définition de ce qu’est une ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la littérature, une ligne de texte a été définie de plusieurs manières, comme présenté sur la Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 (Mechi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Renton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, elle peut être définie uniquement par sa ligne de base (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017), qui correspond à une ligne virtuelle soulignant la plupart des caractères tandis que les descendants restent en dessous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce cas il est nécessaire d’estimer la hauteur de la ligne pour appliquer un reconnaisseur d’écriture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle a également été définie comme étant une boîte englobante Rectangle englobant Ascendant Descendant Polygone englobant X-height Pixels Ligne de base Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Représentation des modélisations d’une ligne de texte proposées dans la littérature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 48 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N (rectangle ou polygone) incluant tous les ascendants et descendants (Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015), comme un ensemble de pixels appartenant au contenu textuel (Simistira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Vo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016) ou encore s’appuyant sur la hauteur en X (X-height).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit de la bande de base de la ligne sans les ascendants et descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Définir une ligne de texte par sa bande de base présente de nombreux avantages par rapport aux autres représentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, elle représente bien les interlignes même lorsque les lignes se chevauchent en raison des ascendants ou des descendants, contrairement à une représentation par boîte englobante incapable de séparer les lignes qui se chevauchent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, l’utilisation de la représentation pixel ou de la ligne de base nécessite un post- traitement avant de pouvoir être transmise à un reconnaisseur, contrairement à la bande de base qui semble plus appropriée à fournir une entrée convenable pour les reconnaisseurs de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous résumons, dans la Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2, les détails d’annotations de chaque jeu de données pré- senté en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette Table, nous indiquons comment les lignes ont été annotées : ligne de base, bande de base, polygone englobant, quadrilatère (ou polygone simple) englo- bant et rectangle englobant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous indiquons également le taux de relâchement des annotations par rapport aux pixels de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous définissons ce taux comme étant la quantité de fond présent autour des pixels de texte dans les annotations (autres que la ligne de base).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les co- lonnes Intersections et Source présentent respectivement la quantité de chevauchements entre les annotations, et la manière dont les annotations ont été obtenues (manuellement, semi-automatiquement ou automatiquement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La dernière colonne indique la présence de transcription des lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette Table montre explicitement que les annotations entre les différents jeux de données sont très variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, il n’y a aucun consensus sur la définition même d’une ligne de texte ni sur la forme que les annotations doivent avoir ou la quantité de fond à intégrer dans les polygones et boîtes englobants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, il n’y a aucune étude, à notre connaissance, comparant les différentes annotations possibles et évaluant leurs impacts sur les résultats de reconnaissance de texte finaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 présente une image de chaque jeu de données associée à son taux de relâchement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les problèmes liés aux annotations des jeux de données ont également été peu étudiés dans la littérature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, Barakat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) montrent les problèmes liés aux lignes de texte qui se touchent et se superposent dans leur jeu de données mais ne traitent pas ces différents problèmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Quelques rares travaux en discutent et proposent quelques solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par exemple, face à des boîtes englobantes qui se touchent, Melnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ont suggéré de supprimer les ascendants et les descendants des lignes de texte en réduisant la hauteur des boîtes annotées de 30 % en haut et en bas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont ensuite redimensionné les polygones à la résolution d’entrée du modèle pour entraîner le système.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même si cette méthode s’est avérée efficace pour réduire le biais d’étiquetage de l’annotation, certains problèmes subsistent lorsqu’il s’agit de lignes verticales et inclinées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le même esprit, Peskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ont proposé différents masques d’annotation (voir Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3) pour la détection et 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 A N N O TAT I O N D E S D O N N É E S 49 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Tableau récapitulatif du type d’annotation des différents jeux de données utilisés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Relâchement indique la quantité de fond présent dans les annotations (important, moyen ou faible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Intersections indique si des lignes se chevauchent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Source indique comment les annotations ont été obtenues : manuellement, semi-automatiquement ou automatiquement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Texte indique la présence de transcriptions des lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Ligne Bande Polygone Quadrilatère Rectangle Relâchement Intersections Source Texte de base de base englobant englobant englobant AN-Index† ✓ ✓ Important Rares Manuelle Balsac Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ✓ ✓ Moyen Rares Manuelle ✓ BNPP† ✓ Moyen Non Manuelle ✓ Bozen Semi- Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) ✓ ✓ ✓ Important Oui automatique ✓ cBAD2019 Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) ✓ ✓ ✓ Variable Oui Manuelle DIVA-HisDB Semi- Simistira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) ✓ ✓ Faible Non automatique HOME-Alcar Stutzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021) ✓ Important Oui Automatique ✓ HOME-NACR Boros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ✓ Important Rares Manuelle ✓ Hugin-Munin Semi Maarand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) ✓ ✓ Variable Oui automatique ✓ Horae Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) ✓ Moyen Non Manuelle ✓ IAM Marti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2002) ✓ Moyen Rares Automatique ✓ RASM Clausner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) ✓ Faible Rares Manuelle ✓ READ Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) ✓ ✓ ✓ Variable Oui Manuelle ScribbleLens Dolfing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ✓ Important Oui Automatique ✓ † Jeux de données privés utilisés durant la thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 50 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N HOME-Alcar ScribbleLens RASM Bozen IAM Horae HOME-NACR AN-Index Balsac BNPP cBAD READ-BAD Hugin-Munin DIVA-HisDB Relˆachement important Relˆachement moyen Relˆachement faible Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Visualisation des différents taux de relâchement détectés dans les jeux de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les taux de relâchement indiquent la quantité de fond présent autour des pixels de texte dans les annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 31 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' paeus Jeeun-die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='cuos sharmm Allacpmoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='er a osull non cuffioyecas pstaaumtoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='o Queng5olitayeonennateim umaneque c cehrerephetnnaum eroitee S faeecm poulersfoa suiplepefn hoc ircam Loghootalye Ligaed Cg emoh fiusoim nmlonem Louicg inlren 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='aecrueyuose nmndFeernun mles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='omeneelo Lts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='nommac wnc Chcoblitum saj-may itusallag-satue mono cem Cnicne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='pcipiencos mmaentf-Afef-umSmt Letues mles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='fanemereisius fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='coainfl ametoum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Aatbae ncrufaemax ng Silcie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ylpino Eheliuramo nr ccalte asupmoir-eeim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='swie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='cjuos6i mahe ek ncuen Arolles gehaiert velek kranl ancksedgin Arere wytrrehhs Zagenasy D27 dwele Leel012029 S3SentenceDatabase N06-169 as of now, apart from a few sacks of gold dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='" He winked at his partners, They all watched us as we ate the beans, Then when we\'d finished and I\'d rolled a cigarette the man called Shorty said, "You were saying when you cane in that 2romep\'n happeued last night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='" heyes hieeLobr EHeAEL eofForheederedLerolleekacroaek A Csemepnhepperedlasr eh Seaemat YYYYYYYYCY ooemceetal Do0somrKODAKColorControlPatches White Kodaloul Jean nhmulevue tntchomcle mahup d aRochelh 0 691mkh13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 737 mReclu 1318 Melens174 Tbanguay Sreat KUC essawmel agenaSlasell utle Coli ltapigaliot Speilt房 nenahkwbwtuleM Cam 3 M Guhellu 340GrimmNr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Ms30 ldeeLalt Best 00 Culpi-oitmneedeansparem Cpadntagmuoam/e uebemgeecodnuabnth duetmnabrammem/monaadathemm Caogpuemh-mmtnediczombu-dntcedmc mampatmatongndemde aogtimdagumdcapoeo idnoumtit-AfatimnoncamnabmusT mtaurmafmaommbnaaxpdae odeayapdooneilmommn Qcodecmmemmagpuconb taggmonnepunoabitronuopuya doptim-etogmamgprahgtiumdopunm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 M É T R I Q U E S D’ É VA L U AT I O N 51 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Masques de segmentation comparés par Peskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' petites marques cen- trales, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' marques centrales plus grandes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' petite marque centrale avec un contour d’un pixel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' petite marque centrale avec un contour de deux pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Schéma issu de Peskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' la classification de formes géométriques à partir d’images en niveaux de gris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont suggéré d’annoter les objets (cercles, rectangles et triangles) de quatre façons : avec de petites marques centrales, avec de grandes marques centrales, avec de petites marques centrales et un contour d’un pixel, et avec de petites marques centrales et un contour de deux pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Concernant les problèmes de localisation, ils ont montré que les petites marques centrales donnent les meilleures performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, de meilleurs résultats de classification sont obtenus avec les petites marques centrales avec contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, trouver l’annotation la plus adaptée n’est pas un problème trivial et les discussions sont toujours ouvertes, mais nous proposons, dans le chapitre 5, une solution pour l’unification des différents types d’annotations pour entraîner des modèles de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 M É T R I Q U E S D’ É VA L U AT I O N En plus des problèmes liés aux différents types d’annotations se pose le problème de re- cherche de métriques appropriées pour évaluer et comparer correctement les résultats de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, afin d’évaluer la qualité d’un modèle, une métrique d’évaluation doit être définie et calculée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette métrique doit avoir un sens vis-à-vis des données utilisées et de l’application future des résultats obtenus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les différentes métriques permettant d’évaluer un modèle de détection d’objets peuvent être regroupées en plusieurs catégories : les métriques basées sur les pixels, les métriques objets et les métriques orientées vers la tâche finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celles- ci sont détaillées dans les paragraphes suivants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, Hemery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2010) ont étudié les propriétés clés qu’une métrique doit avoir pour une tâche de localisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À partir de l’analyse de 33 métriques existantes, ils ont établi les plus appropriées pour cette tâche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En suivant cette idée, nous montrons, dans cette section, que les principales métriques actuellement utilisées ne sont pas suffisantes pour évaluer et comparer les modèles de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 métriques basées sur les pixels Les métriques calculées au niveau des pixels sont principalement basées sur l’intersection entre les pixels d’une région prédite et ceux d’une région annotée manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme le montre la Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3, la majorité des systèmes de détection existants sont évalués à l’aide de métriques pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les mesures de précision et de rappel, détaillées dans le Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, 52 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Métriques d’évaluation utilisées dans les récents travaux liés à la détection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' P et R représentent respectivement les métriques précision et rappel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les métriques R@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 et mAP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='65 représentent respectivement le rappel pour un seuil d’IoU de 0,85 et la précision moyenne pour un seuil d’IoU de 0,65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Système Pixel Objet IoU P/R F1 P/R R@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95 mAP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='65 mAP Tensmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) ✓ Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) ✓ ✓ Barakat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) ✓ ✓ Renton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) ✓ ✓ dhSegment Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) ✓ ✓ ✓ ✓ Mechi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) ✓ ✓ ✓ Tarride et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) ✓ ✓ ✓ ✓ Soullard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ✓ Melnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ✓ ✓ Mechi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021) ✓ ✓ sont largement utilisées, ainsi que l’Intersection-sur-Union (IoU) (Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2) et le score F1 (Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas d’une détection à plusieurs classes et afin de calculer une unique valeur d’évaluation, les métriques sont calculées pour chaque classe et la moyenne arithmétique des valeurs obtenues est calculée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Alberti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) ont d’ailleurs développé un outil permettant d’évaluer des modèles à partir des images de vérité terrain et des prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet outil permet de calculer différentes valeurs dont l’IoU, mais également de visualiser les résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces métriques sont basées sur le nombre de pixels correctement prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, elles ne donnent aucune information sur le nombre d’objets correctement prédits et manqués ou divisés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces métriques sont également biaisées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, comme le montre la Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4, plusieurs prédictions de qualités différentes peuvent être caractérisées par les mêmes valeurs d’IoU et de F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, les Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4b et 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4c présentent deux prédictions pour la même image, en haut, et leur superposition, en bas, avec l’image de vérité terrain présentée sur la Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La première prédiction montre des lignes divisées et fusionnées, une ligne manquante (en rouge) et quelques faux positifs (en cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au contraire, la seconde prédiction montre des lignes plus épaisses mais pas de ligne manquante ni de faux positifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, la seconde prédiction semble meilleure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, les valeurs d’IoU et de score F1 sont égales 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 M É T R I Q U E S D’ É VA L U AT I O N 53 (a) Image de label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (b) Première prédiction : IoU = 0,72 F1 = 0,84 P = 0,81 R = 0,87 AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 = 0,68 (c) Seconde prédiction : IoU = 0,72 F1 = 0,84 P = 0,75 R = 0,95 AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 = 0,94 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Deux détections de lignes différentes obtenues pour une même image et obtenant les mêmes scores d’IoU et de F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les superpositions sont générées avec l’outil DIVA (Alberti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le vert et le noir correspondent respectivement aux pixels d’arrière-plan et d’avant-plan correctement prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le cyan représente les pixels fausse- ment positifs et le rouge les pixels faussement négatifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ici, seul le score AP au niveau objet (avec un seuil IoU de 50 %) permet d’évaluer et de comparer les prédictions avec précision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' pour les deux prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela illustre le fait que ces métriques ne sont pas les plus adaptées pour évaluer les systèmes de détection d’objets, d’où le développement de nouvelles métriques basées sur les objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, le calcul de ces valeurs de précision et de rappel au niveau de la ligne ou de l’objet n’est pas directement applicable car la décision qu’un objet soit bien ou mal détecté est plus complexe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un autre aspect des faiblesses des métriques pixel a été étudié par Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils mettent en évidence le fait que la métrique IoU ne se concentre pas suffisamment sur les contours des objets qui sont les positions les plus importantes à détecter dans des tâches de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils présentent Boundary IoU, une nouvelle mesure d’évaluation de la segmentation axée sur la qualité des frontières.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils analysent différents types d’erreurs sur différentes tailles d’objets et montrent que leur métrique est significativement plus sensible aux erreurs de frontière pour les objets de grande taille, sans pénaliser les erreurs sur les petits objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De la même manière, ils proposent une adaptation de la précision moyenne se basant sur le Boundary IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, lorsqu’un objet prédit ne possède pas d’intersection avec un objet réel, leur IoU est égale à zéro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, la prédiction peut-être plus ou moins proche de la vérité, or l’IoU ne peut pas refléter ce phénomène.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, Rezatofighi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) ont proposé la GIoU (Generalized IoU) qui permet de considérer l’espace entre les deux objets dans l’évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 54 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – PRÉCISION ET RAPPEL Définition Dans le cadre de la détection d’objets, la précision est le nombre d’éléments (pixels ou objets) pertinents détectés, d’une classe considérée, rapporté au nombre total d’éléments détectés de cette classe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle tente de répondre à la question « Quelle proportion d’identifications positives est correcte ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le rappel est défini par le nombre d’éléments pertinents détectés rapporté au nombre total d’éléments annotés de la classe considérée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il montre donc la proportion de positifs réels qui ont été correctement identifiés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Équations P = TP TP + FP R = TP TP + FN (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1) avec : — TP : nombre de pixels ou d’objets positifs correctement prédits ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — FP : nombre de pixels ou d’objets négatifs prédits comme positifs ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — FN : nombre de pixels ou d’objets positifs prédits comme négatifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – INTERSECTION-SUR-UNION Définition La métrique Intersection-sur-Union (IoU) évalue la division entre la zone de chevauchement et la zone d’union entre deux régions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En d’autres termes, elle évalue le degré de chevauchement entre la vérité terrain et les prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle est comprise entre 0 et 1, où 1 correspond à un chevauchement parfait entre la vérité terrain et la prédiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Équation IoU = TP TP + FP + FN (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2) avec : — TP : nombre de pixels positifs correctement prédits ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — FP : nombre de pixels négatifs prédits comme positifs ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — FN : nombre de pixels positifs prédits comme négatifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – F1-SCORE Définition La F-mesure, aussi appelé F1-score, est la moyenne harmonique entre la précision et le rappel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 M É T R I Q U E S D’ É VA L U AT I O N 55 Équation F1-score = 2 × TP 2 × TP + FP + FN = 2 × P × R P + R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3) avec : — TP : nombre de pixels ou d’objets positifs correctement prédits ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — FP : nombre de pixels ou d’objets négatifs prédits comme positifs ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — FN : nombre de pixels ou d’objets positifs prédits comme négatifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 métriques orientées objets Bien que des évolutions des métriques usuelles au niveau pixel aient été proposées, elles ne fournissent toujours pas d’évaluation au niveau objet et ne permettent pas d’indiquer le nombre d’objets correctement détectés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un des problèmes pour la mise en place d’une métrique objet est la difficulté à déterminer si un objet est correctement détecté ou non.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour résoudre ce problème, des métriques conçues à l’origine dans la communauté de la recherche d’information (Information Retrieval) ont été adaptées aux images, et utilisées lors du PASCAL VOC Challenge 2012 pour calculer la précision au niveau des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Lors de cette compétition, la tâche de détection a été évaluée sur la base de la courbe Précision- Rappel au niveau objet, où les détections sont considérées comme de vrais ou de faux positifs en fonction de leur zone de recouvrement avec les objets de vérité terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Suivant cette approche, Tarride et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) associent d’abord les objets prédits à ceux annotés et considèrent une prédiction comme un vrai positif si son IoU est supérieure à un seuil choisi t = 0,65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, ils peuvent calculer la précision (P@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='65), le rappel (R@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='65) et la précision moyenne (mAP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='65) au niveau des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les calculs de précision moyenne sont détaillés dans le Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Soullard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) utilisent la mean Average Precision (mAP), c’est-à-dire la précision moyenne calculée pour différents seuils d’IoU, afin d’évaluer leur modèle de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2006) ont montré l’importance de la qualité de détection (précision des objets détectés) et de la quantité de détections (nombre d’objets) lors de l’évaluation d’un système.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La mesure mAP, qui est l’aire sous la courbe Précision-Rappel, permet d’évaluer la quantité de détections en fonction d’un critère de qualité donné : le seuil d’IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Afin de pouvoir mesurer autant la qualité que la quantité de détections, l’utilisation de la mAP moyennée sur une gamme de seuils d’IoU a émergé pour la détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, Soullard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ont utilisé cette moyenne mAP pour évaluer leur modèle de détection appliqué aux journaux historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La métrique ZoneMap proposée par Galibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2015) évalue également les systèmes de détection au niveau objet et ne repose sur aucun seuil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle est basée sur les liens entre les zones d’hypothèse et de référence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les forces des liens sont d’abord calculées : si une zone prédite est correcte, alors la force avec une zone de référence sera élevée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au contraire, toutes les forces pour une zone faussement positive seront faibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, les zones sont regroupées 56 E N T R A Î N E M E N T E T É VA L U AT I O N D E S S Y S T È M E S D E D É T E C T I O N en fonction de ces liens et chaque groupe reçoit une erreur de segmentation et une erreur de classification, calculées en fonction du type de groupe (match, miss, false alarm, merge ou split).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces deux erreurs sont ensuite combinées pour donner une seule valeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même si cette métrique s’est avérée complémentaire de la métrique IoU dans l’évaluation du projet Maurdor (Oparin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2014), elle n’est pas réellement utilisée à l’heure actuelle en raison de la complexité de ses calculs et de sa difficile applicabilité aux images comportant de nombreux objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – PRÉCISION MOYENNE / AVERAGE PRECISION Définition Le concept de la métrique d’évaluation de la précision moyenne est principalement lié aux compétitions PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Basé sur un seuil défini d’IoU, elle considère les objets prédits comme vrais ou faux positifs, et calcule la précision moyenne grâce à l’aire sous la courbe de la précision par rapport au rappel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La mean Average Precision (mAP) est définie de plusieurs manières selon la com- pétition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas d’un problème à plusieurs classes, la mAP est définie comme étant la valeur moyenne des AP calculées pour chaque classe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle peut aussi correspondre à la moyenne arithmétique réalisée sur plusieurs seuils d’IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, pour s’abstenir d’un seuil prédéfini, la mAP est la moyenne des AP calculées pour plusieurs seuils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas des compétitions PASCAL VOC, la moyenne est calculée sur des valeurs de seuil allant de 0,5 à 0,95 avec un pas de 0,05 (mAP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='05 :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95] ou mAP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Mise en oeuvre — Toutes les prédictions sont ordonnées par leur confiance moyenne décroissante ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Les prédictions dont l’IoU est supérieur ou égal à un seuil t sont considérées comme vrais positifs ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — La courbe Précision-Rappel est construite à partir des prédictions ordonnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette courbe Précision-Rappel permet d’évaluer les performances d’un détec- teur d’objets en fonction d’un seuil sur la confiance associée à la prédiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il existe une courbe pour chaque classe d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un détecteur d’objets d’une classe particulière est considéré comme bon si sa précision reste élevée alors que le rappel augmente, ce qui signifie que si le seuil de confiance varie, la préci- sion et le rappel resteront élevés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Habituellement, la courbe Précision-Rappel commence par des valeurs de précision élevées, qui diminuent à mesure que le rappel augmente ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — La courbe est interpolée de telle sorte que la précision p pour un rappel r prenne la valeur de la précision maximale des rappels supérieurs à r : pinterp(r) = max ˜r⩾r p(˜r) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4) — La précision moyenne (AP@t) est égale à l’aire sous la courbe Précision-Rappel interpolée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 M É T R I Q U E S D’ É VA L U AT I O N 57 Équations Pour une classe donnée c et un seuil d’IoU t, nous avons : AP@tc = � 1 0 pc t(rc t)drc t (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5) Pour un seuil d’IoU t, la AP moyennée sur toutes les classes est calculée comme suit : mAP@t = �C c=1 AP@tc C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6) Pour une classe donnée c, la AP moyennée sur plusieurs seuils est calculée comme suit : mAP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95]c = �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 AP@tc 10 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7) Enfin, la AP moyennée sur plusieurs seuils et l’ensemble des classes est calculée comme suit : mAP = �C c=1 mAP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95]c C ou mAP = �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 mAP@t 10 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8) avec : — C : le nombre de classes ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — p : la précision ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — r : le rappel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 métriques orientées vers la tâche finale Trier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (1995) ont montré l’importance d’une évaluation orientée vers la tâche finale pour les méthodes de binarisation, puisque l’évaluation par un expert humain dépend de ses critères visuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont appliqué onze méthodes de binarisations adaptatives locales à des images de test avant de transmettre les résultats à un module de reconnaissance OCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les méthodes de binarisation ont ensuite été comparées avec les taux de reconnaissance, d’erreur et de rejet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont montré que les classements des méthodes en fonction de la qualité de binarisation et des taux d’erreurs obtenus après le module OCR étaient différents, insistant sur l’importance d’utiliser des métriques orientées vers la tâche finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De même, Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2006) ont montré l’importance de ces mesures pour la détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un cadre de prédiction pixel, les objets sont reconstruits en regroupant les pixels connectés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Or, les objets extraits sont similaires même si quelques pixels ont été mal prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce sens, une de nos propositions vise à utiliser des métriques de reconnaissance de texte (HTR) pour évaluer les systèmes de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' A notre connaissance, il n’existe pas de travaux antérieurs dans la littérature à ce sujet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4 D É T E C T I O N D ’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S La compréhension automatique de documents, et plus particulièrement l’analyse de la mise en page de documents historiques, est toujours un domaine de recherche actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette tâche consiste à diviser un document en différentes régions en fonction de leur contenu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La grande variété des documents existants rend cette tâche très complexe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour détecter des objets dans des images de documents, de nombreux systèmes ont été proposés, la plupart assignant une classe à chaque pixel de l’image donnée en entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que ces systèmes aient montré des performances intéressantes, ils nécessitent un grand nombre de données d’apprentissage annotées et présentent des temps d’inférence longs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce chapitre, nous présentons deux architectures à l’état de l’art et comparons leurs avantages et inconvénients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par la suite, nous proposons, en section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3, un système appelé Doc-UFCN mis au point afin de dépasser les limitations mises en évidence par la comparaison des systèmes à l’état de l’art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons enfin une comparaison expérimentale des systèmes pour la détection des lignes de texte, en section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4, et la détection d’actes, en section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 P R É S E N TAT I O N D U P R O B L È M E L’entraînement de modèles de détection d’objets dans les images de documents requiert un grand nombre de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, dans le cas de documents historiques, ces données annotées sont rarement disponibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour pallier ce problème, des systèmes utilisant des poids pré-entraînés tels que dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) ont été proposés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela permet d’utiliser des réseaux avec plus de paramètres sur des jeux de données de tailles réduites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, l’utilisation du pré-entraînement a montré de nombreux avantages tels que la diminution du temps d’apprentissage et l’amélioration de la précision du modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cepen- dant, ces poids sont souvent appris sur des images de scènes naturelles (ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2009)), puis appliqués à des images de documents, ce qui pose un problème d’adaptation des modèles à un nouveau domaine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, bien qu’ils obtiennent des performances élevées, les systèmes actuellement à l’état de l’art peuvent présenter des temps d’inférence longs qui peuvent avoir de grands impacts financiers et écologiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un cadre industriel, l’utilisa- tion de tels systèmes ne semble pas appropriée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi, nous proposons un nouveau modèle, appelé Doc-UFCN, dans l’optique de répondre à ces problématiques de temps de traitement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les contraintes étant que ce système possède un nombre réduit de paramètres et présente un temps d’inférence réduit par rapport aux systèmes existants, tout en obtenant des performances à l’état de l’art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 59 60 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 S Y S T È M E S À L’ É TAT D E L’ A RT De nombreux modèles (Barakat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Mechi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Renton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) ont été proposés pour la détection d’objets dans les images de documents, notamment pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces modèles ont des architectures similaires, suivant une architec- ture U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle dhSegment suit également l’architecture U-Net, mais, contrairement aux autres systèmes, il intègre une partie pré-entraînée et a été testé sur de nombreuses tâches telles que la détection de pages, de décorations ou encore de photographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi, nous avons choisi de nous comparer à ce système.