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∥bβ−β(S†)∥2 Σ λmin(Σ)(a) <√ 0.25d−ϵ≤1 2k−1 for some S†∈ Sy. Here ( a) follows from the the error guarantee in Problem A.3, and the fact λmin(Σ)≥1 derived in the proof of Theorem A.2. This further indicates eS=S†by the fact that S†⊂[7k+ 1] and β(S†) j= (k−1)1{j∈S†}for any j∈[7k+ 1] derived in the proof of Theorem A.2. C... | https://arxiv.org/abs/2501.17354v1 |
implies the i-th clause and the i′-th clause have shared variable. Since each variable appears no more than 15 times, one clause shares common variables with up to 3 ×15 other clauses. Then we can conclude that each row of matrix Ahas no more than 7 ×(3×15 + 1) = 322 non-zero elements. Combining with the fact that ther... | https://arxiv.org/abs/2501.17354v1 |
⇐, it follows from the proof of (2.3)(c) and the fact that ˚S={7(i−1) +ai}k′ i=1with ai∈[7] naturally implies |˚S∩ {7i− 6, . . . , 7i}|= 1 for i∈[k′]. C.6 Proof of Theorem B.1 It suffices to construct a polynomial-time reduction from 3Sat to Problem B.1. Let xbe any 3Sat instance with input size k, following the notati... | https://arxiv.org/abs/2501.17354v1 |
the identity Σ(¯β−β⋆) =1 |E|X e∈En E[X(e)Y(e)]−E[X(e)(X(e))⊤β⋆]o =1 |E|X e∈EE[X(e)ε(e)]. Therefore, we have Qk,γ(β)−Qk,γ(β⋆)≥0 if γ≥max j∈G 1 |E|P e∈EE[ε(e)X(e) j] wk(j):=γ⋆ k, this completes the proof. 44 We finally establish the upper bound on γ⋆ k. It follows from the definition of wkthat (γ⋆ k)2= max j∈G 1 |E|P e∈E... | https://arxiv.org/abs/2501.17354v1 |
≤2e−t. The next proposition provides upper bounds for |v(S)−bv(S)|. Proposition E.1 (Instance-dependent Error Bounds on |v(S)−bv(S)|).Suppose Condition 3.4 hold. There exists some universal constant Csuch that, for any t >0andϵ >0, ifCσ4 xρ(k, t)≤1, then the following event ∀S⊆[d],|S| ≤k|v(S)−bv(S)| ≤Cnq v(S)·σ4xσ2ybρ... | https://arxiv.org/abs/2501.17354v1 |
|√x|+|√y| ≤(√ 2C+ 1)δ. Case 2. y≥δ2.In this case, it follows from the upper bound on |x−y|and the assumption y≥δ2that |√x−√y|=|x−y|√x+√y≤|x−y|√y≤Cδ2+δ√y√y=δ+Cδ2 y≤(1 +C)δ. Combining the above two cases completes the proof. E.3 Proof of Theorem 3.5 The next several lemmas are standard in high-dimensional linear regressi... | https://arxiv.org/abs/2501.17354v1 |
λ(∥b∆S⋆∥1− ∥b∆Sc⋆∥1)≥ −∥b∆∥1 Cp ♣ ·γr klogd+ (c1+ 1) log n n+C♠s logd+ log n n· |E|! | {z } λ⋆. This immediately implies ∥b∆Sc⋆∥1(λ−λ⋆)≤(λ+λ⋆)· ∥b∆S⋆∥1, then the following holds ∥b∆Sc⋆∥1≤3∥b∆S⋆∥1 (E.11) provided λ≥2λ⋆. Given (E.11), we can apply the restricted strong convexity derived from Lemma E.6 with α= 3 and combi... | https://arxiv.org/abs/2501.17354v1 |
that bv(S) = minu∈R|S|bqS(a) with bqS(a) = (a−β(S) S)⊤bΣ(a−β(S) S)−2(a−β(S) S)⊤( 1 |E|X e∈EbE[R(e,S)X(e) S]) +1 |E|X e∈E (bΣ(e))−1/2bEh X(e) SR(e,S)i 2 2 53 provided bΣ(e) S≻0 for any e∈ E. We can claim that bqS(a) can be minimized by ba=bβ(S)=β(S) S+ (bΣ)−1( 1 |E|X e∈EbE[R(e,S)X(e) S]) =β(S) S+ (bΣ)−1( 1 |E|X e∈EbE[U(... | https://arxiv.org/abs/2501.17354v1 |
F= {β:∥β−¯β∥2≤(1/λmin(Σ))1/2¯β⊤Σ¯β}given 2Qk,γ(β)≥(β−¯β)⊤Σ(β−¯β)≥¯β⊤Σ¯β= 2Qk,γ(0) ∀β∈Fc. The uniqueness will be established using the proof-by-contradiction argument. Let β′andβ†be two optimal solutions with β′̸=β†, then Qk,γβ′+β† 2 =Rβ′+β† 2 +dX j=1γwk(j) β′ j+β† j 2 (a) <1 2 R(β′) +R(β†) +1 2 dX j=1γwk(j)(|β... | https://arxiv.org/abs/2501.17354v1 |
S)1/2β(S) 2(a) ≤σy+σx (Σ(e) S)1/2Σ−1/2 S 2 Σ−1/2 S 1 |E|X e∈EE[X(e) SY(e)]! 2 (b) ≤σy+σx (Σ(e) S)1/2Σ−1/2 S 2s 1 |E|X e∈EE[(