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il en est de même pour le modèle de Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) qui intègre l’information textuelle afin d’aider la détection des objets et qui a obtenu de bonnes performances sur des images de documents modernes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces deux systèmes sont détaillés dans les Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 et 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 A R C H I T E C T U R E D U S Y S T È M E P R O P O S É : D O C- U F C N Nous présentons, dans cette section, l’architecture du modèle que nous proposons Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre objectif est de concevoir un modèle comportant peu de paramètres afin d’être entraîné sur des jeux de données restreints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, celui-ci devra montrer des temps d’inférence réduits par rapport aux modèles à l’état de l’art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le développement de ce modèle étant réalisé dans un cadre industriel, il est nécessaire d’avoir un modèle rapide en inférence, capable de traiter des millions d’images de documents dans des délais raisonnables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En- fin, ce modèle doit répondre à ces différents points tout en obtenant des performances élevées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour concevoir notre système, Doc-UFCN, nous avons choisi d’utiliser le cœur du réseau de Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) car il possède un nombre réduit de paramètres et ne contient pas de parties pré-entraînées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour réduire davantage le nombre de paramètres et être capable d’entraîner notre modèle sur peu de données, seul le contenu visuel est utilisé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, la carte d’incorporation de texte, le pont et le second décodeur pour la tâche de reconstruction ne sont pas intégrés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre architecture est donc un réseau entièrement convolutif (FCN) en forme de U composé d’un encodeur (blocs rouges sur la Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1) suivi d’un décodeur (blocs bleus) et d’une couche de convolution finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’encodeur de Doc-UFCN diffère de celui de dhSegment puisqu’il ne suit pas l’architecture ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre encodeur possède beaucoup moins de paramètres et est entièrement entraîné sur des images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, les deux systèmes ont des décodeurs similaires avec des connexions résiduelles où les cartes de caractéristiques calculées pendant l’encodage, à une échelle donnée, sont utilisées pendant l’étape de décodage de cette même échelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’utilisation d’un FCN sans couche dense présente de nombreux avantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, il réduit fortement le nombre de paramètres puisqu’il n’y a aucune connexion dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, cela permet au réseau de traiter des images de taille variable et de conserver les informations spatiales telles quelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 A R C H I T E C T U R E D U S Y S T È M E P R O P O S É : D O C- U F C N 61 f W H Dilated Block 1 2f W 2 H 2 Dilated Block 2 4f W 4 H 4 Dilated Block 3 8f Dilated Block 4 4f 4f W 4 H 4 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Block 1 || 2f 2f W 2 H 2 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Block 2 || f f W H Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Block 3 || c c W Softmax Dilated convolution Convolution Max pooling Upscaling Softmax || Concatenation c Number of classes Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Schéma de l’architecture du modèle Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’encodeur est représenté en rouge et le décodeur en bleu avec respectivement H et W les hauteur et largeur de l’image d’entrée et f le nombre de cartes de caractéristiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 encodeur L’encodeur vise à extraire les caractéristiques importantes de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il consiste en quatre blocs dilatés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceux-ci sont légèrement différents de ceux présentés par Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) puisqu’ils consistent en cinq convolutions dilatées consécutives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’utilisation de convo- lutions dilatées au lieu de convolutions standards permet au champ réceptif d’être plus grand et au réseau d’avoir plus d’informations contextuelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, exécuter ces convolutions successivement plutôt qu’en parallèle permet d’agrandir le champ réceptif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque bloc est suivi d’une couche de max-pooling, sauf le dernier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 décodeur L’objectif du décodeur est de reconstruire l’image d’entrée avec un étiquetage pixel par pixel à la résolution de l’image d’entrée originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette déconvolution est généralement effectuée à l’aide de convolutions transposées ou d’une mise à l’échelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme suggéré par Mechi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019), nous avons décidé de remplacer les couches de déconvolution du système de Yang par des convolutions transposées afin de conserver la même résolution en entrée et en sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, le chemin de décodage est composé de trois blocs convolutifs, chacun consistant en une convolution standard suivie d’une convolution transposée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les caractéristiques calculées lors de l’étape d’encodage sont concaténées avec celles de l’étape de décodage (flèches violettes sur la Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 62 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S La dernière couche convolutive produit des cartes de caractéristiques en pleine résolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle renvoie C cartes de caractéristiques de la même taille que l’image d’entrée, C étant le nombre de classes concernées dans l’expérience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une couche softmax est ensuite appliquée pour transformer ces cartes de caractéristiques en cartes de probabilités.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 détails d’implémentation Nous donnons, dans cette section, des détails techniques sur l’implémentation utilisée lors de nos expériences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' taille des images en entrée Nous avons décidé d’utiliser la même taille d’image d’entrée que celle de Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les images d’entrée ainsi que leurs vérités terrain sont donc redimensionnées en images plus petites telles que la plus grande dimension de l’image soit égale à 384 pixels tout en conservant le ratio de l’image originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela permet de réduire le temps d’apprentissage et de traitement sans perdre trop d’informations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons également entraîné le modèle sur une taille d’entrée plus grande, de 768 pixels, pour voir l’impact de ce choix (voir en section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' blocs dilatés Comme indiqué précédemment, tous les blocs dilatés sont composés de cinq convolutions dilatées consécutives avec des taux de dilatation de 1, 2, 4, 8 et 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les blocs comportent respectivement 32, 64, 128 et 256 filtres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque convolution a un noyau de taille 3×3, un stride de 1 et un padding adapté pour garder la même taille de tenseur dans tout le bloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Toutes les convolutions des blocs sont suivies d’une couche de batch normalization, d’une activation ReLU et d’une couche de dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle comportant peu de paramètres, les couches de dropout permettent d’éviter le sur-apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' blocs convolutifs Les blocs convolutifs sont utilisés pendant l’étape de décodage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le décodeur est composé de trois blocs convolutifs et chaque bloc est composé d’une convolution standard suivie d’une convolution transposée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les blocs ont respectivement 128, 64 et 32 filtres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque convolu- tion standard a un noyau de taille 3×3, un stride et un padding de 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque convolution transposée a un noyau de taille 2×2 et un stride de 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme pour les blocs dilatés, toutes les convolutions standards et transposées sont suivies d’une couche de normalisation, d’une activation ReLU et d’une couche de dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La dernière couche de convolution est paramétrée comme suit : C (nombre de classes) filtres, noyau 3×3, stride et padding de 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle est suivie d’une couche softmax qui calcule les probabilités de chaque pixel d’appartenir aux C classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E 63 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Statistiques des jeux de données utilisés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Images Lignes train valid test train valid test Balsac Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) 730 92 91 36 941 4 592 4 323 DIVA-HisDB Simistira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) 60 30 30 6 037 2 999 2 897 Horae Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) 522 20 30 12 568 270 958 READ-BAD Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) 388 49 49 22 885 2 699 2 481 post-traitement Comme étape de post-traitement, nous appliquons les mêmes opérations que celles appli- quées par dhSegment : les pixels ayant une probabilité supérieure à un seuil t sont conservés comme appartenant à la classe correspondante, les autres étant assignés au fond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les compo- santes connexes composées de moins de min_cc pixels sont supprimées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E Dans cette section, nous comparons Doc-UFCN et dhSegment sur une tâche de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous montrons que Doc-UFCN obtient de meilleures performances tout en ayant moins de paramètres et un temps de prédiction réduit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons tout d’abord les données utilisées, puis nous discutons des entraînements et des résultats obtenus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 jeux de données Les systèmes sont comparés sur quatre jeux de données annotés pour la détection de lignes de texte : Balsac (Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020), DIVA-HisDB (Simistira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016), Ho- rae (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) et READ-BAD (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 présente les détails de ces bases et la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 en présente les statistiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces jeux de données sont très différents, ce qui permet de tester les systèmes sur des tâches à complexité variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, la base Balsac contient des pages avec uniquement du texte réparti en actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit de documents structurés semblables les uns aux autres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les pages de la base DIVA-HisDB ne contiennent également que du texte, mais les mises en page sont plus complexes avec des commentaires dans les marges et entre les lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les images de la base Horae présentent des pages hétérogènes qui peuvent contenir des illustrations, une quantité variable de lignes de texte et d’initiales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, READ-BAD comporte deux sous-ensembles, l’un dit "simple" et l’autre "complexe", qui permettent d’évaluer et de comparer les systèmes sur une grande diversité de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 64 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 résultats et discussion Nous avons entraîné Doc-UFCN et dhSegment dans les mêmes conditions sur les quatre jeux de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette section détaille les entraînements et présente les résultats obtenus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' détails des entraînements Doc-UFCN est implémenté à l’aide du framework PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous l’avons entraîné avec un learning rate initial de 5e − 3, l’optimiseur Adam (Kingma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) et la fonction de coût d’entropie croisée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les poids sont initialisés grâce à l’initialisation Glorot (Glorot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous avons utilisé des mini-batchs de taille 4 pour réduire le temps d’apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons testé différentes probabilités de dropout et décidé de conserver une probabilité de 0,4, car elle correspond au modèle ayant donné les meilleures performances, en moyenne, sur l’ensemble de validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est entraîné sur un maximum de 200 époques et nous gardons le meilleur modèle en validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons également entraîné dhSegment sur ces mêmes données pour un maximum de 60 époques puisque le modèle est pré-entraîné et converge plus rapidement que le nôtre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons utilisé des mini-batchs de taille 4 et des patchs de taille 400×400 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le learning rate initial est de 5e − 5 et nous avons choisi d’utiliser un ResNet-50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016) comme encodeur pré-entraîné puisque les bonnes performances présentées dans Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) ont été obtenues avec ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme pour Doc-UFCN, le meilleur modèle obtenu pendant l’apprentissage sur l’ensemble de validation est sélectionné.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les deux modèles ont la même étape de post-traitement avec les mêmes hyper-paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Après avoir comparé des valeurs de seuil allant de 0,5 à 0,9, nous avons conservé le seuil t = 0,7 qui permet d’obtenir les meilleurs résultats sur l’ensemble de validation, avec une bonne capacité d’acceptation des pixels attendus comme des lignes de texte et de rejet des pixels appartenant au fond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, les composantes connexes de moins de min_cc = 50 pixels sont écartées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Plusieurs valeurs ont également été comparées pour ce paramètre, cependant, il n’a que peu d’impact sur les résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' évaluation des modèles La plupart des méthodes existantes sont évaluées avec la métrique IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette métrique mesure la similarité moyenne entre les pixels prédits et les pixels de la vérité terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Alberti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) ont conçu un outil pour évaluer la performance d’un modèle en calculant l’IoU, la précision, le rappel et la F-mesure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons utilisé cet outil pour évaluer les modèles car il permet d’obtenir plus d’informations concernant les performances du modèle au niveau du pixel que l’IoU seule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, pour évaluer les modèles, nous avons calculé différentes métriques au niveau du pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous rapportons l’IoU ainsi que la précision (P), le rappel (R) et le score F1 dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour être comparables, les images prédites par dhSegment sont redimensionnées de sorte que leur plus grande dimension soit égale à 384 pixels avant de calculer les métriques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les valeurs ne sont présentées que pour la classe des lignes de texte (le fond n’est pas considéré ici).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E 65 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Résultats obtenus par Doc-UFCN et dhSegment au niveau pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Résultats donnés sur les ensembles de test pour la tâche de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Système IoU P R F1-score Doc-UFCN 0,84 0,95 0,88 0,91 Balsac dhSegment 0,74 0,92 0,79 0,85 Doc-UFCN 0,76 0,92 0,81 0,86 DIVA-HisDB dhSegment 0,74 0,92 0,79 0,85 Doc-UFCN 0,64 0,78 0,80 0,85 Horae dhSegment 0,65 0,72 0,89 0,82 Doc-UFCN 0,64 0,82 0,76 0,77 READ-Simple dhSegment 0,65 0,85 0,72 0,77 Doc-UFCN 0,54 0,84 0,62 0,73 READ-Complex dhSegment 0,53 0,79 0,59 0,69 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Détections de lignes produites par dhSegment, au centre, et notre modèle Doc-UFCN, à droite, sur l’image de la page 5 verso du Livre d’heures Horae 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats obtenus par notre méthode sont en moyenne supérieurs à ceux obtenus par dhSegment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur le jeu de données Balsac, notre modèle surpasse dhSegment jusqu’à 6 points pour la métrique F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous observons également des gains respectifs de 3 et 4 points de F1-score pour les bases Horae et READ-Complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces résultats s’expliquent par une meilleure séparation des lignes de texte proches qui sont souvent fusionnées par dhSegment (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre modèle aide à séparer ces lignes là où dhSegment échoue, mais permet également d’obtenir des contours plus lisses et plus précis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, dhSegment semble avoir plus de difficultés à détecter les lignes verticales et prédit parfois de très petites lignes dans les miniatures, contrairement à Doc-UFCN qui semble plus robuste sur ce type d’images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='digitale-sammlungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='de/en/view/bsb00070331 nichil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='DuoofactumetinipfoBitaetatierBita etatfup fominumset up intenebrielucetatef neBieeam noncomprefenjerit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='fuicfomomi ueaSeotcuinomenetatiofannes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='DicBenitin tefimoniumBtteftimonii petBieretSelumi ne:Btomneecreberentperilfum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Donetatifle faitefitage,fepfue fup fesBtteftimoniumpetfibetetBelumine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' EratfupBeraqueifluminatomnem fomine quifutoncaperceu BenienteminfuncmunJum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='nmunSoetata munJueperipfumfactueeft:etmunoueeum non cognouit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Fn piopia Benitetfuieum non teceperunt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Duotquotautem tecepetunteum Bediteie potefatem filios Jeifieri fiequtcte ountinnomineeiue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Duinonepfanguinifue neg3epBoluntatecatnie:neqsepBoluntateBi tifeo epDeonatifuntEtBetbumcatofactu efetfabBitauitinnobie,EtBiimuegforiam eius gloziam quafi Bnigenitia pafre plenum gtatieet Beritatie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Deo gtafiae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Detreuange ceft par pofle Sun ficaSicta Defeantut noftra Belicta:lmen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='fufue bune Ceinuocamue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='teadoramue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='telaubamug DBeataetgfortofafanctatrinitab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='bfub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Sit nomenSomini BeneSictum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='Eofoc nic etBfg3infecuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Rofector in tefpetantium Beue finequo nicfileftBalidum/nicilfanctum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='mul tipficafupetnoemifericoriamtuam:Bttete Dteniceu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='alectopone ScBetitequetufue rumpre Bitginite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Birgeconceu,feft Brap noue lauone66 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Temps d’inférence, en secondes par image, rapportés pour Doc-UFCN et dhSegment, et calculés sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Ratio contient les ratios d’amélioration (dhSegment/Doc-UFCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Temps d’inférence† Ratio Doc-UFCN dhSegment Balsac 0,41 2,95 7,20 DIVA-HisDB 0,80 12,90 16,13 Horae 0,97 7,87 8,11 READ-Simple 0,45 3,73 8,29 READ-Complex 0,59 4,70 7,97 † Prédictions faites sur une carte graphique GeForce RTX 2070 8G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' comparaison des modèles Jusqu’à présent, notre modèle a obtenu, en moyenne, de meilleures performances que dhSegment bien qu’il ne bénéficie pas d’un encodeur pré-entraîné.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un autre point intéressant est que notre modèle comporte moins de paramètres à apprendre que dhSegment : 4,1 millions pour Doc-UFCN contre 32,8 millions pour dhSegment, dont 9,36 millions, correspon- dant au décodeur non pré-entraîné, qui doivent être entièrement entraînés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette diminution du nombre de paramètres conduit à une réduction significative du temps de prédiction : Doc-UFCN est jusqu’à 16 fois plus rapide que dhSegment, comme illustré dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette réduction du temps d’inférence peut également s’expliquer par le fait que dhSegment utilise des patchs de taille 400×400 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, pour la base DIVA-HisDB, il devra prédire en moyenne 117 patchs de cette taille, là où notre modèle ne fera qu’une seule prédiction de taille moyenne 768×512 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Grâce à ces premières expériences, nous avons montré que notre modèle Doc-UFCN obtient de meilleures performances que dhSegment tout en étant plus rapide en inférence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous allons maintenant étudier l’impact du pré-entraînement sur les résultats finaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 impact du pré-entraînement Nous analysons maintenant l’impact du pré-entraînement sur des images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' dhSegment est pré-entraîné sur des images de scènes naturelles (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2009), ce qui lui permet d’obtenir des résultats satisfaisants, même avec peu de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous nous questionnons donc sur l’intérêt que pourrait avoir un pré-entraînement de Doc-UFCN sur des images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, nous avons entraîné Doc-UFCN ainsi que dhSegment sur l’ensemble des quatre jeux de données présentés en section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces modèles ont été entraînés avec 1700 images d’entraînement et 191 de validation, dans les mêmes conditions que les expériences précédentes, puis évalués sur chaque base indépendamment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les deux modèles ainsi obtenus sont dits "génériques" dans la suite de cette section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats obtenus sont résumés dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour plus de lisibilité, seules les valeurs d’IoU et de F1 sont présentées dans la Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En plus des résultats des modèles pré-entraînés (génériques), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E 67 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Résultats obtenus par Doc-UFCN et dhSegment au niveau pixel pour la tâche de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats montrent les performances des modèles génériques sur les ensembles de test avec et sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Système IoU F1-score Générique Adapté Générique Adapté Doc-UFCN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='86 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='92 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='92 Balsac dhSegment 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='74 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 Doc-UFCN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 DIVA-HisDB dhSegment 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='74 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='84 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 Doc-UFCN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='69 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='89 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='88 Horae dhSegment 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='61 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='63 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='82 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='80 Doc-UFCN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='79 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='79 READ-Simple dhSegment 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='65 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='81 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='77 Doc-UFCN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='60 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='60 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='78 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='78 READ-Complex dhSegment 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='53 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='53 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='69 nous montrons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' dans cette table,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' les résultats après adaptation (fine tuning) des modèles génériques sur chaque jeu de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' comparaison des systèmes Les résultats obtenus par les modèles Doc-UFCN et dhSegment génériques confirment ceux obtenus par les modèles spécifiques présentés en section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, avec et sans adaptation, Doc-UFCN obtient quasiment toujours de meilleurs résultats que dhSegment puisqu’il obtient des valeurs d’IoU et de F1 plus élevées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' pré-entraînement La Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 compare les résultats obtenus par le modèle Doc-UFCN générique par rapport aux modèles spécifiques présentés précédemment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous pouvons noter que, pour les deux métriques, le modèle générique obtient de meilleures performances sur toutes les bases sauf sur DIVA-HisDB où il perd un point d’IoU et de F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur les autres jeux de données, nous observons des gains allant jusqu’à 6 points d’IoU, ce qui indique que le modèle a réussi à apprendre des caractéristiques suffisamment génériques pour représenter des données très variées, et qu’il a donc réussi à tirer profit de chaque jeu de données, malgré des données de pré-entraînement non équilibrées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' adaptation Nous observons également sur la Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 qu’adapter le modèle générique à chaque jeu de données n’apporte que très peu voire aucun gain de performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre hypothèse est que le modèle générique avait déjà appris au mieux sur ces données, l’adaptation n’apportant pas de nouvelles données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce comportement peut être expliqué par le faible nombre de paramètres que comporte le modèle et la taille réduite des jeux de données annotés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 68 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Impact du pré-entraînement de Doc-UFCN, évalué sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats montrent les performances des modèles génériques avec et sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour conclure, nous avons montré qu’utiliser un modèle générique permet d’améliorer la qualité des détections par rapport à un modèle spécifique, même avec une quantité limitée de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le chapitre 5, nous cherchons à aller plus loin dans cette idée en analysant et levant les défis liés à l’obtention d’un modèle encore plus générique et robuste, obtenant de bonnes performances sur de nouvelles données sans aucune adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 étude ablative Après avoir démontré l’intérêt de notre modèle Doc-UFCN pour la détection de lignes de texte, nous synthétisons ici les expérimentations réalisées afin de valider nos choix d’architec- ture et d’évaluer l’impact de certains composants et hyper-paramètres sur les performances finales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les paramètres étudiés dans les paragraphes suivants sont l’utilisation de la normali- sation des batchs, l’utilisation du dropout, les taux de dilatation dans les blocs dilatés et la taille des images en entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les Tables 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 et 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 regroupent les résultats de cette étude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S D E D É T E C T I O N D E L I G N E S D E T E X T E 69 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – Étude ablative de Doc-UFCN sur la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Résultats donnés pour les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' "BN" indique l’utilisation de la batch normalization durant l’entraî- nement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Version IoU F1-score ∅ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='79 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='88 BN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='81 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='89 Balsac BN + dropout 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='84 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='91 ∅ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='41 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='56 BN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='74 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 DIVA-HisDB BN + dropout 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='76 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='86 ∅ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='56 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='78 BN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='81 Horae BN + dropout 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='85 ∅ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='59 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 BN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='58 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='72 READ-Simple BN + dropout 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='77 ∅ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='39 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='56 BN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='50 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='69 READ-Complex BN + dropout 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='54 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 batch normalization La normalisation des batchs appliquée durant l’entraînement de modèles neuronaux est souvent utilisée,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' car elle a un impact positif sur la vitesse de convergence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' mais parfois aussi sur les performances (Ioffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’entraînement de Doc-UFCN avec cette normalisation confirme ces résultats puisque le modèle a convergé plus de deux fois plus rapidement par rapport au modèle sans normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’après la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, la normalisation a également un réel impact sur les valeurs de F1, en particulier pour Diva-HisDB, Horae et READ-Complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En plus de ces valeurs quantitatives, nous avons noté que les résultats visuels sont meilleurs avec normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle aide à séparer les objets proches mais aussi à joindre les parties d’objets qui, sans normalisation, étaient sur-segmentés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, les contours des objets prédits sont souvent plus précis et plus lisses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' dropout Le dropout (Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2014) est également souvent utilisé dans les réseaux de neurones profonds, car il permet notamment de limiter le sur-apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nos expériences, présentées en Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, montrent qu’entraîner avec dropout permet, en effet, d’obtenir de meilleures performances sur toutes les bases, d’autant plus que notre modèle comporte assez peu de paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' taux de dilatation Lors de la conception de notre modèle, nous avons choisi d’utiliser une version modifiée du bloc dilaté proposé par Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) dans le but de prendre en compte davantage de 70 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 – Impact du taux de dilatation dans les blocs d’encodeur de Doc-UFCN sur la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Résultats donnés pour l’ensemble de test du jeu de données Balsac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dilatation IoU F1-score [1] 0,76 0,86 [1, 1, 1, 1, 1] 0,80 0,89 [16] 0,77 0,87 [1, 2, 4, 8, 16] 0,84 0,91 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 – Impact de la taille des images en entrée de Doc-UFCN sur la détection des lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Résultats donnés pour les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Taille IoU F1-score 384 0,84 0,91 Balsac 768 0,87 0,93 384 0,76 0,86 DIVA-HisDB 768 0,77 0,87 contexte pour prédire les lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour justifier nos choix de taux de dilatation, nous avons testé quatre configurations sur le jeu de données Balsac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, nous avons entraîné des modèles avec des blocs dilatés configurés comme suit : — 1 convolution et un taux de dilatation de 1 : [1] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — 1 convolution et un taux de dilatation de 16 : [16] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — 5 convolutions et des taux de dilatation de 1 : [1, 1, 1, 1, 1] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — 5 convolutions et des taux de dilatation de 1, 2, 4, 8 et 16 : [1, 2, 4, 8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats obtenus sont présentés dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats de la dernière configu- ration sont meilleurs que ceux des autres puisque le champ réceptif est beaucoup plus grand et que le modèle considère davantage de contexte pour prédire les lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le fait d’avoir des convolutions dilatées au lieu de convolutions standards a un réel impact sur la taille du champ réceptif, ce qui permet d’utiliser plus de contexte pour prédire les lignes de texte et d’obtenir de meilleures performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' taille des images d’entrée Comme indiqué précédemment, nous avons choisi, pour Doc-UFCN, d’utiliser la même taille d’image en entrée que celle utilisée par Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi, pour tous les résultats présentés jusqu’à maintenant, les images étaient redimensionnées de sorte que leur plus grande dimension soit égale à 384 pixels, tout en conservant l’aspect de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, il est important de connaître l’impact de ce choix sur les résultats du modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette optique, nous avons entraîné un modèle sur les données du jeu Balsac et un autre sur celles de DIVA-HisDB sur des images redimensionnées à 768 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 montre que l’entraînement sur des images plus grandes améliore les résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les lignes étant souvent de petite hauteur sur ces bases, agrandir les images d’entrée permet au modèle de mieux séparer les lignes proches et de les prédire avec une plus grande précision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S 71 Grâce à ces premières expériences, nous avons montré que notre modèle Doc-UFCN obtient de meilleures performances sur la tâche de détection de lignes de texte dans des images de document que dhSegment, tout en comportant moins de paramètres et en étant plus rapide en inférence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous avons montré qu’entraîner un modèle sur plusieurs bases permet d’obtenir un modèle plus générique et d’améliorer la qualité des prédictions par rapport à des modèles spécifiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous évaluons, dans la section suivante, le modèle Doc-UFCN sur une tâche plus complexe de détection et de classification d’actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S Les registres sont des types très courants de documents historiques qui contiennent des listes d’enregistrements, appelés "actes", se rapportant à des personnes, des objets ou des événements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils peuvent être présentés sous forme de tableaux ou de séquences de textes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas des registres royaux, des cartulaires médiévaux ou des registres paroissiaux et civils, les actes sont des segments textuels composés d’un ou plusieurs paragraphes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour traiter le problème de la détection d’actes, la plupart des méthodes existantes utilisent soit le contenu textuel des documents, soit le contenu visuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les systèmes récents basés sur des règles heuristiques ou des réseaux neuronaux se basent uniquement sur les caractéristiques visuelles des images pour détecter les actes, en ignorant le texte des documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, Tarride et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) combinent des règles et un réseau neuronal pour segmenter les registres paroissiaux français en actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils détectent d’abord les signatures des prêtres situées à la fin de chaque acte à l’aide d’un réseau neuronal (dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) ou ARU-Net (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019)) avant d’utiliser un système à base de règles pour générer les actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même s’ils ont obtenu un système avec 80 % de rappel au niveau des actes, leur méthode repose principalement sur l’hypothèse que chaque acte se termine par une signature, ce qui n’est pas toujours le cas, et si elle est effectivement présente, celle-ci n’est pas toujours détectée par le système automatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La méthode que nous proposons comprend également des caractéristiques basées sur des règles, mais combinées avec l’image originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les documents historiques peuvent avoir un contenu textuel riche qui peut per- mettre un meilleur processus de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) ont étudié le cas où l’aspect graphique des images n’est pas suffisant pour segmenter les chartes médiévales en actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ne visent pas seulement à détecter les actes mais cherchent également à les classer comme début, milieu, fin d’acte ou acte complet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, ils utilisent une carte d’indexation probabiliste pour construire des caractéristiques supplémentaires fondées sur le contenu tex- tuel, puis les caractéristiques graphiques et textuelles sont fusionnées afin d’obtenir une seule entrée pour le système de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils montrent que l’ajout de contenu textuel peut faciliter la détection des actes et que l’ajout de connaissances a priori permet d’améliorer encore les performances (73 % à 88 % de la F-mesure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Inspiré par cette idée et celle proposée par Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017), notre travail se concentre sur l’utilisation des deux modalités en entrée d’un système de détection par apprentissage profond pour améliorer la détection des actes 72 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 – Statistiques des jeux de données utilisés pour la détection d’actes : nombre de pages, lignes transcrites et actes par type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Images Lignes Actes complet début milieu fin Balsac train 730 36 941 1 474 503 2 487 Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) valid 92 4 592 181 66 1 58 test 91 4 323 173 62 0 52 train 132 – 144 46 21 40 Himanis-Act valid 19 – 29 3 3 2 Bluche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) test 411 – 172 203 115 196 Himanis-GMV train – 18 504 – – – – valid – 2 367 – – – – test – 2 241 – – – – Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Annotations manuelles pour la détection et la classification d’actes sur une image du jeu de données Balsac, à gauche, et Himanis-Act, à droite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' présents dans des documents historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette section, nous présentons tout d’abord les données utilisées, notre approche pour résoudre la tâche de détection d’actes, puis nous discutons des entraînements et des résultats obtenus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 jeux de données Pour nos expériences, nous avons utilisé deux jeux de données, Balsac (Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) (présenté en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1) et Himanis-Act (Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour traiter les actes répartis sur plusieurs pages, les actes sont annotés avec quatre classes : acte complet, début d’acte, milieu d’acte et fin d’acte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les statistiques de ces ensembles de données sont présentées dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 présente un exemple d’annotation pour chaque jeu de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" Fin d'acte 091 et Mr lglCom uJiehulaluta Acte complet Nhin ehh mari J hleak Acte complet as thut 92 Début d'acteFin d'acte HAMD Acte complet nammeppeneplamcoep gobonu3gennmg Débutd'acte4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S 73 D´etection lignes HTR "Premi`ere ligne" "Autre ligne" Classification Classification Incorporation des lignes Incorporation des classes D´etection et classification d’actes Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – Chaîne de traitement proposée pour la détection et la classification d’actes avec l’utili- sation du contenu textuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le jeu de données Himanis est extrait du corpus Chancery, une grande collection de registres produits par la chancellerie royale française.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est composé de 80 000 images contenant des chartes promulguées par les rois de France aux 14e et 15e siècles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces documents consignent les décisions royales comme les donations ou les grâces, et sont organisés en actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le jeu de données Himanis-Act consiste en un échantillon de 739 images extraites des données Himanis et est annoté au niveau des actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour réaliser nos expériences, nous avons utilisé la répartition proposée par Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020), obtenue après avoir éliminé les pages ne contenant aucune information telles que les pages blanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce jeu de données est uniquement utilisé pour entraîner et évaluer le système de détection d’actes puisque les annotations de lignes de texte et de transcription ne sont pas disponibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La répartition finale est présentée dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le jeu de données annoté Himanis-GMV est composé de 1 435 images extraites des données Himanis, alignées automatiquement au niveau des lignes, des éditions imprimées (Guérin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 1881-1958 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Guyotjeannin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2005 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Viard, Jules, 1899) avec Transkribus (Kahle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Après le processus d’alignement, 23 112 lignes de texte sont disponibles pour l’entraînement et l’évaluation des systèmes de reconnaissance de l’écriture manuscrite, dont la répartition est présentée dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce jeu de données est uniquement utilisé pour entraîner un système de reconnaissance HTR puisque la segmentation en actes n’est pas disponible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 approche proposée La Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 détaille l’approche proposée pour résoudre la tâche de détection et de classi- fication d’actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’approche consiste en plusieurs traitements successifs de l’image d’entrée : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’image d’entrée est d’abord traitée par un détecteur de lignes de texte ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les lignes prédites sont extraites et traitées par un reconnaisseur de texte manuscrit ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Thauuc tth giniLn B贝人 Bleo Bge cloghLen dmuhLe neuf fevrier mil neuf centun,nouspretresoussigneavons74 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 – Résultats du modèle générique de détection de lignes de texte sur l’ensemble de test du jeu de données Balsac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Système IoU AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 mAP Balsac Doc-UFCN 0,87 0,98 0,91 0,76 dhSegment 0,74 0,94 0,54 0,51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque ligne est classifiée selon le texte qu’elle contient en "première ligne" (s’il est probable que la ligne soit la première de l’acte) et "autre ligne" ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’image d’entrée est enrichie par ces lignes classifiées en dessinant les lignes de texte de couleurs différentes selon leurs classes ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, les actes sont détectés et classifiés à partir de cette image enrichie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les paragraphes suivants présentent et discutent chacune de ces étapes en détail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' détection des lignes de texte La première étape du traitement des images consiste à détecter les lignes de texte sur les images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cette tâche, nous avons utilisé Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Afin de créer un modèle générique pouvant être utilisé sur différents jeux de données, le modèle a été entraîné sur neuf jeux de données historiques dont Balsac et à l’exception d’Himanis, puisqu’il ne contient aucune annotation au niveau des lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les images ont été redimensionnées de manière à ce que leur plus grande dimension soit égale à 768 pixels, tout en conservant le rapport d’aspect original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les annotations originales ont également été uniformisées grâce au processus détaillé dans la section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) a également été entraîné afin de fournir une comparaison de référence avec Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les deux modèles de détection de lignes de texte ont été évalués à l’aide de l’IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les modèles ont également été évalués à l’aide de l’AP, qui quantifie le nombre d’objets correcte- ment détectés, alors que la mesure IoU considère uniquement le nombre de pixels correctement prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Plus de détails sur le calcul de la mAP sont donnés dans le Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats sont présentés dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9, pour le jeu de données Balsac uniquement, car aucune annotation manuelle n’est disponible pour les jeux de données Himanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résul- tats montrent que Doc-UFCN surpasse dhSegment pour toutes les métriques et obtient des performances très satisfaisantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' reconnaissance de texte La reconnaissance de texte manuscrit (HTR) est appliquée aux lignes de texte détectées et produit le texte correspondant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle de reconnaissance est construit avec la librairie Kaldi 2, basée sur un modèle DNN-HMM (Deep Neural Network - Hidden Markov Model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre modèle est comparable à celui décrit dans Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://kaldi-asr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='org/ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S 75 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 – Résultats de reconnaissance de textes manuscrits sur les jeux de données Balsac et Himanis-GMV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats d’un modèle HTR+ entraîné avec Transkribus sont donnés à titre de référence mais ne sont pas directement comparables à notre système puisque les séparations train/valid ne sont pas identiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Système CER (%) WER (%) train valid test train valid test Balsac Kaldi 4,1 6,2 6,4 12,4 17,1 17,4 Transkribus 12,2 9,5 – – – – Kaldi 5,4 9,4 8,0 11,9 19,3 18,1 Himanis-GMV Transkribus 9,5 5,3 – – – – Nous avons entraîné un modèle Kaldi sur le jeu de données Balsac en suivant la répartition présentée dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Aucune donnée supplémentaire n’a été utilisée pour le modèle linguistique ni pour le modèle optique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour Himanis, nous avons entraîné un modèle sur le jeu de données annoté Himanis-GMV présenté ci-dessus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les deux modèles, les lignes d’entrée ont été redimensionnées à une hauteur de 40 pixels tout en conservant le rapport d’aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les lignes ayant des largeurs similaires ont été regroupées pour un entraînement plus efficace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À titre de comparaison, un modèle HTR+ a été entraîné à l’aide de la plateforme Transkribus (Kahle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les performances des modèles HTR sont décrites dans la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Concernant notre système HTR (Kaldi), les résultats obtenus pour les deux jeux de données sont similaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même si les modèles présentent des taux d’erreurs de mots (Word Error Rate (WER)) rela- tivement élevés, nous pensons qu’ils sont capables de prédire une transcription suffisamment correcte pour la détection de mots-clés et la détection des actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’interface d’entraînement de Transkribus ne fournit qu’une évaluation pour le Character Error Rate (CER) et ne peut pas être directement comparée à la nôtre puisque les répartitions train/valid/test ne sont pas identiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, les résultats sont du même ordre de grandeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle de Trans- kribus présente un CER plus élevé sur l’ensemble d’entraînement en raison de son processus d’augmentation des données, processus non utilisé dans notre système Kaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' classification des lignes de texte La classification des lignes de texte est effectuée à l’aide de règles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle utilise les transcriptions prédites en entrée et prédit si une ligne de texte est la première ligne d’un acte en se basant sur la présence de phrases clés définies a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu de données Balsac, la plupart des actes commencent par une date telle que "Le trente un janvier, mil neuf", et il n’y a souvent pas d’autres dates dans la suite des actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, la règle est donc de compter le nombre de mots qui sont des chiffres ou des mois pour que la ligne soit considérée comme une date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Expérimentalement, trois mots semblent être suffisants pour que la ligne soit considérée comme contenant une date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les actes de Himanis-Act, la tâche est plus complexe car ils ne commencent pas toujours par les mêmes mots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons donc analysé les premières lignes des actes 76 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 – Résultats de classification des lignes de texte en première ligne d’un acte / autre ligne sur les jeux de données Balsac et Himanis-Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Précision Rappel F1-score Balsac train 0,69 0,87 0,77 valid 0,72 0,87 0,79 test 0,68 0,86 0,76 train 0,79 0,65 0,71 valid 0,81 0,53 0,64 Himanis-Act test 0,68 0,86 0,76 d’entraînement et avons conservé les phrases clés les plus fréquentes (par exemple "dei gratia francorum rex" ou "par la grâce de dieu roys de france").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Si une phrase clé est incluse dans une ligne, elle est considérée comme se trouvant au début d’un acte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 montre la précision, le rappel et le F1-score de la classe "première ligne".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Seule la classe "première ligne" est donnée car c’est la seule classe apportant des informations à la détection des actes, d’autant plus que la distribution des classes est très déséquilibrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu de données Balsac, les résultats sont stables entre les trois ensembles et le rappel est élevé, ce qui est favorable à l’inclusion de cette information dans le système visuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu de données Himanis-Act, le rappel est plus faible et les résultats varient entre les ensembles, ce qui montre bien que la tâche est plus complexe et que la détection des phrases clés est plus difficile, et donc moins fiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 résultats et discussion La Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12 présente les résultats des différents modèles de détection des actes : Doc- UFCN entraîné sur des images brutes (visuel) et Doc-UFCN entraîné sur les images brutes avec les polygones des lignes de texte dessinés de deux couleurs dépendant de la classe de la ligne (visuel + textuel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu de données Balsac, nous ne présentons pas les résultats de la classe milieu d’acte car il n’y en a pas dans le jeu de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’après la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12, les résultats sont en moyenne meilleurs pour le modèle utilisant les contenus visuels et textuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les classes de début et de fin d’acte, les deux systèmes sont presque équivalents pour toutes les métriques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En revanche, nous constatons que l’ajout de la date directement à l’image d’entrée améliore les performances de la classe d’actes complets de 36 points de pourcentage de mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela conduit à une meilleure séparation des actes complets consécutifs dans les prédictions, ce qui était l’un de nos principaux objectifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu de données Himanis-Act, les résultats sans contenu textuel sont significative- ment meilleurs que ceux l’incorporant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous pensons que ces résultats sont dûs aux raisons suivantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, le contenu textuel des documents de la base Himanis-Act est plus complexe et diversifié que celui de la base de données Balsac, qui est très standardisé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous avons pu trouver de nombreux actes imbriqués dans le jeu de données (Vidimus), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 E X P É R I E N C E S D E D É T E C T I O N D’ A C T E S 77 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12 – Résultats de détection d’actes sur les ensemble de test des jeux de données Balsac et Himanis-Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Système Actes IoU AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 mAP Visuel complet 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='84 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='57 0,' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='63 complet 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='61 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='70 début 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='76 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='84 0,' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='65 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='62 complet 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='54 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='51 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='49 Visuel + début 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='69 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='60 textuel milieu 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='84 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='80 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='80 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='80 Himanis-Act fin 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='70 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='63 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='58 ce qui peut ajouter de la confusion au système.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La définition des phrases clés s’est avérée plus complexe et la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 montre que le rappel est faible, même pour l’ensemble d’entraînement, ce qui conduit à des caractéristiques textuelles peu fiables pour entraîner le modèle de détection d’actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les modèles de détection de lignes et d’HTR n’ont pas été entraînés sur l’ensemble de données Himanis-Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle de détection de lignes de texte a été entraîné sur des données similaires mais sans images venant du jeu de données Himanis-Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il en est de même pour le système HTR qui n’a pas été entraîné directement sur les mêmes volumes, ce qui peut créer un décalage entre les conditions d’entraînement et de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En plus de ces expériences, nous avons comparé nos résultats avec ceux de l’état de l’art de Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont testé différentes configurations avec et sans le contenu textuel pour détecter où se terminent les actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour obtenir une comparaison juste, nous avons utilisé le même protocole d’évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’évaluation est effectuée à l’aide du Transkribus Baseline Evaluation Scheme (TBES) (Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet outil a été conçu pour évaluer la détection de la ligne de base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, pour l’utiliser, la ligne de base des actes est définie comme étant la ligne droite horizontale à la fin d’un acte complet ou d’une fin d’acte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour être en accord avec leurs résultats, nous avons utilisé la même valeur de tolérance de 128 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’après la Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13, nous pouvons voir que Doc-UFCN utilisant uniquement l’image améliore les résultats de l’état de l’art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, en comparaison au système visuel de Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020), notre méthode obtient des performances supérieures de 10 points de pourcentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, nous constatons que les deux modèles utilisant le contenu textuel se comportent de la même manière et sont moins performants que notre système basé 78 D É T E C T I O N D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13 – Résultats obtenus par Doc-UFCN et Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) sur le jeu de données Himanis- Act avec et sans l’information textuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Système Visuel Visuel + textuel Doc-UFCN train 0,96 0,96 valid 0,96 0,91 test 0,90 0,88 Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) test 0,80 0,88 uniquement sur les informations visuelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour ce jeu de données, il semble préférable de se concentrer sur les caractéristiques visuelles avec un système d’apprentissage profond robuste, plutôt que d’ajouter un contenu textuel trop peu fiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette partie, nous avons présenté une chaîne de traitement simple permettant d’en- richir des images d’entrée avec le contenu textuel de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces images enrichies per- mettent d’effectuer une tâche de détection d’actes en utilisant simultanément le contenu visuel et la position des lignes de texte contenant des phrases clés définies manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons montré que l’utilisation de ces images peut améliorer la détection des actes, en particulier des actes consécutifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur le jeu de données Balsac, pour lequel des règles de détection de phrases clés ont pu être définies de manière fiable, l’utilisation de ces images augmente la détection d’actes de 38 % à 74 % de mAP par rapport à un modèle standard se basant uniquement sur le contenu visuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 C O N C L U S I O N Dans ce chapitre, nous avons présenté Doc-UFCN, un nouveau système de détection d’ob- jets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons montré que ce système permet d’entraîner des modèles plus performants, plus rapides et comportant moins de paramètres que ceux de l’état de l’art pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le code de ce système est disponible publiquement 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les expérimentations décrites dans ce chapitre ont permis d’amorcer une analyse sur l’intérêt des modèles génériques, qui seront l’objet du prochain chapitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, nous nous sommes intéressés aux méthodes combinant l’image et le texte pour la détection et la classification d’actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un cadre dans lequel la séparation visuelle des actes suit la séparation logique du texte, nous avons montré que l’incorporation du contenu textuel dans l’image d’entrée permet de réellement améliorer la détection des actes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='org/project/doc-ufcn/ 5 E N T R A Î N E M E N T E T É VA L U AT I O N D ’ U N M O D È L E R O B U S T E D E D É T E C T I O N D ’ O B J E T S La littérature montre que des systèmes compétitifs et robustes ont été développés pour résoudre le problème de la détection des lignes de texte, obtenant des performances satisfaisantes sur des jeux de données individuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, leurs performances sont souvent insuffisantes sur d’autres documents hors échantillon, car ils manquent de ca- pacités de généralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Or, dans un cadre industriel avec de nombreux projets et des données très différentes, il est nécessaire de développer des modèles plus génériques, ob- tenant des performances élevées sur des documents très variés provenant de différents projets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’entraînement de systèmes sur des données très diverses permettrait d’obtenir de tels modèles, montrant de meilleures capacités de généralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, il est nécessaire de combiner plusieurs ensembles de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, les schémas d’annotation ne sont pas toujours compatibles entre les jeux de données publics (comme décrit en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2), ce qui rend difficile leur combinaison dans un seul ensemble d’entraînement unifié.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces différents schémas ne permettent pas une comparaison équitable des approches de détection d’un jeu de données à l’autre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, dans la littérature, aucune comparaison de systèmes n’a été effectuée sur un jeu de données large et diversifié, tant en entraînement qu’en évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, dans la littérature, les études sur la détection des lignes de texte manuscrites se concentrent généralement sur le développement d’une architecture de réseau neuronal spéci- fique, ainsi que sur une bonne stratégie d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, elles omettent souvent d’analyser les annotations utilisées lors de l’entraînement et de l’évaluation, alors qu’elles sont aussi importantes que l’architecture du réseau elle-même, puisqu’elles guident la phase d’entraînement et les résultats finaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, le biais d’annotation est particulièrement important lorsque nous voulons analyser l’impact de l’étape de détection sur l’étape de recon- naissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, il est rarement étudié dans les études se concentrant sur la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il n’est pas non plus considéré dans les études portant sur la reconnaissance de l’écriture manuscrite, car le processus de détection n’entre pas dans leur champ d’application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par exemple, la première lettre dans les documents historiques est parfois ornée, l’ajouter ou non dans les lignes de texte pendant le processus d’annotation peut avoir un réel impact sur les résultats de la reconnaissance finale, d’où l’importance de créer et d’analyser soigneu- sement les annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un autre problème se pose lors de la détection des lignes de texte lorsque deux boîtes de délimitation annotées se touchent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans une telle situation, le réseau fournit généralement des lignes fusionnées qui ne seront pas reconnues par le système de re- connaissance HTR, alors que généralement les métriques d’évaluation ne tiennent pas compte 79 80 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S de ces problèmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, la métrique IoU, très souvent utilisée, est incapable de détecter ces situations et de compter correctement les séparations de lignes correctes et incorrectes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce chapitre, nous abordons ce problème, en section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, en introduisant une stratégie d’uniformisation d’étiquetage des jeux de données, qui réduit les chevauchements des polygones englobants afin d’obtenir un résultat cohérent avec l’entrée requise des systèmes de reconnaissance (HTR ou OCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Toujours dans un cadre industriel où l’objectif est la reconnaissance du texte des documents, et pas uniquement la détection des lignes, il est nécessaire d’avoir une bonne évaluation des modèles de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, l’évaluation de tels modèles est complexe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, les métriques d’évaluation au niveau pixel ne sont pas toujours représentatives de l’impact réel de l’étape de détection de lignes de texte sur l’étape de reconnaissance de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les comparaisons des systèmes de détection de lignes de texte en termes de taux d’erreur de reconnaissance sont rarement rapportées en raison de la complexité de cette évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La plupart des méthodes de détection de lignes de texte existantes sont évaluées et comparées à l’aide de métriques au niveau pixel telles que l’IoU, la précision, le rappel et le score F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même si ces mesures indiquent les performances du modèle, elles ne donnent aucune information réelle sur la quantité d’informations détectées, comme le nombre de lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certaines mesures au niveau de l’objet, telles que la précision moyenne (AP), ont été proposées pour surmonter ce problème, mais elles reposent toujours sur un seuil d’IoU fixe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2, nous analysons donc ces limitations et introduisons la métrique mAP déjà utilisée dans les défis de détection COCO en vision, qui ne nécessite pas de seuil de détection pour être mise en œuvre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, comme la détection des lignes manuscrites est la première étape de l’ensemble du processus de reconnaissance, il devrait être plus réaliste d’évaluer son véritable impact sur les résultats de reconnaissance finaux (taux d’erreur caractères et mots), en effectuant une évaluation orientée vers la tâche finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À cet égard, nous proposons, en section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, une stratégie d’évaluation fondée sur les résultats de reconnaissance obtenus après un système de reconnaissance de texte (HTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette partie, nous fournissons une évaluation juste et approfondie de trois approches pour détecter les lignes de texte, Doc-UFCN (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021a), dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) et ARU-Net (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019), sur une large collection de jeux de données historiques et avec plusieurs métriques, y compris une métrique orientée recon- naissance de texte (HTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous analysons les métriques de détection de lignes de texte de la littérature par rapport à de multiples jeux de données publiquement disponibles et mon- trons certaines incohérences entre les jeux de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous proposons, en section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, une stratégie d’uniformisation des annotations des jeux de données qui évite le biais d’étiquetage pour la tâche de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet étiquetage modifié permet de considérer la variabilité des annotations et d’entraîner des modèles avec une plus grande capacité de généralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un second temps, nous effectuons une évaluation de l’état de l’art grâce à différentes métriques et protocoles d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces protocoles permettent de construire des modèles de détection plus génériques qui considèrent les limitations mentionnées ci-dessus 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 U N I F O R M I S AT I O N D E S A N N O TAT I O N S 81 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Statistiques des jeux de données utilisés pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Images Lignes train valid test train valid test AN-Index† 19 3 12 433 62 171 Balsac Vézina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) 730 92 91 36 941 4 592 4 323 BNPP† 7 2 3 705 218 358 Bozen Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) 350 50 50 8 366 1 040 1 138 cBAD2019 Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) 720 716 45 266 42 672 DIVA-HisDB Simistira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016) 60 30 30 6 037 2 999 2 897 HOME-NACR Boros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) 338 40 42 6 590 602 909 Horae Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) 522 20 30 12 568 270 958 READ-Complex Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) 216 27 27 17 768 2 160 1 758 READ-Simple Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) 172 22 22 5 117 539 723 HOME-Alcar Stutzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021) – – 55 – – 2 727 ScribbleLens Dolfing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2020) – – 21 – – 563 † Jeux de données privés utilisés durant la thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' et obtiennent des résultats similaires, voire meilleurs, que les modèles entraînés sur des en- sembles de données uniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 U N I F O R M I S AT I O N D E S A N N O TAT I O N S L’un de nos objectifs étant de développer un détecteur générique de lignes de texte sur des documents historiques, nous avons collecté neuf jeux de données historiques dont sept jeux publics pour mener les expérimentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces jeux de données sont présentés en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 et une description est donnée dans la Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, comme nous souhaitons évaluer la capacité de généralisation des modèles obtenus, nous avons collecté deux jeux de données supplémentaires : ScribbleLens (Dolfing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) et HOME-Alcar (Stutzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021) utilisés uniquement pendant l’étape d’évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 82 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S Tous ces ensembles de données ont été choisis pour leur diversité en termes de tailles, d’écritures et de mises en page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 présente la variété des documents en montrant un exemple d’image de chaque ensemble de données avec ses annotations lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La réparti- tion utilisée pour entraîner les modèles a été obtenue en regroupant simplement les données d’entraînement et de validation respectives des sous-ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, puisque le jeu de données HOME-Alcar ne contenait pas d’ensembles d’entraînement, de validation et de test officiels au moment des expériences présentées ci-après nous avons rassemblé, pour générer un ensemble de test, 55 pages annotées au niveau ligne avec leurs transcriptions correspondantes parmi tous les manuscrits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 analyse des annotations Tous les ensembles de données cités ci-dessus ont été utilisés pour entraîner des modèles génériques de détection de lignes de texte avec les modèles Doc-UFCN (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021a), dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) et ARU-Net (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces modèles nécessitent des images annotées au niveau pixel pour l’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons les défis rencontrés pour générer un ensemble d’entraînement annoté unifié.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' diversité dans les annotations Pour générer les images d’annotations au niveau pixel, les polygones englobants sont ex- traits de la vérité terrain et dessinés sur une image de fond noir de même taille que l’image originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme nous pouvons le voir sur la Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2, les annotations sont très variées parmi les ensembles de données : — Dans la plupart des jeux de données (AN-Index, Balsac, Bozen, BNPP, HOME et Horae), les images sont annotées à l’aide de simples polygones incluant les ascendants et descendants ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Dans les bases cBAD2019, READ-Simple et READ-Complex, les ascendants et descen- dants ne sont généralement pas inclus dans les polygones, qui sont très fins par rapport au premier cas ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Dans les images de la base ScribbleLens, les annotations sont de larges rectangles qui incluent de nombreux pixels appartenant au fond ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Dans la base DIVA-HisDB, les lignes de texte sont annotées à l’aide de polygones plus complexes qui suivent précisément le contour de chaque lettre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les polygones des lignes de texte du jeu HOME-Alcar sont également précis mais légèrement moins que ceux de DIVA-HisDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette diversité dans les annotations nous empêche d’entraîner directement un modèle gé- nérique qui pourrait être appliqué à de nouveaux jeux de données, car l’annotation serait parfois incohérente entre deux exemples provenant de deux jeux de données différents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De telles incohérences dégraderaient considérablement les performances du système.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Corriger les incohérences d’annotation entre les ensembles de données est donc une nécessité pour permettre l’unification des ensembles d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 U N I F O R M I S AT I O N D E S A N N O TAT I O N S 83 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Processus de génération d’annotations pour une image du jeu de données de Bozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À gauche : génération d’annotations à la taille originale de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au centre : redi- mensionnement de l’image à la taille de l’entrée du réseau (768 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À droite : redi- mensionnement des polygones englobants à la taille d’entrée du réseau, atténuation des chevauchements et génération de l’image d’annotations à la taille d’entrée du réseau (768 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' chevauchement des polygones Un autre problème qui accentue les incohérences des annotations est le chevauchement des polygones englobants annotés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même si certains ensembles de données ont été annotés de telle sorte que jamais les polygones ne se touchent ni ne se chevauchent, d’autres ont été annotés moins précisément, ce qui entraîne des polygones qui se touchent et se superposent (page ScribbleLens sur la Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2), comme le montre l’image de gauche de la Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Évidemment, une telle vérité terrain ne peut pas servir à une évaluation précise de la capacité d’un système à détecter chaque ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, Doc-UFCN et ARU-Net utilisent des sous-résolutions des images d’entrée, ce qui peut entraîner des fusions supplémentaires dans les images d’annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 présente cet effet indésirable en montrant l’image d’annotation originale, à gauche, et sa version redimensionnée à 768 pixels, au centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la littérature, la plupart des études utilisent la vérité terrain telle quelle, sans prêter beaucoup d’attention à ce problème de fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La raison principale à cela est probablement que les mesures d’évaluation utilisées comptent uniquement la précision des pixels, sans évaluer plus précisément le processus de détection des lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, un nombre important de fusions dans l’ensemble de données d’entraînement va certainement faire dévier le réseau vers la fusion de plus de lignes que souhaité, avec un effet induit négatif sur le système HTR incapable de reconnaître les lignes fusionnées verticalement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Lors de nos expériences, nous avons atténué ce problème en supprimant, autant que pos- sible, les chevauchements entre les lignes, tout en perdant le moins d’informations possible, comme on peut le voir sur l’image de droite de la Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 84 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S stratégie d’uniformisation Pour unifier les annotations, nous avons choisi d’utiliser uniquement des polygones simples pendant l’étape d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, les rectangles englobant les polygones de DIVA-HisDB ont été utilisés pendant l’apprentissage au lieu des polygones complexes origi- naux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, pour résoudre le problème du chevauchement des polygones englobants de certains jeux de données, nous avons utilisé la stratégie suivante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour une image donnée, nous recherchons les paires de lignes qui se touchent et se chevauchent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, trois cas ont été identifiés pour chaque paire : — Les polygones se touchent : nous les érodons de 1 pixel ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Les polygones se chevauchent de moins de 20 % chacun : nous appliquons le processus de scission des polygones décrit ci-dessous ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Les polygones se chevauchent de plus de 20 % : nous les gardons tels quels car la scission peut entraîner la perte de trop d’informations (perte d’un polygone ou séparation en deux polygones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas d’un petit chevauchement (moins de 20 % de l’aire des polygones), le processus suivant est appliqué : la ligne ayant le plus petit ratio de surface de chevauchement par rapport à sa surface totale est détectée, et son intersection avec l’autre ligne est supprimée, l’autre ligne étant conservée telle quelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, tous les polygones sont dessinés sur une image de fond noir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce seuil de 20 % a été choisi car il correspond, à peu près, à la hauteur des ascendants et descendants des lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme le redimensionnement de l’image d’annotation peut provoquer des fusions indési- rables, les polygones englobants sont d’abord redimensionnés à la taille de l’image d’entrée du réseau, puis la scission est appliquée à cette échelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, l’image d’annotation est di- rectement générée à la résolution souhaitée de l’entrée du réseau, ce qui empêche la fusion de certaines lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’image de droite de la Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 présente le résultat de ce processus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme nous pouvons le voir, l’image d’annotation produite contient des polygones mieux séparés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même si certaines lignes se chevauchent encore sur certaines pages, nous espérons avoir généré une vérité terrain plus appropriée qui aidera à entraîner le modèle de détection et à améliorer sa capacité à prédire des lignes de texte séparées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le code permettant de générer ces annotations modifiées et les images de labels utilisées dans les expériences sont accessibles publiquement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 C O M PA R A I S O N D E S A P P R O C H E S D E D É T E C T I O N Pour nos expériences, nous avons choisi d’étudier trois systèmes à l’état de l’art : Doc-UFCN (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021a), dhSegment (Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2018) et ARU-Net (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doc-UFCN a été choisi pour ses bonnes performances sur des jeux de données historiques et son nombre réduit de paramètres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous avons sélectionné les systèmes dhSegment et ARU-Net car ils sont open-source, faciles à entraîner et ont obtenu des perfor- mances satisfaisantes sur des tâches de détection sur des documents historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ARU-Net est 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='com/teklia/dla/arkindex_document_layout_training_label_normalization 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 C O M PA R A I S O N D E S A P P R O C H E S D E D É T E C T I O N 85 également le modèle de détection de lignes de texte utilisé dans Transkribus (Kahle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons maintenant les systèmes et les détails d’entraînements puisque nous les avons tous entraînés afin de pouvoir comparer équitablement leurs performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 doc-ufcn Le système Doc-UFCN est le même que celui présenté en section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La seule différence est la taille d’entrée du réseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, nous avons montré dans le chapitre précédent, que de meilleures performances sont obtenues avec des images d’entrée plus grandes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, les images sont redimensionnées telles que leur plus grande dimension soit de 768 pixels tout en conservant leur rapport d’aspect original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, pour entraîner Doc-UFCN, les annotations sont directement générées à 768 pixels grâce au processus présenté dans la section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les expériences suivantes, Doc-UFCN est entraîné avec un taux d’apprentissage initial de 5e − 3, des mini-batchs de taille 2, l’optimiseur Adam, la fonction de coût Dice et l’arrêt anticipé (early stopping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 dhsegment L’encodeur pré-entraîné présent dans le système dhSegment nécessite que les images d’en- trée soient de taille fixe de 400×400 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, contrairement aux deux autres systèmes, dhSegment est entraîné sur des patchs de 400×400 pixels d’images en taille réelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le proces- sus de scission est donc appliqué sur les polygones de taille originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est entraîné avec un arrêt anticipé et des mini-batchs de taille 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous avons conservé le post- traitement proposé dans Ares Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2018) en seuillant les probabilités avec t = 0,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Différentes valeurs ont été testées pour ce paramètre et le seuil de 0,7 a donné les meilleurs résultats sur l’ensemble de validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 aru-net Le système ARU-Net (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) est une version étendue du modèle U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Deux concepts ont été ajoutés : une attention spatiale et une structure résiduelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’attention spatiale (A-Net) est un CNN multicouche et est utilisée pour gérer différentes tailles de police sur une même page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les blocs résiduels sont introduits pour permettre la rétro propagation des erreurs sur les couches basses du réseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les annotations ARU-Net, nous utilisons le même processus que pour Doc-UFCN mais sur des polygones redimensionnés à 33 % de leur taille originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est entraîné en utilisant l’arrêt anticipé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons utilisé la fonction de coût d’entropie croisée et un taux d’apprentissage initial de 1e − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme pour dhSegment, il faut seuiller les probabilités pour obtenir les prédictions finales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, le choix du seuil pour ARU-Net n’a pas été une tâche facile car il a un impact réel sur les résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Finalement, nous avons choisi un seuil bas de t = 0,3 car une valeur plus élevée éliminerait une quantité importante de pixels de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 86 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Comparaison des systèmes Doc-UFCN, dhSegment et ARU-Net : nombre de paramètres, en millions, et temps d’inférence moyen, en secondes par image, mesuré sur l’ensemble de test du jeu de données Balsac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' dhSegment possède 32,8 millions de paramètres mais comme l’encodeur est pré-entraîné, seulement 9,36 millions doivent être entraînés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Système Temps d’inférence† Paramètres Doc-UFCN 0,41 4,09 dhSegment 2,95 32,8 (9,36) ARU-Net 1,39 4,14 † Prédictions faites sur une carte graphique GeForce RTX 2070 8G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour toutes les architectures, nous conservons les meilleurs modèles en validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, les éléments détectés de taille inférieur à 50 pixels sont supprimés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette paramétrisation est optimisée sur l’ensemble de validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 montre le nombre de paramètres et les temps d’inférence des trois systèmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doc-UFCN et ARU-Net ont des poids similaires en nombre de paramètres tout en étant beaucoup plus légers que dhSegment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les temps d’inférence, dhSegment et ARU-Net sont peu compétitifs, étant bien plus lents que Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 É VA L U AT I O N D E S D É T E C T I O N S Les trois systèmes ont été entraînés sur toutes les images d’entraînement afin de dispo- ser de modèles génériques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette section, nous présentons les résultats au niveau des pixels et des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour une comparaison équitable, toutes les prédictions sont d’abord redimensionnées à la taille de l’image originale avant de procéder à l’évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, pour être comparables à d’autres résultats publiés dans la littérature, les modèles sont éva- lués avec les lignes de vérité terrain originales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est intéressant de noter que, malgré des résultats visuels solides, cette évaluation basée sur les annotations originales est en défaveur des systèmes testés puisque les polygones d’entraînement sont beaucoup plus fins que ceux de la vérité terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Puisque notre objectif est de développer un modèle historique générique obtenant des performances satisfaisantes sur des jeux de données hors échantillon, nous rap- portons également les résultats sur les jeux de données ScribbleLens (Dolfing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) et HOME-Alcar (Stutzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2021), deux jeux de données qui n’ont pas été utilisés pour l’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, nous montrons l’impact de l’unification des annotations sur les résultats de détection lors de l’entraînement du système Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 métriques niveau pixel Dans cette section, nous présentons les résultats d’évaluation des systèmes par les mé- triques pixel IoU et F1-score, souvent utilisées dans la littérature pour évaluer les systèmes de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 É VA L U AT I O N D E S D É T E C T I O N S 87 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Résultats au niveau pixel obtenus par les systèmes Doc-UFCN, dhSegment et ARU-Net sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats présentent les performances des modèles génériques sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ScribbleLens* rapporte les résultats des modèles spécifiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données IoU F1-score Doc-UFCN dhSegment ARU-Net Doc-UFCN dhSegment ARU-Net AN-Index 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='69 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='82 0,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='77 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='67 Horae 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='63 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='87 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='79 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='79 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 READ-Complex 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='49 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='58 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='81 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='70 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 READ-Simple 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='60 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='57 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='88 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='71 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='71 HOME-Alcar 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='35 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='49 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='58 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='49 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='60 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='70 ScribbleLens 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='35 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='36 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='41 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='51 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='51 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='58 ScribbleLens* 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='80 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95 – 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='89 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='97 – comparaison des systèmes sur les ensembles de test des jeux d’entraîne- ment Les résultats obtenus par les trois réseaux sur les ensembles de test des jeux d’entraînement sont présentés en haut de la Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le réseau ARU-Net semble plus performant en termes d’IoU, alors qu’il l’est moins que les autres systèmes si nous considérons le score F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, le score F1 repose réellement sur les mesures de précision et de rappel (non présentées ici), en les résumant de manière précise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceci peut expliquer les faibles résultats obtenus par ARU-Net puisque ses scores en précision ne sont jamais supérieurs à 60 % (sauf pour Balsac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au contraire, le score IoU étant moins focalisé sur les pixels correctement prédits (TP est considéré deux fois pour le score F1 et seulement une fois pour IoU), les scores IoU sont plus élevés, ce qui conduit à un meilleur classement dans le tableau des résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces valeurs de précision faibles mais de rappel élevées obtenues par ARU-Net suggèrent que le modèle a correctement prédit la majorité des pixels de lignes de texte alors que, par ailleurs, beaucoup de pixels d’arrière-plan ont été classés comme lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela reflète la présence de fusions dans les lignes détectées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 montre les prédictions obtenues par les modèles sur une image tirée au hasard dans l’ensemble de test Horae (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle confirme notre hypothèse selon laquelle ARU-Net a fusionné certaines lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, comme indiqué précédemment, l’utilisation d’un seuil plus élevé aurait conduit à manquer un grand nombre de pixels de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous pensons qu’ARU-Net n’est peut-être pas le système le plus approprié pour détecter des objets proches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, il a souvent obtenu de très bonnes performances lorsqu’il était entraîné avec des lignes de base, où les objets sont plus espacés et plus fins que les polygones englobants des lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 88 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Détections de lignes produites sur une image du jeu de données Horae : Doc-UFCN à gauche, dhSegment au centre et ARU-Net à droite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doc-UFCN et dhSegment produisent des résultats similaires, tandis que ARU-Net surestime l’épaisseur des lignes et fusionne plusieurs lignes (l’une d’elles est mise en évidence en vert foncé).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comparer Doc-UFCN et dhSegment est un peu plus facile car ils se comportent de la même manière pour les scores IoU et F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doc-UFCN surpasse dhSegment sur la majorité des jeux de données pour les deux mesures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est cependant moins bon sur le jeu de données READ-Complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous supposons que cela est dû au nombre élevé de petits objets dans les images des documents de ce jeu qui peuvent avoir été manqués par Doc-UFCN puisqu’il travaille à une faible résolution, contrairement à dhSegment et ARU-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’évaluation et la comparaison des trois modèles sur la base de l’IoU uniquement condui- raient à choisir ARU-Net comme étant le meilleur modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Or, nous avons montré que ses faibles précisions peuvent conduire à une faible capacité à distinguer des lignes proches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les mesures au niveau objet, qui peuvent rendre compte des lignes fusionnées, devraient être utilisées en complément de ces valeurs de pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' évaluation hors échantillon La Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 présente également les résultats des modèles génériques appliqués aux jeux de données ScribbleLens et HOME-Alcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons également entraîné Doc-UFCN et dhSeg- ment sur ScribbleLens afin de disposer de modèles spécifiques pour la comparaison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu de données HOME-Alcar, nous ne disposons pas d’images d’entraînement pour la détection de lignes, donc seuls les résultats génériques sont présentés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les performances obtenues par les trois systèmes sur les jeux de données ScribbleLens et HOME-Alcar sont bien inférieures à celles obtenues sur les jeux de données d’entraînement, et également à celles obtenues par les modèles spécifiques de ScribbleLens*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu de test ScribbleLens, la précision est égale à 97 % pour les modèles génériques dhSegment et Doc-UFCN alors qu’elle se situe entre 82 et 85 % pour HOME-Alcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela suggère que htretrneiru ermtrateaste moeumtr 防· HHUS e KOJIRIOhmraememmm toot:iairemtre ars-ne iomttmrmf·oremm tortmmm:mohtm mmtotottmmtirgtrrg toot metm m2 1smm·oeumtn mmon mmmy temrrommt H XU cuott5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 É 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='60 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='72 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 HOME-Alcar 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='51 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='35 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='63 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='49 ScribbleLens 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='42 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='35 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='59 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='51 presque tous les pixels prédits étaient corrects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' alors qu’un grand nombre de pixels de la vérité terrain n’ont pas été détectés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre hypothèse est que les modèles ont prédit de bons polygones mais très fins par rapport aux polygones annotés très larges des pages ScribbleLens, ce qui a conduit à des valeurs d’IoU dégradées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il en est de même pour les images HOME-Alcar, où des polygones fins rectangulaires comprenant uniquement quelques pixels d’arrière-plan ont probablement été prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur la base de ces métriques, nous ne pouvons pas être certains que les systèmes ne par- viennent pas à généraliser sur les deux nouveaux ensembles de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’autres mesures pourraient donner un meilleur aperçu des capacités de généralisation réelles des modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' impact de l’unification des annotations Pour évaluer l’impact de l’unification des annotations sur les résultats, nous avons entraîné Doc-UFCN sur tous les jeux de données avec des annotations non uniformisées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Selon la Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4, l’entraînement avec les annotations uniformisées améliore les performances au niveau pixel jusqu’à +16 points de pourcentage d’IoU sur Balsac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, comme expliqué pour ARU-Net dans la section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, les mesures de rappel sont plus élevées sans le processus d’unification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, les pixels entre les lignes consécutives et ceux le long des bords des lignes sont plus souvent prédits comme des lignes de texte, ce qui augmente les valeurs de rappel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, certains de ces pixels ne sont pas censés faire partie des lignes de texte (puisqu’ils créent des fusions), ce qui diminue les valeurs de précision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur la base de ces métriques, la scission des lignes proches semble être nécessaire pour aider le modèle à les distinguer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 90 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S limitation des métriques pixel Même si ces mesures au niveau pixel peuvent donner une première idée de la performance d’un modèle, nous présentons, sur la Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4, deux exemples prouvant qu’elles peuvent ne pas être suffisantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, deux prédictions différentes peuvent être qualifiées par les mêmes valeurs d’IoU et de score F1 malgré une différence importante de qualité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Or, dans la littérature, les systèmes sont souvent comparés par leurs valeurs d’IoU et de F1-score, nous montrons donc ici que ces métriques ne sont pas appropriées pour choisir le meilleur modèle car elles ne prennent pas en compte le nombre d’objets détectés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En conclusion, ces métriques ne nous permettent pas de déterminer la capacité de généralisation des modèles entraînés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour surmonter ces problèmes, la section suivante présente et analyse les résultats des métriques au niveau objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 métriques niveau objet Nous avons montré, dans la section précédente, que les métriques au niveau pixel peuvent ne pas être suffisantes pour une évaluation et une comparaison approfondies des modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons maintenant les métriques au niveau objet et montrons qu’elles sont complémentaires aux métriques précédentes, et peuvent donner des informations plus parlantes sur la qualité d’un résultat de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme indiqué précédemment, déterminer si un objet doit être considéré comme positif ou négatif est complexe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En se basant sur l’idée proposée dans les compétitions PASCAL VOC, il est possible de calculer la précision, le rappel et la précision moyenne (AP) au niveau de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour ce faire, les objets prédits et les objets annotés sont d’abord appariés en fonction de leurs scores IoU, de sorte qu’un seul objet prédit soit apparié à un objet annoté et inversement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, les objets appariés sont classés par score de confiance décroissant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour chaque objet prédit i, les mesures de précision Pi et de rappel Ri sont calculées en considérant uni- quement les objets ayant des scores de confiance supérieurs ou égaux à celui de l’objet courant i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces mesures sont calculées en fonction d’un seuil d’IoU choisi t, à l’aide des équations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 suivantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pi = TPi Totali Ri = TPi TotalGT (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1) Ces équations s’appliquent avec : — TPi : nombre d’objets positifs correctement prédits ayant une confiance supérieure ou égale à celle de l’objet i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Totali : nombre d’objets prédits ayant une confiance supérieure ou égale à celle de l’objet i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — TotalGT : nombre d’objets annotés à retrouver ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' où un objet est considéré comme positif si son IoU est supérieur au seuil choisi t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 É VA L U AT I O N D E S D É T E C T I O N S 91 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – Résultats au niveau ligne obtenus par les systèmes Doc-UFCN, dhSegment et ARU-Net sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats présentent les performances des modèles génériques sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ScribbleLens* rapporte les résultats des modèles spécifiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 AP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95] Doc-UFCN dhSegment ARU-Net Doc-UFCN dhSegment ARU-Net AN-Index 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='76 0,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 ScribbleLens 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='06 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='02 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='02 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='02 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 ScribbleLens* 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='94 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 – 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='61 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 – La courbe Précision-Rappel est ensuite calculée et interpolée et la précision moyenne (AP) est définie comme l’aire sous cette courbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette AP est calculée pour toutes les classes d’une expérience, puis la moyenne est calculée pour toutes les classes, ce qui donne la précision moyenne (mAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour la détection des lignes de texte, nous n’avons qu’une seule classe d’objets, la mAP est donc égale à l’AP et est notée AP@t dans la suite, t étant toujours le seuil IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' comparaison des systèmes sur les ensembles de test des jeux d’entraîne- ment La Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 présente les résultats d’AP obtenus sur les ensembles de test pour un seuil d’IoU de 50 % (AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, et afin de s’affranchir de tout seuil, la moyenne des AP sur une plage de valeurs d’IoU (50 % – 95 %) est également calculée et présentée comme AP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats présentés ici renforcent notre hypothèse précédente selon laquelle ARU-Net ne parvient pas à séparer les objets proches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, tous les résultats de ARU-Net sont très inférieurs à ceux des deux autres systèmes, sauf pour le jeu de données Balsac où les polygones des lignes de texte sont vraiment espacés dans les annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous observons que, pour un seuil bas de 50 %, Doc-UFCN surpasse légèrement dhSegment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En passant de 50 % à la moyenne des AP, nous constatons que les résultats des deux modèles se dégradent, ce qui signifie que, avec des seuils plus élevés, certaines lignes deviennent considérées comme fausses positives car leur localisation n’est pas assez précise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, cette dégradation est plus faible pour Doc-UFCN que pour dhSegment, ce qui signifie que la localisation des objets par dhSegment est moins précise que celle de Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 présente les résultats des 92 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Détections de lignes produites sur une image du jeu de données Bozen : Doc-UFCN à gauche, dhSegment au centre et ARU-Net à droite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doc-UFCN prédit des lignes bien séparées alors que ARU-Net prédit des lignes fusionnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' dhSegment ne produit pas de lignes fusionnées mais elles sont plus proches que celles produites par Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les polygones de dhSegment incluent plus d’espace en haut des lignes, ce qui peut avoir un impact négatif sur la reconnaissance du texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' trois modèles sur une image tirée au hasard dans le jeu de données Bozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces prédictions confirment l’intérêt des mesures AP pour évaluer les prédictions de détection puisqu’elles mettent en évidence les mauvais comportements, comme ceux montrés par ARU-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' évaluation hors échantillon La Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 présente également les résultats des modèles génériques appliqués à Scribble- Lens et HOME-Alcar et les modèles spécifiques de ScribbleLens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats obtenus par les modèles spécifiques au niveau objet sont totalement opposés à ceux obtenus au niveau pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats obtenus par Doc-UFCN confirment que le modèle fonctionne bien lorsqu’il est entraîné directement sur ScribbleLens, sauf sur quelques images comme montré sur la Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au contraire, alors que dhSegment a obtenu de bonnes mesures au niveau des pixels, ses valeurs d’objet sont toutes à 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, comme pour les résultats précédents obtenus par ARU-Net, nous observons de nombreuses lignes fusionnées prédites par le modèle spécifique dhSegment, ce qui signifie que ce dernier n’a pas réussi à apprendre directement à partir des images ScribbleLens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les faibles scores AP des modèles génériques peuvent être expliqués par la façon dont le jeu de données ScribbleLens a été annoté : des rectangles englobants très larges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les modèles ayant été entraînés sur des polygones bien divisés et beaucoup plus fins, seuls quelques polygones réels ont été appariés aux polygones prédits lors du calcul de l’AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La même observation s’applique aux résultats du jeu HOME-Alcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 et 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 présentent une visualisation des résultats obtenus sur les images des bases ScribbleLens et HOME-Alcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Malgré des valeurs de métriques peu élevées, les modèles génériques semblent nettement surpasser les modèles spécifiques, d’où l’importance de développer des modèles génériques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, il est nécessaire de les évaluer sur des annotations cohérentes avec celles de l’ensemble d’entraînement des modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 州5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 É VA L U AT I O N D E S D É T E C T I O N S 93 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Détections de lignes produites par les modèles génériques, en haut, et spécifiques, en bas, sur une image du jeu de données ScribbleLens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les images de gauche montrent les résultats produits par Doc-UFCN et celles de droite par dhSegment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' impact de l’unification des annotations Sans surprise, selon la Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6, presque toutes les valeurs sont meilleures lorsque nous utilisons le modèle entraîné sur les annotations unifiées, parfois avec une marge assez impor- tante (+33 points de pourcentage pour Balsac et +37 pour Bozen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu de données DIVA-HisDB, les résultats sont mitigés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous supposons que cela est dû au processus d’unifi- cation qui peut considérablement modifier les annotations en réduisant la hauteur de la ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces métriques au niveau objet ont souligné la nécessité de les utiliser avec celles au ni- veau du pixel pour évaluer et comparer les modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, il est encore difficile de voir l’avantage d’utiliser des modèles génériques sur des documents hors échantillon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les mé- triques orientées vers les objectifs, décrites dans la section suivante, permettront une meilleure comparaison des objets prédits et des objets réels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 89A 13080 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='AA la9e8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' enetalrmycawl moeneeim 4se3 2-0002 9020004 080 00n489840 1380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='A147 enxelaermiek Ae XA& aAg 9020894 80058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' mrceaLon48p8f ongop yhyerayo Ramse Lem G- Coesusi94 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – Détections de lignes produites par les modèles génériques Doc-UFCN, à gauche, et dh- Segment, à droite, sur une image du jeu de données HOME-Alcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 – Résultats au niveau ligne obtenus par Doc-UFCN avec et sans uniformisation des labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats montrent les performances des modèles génériques sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 AP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95] Originaux Uniformes Originaux Uniformes AN-Index 0,69 0,75 0,28 0,34 Balsac 0,95 0,98 0,44 0,76 BNPP 0,81 0,83 0,30 0,31 Bozen 0,77 0,99 0,31 0,69 cBAD2019 0,71 0,86 0,25 0,48 DIVA-HisDB 0,86 0,77 0,40 0,36 HOME-NACR 0,82 0,85 0,33 0,46 Horae 0,84 0,83 0,34 0,38 READ-Complex 0,61 0,60 0,24 0,23 READ-Simple 0,60 0,69 0,19 0,28 HOME-Alcar 0,86 0,16 0,27 0,03 ScribbleLens 0,41 0,06 0,08 0,02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 É VA L U AT I O N O R I E N T É E V E R S L A TÂ C H E D E R E C O N N A I S S A N C E Dans les sections précédentes, nous avons discuté des résultats aux niveaux pixel et objet pour les trois modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} 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aluoguonuemencinuumcecoet retuemnoamctehmommpietett pocatabiptnenotmfmemozattabbrce oienntgaeruzabntaoutcoo tat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='amo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='ot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='wodtw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='nce futeuotnes ftaneoneomtScaauejoh matose qutentficeanta twontaneanonCaacta qmuottetaeueibreocarnp itebetonecotaaetaltoquctmo mttfptenrertttafmmeut mocoapbdcoceptiecmletfoo obutdamttenottonemwolutteLataatut tamfaemqomnaptetzaonintuut concelliero2atobtiteCitaceteao oix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='oepntiecuterseotnettmealle noutttebnbtutieaccitelbtecconttent titeeoqototmteumetmmconem quinggiea toitoog tura caoxm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='qttran onetactenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='aceannoonit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='oscoxav llg pmitoneotttma atyg cohie nnheinozamtbnmat5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 É VA L U AT I O N O R I E N T É E V E R S L A TÂ C H E D E R E C O N N A I S S A N C E 95 Nous avons effectué une évaluation orientée vers la reconnaissance de texte sur les cinq ensembles de données pour lesquels la transcription des lignes de texte est disponible, en calculant le taux d’erreur de caractère (CER) et le taux d’erreur de mot (WER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par souci de clarté, dans les tables suivantes, seuls les CER sont présentés car les WER y sont fortement corrélés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour réaliser cette évaluation axée sur la reconnaissance du texte, nous avons utilisé un reconnaisseur de texte manuscrit (Boros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) basé sur la bibliothèque Kaldi (Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est composé de deux éléments principaux : un modèle optique utilisant un modèle hybride Deep Neural Network-Hidden Markov Model et un modèle de langue fondé sur un modèle n-gram entraîné sur des sous-mots générés par la méthode Byte Pair Encoding (BPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Contrairement au modèle de détection de lignes de texte entraîné sur tous les jeux de données, nous avons entraîné un modèle de reconnaissance spécifique pour chaque jeu de données et utilisé ces modèles pour l’évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les paragraphes suivants présentent et analysent les résultats de la détection à l’aide de deux métriques basées sur le CER au niveau des pages et des lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 cer niveau page Pour commencer l’évaluation, nous avons d’abord choisi de calculer le CER au niveau de la page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les calculs sont détaillés dans l’Algorithme 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, tous les polygones de lignes prédits et annotés d’une image sont triés de haut en bas et de gauche à droite de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En suivant cet ordre, toutes les transcriptions sont concaténées en une seule ligne de texte et le CER@page est calculé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 présente les CER@page obtenus par les systèmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, nous avons calculé le CER obtenu par le système HTR lors de la transcription des polygones annotés manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, la colonne "Manuel" des tables suivantes correspond au meilleur CER réalisable avec le système de détection idéal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit du CER que nous aurions si nous avions 100 % pour toutes les métriques pixel et objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Algorithme 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Calcul du CER@page Entrée: HTR ← modèle de reconnaissance entraîné Entrée: DLA ← modèle de détection de lignes de texte entraîné Entrée: image ← image à évaluer Entrée: transcription ← transcription manuelle de l’image,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' texte ordonné de haut en bas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' gauche à droite 1: lignes ← DLA(image) 2: ord(lignes) {ordonne les lignes de haut en bas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' gauche à droite} 3: prediction ← ”” 4: pour chaque ligne ∈ lignes faire 5: prediction ← concat(prediction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' HTR(ligne)) 6: fin pour 7: cer ← CER(prediction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' transcription) Sortie: cer 96 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 – Résultats de reconnaissance niveau page obtenus par les systèmes Doc-UFCN, dhSeg- ment et ARU-Net sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats présentent les performances des modèles génériques sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ScribbleLens* rapporte les résultats des modèles spécifiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données CER@page (%) Manuel Doc-UFCN dhSegment ARU-Net Balsac 4,3 14,9 15,8 31,5 BNPP 15,5 37,2 38,2 46,5 Bozen 5,8 11,7 13,2 74,9 HOME-NACR 11,9 38,6 22,3 75,2 Horae 10,3 14,8 12,1 31,5 HOME-Alcar 12,5 37,4 43,5 43,3 ScribbleLens 4,4 9,5 21,9 15,4 ScribbleLens* 4,4 25,2 92,6 – comparaison des systèmes sur les ensembles de test des jeux d’entraîne- ment Les résultats de la Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 montrent la faible performance de la reconnaissance de texte sur les lignes détectées par ARU-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce taux d’erreur élevé est la conséquence de la fusion de nombreuses lignes de texte détectées qui ne peuvent pas être correctement reconnues, comme cela a déjà été mis en évidence avec l’évaluation au niveau de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doc-UFCN est plus performant que dhSegment sur trois des cinq jeux de données et l’est légèrement moins que dhSegment pour le jeu de données Horae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces résultats confirment ceux obtenus avec les évaluations au niveau du pixel et de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doc-UFCN est cependant loin derrière dhSegment sur le jeu de données HOME-NACR, contrairement aux résultats obtenus avec les métriques pixel et objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Si nous analysons davantage les résultats de détection obtenus par Doc-UFCN sur le jeu de données HOME-NACR, nous constatons qu’environ la moitié des pages ont été parfaitement segmentées sans aucune fusion, ce qui augmente considérablement les scores AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, les autres pages contiennent des lignes prédites qui sont des fusions de deux, trois lignes, ou même des fusions de lignes de paragraphes entiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela conduit à une légère diminution des scores AP mais à une dégradation drastique des performances en termes de CER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, si une seule fusion n’a qu’un faible impact sur le score AP, elle a un impact direct sur le CER par deux types d’erreurs : — Le CER entre la prédiction et sa ligne annotée correspondante (qui est souvent élevé dans le cas d’une fusion) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Le CER des lignes annotées non appariées, égal à la longueur de chaque ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette seconde erreur n’est pas significative lorsque seules quelques lignes annotées ne sont pas appariées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est le cas pour les quatre premiers ensembles de données où le nombre de fusions est négligeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle est encore moins significative lorsque les lignes non appariées ont un petit nombre de caractères.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, HOME-NACR est le jeu de données avec la plus grande densité de caractères par ligne (jusqu’à six fois plus que les autres jeux de données).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 É VA L U AT I O N O R I E N T É E V E R S L A TÂ C H E D E R E C O N N A I S S A N C E 97 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 – Simulation des scores lorsque deux lignes sont bien séparées, à gauche, et fusionnées, à droite, sur une image du jeu de données HOME-NACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À gauche, AP[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95]=60 % et CER@page=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 % ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' à droite, AP[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95]=51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 % et CER@page=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi, cette seconde erreur a un réel impact sur le CER final de l’ensemble de données HOME-NACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est également la raison pour laquelle les scores AP ne révèlent pas le phénomène.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 illustre ce point : l’image de gauche est correctement segmentée alors que sur celle de droite, deux lignes sont fusionnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce cas, l’introduction d’une fusion dans les prédictions entraîne une diminution relative de la moyenne AP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95] de 15 % (60 % à 51,3 %) tandis que le CER@page se dégrade de 179 % (7,3 % à 20,4 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela prouve qu’une fusion n’affecte pas les différentes métriques de la même manière.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Toujours sur le jeu de données HOME-NACR, dhSegment montre une localisation moins précise des lignes de texte (scores AP plus faibles) par rapport à Doc-UFCN mais très peu de fusions, ce qui conduit à de meilleures performances de reconnaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, HOME-NACR est le jeu de données où l’écriture est la plus dense et où les lignes sont les plus proches les unes des autres parmi les dix jeux de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En raison de la mince hauteur des lignes de texte et du redimensionnement à 768 pixels, nous pensons que Doc-UFCN n’est pas l’architecture la plus adaptée pour travailler avec ces pages, contrairement à dhSegment qui présente une meilleure détection puisqu’il est appliqué sur les images dans leur taille originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette métrique supplémentaire donne de nouveau un aperçu des performances des modèles, en étant complémentaire aux métriques vues précédemment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle peut, en effet, détecter des comportements qui ne sont pas mis en évidence par les mesures de pixels ou d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' évaluation hors échantillon Les résultats de la généralisation sont également présentés dans la Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le jeu ScribbleLens, nous constatons l’avantage d’utiliser un modèle générique : les résultats des modèles spécifiques (Doc-UFCN 25,2 % de CER, dhSegment 92,5 de % CER) sont nettement moins bons que ceux des modèles génériques (Doc-UFCN 9,5 % de CER, dhSegment 21,9 % de CER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les valeurs de CER du jeu HOME-Alcar sont élevées pour tous les systèmes, ce qui peut être dû à la complexité de certaines images de documents : mauvaise qualité de la numérisation, mauvaises conditions de conservation (certaines pages ont été déchirées, par exemple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, ces résultats mettent en évidence les capacités de généralisation du modèle générique Doc-UFCN, donnant de meilleurs résultats sur ScribbleLens que le modèle spécifique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ACK 124ACK 12498 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 – Résultats de reconnaissance niveau page obtenus par Doc-UFCN avec et sans uniformi- sation des labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats présentent les performances des modèles génériques sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données CER@page (%) Originaux Uniformes Balsac 14,4 14,9 BNPP 34,4 37,2 Bozen 27,6 11,7 HOME-NACR 33,5 38,6 Horae 15,1 14,8 HOME-Alcar 43,2 37,4 ScribbleLens 12,9 9,5 impact de l’unification des annotations Comme présenté dans la Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8, les deux Doc-UFCN avec et sans le processus d’unifi- cation ont des résultats assez similaires sur quatre jeux de données sans aucune dégradation significative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, l’impact de l’uniformisation des annotations est plus important sur la base de données Bozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour la même raison que ARU-Net, le modèle entraîné avec les annotations originales prédit un grand nombre de lignes fusionnées, ce qui conduit à un taux d’erreur caractères très élevé par rapport au modèle entraîné avec les annotations uniformes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le processus d’unification n’a pas amélioré les résultats sur Balsac et BNPP, car les anno- tations originales étaient déjà fines et constituaient une entrée correcte pour le système de reconnaissance HTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Même si l’entraînement avec les annotations uniformisées n’a pas montré d’amélioration significative des valeurs de CER pour quatre ensembles de données, il a eu un réel impact sur les prédictions de Bozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Concernant les jeux de données hors échantillon, l’unification des annotations a également un impact positif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle entraîné avec les annotations uniformisées donne un CER de 9,5 % pour ScribbleLens et 37,4 % pour HOME-Alcar, ce qui correspond respectivement à 26 % et à 13 % de diminution relative de l’erreur caractère.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 cer niveau ligne Cette dernière mesure est étroitement liée au CER au niveau de la page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ici, le CER n’est pas calculé sur le texte complet de la page, mais sur chaque ligne de texte prédite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À cet égard, les lignes prédites et les lignes annotées doivent d’abord être appariées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la littérature, elles sont souvent appariées sur la base d’un seuil IoU de t = 50 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme pour l’AP, nous avons calculé le CER pour ce seuil d’IoU de 50 % (CER@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5) ainsi qu’une moyenne sur la plage 50 % – 95 % d’IoU (CER@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les lignes prédites sont appariées avec celles annotées manuellement qui ont l’IoU la plus élevée de sorte qu’une seule prédiction soit appariée avec une annotation et inversement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une fois les lignes appariées, nous calculons le CER pour tous 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 É VA L U AT I O N O R I E N T É E V E R S L A TÂ C H E D E R E C O N N A I S S A N C E 99 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 – Résultats de reconnaissance niveau ligne obtenus par les systèmes Doc-UFCN, dhSeg- ment et ARU-Net sur les ensembles de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats présentent les performances des modèles génériques sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ScribbleLens* rapporte les résultats des modèles spécifiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données CER@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5† (%) CER@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95] (%) Doc-UFCN dhSegment ARU-Net Doc-UFCN dhSegment ARU-Net Balsac 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='64 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 BNPP 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='93 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='83 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='73 44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 53,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 Bozen 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='94 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='93 86,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='12 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 93,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 HOME-NACR 36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='61 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='79 80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='18 61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='8 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 Horae 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='98 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='97 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='90 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 58,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 HOME-Alcar 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='92 27,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='0 45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 ScribbleLens* 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='76 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9/0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 – / – 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 93,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – † CER@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 / Proportion de caractères des lignes annotées appariés à une ligne de prédiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1 signifie que 100 % des caractères de l’annotation ont été appariés à une ligne de prédiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 – Résultats de reconnaissance niveau ligne obtenus, sur les ensembles de test, par les modèles génériques Doc-UFCN, dhSegment et ARU-Net sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' les couples dont l’IoU est supérieur au seuil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, le CER final est pénalisé par toutes les lignes qui ne sont pas appariées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9 et la Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 présentent les résultats obtenus après le système HTR au niveau ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les résultats de CER@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, la Table montre la proportion de caractères des lignes annotées appariés à une ligne prédite pour calculer les valeurs de CER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il aurait été possible de faire cela au niveau de la ligne (proportion de lignes appariées) pour voir sur quelle quantité de lignes les CER ont été calculés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, comme les lignes peuvent contenir un nombre variable de caractères, cela ne refléterait pas précisément le nombre réel de correspondances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 100 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 – Résultats de reconnaissance niveau ligne obtenus par Doc-UFCN avec et sans uniformi- sation des labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats montrent les performances des modèles génériques sans adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données CER@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 (%) CER@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95] (%) Originaux Uniformes Originaux Uniformes Balsac 7,2/0,97 7,2/0,95 17,1 14,2 BNPP 18,8/0,87 22,4/0,93 36,0 44,2 Bozen 27,3/0,67 8,8/0,94 47,3 20,7 HOME-NACR 29,4/0,73 36,1/0,61 46,6 61,8 Horae 15,7/0,98 15,2/0,98 20,6 22,6 HOME-Alcar 31,7/0,79 22,9/0,92 61,3 46,6 ScribbleLens 14,3/0,71 9,8/0,80 54,2 40,3 Comme pour les métriques précédentes, ARU-Net n’est pas compétitif : pas assez de carac- tères appariés et des valeurs de CER très élevées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour comparer Doc-UFCN et dhSegment, il est nécessaire d’étudier le CER et la proportion d’appariement dans leur ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, avoir un CER très bas lorsqu’il n’est calculé que sur une petite partie des lignes prédites n’est pas significatif puisque certaines lignes peuvent être plus faciles à reconnaître.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est préférable d’avoir un bon compromis entre le nombre de caractères appariés et le taux d’erreur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' comparaison des systèmes sur les ensembles de test des jeux d’entraîne- ment Les résultats obtenus au niveau des lignes reflètent réellement ceux obtenus avec les scores AP et CER au niveau des pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À 50 % d’IoU, Doc-UFCN semble meilleur pour les jeux de données Balsac, BNPP et Bozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, si nous considérons les résultats moyens CER@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95], dhSegment et Doc-UFCN ont tous deux des performances similaires sur les jeux de données Balsac, BNPP et Bozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le but d’avoir un modèle historique générique, les deux architectures semblent appropriées, obtenant de bons résultats au niveau des pixels et des objets, et des taux d’erreurs caractères acceptables au niveau des pages et des lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' évaluation hors échantillon Les résultats de l’évaluation hors échantillon sont également présentés dans la Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats pour ScribbleLens confirment ceux obtenus au niveau page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les modèles génériques sont, en effet, meilleurs que les modèles spécifiques puisqu’ils montrent des scores de CER plus bas (Doc-UFCN 9,8 % de CER par rapport à 24,3 %, dhSegment 18,2 % de CER par rapport à 90,9 %) et des proportions d’appariement supérieures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les performances sur les données HOME-Alcar sont comparables à celles obtenues au niveau page avec des taux d’erreurs élevés, malgré de hautes proportions d’appariement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 C O N C L U S I O N 101 impact de l’unification des annotations La Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='10 présente les résultats, au niveau ligne, de reconnaissance obtenus par Doc- UFCN avec et sans uniformisation des annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats présentés dans cette table sont semblables à ceux présentés au niveau page, à savoir des résultats assez similaires sur quatre jeux de données, avec et sans uniformisation, sans aucune dégradation significative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’impact est cependant très important sur le jeu de données Bozen puisqu’il y a un gain de 18,5 points de pourcentage de CER@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 en uniformisant les labels, impact expliqué par les mêmes raisons que celles exposées dans la section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Concernant les jeux de données hors échantillon, l’unification des annotations a également un impact très positif avec des diminutions de CER@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 de 8,8 et 4,5 points de pourcentages respectivement sur les bases HOME-Alcar et ScribbleLens, par rapport aux labels originaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 C O N C L U S I O N Dans ce chapitre, nous avons montré qu’il est possible d’entraîner un modèle générique pour détecter les lignes de texte dans les documents historiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons entraîné trois systèmes à l’état de l’art qui ont obtenu de bonnes performances sur différents ensembles de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceci a été rendu possible par la création d’un large jeu de données d’entraînement, qui est, à notre connaissance, le plus grand et le plus diversifié des jeux de données historiques utilisés pour comparer les systèmes de segmentation de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons également montré que, lors de l’agrégation de différents ensembles de données, l’uniformisation des polygones englobants annotés réduit les incohérences d’annotation entre les jeux annotés et permet d’entraîner de meilleurs modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, les modèles génériques entraînés sur plusieurs ensembles de données peuvent être meilleurs, non seulement sur les ensembles de données individuels, mais également sur les documents hors échantillon, ce qui prouve leurs capacités de généralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour une évaluation pertinente des performances des trois systèmes, ce chapitre compare et analyse également plusieurs métriques de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons montré que les métriques standards au niveau pixel ne sont pas suffisantes car elles ne tiennent pas compte de la qualité des objets prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour pallier cet inconvénient, des métriques au niveau des lignes ont été introduites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celles-ci ont montré que le système ARU-Net n’est pas approprié pour la tâche de détection de lignes de texte lorsqu’il est entraîné avec de telles annotations, le nombre de lignes fusionnées étant important par rapport aux deux autres approches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système est, en effet, souvent utilisé pour détecter les lignes de base des documents, qui sont plus fines et plus espacées que les polygones englobants des lignes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces mesures ont également confirmé les bonnes performances de Doc-UFCN et dhSegment sur la plupart des jeux de données, fournissant une détection précise et exacte des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces résultats n’auraient pas été possibles en utilisant uniquement des mesures au niveau pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous sommes convaincus que l’utilisation des scores de précision moyenne est nécessaire pour évaluer correctement 102 E N T R A Î N E M E N T E T É VA L U AT I O N D’ U N M O D È L E R O B U S T E D E D É T E C T I O N D’ O B J E T S les modèles de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre bibliothèque d’évaluation a été rendue publique 2, elle peut être utilisée sur n’importe quel jeu de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, ce chapitre fournit une évaluation orientée vers la tâche de reconnaissance de texte qui, à notre connaissance, n’a jamais été réalisée jusqu’à présent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les métriques d’évaluation de reconnaissance HTR donnent encore davantage d’informations sur les objets prédits, étant complémentaires aux métriques au niveau des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, elles permettent d’explorer l’impact de la qualité des lignes détectées sur les résultats finaux de reconnaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='com/teklia/dla/document_image_segmentation_scoring 6 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S Malgré les performances remarquables des réseaux de neurones profonds obtenus dans les travaux scientifiques, leur utilisation dans des applications réelles exige qu’ils soient, non seulement performants, mais aussi capables d’évaluer la confiance de leurs décisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceci est particulièrement important pour les applications liées aux images médicales ou à la conduite autonome, par exemple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le problème se pose également dans le cas de l’adaptation d’un modèle à un nouveau domaine, où nous souhaitons fournir au système le minimum de nouveaux exemples étiquetés pour réaliser l’adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le choix des exemples pertinents à soumettre à un annotateur humain est crucial pour optimiser le processus d’adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce cadre, connu sous le nom d’apprentissage actif (active learning), exige qu’un premier système effectue la tâche finale tout en évaluant automatiquement sa confiance sur de nouvelles données non vues, de sorte que les décisions moins confiantes puissent être soumises à un opérateur humain pour une annotation manuelle, tandis que les décisions plus confiantes prises par le système seraient conservées telles quelles pour fournir un étiquetage automatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce chapitre, nous visons à développer des mesures de confiance pour l’adaptation d’un modèle de détection d’objets dans un cadre d’apprentissage actif, afin de réduire au minimum l’effort d’annotation humaine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, notre objectif est de construire un estimateur de confiance pour la détection d’objets dans des images de documents dans un scénario d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce but, nous étudions trois approches afin d’estimer la confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La première consiste à utiliser les probabilités de classe a posteriori du modèle de détection pour estimer la confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La seconde approche proposée est inspirée de la méthode de Monte Carlo (Gal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016) et consiste à construire des estimations de confiance en utilisant la méthode de dropout au moment du test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le principal avantage de cette approche est qu’aucun entraînement supplémentaire n’est nécessaire pourvu que le modèle ait été entraîné avec des couches de dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle peut être appliquée à des modèles déjà entraînés sans aucune modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette approche est cependant coûteuse en calculs, c’est pourquoi notre dernière proposition consiste à construire un système dédié qui peut prédire une estimation de confiance avec une seule prédiction pendant l’inférence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Indépendant du système de prédiction, ce système nécessite cependant une phase d’entraînement spécifique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce chapitre présente tout d’abord, en section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, les estimateurs de confiance que nous proposons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La configuration utilisée pour les expériences (données, détails de l’entraînement 103 104 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S des modèles de détection et ceux des estimateurs de confiance) est ensuite détaillée dans la section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, dans la section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3, nous présentons et discutons les résultats obtenus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 M É T H O D E S D’ E S T I M AT I O N D E L A C O N F I A N C E Comme énoncé dans la section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2, très peu de travaux ont été proposés afin d’estimer la confiance des objets prédits par un modèle de détection d’objets dans les images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certains travaux utilisent le dropout de Monte Carlo (Gal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2016)) et analysent la distribution des prédictions afin d’estimer la confiance de la prédiction sans dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans d’autres travaux, un réseau adverse est entraîné pour estimer la proximité des prédictions avec la vérité terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce qui suit, nous proposons quatre estimateurs de confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le premier se base sur les probabilités a posteriori des classes données par le modèle de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les deux suivants s’inspirent des travaux réalisés sur le dropout de Monte Carlo et sont déduits de la variance des prédictions calculées avec dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, le dernier se base sur des statistiques descriptives des objets attendus et prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la suite de ce chapitre, les objets prédits font référence aux composantes obtenues après l’application d’un modèle de détection au niveau pixel suivi d’un seuillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le seuillage assigne à chaque pixel la classe (ou fond) de plus grande probabilité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 estimateur basé sur les probabilités a posteriori Les réseaux neuronaux de détection d’objets produisent des probabilités au niveau pixel qui sont ensuite seuillées afin de créer des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le premier estimateur que nous proposons, dénoté Posterior probability-based Confidence Estimator (PCE), utilise directement ces pro- babilités a posteriori afin d’estimer la confiance des prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, le modèle de détection est appliqué à une image d’entrée, les probabilités pj obtenues pour chaque pixel sont ensuite seuillées afin d’en extraire les objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour chaque objet prédit sur une image, nous calculons d’abord la moyenne des probabilités des pixels prédits par le modèle de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, le score PCE de l’image est déduit en calculant la moyenne de toutes les probabilités des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le calcul du score PCE est détaillé dans l’équation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les valeurs calculées par cet estimateur sont comprises entre 0 et 1, une valeur de 1 étant interprétée comme un indicateur d’une détection correcte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' PCE = 1 N × N � i=1 \uf8eb \uf8ed 1 Ni × Ni � j=1 pj \uf8f6 \uf8f8 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1) avec : — N : le nombre d’objets prédits sur l’image d’entrée ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Ni : le nombre de pixels composant l’objet i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — pj : la probabilité du pixel j d’appartenir à la classe d’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 M É T H O D E S D’ E S T I M AT I O N D E L A C O N F I A N C E 105 Cet estimateur présente les avantages d’être simple et rapide à calculer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, il ne nécessite aucun entraînement supplémentaire autre que le modèle de détection, et peut ainsi être utilisé pour n’importe quel modèle de détection produisant des probabilités en sortie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 estimateurs basés sur le dropout de monte carlo L’estimation de la confiance d’une prédiction avec le dropout de Monte Carlo consiste à calculer N prédictions de la même observation et à analyser la distribution des prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La variance entre les N prédictions est un indicateur de l’incertitude du modèle et peut donc être considérée comme une estimation de la confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette partie, nous proposons deux scores résumant la variance des prédictions : la précision moyenne (Dropout Average Precision (DAP)) et la variance du nombre d’objets (Dropout Object Variance (DOV)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' dropout average precision Comme démontré dans le chapitre 5, la précision moyenne (mAP) utilisée dans les défis PASCAL VOC et décrite dans Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022b) permet d’évaluer une prédiction au niveau objet par rapport à une annotation manuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’avantage de cette métrique est qu’elle considère la taille et la position des objets prédits puisqu’elle s’appuie sur une correspondance des objets basée sur l’IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Inspirés de cette métrique, nous dérivons l’estimateur DAP qui est calculé en considérant chaque paire de prédictions ((pi, pj) où pi et pj sont deux prédictions distinctes de la même image avec i, j ∈ [1, N] et i ̸= j) et en calculant la mAP (voir Focus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4) pour chaque paire, une des deux prédictions étant considérée comme vérité terrain arbitrairement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, le DAP est la moyenne de tous les scores mAP (voir l’équation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les valeurs calculées par cet estimateur sont comprises entre 0 et 1, un score DAP élevé indique que les N prédictions sont très similaires et est interprété comme un indicateur d’une détection correcte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' DAP = 1 N2 − N × N � i=1,j=1,i̸=j mAP(pi, pj) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2) dropout object variance Le second estimateur que nous proposons est basé uniquement sur la variance du nombre d’objets prédits parmi les N prédictions avec dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Lorsque le modèle est peu confiant, nous avons observé qu’un nombre très variable d’objets est prédit avec de nombreux petits objets autour de l’objet principal (comme le montre l’image de droite de la Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour obtenir une valeur unique, nous calculons la variance du nombre d’objets dans les prédictions avec dropout comme indiqué dans l’équation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3, où ni est le nombre d’objets dans la prédiction pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les valeurs calculées par cet estimateur sont comprises entre 0 et 1, un score DOV de 0 indique que toutes les prédictions ont le même nombre d’objets et est interprété comme un indicateur d’une détection correcte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 106 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Deux images issues du jeu de données Horae, à gauche, avec leurs prédictions, au centre et la variance pour N =10 prédictions avec dropout, à droite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une variance élevée est représentée en jaune alors que les zones sans variance sont en noir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’image de gauche a des estimations de confiance de DOV=0,0, DAP=1,0 et mAP-RFR=1,0 et celle de droite DOV=17,36, DAP=0,0993 et mAP-RFR=0,5553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' DOV = 1 N − 1 × N � i=1 (ni − n)2 avec n = 1 N × N � i=1 ni (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3) Comme l’estimateur PCE, le calcul des scores DAP et DOV ne nécessitent pas d’autre entraînement que celui du modèle de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, ces estimateurs peuvent être utilisés pour tout modèle de détection possédant des couches de dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 estimateur basé sur les statistiques d’objets Pour ce dernier estimateur, nous adoptons une approche basée sur une extraction de ca- ractéristiques pour estimer la confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous concevons un système qui analyse les caracté- ristiques des objets détectés et estime la mAP, car aucune vérité terrain n’est disponible au moment du test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Contrairement à nos premières propositions, le système étant indépendant du détecteur, cette approche peut être appliquée à tout type de détecteur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' statistiques descriptives d’objets Les modèles de détection développés dans nos travaux fournissent, pour chaque pixel, les probabilités d’appartenir à une classe d’objet ou d’arrière-plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les pixels sont d’abord affec- tés à la classe ayant la plus forte probabilité, puis nous détectons les éléments constitués des pixels connexes, ce qui conduit à plusieurs objets prédits pour une image donnée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, nous extrayons les polygones englobants des éléments détectés ainsi que leurs rectangles englo- bants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À partir de ces informations, nous calculons les huit caractéristiques d’objets suivantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour chaque image et chaque objet, nous calculons : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ratio entre la hauteur du rectangle englobant et la hauteur de l’image ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ratio entre la largeur du rectangle englobant et la largeur de l’image ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ratio entre la hauteur et la largeur du rectangle englobant ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ratio entre l’aire du polygone et l’aire de l’image ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ratio entre l’aire du polygone et l’aire du rectangle englobant ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ratio entre l’aire du rectangle englobant et l’aire de l’image ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ELE s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='oao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 C A D R E E X P É R I M E N TA L 107 et pour chaque image : 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Distances en y (hauteur) entre les centroïdes de tous les rectangles englobants, norma- lisées par la hauteur de l’image ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Distances en x (largeur) entre les centroïdes de tous les rectangles englobants, norma- lisées par la largeur de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les distances sont calculées en considérant chaque paire de rectangles englobants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les caractéristiques permettent de décrire les tailles, les formes et les positions des objets détectés dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour une image donnée et chacune des huit caractéristiques, les ratios sont calculés pour chaque objet détecté dont les valeurs résultantes sont regroupées en B intervalles pour fournir un histogramme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les histogrammes de caractéristiques sont ensuite concaténés pour constituer un vecteur de statistiques d’objets de taille 8×B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces statistiques sont ensuite utilisées pour entraîner un modèle de régression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' mean average precision - random forest regressor Pour construire l’estimateur de confiance, nous avons choisi d’estimer la mAP des pré- dictions, car nous avons montré, en section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2, qu’elle est plus significative que l’IoU (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour estimer la mAP d’une prédiction, plusieurs méthodes de ré- gression peuvent être utilisées telles que la régression par vecteur de support (SVR) ou le régresseur Random Forest (RFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans nos expériences, nous avons utilisé RFR, car il a obtenu les meilleurs résultats dans nos travaux préliminaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Après l’application du modèle de régression, aucun traitement supplémentaire n’est nécessaire puisqu’il fournit directement un score unique considéré comme l’estimation de confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce qui suit, cet estimateur est appelé mean Average Precision - Random Forest Regressor (mAP-RFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 C A D R E E X P É R I M E N TA L Nous avons évalué et comparé les estimateurs présentés en 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 sur deux tâches de difficultés différentes : la détection de pages et la détection de lignes de texte manuscrites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La détection de pages correspond au détourage des pages dans des prises de vues de doubles ou de simples pages dont les dimensions ne correspondent pas exactement aux dimensions des images pro- duites par l’imageur (scanner ou caméra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’une tâche assez simple puisqu’il y a souvent un ou deux objets sur une image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La détection de lignes de texte manuscrites est une tâche plus complexe car les pages de documents peuvent contenir un nombre variable, parfois important, de lignes de texte qui ont des formes et des positions très différentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 jeux de données Pour les expériences de détection de pages, nous avons utilisé les jeux de données READ- BAD (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017) et Horae (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre objectif est d’adapter le modèle de détection pré-entraîné sur les données READ-BAD aux images de documents de la base Horae en annotant le moins de données possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour la tâche de détection de 108 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Statistiques des jeux de données utilisés pour la détection de pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Images Pages simple double anormal READ-BAD train 1 635 1 459 171 5 Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) valid 200 179 21 – test 200 179 20 1 train 1 630 1 801 – – READ-BAD* valid 200 221 – – Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017) test 199 219 – – train 522 789 – – Horae valid 20 27 – – Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) test 30 51 – – test-300 300 364 – – lignes de texte, notre objectif est d’adapter un modèle générique pré-entraîné à un nouvel ensemble de documents hors échantillon d’apprentissage, à savoir le jeu de données Hugin- Munin (Maarand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022), détaillé en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les statistiques de ces jeux de données sont présentés dans la Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' jeu de données read-bad Le jeu de données READ-BAD (Grüning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017), présenté en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, contient 2 035 images de documents manuscrits utilisées lors des compétitions READ-BAD pour la détection des lignes de base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le jeu de données a été annoté aux niveaux simple et double pages 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans nos expériences, nous prédisons au niveau simple page, ce qui conduit à détecter deux objets sur les images qui présentent un document en double-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les images ayant été annotées comme "anormales" dans la base ont été supprimées car leurs annotations n’étaient pas assez précises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce qui suit, cette version du jeu de données est appelée READ-BAD* et comprend 1 630 images d’entraînement, 200 images de validation et 199 images de test avec respectivement 1 801, 221 et 219 pages simples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' jeu de données horae Le jeu de données Horae (Boillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) est semblable à celui présenté en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Afin d’avoir des résultats plus significatifs, nous avons étendu l’ensemble de test original qui ne contenait que 30 images en annotant 300 images supplémentaires choisies au hasard parmi les 1 158 livres d’heures, ce qui représente 364 pages simples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet ensemble de test est dénommé Horae-test-300 dans la suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le corpus complet Horae est composé de 1 158 livres d’heures présentant une grande diversité d’images de documents non annotés en termes de types de numérisations, de fonds et de formes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce corpus est utilisé pour comparer les différents estimateurs lorsqu’ils sont utilisés dans un cadre d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='com/ctensmeyer/pagenet 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 C A D R E E X P É R I M E N TA L 109 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 entraînement des systèmes de détection Pour nos expériences, nous avons utilisé le système Doc-UFCN comme détecteur d’objets, car, comme détaillé dans les deux chapitres précédents 4 et 5, il a montré de bonnes performances pour la détection d’objets sur des documents historiques tout en ayant un temps d’inférence réduit par rapport aux autres systèmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour les deux tâches, les modèles Doc-UFCN pré-entraînés (désignés par "référence" dans la suite) sont entraînés avec les images redimensionnées de telle sorte que leur plus grande dimension soit égale à 768 pixels, en conservant leur rapport d’aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un prétraitement est appliqué aux labels d’entraînement afin d’éviter que les zones annotées ne se touchent lors du redimensionnement des images (prétraitement détaillé dans la section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les modèles sont entraînés pendant 150 époques avec un taux d’apprentissage de 5e − 3 et l’optimiseur Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La configuration (poids) qui minimise la fonction de perte sur l’ensemble de validation est conservée à l’issue de l’apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' métriques d’évaluation Outre les métriques de détection standards, les modèles de lignes de texte sont également évalués à l’aide de métriques orientées vers la tâche finale, notamment le CER et le WER au niveau page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À cette fin, un reconnaisseur de texte manuscrit (HTR) basé sur Kaldi (Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) a été entraîné sur les lignes transcrites Hugin-Munin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons choisi cet HTR parce qu’il s’agit d’un outil prêt à l’emploi qui fonctionne généralement assez bien dans la plupart des cas d’utilisation et qui a obtenu des performances compétitives sur les documents Hugin-Munin (Maarand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle de reconnaissance entraîné est appliqué à toutes les lignes prédites par Doc- UFCN, ordonnées par leur centroïde du coin supérieur gauche de la page au coin inférieur droit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les textes prédits sont concaténés dans ce même ordre pour fournir une transcription unique au niveau de la page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les transcriptions manuelles sont ordonnées de la même manière et les CER et WER au niveau de la page sont calculés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle de détection de référence obtient environ 24 % de CER sur les images Hugin-Munin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, nous calculons la WordCountFMeasure (WCFM) (Pletschacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) qui évalue les modèles HTR sur la base du nombre de mots correctement prédits, indépendamment de leur position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons utilisé le PRIMA Text Evaluation Toolkit 2 pour calculer les scores WCFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Kaldi obtient un WCFM de 59 % par rapport aux transcriptions manuelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces valeurs de CER relativement élevées et de WCFM faibles indiquent que les lignes détectées par le modèle de référence ne sont pas de très bonne qualité pour le modèle de reconnaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles peuvent refléter des lignes détectées mal placées (pas de texte), des lignes trop fines (texte coupé) ou des lignes manquées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 110 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Résultats de détection de pages obtenus par le modèle de référence entraîné sur le jeu de données READ-BAD* et évalué sur les jeux de données READ-BAD* et Horae-test-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données IoU F1-score mAP READ-BAD* train 0,97 0,98 0,92 valid 0,97 0,98 0,91 test 0,97 0,98 0,94 Horae test-300 0,90 0,94 0,60 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Résultats de détection de lignes de texte obtenus par le modèle de référence entraîné sur 19 jeux de données et évalué l’ensemble de test du jeu de données Hugin-Munin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données IoU F1-score mAP CER (%) WCFM Hugin-Munin test 0,48 0,63 0,21 24,37 0,59 résultats des systèmes de détection Pour la tâche de détection de pages, le modèle de référence est entraîné sur des images READ-BAD* dont les résultats sont présentés dans la Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il obtient une IoU de 97 % et une mAP de 94 % sur READ-BAD*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, la mAP sur les images de l’ensemble Horae-test-300 est d’environ 60 %, ce qui laisse une marge d’amélioration importante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce qui suit, les images ayant les plus faibles scores de confiance estimés dans le corpus Horae sont annotées afin d’améliorer la détection sur Horae-test-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour la détection des lignes de texte, nous avons entraîné un modèle générique de détection des lignes de texte, différent de celui présenté dans le chapitre 5, ainsi que les estimateurs de confiance sur de nombreux jeux de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À cet égard, nous avons rassemblé 19 bases de données principalement publiques, comprenant des documents historiques et modernes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au total, ce jeu de données contient 9 432 images d’entraînement, 1 907 images de validation et 6 669 images de test, ce qui correspond à 374 316 lignes annotées d’entraînement, 85 208 lignes de validation et 190 502 lignes de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce modèle générique appliqué à l’ensemble de test Hugin-Munin a été évalué à 48 % d’IoU et 21 % de mAP (Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces résultats assez faibles étaient attendus puisque les documents sont beaucoup plus complexes que ceux utilisés lors du pré-entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 entraînement des estimateurs de confiance Aucun apprentissage supplémentaire n’est requis pour les estimateurs basés sur le dropout de Monte Carlo, puisque seuls les modèles de détection d’objets sont utilisés pour estimer la confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En revanche, les régresseurs doivent être entraînés sur les statistiques d’objets décrites en section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, le modèle de détection d’objets est appliqué à toutes les images (READ-BAD* pour la détection de pages et les 19 jeux de données pour la détection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='primaresearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='org/tools/PerformanceEvaluation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 R É S U LTAT S E T D I S C U S S I O N 111 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Courbes de rejet présentant l’évolution des performances du modèle de détection de pages de référence sur l’ensemble de test Horae-test-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Courbes présentées pour les estimateurs DAP, à gauche, et DOV, à droite, en fonction du nombre de prédictions avec dropout N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' de lignes de texte), ce qui permet de calculer les statistiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme les jeux de données sont annotés, le modèle de détection est ensuite évalué sur chaque image séparément, ce qui fournit un IoU et une mAP pour chaque image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces valeurs de mAP sont utilisées comme cible pour l’entraînement des régresseurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour entraîner les modèles de régression, nous avons utilisé le RandomForestRegressor de scikit-learn avec les paramètres par défaut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les modèles de régression présentent de faibles erreurs quadratiques moyennes (MSE) sur les ensembles de données d’entraînement (0,0164 MSE sur l’ensemble de test de READ-BAD*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 R É S U LTAT S E T D I S C U S S I O N Dans cette section, nous évaluons et comparons les estimateurs de confiance à l’aide de courbes de rejet, puis nous comparons leurs performances lorsqu’ils sont intégrés dans un cadre d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 nombre de prédictions avec dropout Pour les expérimentations avec le dropout de Monte Carlo (DAP et DOV), nous devons définir le nombre de prédictions N à calculer pour estimer la qualité des prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 montre la mAP en fonction du taux de rejet pour les estimateurs DAP et DOV calculés pour différentes valeurs de N (2, 5, 10, 25 et 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons choisi ces valeurs car nous recherchons un ordre de grandeur de N plutôt qu’une valeur précise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’idée est de savoir si nous avons besoin d’un nombre important de prédictions pour obtenir une variance suffisamment fiable, ou si quelques prédictions suffisent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous ne sommes pas allés au-delà de 50 prédictions car nous voulons garder un temps de calcul raisonnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 112 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Courbes de rejet présentant l’évolution du score mAP en fonction du taux de rejet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes présentent les résultats du modèle de détection de pages de référence sur l’ensemble de test Horae-test-300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats sont présentés sur Horae-test-300 pour la détection de pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes de rejet sont construites en ordonnant les images en fonction de leur confiance estimée, les exemples ayant une valeur DAP inférieure (ou une valeur DOV supérieure) à un seuil prédéfini sont retirés de l’ensemble d’évaluation et la mAP est calculée sur les exemples restants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour DAP, le seuil varie de 0 à 1 avec un pas de 0,05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour DOV, les valeurs ne sont pas bornées, le seuil varie donc de 10 à 0 avec un pas de -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces graphiques montrent que l’utilisation de N =10 prédictions pour l’estimation avec dro- pout est suffisante et qu’aucune amélioration n’est observée avec N =25 ou N =50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, le coût de calcul est réduit avec seulement 10 prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur la base de cette observation, nous avons utilisé N =10 prédictions avec dropout pour estimer les scores de confiance dans le reste des expériences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 performances des estimateurs en rejet Dans une première expérience, nous évaluons la capacité des estimateurs de confiance à détecter les exemples mal prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour ce faire, nous évaluons les performances du modèle de détection lorsque les images ayant le score de confiance estimé le plus faible sont retirées de l’ensemble d’évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette évaluation est réalisée grâce à des courbes de rejet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur les courbes de rejet, chaque point correspond à un seuil pour lequel les images dont le score estimé est inférieur à ce seuil sont retirées de l’évaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes n’atteignent pas 100 % car, au-dessus d’un seuil donné, il reste uniquement des images ayant le même score, de sorte qu’elles ne peuvent plus être retirées sans que l’ensemble d’évaluation soit vide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par souci de clarté, nous montrons seulement l’évolution de la mAP, les résultats d’IoU suivant la même tendance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 R É S U LTAT S E T D I S C U S S I O N 113 La Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 montre l’évolution des performances du modèle de référence sur Horae- test-300 pour la tâche de détection de pages par rapport au taux de rejet pour différents estimateurs de confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous montrons les courbes médianes ainsi que les intervalles de confiance (10eet 90e percentiles) obtenus en calculant 100 courbes de rejet générées par 100 ré-échantillonnages avec remplacement à partir de l’ensemble de test original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La courbe aléatoire montre les résultats obtenus pour 100 échantillonnages aléatoires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Notre objectif est d’avoir un modèle avec une mAP élevée et un faible taux de rejet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous pouvons constater que les estimateurs basés sur le dropout ne sont pas compétitifs par rapport au régresseur basé sur les statistiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, comme mAP-RFR ne nécessite qu’une seule prédiction en inférence, ce premier résultat montre que l’utilisation de mAP-RFR au lieu du dropout de Monte Carlo est plus intéressante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats de PCE étant comparables à ceux de DAP et DOV, il semble que les estimateurs dropout de Monte Carlo ne fassent pas de meilleurs indicateurs que les probabilités a posteriori des détecteurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela peut s’expliquer par le fait qu’aucune information supplémentaire à part les prédictions du réseau neuronal ne soit fournie à ces trois estimateurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette première expérience montre que notre proposition mAP-RFR a une grande capacité à estimer la confiance des pages prédites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il surpasse DAP et DOV qui sont eux-mêmes à peine meilleurs que PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur la Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, nous montrons deux prédictions obtenues par le modèle de référence pour la tâche de détection de pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À gauche, nous montrons une bonne prédiction où la variance est faible, sauf sur les bords des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les estimations de confiance DOV=0,0, DAP=1,0 et mAP-RFR=1,0 reflètent bien la bonne qualité de la détection de l’image de gauche tandis que les estimations de confiance DOV=17,36, DAP=0,0993 et mAP-RFR=0,5553 de l’image de droite reflètent également la très mauvaise qualité de la détection, qui contient un nombre élevé de petits objets prédits autour du principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 apprentissage actif Dans un cadre d’apprentissage actif, l’objectif est d’entraîner un bon détecteur d’objets tout en minimisant la quantité d’exemples à annoter manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour y parvenir, il est crucial de bien choisir les données à annoter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans nos expériences, nous suivons une configuration standard d’apprentissage actif (Cohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, un modèle Doc-UFCN de référence est entraîné, puis appliqué à des documents non vus et non annotés provenant d’un nouveau jeu de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, ces images sont classées en fonction de leur confiance estimée, celles dont la confiance est la plus faible sont sélectionnées pour une annotation manuelle et utilisées pour entraîner un nouveau modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que de nombreuses stratégies de sélection des données à annoter aient été proposées pour améliorer au mieux les modèles (Settles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2008), nous nous concentrons, dans cette section, sur la sélection des images ayant la plus faible confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, les images ayant une confiance inférieure à un seuil prédéfini sont sélectionnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le seuil varie d’une itération à l’autre en fonction de la distribution des confiances estimées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons une analyse de deux stratégies de sélection dans la section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 114 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Résultats des modèles de détection de pages sur l’ensemble de test Horae-test-300 après apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Itération indique le nombre d’itérations réalisées afin d’obtenir le meilleur modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le nombre d’images annotées est indiqué dans la colonne Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Estimateur Itération Images IoU mAP Référence – 0 0,90 0,60 Aléatoire 5 300 0,93 0,86 PCE 9 90 0,93 0,86 mAP-RFR 8 107 0,94 0,89 DAP 9 129 0,94 0,91 DOV 9 168 0,95 0,92 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Évolution des performances de détection de pages (mAP) sur l’ensemble de test Horae- test-300 pendant les itérations d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque modèle de détection est entraîné dans la même configuration que les modèles de référence décrits dans la section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pendant les itérations d’apprentissage actif, plusieurs stratégies d’initialisation des poids des modèles peuvent être envisagées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils peuvent être initialisés avec les poids des derniers modèles entraînés, ceux du modèle de référence (à chaque itération) ou encore ceux du meilleur modèle entraîné durant les itérations précédentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour nos expériences, nous initialisons les poids avec ceux des derniers modèles entraînés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, pour les expériences suivantes, nous avons calculé un intervalle de confiance sur les modèles de la dernière itération.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, nous avons utilisé le bootstrapping empirique (Wasserman, 2004) avec 100 ré-échantillonnages avec remplacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En outre, les expé- riences avec la sélection aléatoire sont répétées cinq fois et les valeurs moyennes et les écarts types sont présentés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' détection de pages La Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 et la Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 présentent les résultats obtenus pour la tâche de détection de pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À chaque itération, le modèle courant est appliqué aux images du corpus Horae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les images dont le score de confiance estimé est inférieur à un seuil sont annotées manuellement 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 R É S U LTAT S E T D I S C U S S I O N 115 et ajoutées à l’ensemble d’entraînement de l’itération précédente afin d’entraîner un nou- veau modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme pour les courbes de rejet, ces graphiques montrent que les estimateurs sont capables de détecter les mauvaises prédictions afin d’entraîner des modèles plus perfor- mants, avec seulement une petite quantité de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, les estimateurs sont meilleurs qu’une sélection aléatoire puisqu’avec deux fois moins de données, les modèles pré- sentent des augmentations relatives de 6 % de mAP (+5 points de pourcentage) pour DAP, 7 % (+6 points de pourcentage) pour DOV et presque 3,5 % (+3 points de pourcentage) pour mAP-RFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sur la Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4, nous observons également que la courbe correspondant à mAP-RFR est presque toujours supérieure à celles des autres estimateurs, ce qui indique des modèles plus performants avec moins de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces résultats montrent que l’estimateur mAP-RFR est plus performant que les estimateurs basés sur le dropout de Monte Carlo puisqu’il présente une mAP plus élevée tout en ne nécessitant qu’une seule prédiction pendant l’inférence et moins de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une explication possible à ces résultats, que nous avons déjà formulée précédemment, est que les estimateurs DAP et DOV sont non supervisés : ils n’ont aucune connaissance préalable de ce qu’est une prédiction correcte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au contraire, mAP-RFR est entraîné avec les mAPs réelles calculées sur les données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' détection de lignes de texte La Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 et la Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 montrent les résultats obtenus avec les estimateurs mAP-RFR, DAP et PCE pour l’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous ne montrons pas les résultats de DOV car ils sont équivalents à ceux de DAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, le WER n’est pas rapporté ici puisqu’il est fortement corrélé au CER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’après la Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, il apparaît que la sélection aléatoire donne de bons résultats avec seulement 50 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, ces résultats dépendent fortement des données choisies, ce qui conduit à des performances très variables d’une sélection à l’autre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par conséquent, au vu de cette grande variabilité des résultats, nous pensons qu’il est préférable de se concentrer sur un estimateur plus robuste et moins aléatoire qui peut obtenir des résultats tout aussi satisfaisants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’après la Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, l’estimateur DAP se distingue des autres en obtenant des valeurs d’IoU et de mAP plus faibles que les autres estimateurs mais un CER bien moins élevé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' mAP-RFR ne semble pas ici faire un meilleur estimateur que PCE ou que la sélection aléatoire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Malgré des résultats bien moins bons en termes d’IoU et de mAP, DAP présente de meilleures valeurs de CER et de WCFM par rapport à mAP-RFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, mAP-RFR a été conçu pour estimer la mAP de chaque prédiction et ainsi maximiser la mAP des modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, nous avons montré, en section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, que la maximisation de la mAP ne signifie pas nécessairement l’amélioration de l’entrée pour le reconnaisseur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cette tâche, il serait intéressant de sélectionner les images en fonction d’un score de confiance lié à la reconnaissance de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle de détection s’adapterait pour améliorer directement la reconnaissance du texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 116 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – Résultats des modèles de détection de lignes de texte sur l’ensemble de test du jeu de données Hugin-Munin après apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Itération indique le nombre d’itérations réalisées afin d’obtenir le meilleur modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le nombre d’images annotées est indiqué dans la colonne Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Estimateur Itération Images IoU mAP CER (%) WCFM Référence – 0 0,48 0,21 24,37 0,59 Aléatoire 1 50 0,63 0,45 22,18 0,64 PCE 6 83 0,67 0,46 22,79 0,66 mAP-RFR 9 139 0,64 0,44 22,50 0,66 DAP 6 110 0,63 0,40 20,23 0,68 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – Évolution des performances de détection de lignes de texte (mAP) sur l’ensemble de test du jeu de données Hugin-Munin pendant les itérations d’apprentissage actif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 S T R AT É G I E D’ E N T R A Î N E M E N T : S É L E C T I O N E T A N N O TAT I O N D E S D O N N É E S Dans le cadre de l’apprentissage actif, de nombreuses stratégies d’entraînement peuvent être exploitées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que de nombreuses méthodes aient été proposées dans la littérature, aucune ne semble réellement surpasser les autres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi, dans cette section, nous étudions deux stratégies d’entraînement qui concernent la sélection des données ainsi que leur annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La première est la même que celle utilisée dans les expériences précédentes : les exemples avec les confiances estimées les plus faibles sont annotés manuellement puis ajoutés à l’en- semble d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La seconde stratégie sélectionne les exemples avec les confiances les plus élevées et utilise les prédictions du modèle de détection comme labels pour les entraî- nements suivants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette stratégie de sélection permet de réduire le coût d’annotation ma- nuelle au minimum puisqu’aucune donnée n’est annotée manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous présentons tout d’abord les résultats pour la détection de pages, en section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, puis pour la détection de lignes de texte, en section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour l’ensemble des résultats présentés dans ce qui suit, les modèles sont entraînés dans les mêmes conditions que dans les expériences précédentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 S T R AT É G I E D’ E N T R A Î N E M E N T : S É L E C T I O N E T A N N O TAT I O N D E S D O N N É E S 117 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 – Résultats des modèles de détection de pages sur l’ensemble de test Horae-test-300 après apprentissage actif et pour différentes stratégies de sélection de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Ité- ration indique le nombre d’itérations réalisées afin d’obtenir le meilleur modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La sélec- tion "Faible" correspond à la sélection des images avec les confiances les plus faibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La sélection "Élevée" correspond à la sélection des images avec les confiances les plus élevées où leurs prédictions sont directement utilisées comme labels d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les colonnes "Manuelle" et "Auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='" indiquent respectivement les nombres d’images d’entraînement avec annotations manuelles et automatiques permettant d’obtenir le meilleur modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Estimateur Sélection Itération Images IoU mAP Manuelle Auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Référence – – – – 0,90 0,60 Faible 8 107 – 0,94 0,89 mAP-RFR Élevée 9 – 444 0,90 0,84 Faible 9 129 – 0,94 0,91 DAP Élevée 8 – 475 0,90 0,72 Faible 9 168 – 0,95 0,92 DOV Élevée 3 – 163 0,90 0,64 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 – Évolution des performances de détection de pages (mAP) sur l’ensemble de test Horae- test-300 pendant les itérations d’apprentissage actif pour différentes stratégies de sélec- tion de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes "Manuelle" correspondent à la sélection des exemples avec les confiances les plus faibles et une annotation manuelle de ces exemples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes "Automatique" correspondent à la sélection des exemples avec les confiances les plus élevées et l’utilisation des prédictions comme labels d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 détection de pages Les Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 et Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6 présentent les résultats obtenus pour les deux stratégies de sélection pour la tâche de détection de pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes et valeurs correspondant à la sélection basée sur les faibles confiances sont les mêmes que celles présentées en section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’après la Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='6, l’utilisation des prédictions comme labels d’entraînement pour l’estimateur DOV ne permet pas réellement d’améliorer le modèle de détection par rapport au modèle de référence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au contraire, pour les estimateurs DAP et mAP-RFR, l’utilisation des prédictions permet une importante amélioration des performances par rapport au modèle de référence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, le modèle mAP-RFR permet une amélioration de 40 % de mAP (+24 118 E S T I M AT I O N D E L A C O N F I A N C E D E S P R É D I C T I O N S Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 – Résultats des modèles de détection de lignes de texte sur l’ensemble de test du jeu de données Hugin-Munin après apprentissage actif et pour différentes stratégies de sélection de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Iteration indique le nombre d’itérations réalisées afin d’obtenir le meilleur modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La sélection "Faible" correspond à la sélection des images avec les confiances les plus faibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La sélection "Élevée" correspond à la sélection des images avec les confiances les plus élevées où leurs prédictions sont directement utilisées comme labels d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les colonnes "Manuelle" et "Auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='" indiquent respectivement les nombres d’images d’entraînement avec annotations manuelles et automatiques permettant d’obtenir le meilleur modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Estimateur Sélection Iteration Images IoU mAP CER (%) WCFM Manuelle Auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Référence – – – – 0,48 0,21 24,37 0,59 Faible 9 139 – 0,64 0,44 22,50 0,66 mAP-RFR Élevée 4 – 54 0,53 0,28 22,89 0,62 Faible 6 110 – 0,63 0,40 20,23 0,68 DAP Élevée 1 – 38 0,51 0,26 21,98 0,63 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 – Évolution des performances de détection de lignes de texte (mAP) sur l’ensemble de test du jeu de données Hugin-Munin pendant les itérations d’apprentissage actif pour différentes stratégies de sélection de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes "Manuelle" correspondent à la sélection des exemples avec les confiances les plus faibles et une annotation manuelle de ces exemples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes "Automatique" correspondent à la sélection des exemples avec les confiances les plus élevées et l’utilisation des prédictions comme labels d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' points de pourcentage) et le modèle DAP de 20 % (+12 points de pourcentage) par rapport au modèle de référence, sans aucune donnée annotée manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle obtenu avec l’estimateur DAP et les données annotées automatiquement est tout de même bien moins performant que celui obtenu avec les données annotées manuelle- ment (-21 % de mAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour l’estimateur mAP-RFR, l’écart est moins important puisque les performances sont dégradées de seulement 5,5 % de mAP en passant des données annotées manuellement aux labels automatiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, nous constatons que l’estimateur mAP-RFR est le plus robuste pour cette tâche puisqu’il permet de détecter de manière fiable les bonnes ainsi que les mauvaises prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet estimateur permet d’obtenir l’amélioration de perfor- mance la plus intéressante sans annotation manuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 C O N C L U S I O N 119 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 détection de lignes de texte Les Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 et Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7 présentent les résultats obtenus pour les deux stratégies de sélection pour la tâche de détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les courbes et valeurs correspondant à la sélection basée sur les faibles confiances sont les mêmes que celles présentées en section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’après la Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='7, l’utilisation des prédictions comme labels d’entraînement mène à une légère amélioration des performances par rapport au modèle de référence (-6 % de CER pour mAP-RFR et -11 % de CER pour DAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceci permet de valider l’utilité des estimateurs mAP-RFR et DAP dans la sélection des mauvais autant que des bons exemples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme pour la détection de pages, les modèles obtenus avec les données annotées automa- tiquement sont moins performants que ceux obtenus avec les données annotées manuellement (+1,7 % de CER pour mAP-RFR et +8,7 % de CER pour DAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 C O N C L U S I O N Dans ce chapitre, nous avons comparé quatre estimateurs de confiance pour les modèles de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons montré que, dans un contexte d’apprentissage actif, ces estimateurs peuvent être utilisés pour entraîner des modèles atteignant des performances élevées pour la détection d’objets en termes d’IoU et de mAP tout en ne nécessitant qu’un faible effort d’annotation manuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Lorsque les métriques optimisées sont étroitement liées à l’objectif, comme pour la mAP et la détection de pages, nous avons montré que l’estimateur mAP-RFR permet d’obtenir de meilleures performances de détection que celles basées sur le dropout de Monte Carlo, tout en ayant un coût de calcul réduit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, cet estimateur est supervisé et doit être entraîné, ce qui n’est pas le cas pour DAP, DOV et PCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas d’une adaptation à de nouvelles données, il est donc avantageux, dans un premier temps, d’utiliser l’estimateur DAP basé sur le dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Si les résultats n’atteignent pas les performances attendues, il semble alors plus intéressant d’utiliser un estimateur entraîné tel que mAP-RFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’autre part, lorsque les métriques sont moins étroitement liées à l’objectif, comme pour la détection des lignes de texte, les méthodes basées sur le dropout sont plus compétitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' À l’avenir, nous envisageons d’adapter l’estimateur mAP-RFR afin qu’il estime la confiance au niveau de l’objet directement de façon à ne plus rejeter les images mais les objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela permettrait de savoir exactement quels objets posent un problème et de les corriger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, il serait intéressant de créer automatiquement des vecteurs de description d’objets à travers des représentations apprises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, nous avons montré que l’utilisation de métriques orientées vers la tâche finale permet d’évaluer l’impact des modèles de détection sur les résultats finaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il semblerait donc intéressant de sélectionner les images ou les objets en se basant sur les résultats de reconnaissance de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette optique, nous prévoyons de mettre en place un nouvel estimateur qui reflète les résultats de la reconnaissance de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7 D É T E C T I O N S É Q U E N T I E L L E D ’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Les systèmes à base de Transformers proposés récemment, et détaillés dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11, obtiennent désormais les meilleures performances de l’état de l’art tant sur des tâches de traitement de la langue que des tâches de classification d’images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un de leurs avantages réside dans leur capacité à modéliser et à générer des séquences et même des objets structurés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il semble désormais possible de prédire automatiquement la structure complète d’une image de document, avec l’ensemble de ses éléments organisés de manière hiérarchique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, ces systèmes sont capables de prédire séquentiellement les coordonnées des objets à détecter (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022), sans avoir à passer par une prédiction pixel à pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien qu’il n’y ait pas, à notre connaissance, de travaux proposés dans la littérature afin de réaliser une telle tâche, une prédiction directe de coordonnées présente de nombreux avantages comparée à une prédiction standard niveau pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi, dans ce chapitre, nous avons choisi d’explorer les modèles Transformers pour construire un nouveau modèle de détection séquentielle d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un premier point ayant motivé nos travaux dans ce sens est lié à la capacité d’un tel système à passer outre les problèmes liés aux boîtes englobantes qui se touchent et se superposent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, le modèle n’est pas appris avec des images de labels mais directement avec les coordonnées des éléments à détecter, telles que les boîtes englobantes ou les lignes de base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Similairement aux approches par régression de boîtes englobantes, il devient possible de détecter plusieurs objets d’une même classe au même endroit sur l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un autre avantage de cette approche tient au fait qu’elle permet d’apprendre un ordre de lecture implicitement représenté par la séquentialité du processus de détection des éléments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle est, en effet, entraîné à détecter les éléments dans l’ordre imposé par la séquence des objets représentés dans la vérité terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette séquentialité de la vérité terrain définit donc un ordre de détection, et donc de lecture des objets présents dans l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Resitué dans le contexte de la reconnaissance de documents, il devient possible d’apprendre à prédire les lignes de texte dans l’ordre de lecture du texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, comme énoncé plus tôt, cette approche permet d’avoir une sortie structurée des résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Si nous imaginons un problème à deux classes telles que les paragraphes et les lignes de texte, il est possible d’apprendre un modèle qui détecte le début d’un paragraphe, toutes les lignes qu’il contient puis la fin de ce paragraphe avant de passer au suivant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous pouvons donc obtenir directement une détection hiérarchique des éléments sur une image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 121 122 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S La plupart des systèmes proposés traitant la détection d’objets prédisent un masque de probabilités à la résolution de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que cette tâche de détection puisse être traitée à l’aide de Vision Transformers (voir le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='14), qui utilisent un encodeur Transformer suivi d’un décodeur CNN, elles ne bénéficient pas de l’avantage principal des Transformers qui réside en leur capacité à prédire des éléments de manière séquentielle, capa- cité induite par le décodeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De même, dans le domaine de la vision, la plupart des travaux proposés dans la littérature ont exploré les architectures Transformers pour constituer de nou- veaux extracteurs de caractéristiques, des encodeurs Transformers, et ainsi tenter d’améliorer les architectures convolutives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’autres travaux prédisent un nombre fixe d’objets (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020) ou utilisent la sortie du décodeur comme entrée d’un CNN afin d’avoir une prédiction dense (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce domaine, la séquentialité du processus de décision n’est pas non plus exploitée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les architectures Transformers, et plus particulièrement les décodeurs Transformers, fonc- tionnent selon un nouveau paradigme qui traite un élément en entrée séquentiellement, au rythme d’une séquence d’attention visuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela nécessite donc de repenser le type de sorties attendues qui doivent nécessairement être structurées sous forme d’une séquence d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il semble complexe de réaliser une prédiction pixel à pixel de manière séquentielle puisqu’elle induirait des temps et coût de traitement très élevés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, l’application de ce para- digme pour résoudre un problème d’analyse de document est relativement directe puisque, dans la plupart des applications, les sorties du modèle de détection et de reconnaissance ont besoin d’être organisées dans l’ordre naturel de lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La tâche de détection doit donc être reformulée afin de profiter pleinement de la capacité de prédiction séquentielle de ces nouvelles architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi, nous présentons, dans la section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1, une étude et comparaison de différentes modélisations du problème de détection d’objets permettant une prédiction séquentielle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme nous venons de l’évoquer, très peu de systèmes ont été proposés dans la littérature permettant de prédire séquentiellement les objets présents dans des images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le seul modèle réalisant une telle tâche est Pix2Seq (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022), détaillé dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='15, appliqué aux images de scènes naturelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système possédant un grand nombre de paramètres et nécessitant un pré-entraînement sur des milliers d’images, il n’est pas directement applicable à nos jeux de données réduits d’images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, inspiré par Pix2Seq, nous nous sommes intéressés à la mise en place d’un système permettant de prédire séquentiellement les objets tout en possédant un nombre réduit de paramètres afin d’être entraîné sur des jeux de données réduits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système est détaillé dans la section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 M O D É L I S AT I O N D E L A TÂ C H E D E D É T E C T I O N Dans cette section, nous présentons et comparons différentes modélisations possibles de la tâche de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un premier temps, nous comparons plusieurs modélisations de la position et de la forme des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous comparons ensuite plusieurs stratégies de prédiction des coordonnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, nous présentons la modélisation des classes des objets que nous avons retenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 M O D É L I S AT I O N D E L A TÂ C H E D E D É T E C T I O N 123 (x1, y1) (x2, y2) Rectangle englobant (x1, y1) Point d’origine (x1, y1) h Position d´ebut + hauteur (x1, y1) (x2, y2) (xN, yN) Ligne de base (x1, y1) (x2, y2) (xN, yN) Polygone englobant (x1, y1) (x2, y2) (xN, yN) h Ligne de base + hauteur Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Représentation de différentes modélisations de la position et de la forme des objets à détecter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Exemple pour la détection d’une ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 modélisation de la position et de la forme des objets Afin de réaliser une prédiction séquentielle des éléments présents sur une image de document, il est nécessaire de passer d’une prédiction pixel à une prédiction de plus haut niveau, au niveau de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, différentes formulations du problème de détection d’objets ont été proposées dans la littérature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous les présentons, dans le cadre d’une détection de lignes de texte, sur la Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 et détaillons leurs caractéristiques dans la Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le domaine de la vision, la grande majorité des systèmes tels que les modèles R- CNN (Girshick, 2015 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2014 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) et YOLO (Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2016 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2017 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2018) définissent les objets à détecter par leur rectangle englobant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est également le cas de Pix2Seq (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022) qui prédit la séquence suivante : ordonnée du point supérieur gauche, abscisse du point supérieur gauche, ordonnée du point inférieur droit, abscisse du point inférieur droit et classe de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette détection permet une extraction directe de l’objet mais n’est pas correctement applicable à des éléments non rectangulaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Lors de la compétition ANDAR-TL de détection de lignes de texte (Murdock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015), la tâche de détection correspond à l’identification des points d’origine des lignes de texte, à savoir la ligne de base du premier caractère d’une ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' D’un autre côté, dans Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2017), les auteurs proposent une localisation des lignes de texte basée sur des régressions dans lesquelles seules les positions du début des lignes de texte et leurs hauteurs sont prédites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le reconnaisseur de texte est alors chargé de reconnaître le texte de 124 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 – Tableau récapitulatif de différentes modélisations de la position et forme des objets à détecter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne Prédiction indique les valeurs à prédire pour un objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne "Extraction directe" indique si l’objet peut directement être extrait en sortie du détecteur ou si des traitements supplémentaires sont nécessaires tels que l’estimation de la largeur et/ou de la hauteur de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La colonne "Optimisation mémoire" indique si la quantité de mémoire nécessaire pour prédire un objet est importante ou non, cette quantité étant directement corrélée au nombre de valeurs à prédire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La dernière colonne indique si la détection est applicable à des objets non rectangulaires ainsi qu’à des lignes inclinées ou incurvées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Modélisation Prédiction Extraction Optimisation Objets non- directe mémoire rectangulaires Rectangle englobant (x1, y1, x2, y2) Pix2Seq (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022)) → 4 valeurs ✓ ✓ ✗ Point d’origine (x1, y1) (Murdock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2015) → 2 valeurs ✗ ✓ ✗ Position du début + hauteur (x1, y1, h) (Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017) → 3 valeurs ✗ ✓ ✗ Ligne de base (x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='xN, yN) (Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) → N valeurs ✗ ✗ ✓ (x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='xN, yN) Polygone englobant → N valeurs ✓ ✗ ✓ (x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='xN, yN, h) Ligne de base + hauteur → N+1 valeurs ✗ ✗ ✓ la ligne et de s’arrêter lorsqu’il n’y a plus de texte à reconnaître.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces propositions permettent d’envisager une détection optimisée des éléments puisque, pour chaque objet, seules deux ou trois valeurs sont à prédire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, elles ne permettent pas une détection complète de l’objet puisque la largeur est inconnue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il serait donc nécessaire d’avoir des traitements supplémentaires afin d’extraire les objets de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sans cela, il serait impossible d’appliquer un reconnaisseur de texte standard sur les lignes de texte par exemple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, la détection basée sur les lignes de base (Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2017 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Diem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2019) ou les polygones englobants présente l’avantage de localiser précisément les contours d’objets rectangulaires ou non, tels que des lignes inclinées et incurvées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, ces propositions sont très coûteuses en mémoire puisque le système doit prédire un nombre de points inconnu à l’avance, et qui peut-être très variable d’un objet à l’autre, en fonction de la taille et de la forme des objets à localiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette représentation pose également un problème de paramétrage du Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, dans un Transformer, la taille maximale de la séquence pouvant être générée pour une image est fixée durant la phase d’entraînement afin de réduire la mémoire utilisée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Durant l’inférence, il est donc impossible de prédire plus de valeurs que cette limite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que celle-ci puisse être fixée à plusieurs milliers de valeurs, il est toujours possible de rencontrer un document avec un très grand nombre d’objets, menant à une séquence plus longue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans le cas d’une détection de rectangles englobants, cette limite est facile à fixer puisque seules quatre valeurs sont à prédire pour chaque objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, le problème est réduit à 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 M O D É L I S AT I O N D E L A TÂ C H E D E D É T E C T I O N 125 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Stratégies de prédiction séquentielle des rectangles englobants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour un rectangle donné i de coordonnées (xi 0, yi 0, xi 1, yi 1), à chaque pas t, une coordonnée unique, un point ou le rectangle complet peut être prédit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Séquence t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Coordonnée x0 0 y0 0 x0 1 y0 1 x1 0 y1 0 x1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Point (x0 0, y0 0) (x0 1, y0 1) (x1 0, y1 0) (x1 1, y1 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Rectangle (x0 0, y0 0, x0 1, y0 1) (x1 0, y1 0, x1 1, y1 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' quelques dizaines de valeurs à prédire par image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le problème se complexifie lorsque nous souhaitons prédire des polygones plus précis puisque nous ignorons à l’avance combien de valeurs sont nécessaires pour prédire chaque objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il serait possible de simplifier les polygones englobants afin de fixer le nombre de coordonnées les définissant, cependant, cela mènerait à un traitement supplémentaire et à une perte de précision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, de nombreuses questions en découlent telles que le nombre de points à utiliser pour décrire un polygone an- noté, leurs espacements, l’évaluation des points obtenus pour le calcul de la fonction de perte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour toutes ces raisons, nous pensons que la détection des rectangles englobants par la prédiction du point supérieur gauche et du point inférieur droit, semblable à Pix2Seq, repré- sente un bon compromis entre performance et précision de la localisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette formulation est assez simple et rapide à appliquer, et peut permettre à un système d’apprendre malgré la quantité relativement faible de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, plus le système doit prédire de points, plus il sera en difficultés et nécessitera un grand nombre de données d’apprentissage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 stratégie de prédiction des coordonnées : singleton vs n-uplet La modélisation de la tâche de détection explicitement définie, il est maintenant nécessaire de choisir la stratégie de prédiction des rectangles englobants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, le système devra être capable de prédire séquentiellement les coordonnées des rectangles englobants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, la Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 présente trois stratégies différentes afin de réaliser cette tâche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, pour un rectangle donné i de coordonnées (xi 0, yi 0, xi 1, yi 1) avec (xi 0, yi 0) les coordonnées du point supérieur gauche et (xi 1, yi 1) les coordonnées du point inférieur droit, il est possible de prédire, à chaque pas de la séquence : — Un singleton correspondant à une coordonnée d’un des deux points du rectangle ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Un couple de valeurs correspondant à un des deux points du rectangle ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Un quadruplet correspondant aux coordonnées du rectangle complet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La première stratégie, qui consiste à prédire un singleton à chaque pas dans la séquence, permet d’avoir un modèle possédant quelques paramètres en moins par rapport à la prédic- tion de couples qui elle-même nécessite moins de paramètres que la prédiction de quadruplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, la dernière couche du modèle produisant les coordonnées finales sera différente d’une stratégie à l’autre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, la prédiction de singletons requiert davantage d’itérations puis- 126 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S qu’elle nécessite deux fois plus d’itérations que la prédiction de couples, elle-même nécessitant deux fois plus d’itérations que la prédiction du quadruplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Pix2Seq, les auteurs ont choisi de traiter cette tâche de telle sorte qu’à chaque pas, une seule valeur de coordonnée est prédite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, quatre prédictions sont nécessaires afin de prédire un objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la suite de ce chapitre, nous décidons d’adopter la même stratégie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 stratégie de prédiction des coordonnées : classification vs régres- sion Une autre stratégie à étudier dans la conception du modèle concerne le type de prédiction souhaité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, la prédiction de coordonnées peut être réalisée de deux manières.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La première consiste à réaliser une régression où le but est de prédire une coordonnée de la boîte sur l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La seconde consiste à considérer chaque pixel de l’image d’entrée comme étant une classe distincte et à réaliser une classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le but est alors de maximiser les probabilités de la classe correspondant à la coordonnée de la boîte dans l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’avantage de la régression est qu’elle nécessite légèrement moins de paramètres que la clas- sification puisqu’une seule valeur est produite par le modèle, directement considérée comme la coordonnée finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, la dernière couche du modèle ne produira qu’une seule valeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, les valeurs prédites ne sont pas bornées, il est donc possible que le modèle pré- dise des valeurs en dehors de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour pallier ce problème, les coordonnées peuvent être normalisées, ce qui permet également d’obtenir une cohérence des labels entre les images pouvant être de tailles variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La classification est quant à elle plus simple à mettre en œuvre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Pix2Seq, les auteurs choisissent de traiter les images ainsi, en redimensionnant les images dans une taille fixe et en considérant une classe pour chaque valeur possible en abscisse et en ordonnée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans un premier temps, nous avons choisi d’utiliser cette même stratégie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022), les auteurs considèrent les classes permettant de représenter les positions des objets en abscisse et en ordonnée comme appartenant à un "vocabulaire".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela leur permet de distinguer les positions des objets des "classes", utilisées pour représenter les classes des objets à détecter telles que, dans leur application, les classes "chaise" ou "voiture".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous utilisons ces mêmes termes dans la suite de ce chapitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans Pix2Seq, les auteurs utilisent un vocabulaire partagé pour les deux axes et pour toutes les classes de position, la taille du vocabulaire est donc égale au nombre maximal de pixels sur les deux axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour une image de taille 600×600 pixels, le vocabulaire a donc une taille de 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De la même manière, nous disposons, dans nos expériences, d’un vocabulaire V de taille TV = max(H, W) avec H et W respectivement les hauteur et largeur de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 stratégie de prédiction de la classe des objets En plus des coordonnées des objets présents dans les images, il est nécessaire de prédire leurs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, dans les chapitres précédents, nous avons principalement abordé la tâche de détection de lignes de texte uniquement, ainsi une seule classe est à prédire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 M O D É L I S AT I O N D E L A TÂ C H E D E D É T E C T I O N 127 Ordre de pr´ediction : y0 x0 cp 0 y0 x0 cl 0 y1 x1 cl 1 y0 x0 cl 0 y1 x1 cl 1 y0 x0 cl 0 y1 x1 cl 1 y1 x1 cp 1 eos Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 – Exemple de séquence à deux classes : paragraphe et ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’ordre de prédiction préserve la hiérarchie des objets : point supérieur gauche du paragraphe, point supérieur gauche de la première ligne de texte, point inférieur droit de la première ligne de texte, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', point inférieur droit du paragraphe, fin de séquence (eos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' certaines tâches plus complexes considèrent davantage de classes d’objets qui doivent être prédites par le modèle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces classes peuvent être représentées de différentes manières.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, dans Pix2Seq, le modèle prédit la classe de l’objet après les quatre coordonnées du rectangle englobant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque objet est donc défini par cinq prédictions successives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette représentation ne permet cependant pas de représenter la hiérarchie des objets présents sur une image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Afin de représenter au mieux ces informations, nous proposons de représenter un objet par ses deux points supérieur gauche et inférieur droit avec, pour chacun de ces points, une classe indiquant s’il s’agit du premier ou du second point de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette représentation permet au modèle d’apprendre, en plus de l’ordre de lecture, la hiérarchie des objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 présente un exemple de séquence construite pour deux classes d’objets : paragraphe et ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque ligne représente un point avec son ordonnée, son abscisse et la classe correspondante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour chaque classe d’objet, deux classes sont définies : une classe indiquant le début de l’objet et une classe indiquant la fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, un objet paragraphe est défini par deux points, le premier avec la classe cp 0 et le second avec la classe cp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De la même manière, les lignes de texte sont définies par les classes cl 0 et cl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette représentation permet également de reconstruire les boîtes englobantes de manière plus fiable, même dans le cas d’une prédiction manquante ou supplémentaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, si le modèle prédit deux points de début de ligne à la suite, lors de la reconstruction des objets, un des deux points devra être mis de côté.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Sans ces indicateurs de début et de fin d’objet, il serait impossible de détecter ce phénomène.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le processus étant séquentiel, l’ensemble des boîtes reconstruites après le point erroné seraient fausses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 0 Sentence Database M04-251 100 ThmauSem,cndRe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='iunds 200 - locofal 300 400 - 009 600 002 0 100 200 300 400 500128 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S En conclusion, nous avons choisi de représenter un objet par les deux séquences "y0, x0, c0" et "y1, x1, c1" avec : — y0 et y1, respectivement les ordonnées des points supérieur gauche et inférieur droit du rectangle englobant de l’objet ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — x0 et x1, respectivement les abscisses des points supérieur gauche et inférieur droit du rectangle englobant de l’objet ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — c0 et c1, les jetons de début et de fin de la classe de l’objet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 A R C H I T E C T U R E D U S Y S T È M E P R O P O S É : D O C2 S E Q Dans cette section, nous présentons le modèle que nous avons développé, appelé Doc2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous détaillons l’architecture ainsi que les choix que nous avons faits lors de sa conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Très peu de modèles ont été proposés pour la détection séquentielle d’objets dans les images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Seul Pix2Seq (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2022) a été proposé, appliqué aux images de scènes naturelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’un modèle comportant un très grand nombre de paramètres (341 millions pour le meilleur modèle) qui montre des résultats satisfaisants lorsqu’il est pré-entraîné sur des milliers d’images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Or, nous ne disposons pas d’une telle quantité d’images annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est pourquoi, il est nécessaire que notre système comporte moins de paramètres afin de pouvoir être entraîné sur les jeux de données d’images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est dans cet objectif que nous avons conçu Doc2Seq, dont l’architecture est présentée en Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il s’agit d’un modèle hybride encodeur-décodeur où l’encodeur extrait les caractéristiques importantes de l’image d’entrée et le décodeur prédit séquentiellement les éléments à partir de l’image encodée et des prédictions précédentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’encodeur génère une matrice de caractéristiques 2D de l’image d’entrée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Un encodage positionnel 2D est additionné à cette matrice afin de conserver l’information spatiale, avant d’être aplani en une séquence 1D de caractéristiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme pour un FCN, cette représentation est calculée une seule fois et sert d’entrée au décodeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le décodeur suit un processus récurrent : étant donné l’image encodée et les éléments prédits précédemment ( ˆy0, ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', ˆyt−1), il produit les caractéristiques de l’élément suivant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, la branche de classification produit des probabilités à partir de la sortie du décodeur et l’élément prédit ˆyt est celui qui a la plus forte probabilité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chacun de ces composants est détaillé dans les paragraphes suivants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 encodeur doc-ufcn L’encodeur de Doc2Seq est identique à l’encodeur de Doc-UFCN présenté dans le chapitre 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est donc composé de quatre blocs dilatés comportant chacun cinq convolutions dilatées consécutives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque bloc est suivi d’une couche de max-pooling, sauf le dernier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons choisi d’utiliser l’encodeur de Doc-UFCN car, d’après les expériences présentées précédemment, il a démontré de bonnes capacités d’extraction de caractéristiques sur les images tout en possédant un nombre réduit de paramètres, ce qui nous permet d’entraîner le 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 A R C H I T E C T U R E D U S Y S T È M E P R O P O S É : D O C2 S E Q 129 W H Doc-UFCN Encoder 8f W 8 H 8 + 8f W 8 H 8 2D positional encoding H 8 × W 8 8f Image features Flatten + transpose ˆy0 ˆy1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' ˆyt−1 t − 1 8f + t − 1 8f 1D positional encoding t − 1 8f Embedding Masked Self-Attention Add & Norm Multi-Head Attention Add & Norm Feed Forward Add & Norm 4× 8f C ˆyt Linear Argmax Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Schéma de l’architecture du modèle Doc2Seq avec respectivement H et W les hauteur et largeur de l’image d’entrée, f le nombre de cartes de caractéristiques et ˆyi les prédictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' système complet sur des jeux de données restreints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous avons opté pour un FCN comme encodeur car ces modèles peuvent traiter des entrées de tailles variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Certains systèmes tels que les Vision Transformers remplacent les encodeurs convolutifs par des encodeurs Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces systèmes sont appliqués sur des patchs d’image à la résolution originale projetés dans la dimension dmodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que cet encodeur ait montré de légèrement meilleures performances dans Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2022) par rapport à un encodeur CNN, il augmente significativement le nombre de paramètres et nécessite donc une plus grande quantité de données d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans nos expériences, l’encodeur prend en entrée une image de document de taille (H × W × 3) avec H la hauteur, W la largeur et 3 le nombre de canaux de l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il produit une matrice de caractéristiques de taille ( H 8 × W 8 × 8f) avec f = 32 comme pour Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 130 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 encodage positionnel 2d Une fois l’image encodée, sa matrice de caractéristiques est additionnée à une matrice de codage positionnel 2D afin de garder en mémoire à quelle partie de l’image chaque pixel correspond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le Transformer original a été conçu pour traiter des séquences en entrée en 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour lui donner des représentations 2D en entrée, il suffit de transformer les cartes de caractéristiques en les sérialisant ligne par ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, il est important d’associer à ces représentations un encodage positionnel cohérent avec l’information originale, c’est-à-dire un encodage 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est ce que nous avons réalisé comme proposé par Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, il s’agit toujours d’un codage fixe basé sur les fonctions cosinus et sinus, mais, au lieu de coder une position 1D sur tous les canaux, la première moitié est dédiée à l’encodage positionnel vertical et la seconde à l’encodage positionnel horizontal (voir équations 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' PE(posx, posy, 2i) = sin(wi · posy) ∀i ∈ � 0, dmodel 4 � PE(posx, posy, 2i + 1) = cos(wi · posy) ∀i ∈ � 0, dmodel 4 � PE(posx, posy, dmodel 2 + 2i) = sin(wi · posx) ∀i ∈ � 0, dmodel 4 � PE(posx, posy, dmodel 2 + 2i + 1) = cos(wi · posx) ∀i ∈ � 0, dmodel 4 � (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1) avec : wi = 1 10000 2i dmodel La position de l’élément dans la séquence 2D est donnée par posx et posy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' dmodel correspond à la dimension d’encodage de l’image d’entrée et des éléments au sein du Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans notre cas, dmodel = 8f = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La matrice de caractéristiques ainsi enrichie de la position des éléments est ensuite aplanie afin de pouvoir être utilisée lors du décodage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 décodeur transformer Pour le décodeur, nous utilisons un Transformer standard puisqu’il permet la prédiction de séquences de longueurs variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celui-ci est constitué d’un empilement de quatre couches de décodeur Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque couche suit une architecture standard avec un mécanisme d’auto-attention et un mécanisme d’attention croisée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’auto-attention modélise les dépendances entre les éléments de la séquence prédite, contrairement à l’attention croisée, utilisée pour extraire des informations visuelles de l’encodeur, sur la base des prédictions précédentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En d’autres termes, étant donné les prédictions précédentes, elle indique où le modèle doit regarder pour prédire le prochain élément.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 D É TA I L S D’ I M P L É M E N TAT I O N E T S T R AT É G I E S D’ E N T R A Î N E M E N T 131 Le décodeur suit donc un processus récurrent où à chaque itération, il prend en entrée les caractéristiques visuelles aplanies et les éléments prédits précédemment ( ˆy0, ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', ˆyt−1) et produit un vecteur de caractéristiques pour la prédiction de l’élément au pas de temps t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque prédiction précédente est encodée dans un vecteur de taille 8f grâce à une couche d’embedding apprise, puis les vecteurs sont concaténés afin de former une matrice de taille (t − 1 × 8f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, les caractéristiques visuelles et les vecteurs des prédictions précédentes ont la même dimension dmodel = 8f = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme pour un Transformer standard, la matrice des embeddings est additionnée à une matrice de codage positionnel 1D de même taille qui permet de savoir où se trouve cette prédiction dans la séquence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cet encodage positionnel 1D est détaillé dans le Focus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 branche de classification La branche de classification génère des probabilités à partir de la sortie du décodeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elle est composée d’une couche linéaire qui permet de modifier la dimension de la sortie de dmodel = 256 à TC = TV + 2 × C + 2, C étant le nombre de classes considérées dans l’expérience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Deux sorties supplémentaires sont ajoutées afin de représenter les jetons de fin de séquence (EOS) et de padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le jeton de padding est utilisé afin de permettre un entraînement par batchs dans lesquels les séquences doivent être de même longueur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette couche linéaire est suivie d’une fonction argmax qui assigne à ˆyt la position ou la classe d’objet pour laquelle la probabilité est maximale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 D É TA I L S D’ I M P L É M E N TAT I O N E T S T R AT É G I E S D’ E N T R A Î N E M E N T Dans cette section, nous donnons des détails techniques sur l’implémentation utilisée lors de nos expériences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doc2Seq est implémenté à l’aide du framework PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est entraîné avec un learning rate initial de 5e − 5, l’optimiseur Adam et la fonction de coût d’entropie croisée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les poids sont initialisés grâce à l’initialisation Glorot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous utilisons le teacher forcing, qui est une stratégie d’apprentissage de modèle qui utilise la vérité terrain comme entrée au lieu de la sortie du modèle de l’itération précédente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela permet de paralléliser les calculs en prédisant en parallèle tous les éléments de la séquence de sortie, et donc de réduire le temps d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au total, le modèle comporte 6,6 millions de paramètres répartis comme suit : — 3,5 millions venant de l’encodeur Doc-UFCN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — 0,2 million pour la couche d’embedding ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — 2,6 millions venant du décodeur Transformer ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — 0,2 million pour la branche de classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces valeurs sont données dans un contexte dans lequel une seule classe d’objets est considérée (C = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les nombres de paramètres venant de la couche d’embedding et de la branche de classification varient légèrement en fonction de ce nombre de classes C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 132 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 taille des images en entrée Puisque nous utilisons l’encodeur du modèle Doc-UFCN, nous avons décidé d’utiliser la taille d’image en entrée ayant obtenu les meilleurs résultats dans les expériences présentées dans les chapitres précédents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, les images d’entrée sont redimensionnées en images plus petites telles que la plus grande dimension de l’image soit égale à 768 pixels tout en conservant le ratio de l’image originale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les coordonnées des objets présents sur les images sont également mises à l’échelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, nous disposons d’un vocabulaire de taille TV = 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 augmentation de données Durant l’entraînement des modèles, nous utilisons une stratégie d’augmentation des don- nées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Tout d’abord, des augmentations sont appliquées à l’image d’entrée telles qu’un ajout de bruit et de flou gaussien, un changement de luminosité, une inversion des canaux de cou- leur ou une mise en niveaux de gris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, des transformations linéaires, telles que des translations et des rotations, sont appliquées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, nous augmentons les séquences d’entrée pendant l’apprentissage pour inclure des jetons bruités.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela améliore la robustesse du modèle contre les prédictions bruitées et dupli- quées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les séquences sont augmentées de trois manières : — Ajout de bruit sur les coordonnées des boîtes englobantes (translation et redimension- nement aléatoires avec une probabilité de 0,3) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Suppression aléatoire de 20 % des boîtes ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Inversion des jetons de classes de début et de fin avec une probabilité de 0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 décodeur transformer Nous utilisons quatre couches de décodage avec la dimension dmodel = 8f = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Chaque couche de décodage possède quatre têtes d’attention et utilise une activation ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme les différentes images comportent souvent un nombre différent d’objets, les sé- quences générées auront des longueurs différentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour indiquer la fin d’une séquence, nous incorporons donc un jeton de fin de séquence (EOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, le processus de prédiction se termine lorsque le jeton EOS est prédit ou après un nombre prédéfini de valeurs prédites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 choix du meilleur modèle Dans le cadre d’une application de reconnaissance de document, l’objectif principal est d’obtenir le texte contenu dans celui-ci ainsi que sa position sur l’image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme montré dans le chapitre 5, il est nécessaire d’évaluer les modèles de détection de lignes de texte grâce aux métriques de reconnaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, nous pouvons évaluer l’impact des résultats de détection sur les résultats finaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans nos expériences, nous avons souhaité poursuivre dans cette direction en intégrant un reconnaisseur pré-entraîné dans les processus de sélection des meilleurs modèles et d’évalua- 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S E T R É S U LTAT S 133 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 – Statistiques du jeu de données IAM utilisé pour la détection de lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Jeu de données Images Lignes train valid test train valid test IAM Marti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2002) 747 220 232 6 482 1 926 1 965 tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, chaque expérience impliquant une détection de lignes de texte intègre un modèle de reconnaissance de texte niveau ligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, un modèle de reconnaissance est tout d’abord entraîné sur les lignes transcrites provenant du même jeu de données que celui consi- déré dans l’expérience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ensuite, à partir de la 500e époque et toutes les cinq époques, le modèle de reconnaissance est appliqué à l’ensemble des lignes prédites sur l’ensemble de vali- dation et un CER@page est calculé (voir algorithme 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle final est celui obtenant le CER le plus bas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette stratégie de sélection du meilleur modèle est comparé à une sélection standard basée sur la fonction de coût dans la section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, ce même modèle de reconnaissance est appliqué durant la phase d’évaluation afin d’obtenir le CER@page sur l’ensemble de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S E T R É S U LTAT S Nous décrivons, dans cette section, les résultats préliminaires obtenus avec Doc2Seq pour la détection de lignes de texte sur le jeu de données IAM (Marti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 jeu de données La Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 présente les statistiques du jeu de données issu de la base IAM utilisé pour l’entraînement et l’évaluation du modèle Doc2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons choisi ce jeu car il est annoté au niveau ligne et nous disposons des transcriptions pour chaque ligne de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, il s’agit d’un jeu de données assez simple, annoté en rectangles englobants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Durant l’entraînement, les lignes de texte sont ordonnées de haut en bas afin que le modèle apprenne cet ordre de lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 entraînement des modèles de détection Le modèle est entraîné avec des mini-batchs de taille 12 pour réduire le temps d’apprentis- sage sur un maximum de 1500 époques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, dans un processus d’entraînement standard, le meilleur modèle est choisi comme étant celui obtenant les meilleures performances sur l’en- semble de validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est choisi selon les valeurs de la fonction de coût ou d’une métrique directement liée à la tâche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans nos expériences, nous choisissons le meilleur modèle comme étant celui obtenant le plus faible CER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous comparons l’impact de ce choix par rapport à une sélection standard basée sur la valeur de la fonction de coût dans les paragraphes suivants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 134 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Résultats de reconnaissance de textes manuscrits sur le jeu de données IAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous pré- sentons également les résultats du modèle de Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019), modèle obtenant les résultats à l’état de l’art sans modèle de langue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Système CER (%) WER (%) train valid test train valid test PyLaia 0,32 6,50 7,68 1,26 19,12 19,82 Moysset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' (2019) – 4,62 7,73 – 17,31 25,22 modèle de reconnaissance pylaia Un modèle de reconnaissance PyLaia (Puigcerver, 2017) est entraîné au préalable sur les mêmes données et en suivant la même répartition dans les ensembles d’entraînement, de validation et de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est ensuite intégré à l’entraînement de Doc2Seq et appliqué aux boîtes prédites sur l’ensemble de validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle PyLaia a été choisi car il obtient des résultats satisfaisants sur les textes manuscrits tout en étant assez rapide ce qui permet de l’intégrer à l’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, le système est facilement interfaçable avec le code PyTorch de Doc2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats du modèle de reconnaissance niveau ligne sont présentés dans la Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette table montre des performances assez satisfaisantes que nous considérons suffisantes afin d’évaluer et de comparer les modèles de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='3 résultats et discussion Dans cette section, nous présentons, tout d’abord, les résultats quantitatifs du modèle Doc2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous visualisons ensuite les prédictions et analysons les erreurs obtenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Durant l’inférence, les valeurs prédites sont regroupées par six (les quatre coordonnées et les deux classes de points) afin de créer les rectangles englobants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle a très bien appris sur le jeu de données IAM puisque qu’aucune valeur n’est prédite en plus et que les jetons de classes ont tous été correctement prédits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour comparaison, nous avons entraîné un modèle Doc-UFCN sur les mêmes données IAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle a été entraîné durant 150 époques dans les mêmes conditions que celles décrites dans les chapitres précédents : — Images redimensionnées telles que la plus grande dimension de l’image soit égale à 768 pixels ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Taux d’apprentissage initial de 5e − 3, mini-batchs de taille 4, optimiseur Adam, fonc- tion de coût Dice et arrêt anticipé (early stopping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, afin d’avoir des résultats comparables, les rectangles englobants des composantes connexes prédites par Doc-UFCN sont extraites et le même schéma d’évaluation que Doc2Seq est appliqué.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S E T R É S U LTAT S 135 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 – Résultats des modèles de détection de lignes sur le jeu de données IAM, donnés en fonction du critère de sélection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les colonnes "Manuel" présentent les résultats du reconnaisseur sur les lignes annotées manuellement contrairement aux colonnes "Prédit" qui présentent les résultats du reconnaisseur sur les lignes prédites automatiquement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Système Critère de Set AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 mAP CER (%) WER (%) sélection Manuel Prédit Manuel Prédit train 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='98 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='83 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='31 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='21 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='43 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='82 Doc2Seq CER valid 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='98 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='60 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='72 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='98 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='11 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='67 test 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='98 0,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='82 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='69 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='31 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='17 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='43 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='86 Doc2Seq Entropie valid 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='97 0,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='61 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='57 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='31 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='37 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='43 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='13 Doc-UFCN CER valid 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='97 0,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='89 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='71 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='31 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='83 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='43 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='69 Doc-UFCN DICE valid 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='92 0,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='83 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='65 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='59 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='97 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='68 résultats quantitatifs La Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 présente les performances des modèles obtenus pour le jeu de données IAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette Table, nous montrons les résultats de deux modèles issus du même entraînement mais sélectionnés selon un critère différent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les modèles sont évalués par différentes métriques : — Les métriques objet fournies par COCO 1, notamment la précision moyenne (AP) pour différentes valeurs de seuil : AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5, AP@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='75 et AP@[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='95] (mAP) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' — Les métriques de reconnaissance niveau page : CER et WER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme dans les chapitres précédents, les colonnes "Manuel" présentent les résultats du reconnaisseur sur les lignes annotées manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Elles correspondent donc au CER entre les transcriptions et les résultats d’HTR appliqué sur les mêmes lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, ces valeurs représentent les meilleures atteignables dans le cas d’un détecteur de lignes idéal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les deux modèles Doc2Seq présentent des résultats très satisfaisants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, les valeurs d’AP sont relativement élevées pour de la détection de lignes de texte dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les valeurs de CER sur les ensembles de validation et de test sont très proches des valeurs obtenues sur les lignes annotées manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela signifie que les lignes obtenues sont localisées avec précision sur les images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, pour le modèle ayant obtenu le CER le plus faible, nous perdons moins d’un point de pourcentage de CER (de 6,65 % à 7,58 %) en passant des lignes manuelles aux lignes prédites sur l’ensemble de test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, les deux modèles Doc2Seq obtiennent des résultats similaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, pour les métriques AP, les deux modèles obtiennent, en moyenne, un point de pourcentage d’écart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle sélectionné sur la base du CER présente un faible gain de performances en CER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='com/cocodataset/cocoapi 136 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S et WER sur l’ensemble de test, ce qui était attendu puisqu’il a été choisi afin de minimiser le CER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, le modèle sélectionné sur la base de l’entropie croisée présente des résultats très satisfaisants, ce qui valide l’utilisation de la classification comme type de prédiction couplé à la fonction de perte d’entropie croisée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le calcul du CER durant l’entraînement, qui nécessite le texte des documents transcrits ainsi qu’un modèle de reconnaissance entraîné, ne semble donc pas nécessaire pour obtenir un modèle très performant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que cela permette d’optimiser les résultats de reconnaissance, il est envisageable d’entraîner un modèle performant sans utilisation de reconnaisseur de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les résultats obtenus par le modèle Doc-UFCN sont légèrement meilleurs en termes de précision moyenne par rapport aux modèles Doc2Seq, notamment sur l’ensemble de validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, les résultats finaux de reconnaissance niveau page sont bien moins bons que ceux des modèles Doc2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, pour le critère de sélection basé sur la fonction de coût, nous notons une augmentation de +5,70 points de pourcentage de CER sur l’ensemble de validation et de +5,75 points de WER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces écarts sont plus faibles lorsque nous comparons les modèles sélectionnés sur la base du CER, cependant, le modèle Doc-UFCN reste bien moins bon que le modèle Doc2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, pour des résultats niveau objet équivalents, le modèle Doc2Seq obtient de bien meilleures performances en reconnaissance de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela s’explique, entre autres, par sa capacité à apprendre l’ordre de lecture, ordre non disponible dans les prédictions de Doc- UFCN pour lequel nous avons dû ordonner les boîtes de haut en bas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces résultats montrent une nouvelle fois l’intérêt des métriques orientées vers la tâche finale dans l’évaluation et la comparaison de modèles de détection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le modèle Doc2Seq présente un temps de prédiction moyen de 284 ms par image sur une carte graphique Tesla V100-SXM2-16GB pour le jeu de données IAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans les mêmes conditions, Doc-UFCN est lui deux fois plus rapide, avec un temps moyen de 134 ms par image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le temps d’inférence présenté par Doc2Seq est très raisonnable sachant que le modèle permet l’extraction directe des objets dans l’ordre de lecture demandé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' visualisation des prédictions La Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 montre les résultats visuels sur quatre images de l’ensemble de test du jeu de données IAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les boîtes annotées manuellement sont représentées en bleu et les boîtes prédites par le modèle sont en rouge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Visuellement, les prédictions sont très proches des boîtes annotées pour les trois images en haut et en bas à gauche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous remarquons très peu de problèmes liés à la largeur des boîtes et à leur position selon l’axe des abscisses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous notons que les principales erreurs viennent de la hauteur des boîtes ainsi que leur position selon l’axe des ordonnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, sur les images en haut à droite et en bas à gauche, nous voyons que certaines boîtes sont trop hautes par rapport aux boîtes annotées correspondantes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' L’image en bas à gauche montre des boîtes mal localisées, trop basses selon l’axe des ordonnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 E X P É R I E N C E S E T R É S U LTAT S 137 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='4 – Détections de lignes produites par le modèle Doc2Seq, sélectionné sur les valeurs du CER, sur quatre images de l’ensemble de test du jeu de données IAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les rectangles englobants annotés manuellement sont représentés en bleu et les boîtes prédites en rouge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' P03-185 Sentence Database Thus had they parted the previous evening and now Diana was trailing up the grav- elled drive to the hospital alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" Of course one couldn't say for certain when a doctor would be free during the day;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" tea was served from four until five-thirty in the residents' common-room, which proved the elasticity of medical commitments." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" Something had cropped up which required Nigel's attention, she was convinced, or he would have granted her small request to be met at the gates." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' s hadfheypavledthe prevlous evenins aud nou Diany was railing up Hhe graveled drve fo He osofta alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content="OfCouvse eowe couldu'f say fo cerhuin whe neduri lochov r)nom beee duvn fea WaS gevee 4m ve-trhnenesdenhs may Ymm proved He elasdahy ofwedical commitreruhs 1m00 Somethly laedCropped upwmuvegnineeNil's aHenHor SMME convinced or would have gfaedlersalle b meFat He geules Name:SentenceDatabase M01-131 When he finally beckoned to them to enter, the action gave the impression of having the floor." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' He seemed completely unaware of their presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' They just stared at him, turning their heads like tennis spectators as he walked up and down, up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' When he Bualy beduoned to Hhin to enles, thu acha gaveHheimpresscmap benhougktoufand deaded upan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='lnside they sat down un bidd,whie Dan paced Hhefor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content="lle Seemed Complelely unawae mun Name:Sentence Database N04-048 So he put up for the night at The Admiral's Head, that famous Portsmouth hostelry, during the last war." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Having deposited his baggage and unpacked his overnight-bag he went in search of a drink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' The lower bar was empty, save for the lady known by all habitue*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content="2s as 'Seaweed', and a youngish, sharp-eyed man who was staring moodily into a gin and tonic." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' So kepuf up Perthemightat he Aimiral HeadtMe lamirrotmeihhoytelny,Olisndonlymhimeri mitiretoThebeepee,lnhappiluydeokruyeel NyCemar lemlr dunne Nu lafar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' lhanng depaotd lu lriggaeg and cinpaihed his oemglt-lug etnsearR aladmnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='DheXoerlarws emphy,Mve for thu lady hnom luyall haluilue X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='ls a Seumeed , anila youngsh, yhanp-eyel manAho ansiorng moodilys inkea gin inelenie Name:Sentence Database P02-115 Doc gave her hand a shake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' "Wake up Gay, and don\'t even contemplate throwing yourself away on a chap like that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" You're a fine girl, intelligent, and pretty, and I had thought you were sensible too." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" Don't make a fool of yourself over someone who doesn't care two jots for your feelings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' If he behaves like this now what is your married lifegoing tobelike?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' hana gae May qshahe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=" throwing pup een don yoursel that You're qway on Chao Tike Pine inteligent gir/ prel anc bpy thoughl you were sensibk too Don't make 100/ 0f yourselp over who doesnt two 2015 your he behaves this whal IS youl lihe buob fo be Name:138 D É T E C T I O N S É Q U E N T I E L L E D’ O B J E T S D A N S D E S I M A G E S D E D O C U M E N T S Le modèle prédit, tout d’abord, l’ordonnée puis l’abscisse de chaque point." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, notre hypothèse est que, pour prédire l’abscisse d’un point, le modèle a déjà connaissance de son ordonnée, il sait donc exactement où regarder sur l’image (sur quelle ligne d’ordonnée) pour se positionner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les coordonnées prédites selon l’axe des abscisses sont donc plus précises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Au contraire, afin de prédire la première coordonnée, le modèle peut regarder partout sur l’image, ou du moins en dessous des lignes précédemment prédites, ce qui peut mener à des localisations moins précises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette hypothèse s’applique également au second point à prédire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Afin de vérifier cette hypothèse, il serait nécessaire d’entraîner un second modèle à prédire d’abord l’abscisse puis l’ordonnée de chaque point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, nous pourrions voir si les problèmes seraient désormais liés à la largeur des boîtes et leur localisation selon l’axe des abscisses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces expérimentations sont actuellement en cours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='5 C O N C L U S I O N Dans ce chapitre, nous avons présenté un nouveau système de détection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celui-ci se base sur les Transformers, algorithmes les plus performants actuellement dans de nombreux domaines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons montré que ce système pouvait être entraîné sur un jeu de données restreint d’images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il permet d’obtenir des premiers résultats prometteurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, il montre des temps d’inférence raisonnables tout en obtenant des performances à l’état de l’art et en permettant une prédiction séquentielle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons également proposé une modélisation complète du problème de détection d’ob- jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette modélisation permet de représenter les objets de manière simple et une détection efficace de ceux-ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette modélisation a le potentiel à se généraliser à d’autres tâches plus complexes, qui feront l’objet de nos futures recherches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 8 C O N C L U S I O N S E T P E R S P E C T I V E S 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='1 C O N C L U S I O N S Dans cette thèse, nous avons proposé deux systèmes à base de réseaux neuronaux afin de résoudre la tâche de détection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le premier mo- dèle, Doc-UFCN, a présenté de grandes capacités de détection sur de nombreux jeux de données manuscrits hétérogènes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il a également montré une grande robustesse en obtenant des résultats très compétitifs sur de nouveaux documents hors échantillon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le second modèle, Doc2Seq, a également obtenu des premiers résultats encourageants qui permettent de traiter la détection d’objets à un plus haut niveau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le développement et l’application de ces modèles ont mené à des études plus globales concentrées sur les données et leurs annotations, les métriques d’évaluation et les scores de confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons répondu aux principales problématiques liées à la détection d’objets dans les images de document dans un cadre industriel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La première problématique concerne le déve- loppement de modèles avec de grandes capacités de généralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, dans ce cadre dans lequel de nouveaux projets sont régulièrement mis en place, il n’est pas envisageable d’annoter de nombreuses données pour chacun de ces projets, d’où la nécessité de dévelop- per des modèles plus génériques, montrant des performances élevées sur des documents très hétérogènes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette optique, nous avons entraîné plusieurs systèmes sur un grand volume de données très différentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces entraînements ont mené à des modèles plus robustes, obtenant de meilleures performances que des modèles spécifiques entraînés sur un jeu de données unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme la plupart des modèles de type FCN, ces modèles ont cependant montré des difficultés à prédire des éléments qui se touchent ou se chevauchent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cette raison, nous avons proposé une uniformisation des annotations, ainsi qu’une scission des boîtes afin de réduire ces chevauchements dans les annotations, utilisées durant la phase d’en- traînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces traitements permettent la prédiction de boîtes plus précises et non fusionnées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une seconde problématique induite par le cadre de production dans lequel se situe cette thèse concerne l’efficacité des modèles de détection : ceux-ci doivent fournir des détections de grande précision tout en montrant des temps de traitement réduits et en pouvant être entraînés sur des jeux de données restreints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans la littérature, de nombreux modèles ont été proposés pour la détection d’objets dans les images de documents, cependant, la plupart requièrent un grand nombre de données annotées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour pallier ce problème, des systèmes 139 140 C O N C L U S I O N S E T P E R S P E C T I V E S utilisant des poids pré-entraînés sur des images de scènes naturelles, tels que dhSegment, ont été proposés, mais ces systèmes montrent des temps d’inférence encore trop élevés pour une utilisation à l’échelle industrielle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour répondre à ces problématiques, nous avons proposé le système Doc-UFCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système a montré des temps d’inférence réduits et obtenu des performances à l’état de l’art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Celui-ci peut être entraîné sur peu de données en comparaison avec les systèmes dédiés à la détection d’objets dans les images de scènes naturelles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans de nombreux projets, il y a peu, voire aucune donnée annotée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Comme énoncé plus tôt, il est nécessaire d’avoir un détecteur assez générique afin de traiter ces documents plus facilement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cependant, les modèles génériques peuvent parfois ne pas être suffisamment per- formants sur ces nouveaux documents, qui peuvent avoir une mise en page très différente de celles des documents que le modèle a rencontrés durant sa phase d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour cela, l’apprentissage actif a été introduit, permettant d’entraîner itérativement des modèles en ajoutant, à chaque itération, de nouvelles données annotées sélectionnées dans le but d’améliorer les résultats du modèle de détection tout en réduisant le coût d’annotation ma- nuelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce cadre, il est nécessaire que le modèle fournisse les détections tout en estimant automatiquement leur qualité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons proposé et évalué quatre estimateurs de confiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceux-ci ont permis d’entraîner des modèles atteignant des performances élevées pour la détection d’objets tout en ne nécessitant qu’un faible nombre d’images annotées manuellement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons également démontré que deux de ces estimateurs permettent de sélectionner les détections les plus précises afin d’être utilisées dans un entraînement autosupervisé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Les modèles ainsi entraînés ont permis d’obtenir des gains significatifs de performances par rapport aux modèles génériques tout en ne nécessitant aucune donnée annotée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La reconnaissance de documents consiste généralement en l’application successive de diffé- rents modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans ce cadre, les lignes de texte produites par un modèle de détection sont généralement fournies à un modèle de reconnaissance de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, l’amélioration de la détection des lignes de texte doit permettre d’améliorer les résultats de reconnaissance, or les deux tâches ne sont pas étroitement liées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il est donc nécessaire d’évaluer, à chaque étape du traitement, son impact sur les résultats de l’étape suivante ou finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cette problématique a été très peu étudiée dans la littérature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Afin d’avoir une évaluation de la détection de lignes de texte davantage cohérente avec la tâche finale, nous avons donc proposé des métriques liées à la reconnaissance de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces métriques permettent de voir directement l’impact de la tâche de détection sur les résultats finaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous avons constaté que l’utilisation de ces métriques durant l’entraînement de modèles de détection permet d’optimiser les résultats du reconnaisseur de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Enfin, les modèles à base de réseaux de neurones profonds de type Transformers ont récem- ment été proposés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ceux-ci ont été introduits pour des tâches de traitement de la langue et notamment la tâche de traduction de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ont été initialement établis afin de pallier le 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 P E R S P E C T I V E S 141 trop faible contexte disponible pour traiter les longues séquences de texte par les réseaux ré- currents, systèmes largement utilisés jusqu’alors pour traiter ces tâches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Par la suite, certains travaux ont cherché à adapter ces modèles aux tâches de vision, motivés par leur capacité de modélisation des dépendances des éléments en entrée réalisée à l’aide du mécanisme d’at- tention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que ces travaux aient montré des avancées significatives pour les tâches de classification d’images, très peu se sont intéressés à leur application aux tâches de détection d’objets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette thèse, nous nous sommes intéressés à adapter ces modèles à base de Transfor- mers à la tâche de détection d’objets dans les images de documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous avons donc proposé Doc2Seq, un modèle hybride combinant un encodeur convolutif et un décodeur Transfor- mer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce modèle a permis d’obtenir des premiers résultats encourageants, tout en respectant l’ensemble des contraintes évoquées précédemment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il permet de modéliser correctement les dépendances entre les différentes parties de l’image d’entrée mais aussi celles entre l’image d’entrée et les coordonnées prédites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ce système apporte également d’autres avantages tels que sa capacité à produire des résultats séquentiels et structurés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='2 P E R S P E C T I V E S Dans de nombreux domaines d’application des réseaux de neurones profonds, les modèles sont entraînés sur des milliers, voire des millions d’exemples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, le premier Vision Trans- former proposé, pour la tâche de classification d’images, a nécessité un pré-entraînement sur 303 millions d’images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De la même manière, Pix2Seq a été entraîné sur le jeu de données de référence MS-COCO 2017 comportant 118 000 images d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, leur meilleur modèle a été obtenu grâce à un pré-entraînement sur les données Objects365, qui représentent 600 000 images d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ces modèles ont montré des performances très élevées, mon- trant l’intérêt d’utiliser de telles quantités de données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Dans cette optique, une perspective de cette thèse est de collecter encore davantage de jeux de données, toujours plus divers, et d’étudier la capacité du modèle à apprendre à partir de ces données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' La plupart de ceux utilisés jusqu’ici étaient principalement historiques, il serait également envisageable de collecter des documents modernes afin d’obtenir un détecteur de lignes de texte très générique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous souhaiterions également comparer cette approche à une stratégie de collecte dans laquelle un nombre plus restreint d’exemples serait utilisé, mais qui chercherait à maximiser la diversité des mises en page et des contenus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Durant cette étude, il serait également intéressant d’évaluer l’impact du balancement des différentes données dans l’ensemble d’entraînement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, nous avons proposé, durant cette thèse, quatre estimateurs de confiance permet- tant de sélectionner les images à annoter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Pour le moment, ceux-ci sont utilisés dans le cadre d’une adaptation d’un modèle générique à un nouveau domaine, afin de sélectionner les images à annoter pour mettre en place un nouveau système.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Une perspective serait d’utiliser les confiances estimées afin de suivre la qualité des résultats en production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, dans un 142 C O N C L U S I O N S E T P E R S P E C T I V E S cadre non supervisé, les confiances estimées par le modèle de détection peuvent permettre de vérifier que celui-ci s’améliore, durant les différentes itérations, grâce à sa confiance moyenne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De nombreux modèles de détection d’objets traitent les images de documents à partir de sous-résolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' C’est également le cas des deux modèles que nous avons proposés, Doc-UFCN et Doc2Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bien que, dans la plupart des cas, cette sous-résolution soit suffisante pour obtenir des résultats satisfaisants, dans le cas où les objets sont très petits et très proches, la détection est impossible puisque de nombreuses fusions sont produites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il serait intéressant de comparer différentes sous-résolutions mais aussi une approche par patchs, bien que celle-ci soit beaucoup plus coûteuse en ressources et en temps d’inférence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De même, nous pourrions estimer la sous-résolution optimale pour un jeu de données ou pour une image de manière automatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous souhaitons également évaluer le modèle Doc2Seq sur d’autres jeux de données et d’autres tâches, et notamment des problèmes plus complexes avec des objets imbriqués tels que des paragraphes et lignes de texte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Il serait intéressant de le tester sur d’autres tâches telles que l’analyse de mise en page de tableaux avec la détection des éléments structurels tels que les lignes et colonnes de titre et les pieds de tableaux 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' De plus, l’ensemble des modèles de détection traitent les images de manière isolée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ils ne possèdent aucune mémoire quant aux prédictions réalisées sur les images précédentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Or, dans le cadre du traitement de séries (ouvrages ou collections), il pourrait être bénéfique de considérer, lors du traitement d’une nouvelle image, des propriétés établies sur d’autres images ou au niveau de la série.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Le système Doc2seq permet d’envisager un tel traitement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' En effet, l’utilisation des prédictions précédentes dans le décodeur Transformer permet d’imagi- ner des tâches de plus haut niveau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ainsi, lors du traitement d’un livre ou d’une collection, la prédiction d’une image pourrait être initialisée par les éléments prédits sur l’image précédente ou par une moyenne des positions prédites sur l’ensemble des pages précédentes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Cela permet- trait de transférer des informations d’une image à l’autre et d’avoir des résultats homogènes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Nous souhaitons étudier cette possibilité dans de futurs travaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='socface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content='org/ B I B L I O G R A P H I E Agrawal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' et D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Doermann (juill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Voronoi++ : A Dynamic Page Segmentation Approach Based on Voronoi and Docstrum Features ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : 10th International Conference on Document Analysis and Recognition (ICDAR), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1011-1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Akindele, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' et A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Belaid (oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Page Segmentation by Segment Tracing ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : 2nd International Conference on Document Analysis and Recognition (ICDAR), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 341-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Alberti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Bouillon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Ingold et M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' Liwicki (nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' « Open Evaluation Tool for Layout Ana- lysis of Document Images ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' In : 14th International Conference on Document Analysis and Recognition (ICDAR), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFKT4oBgHgl3EQfNC2a/content/2301.11753v1.pdf'} +page_content=' 43-47.' metadata={'source': 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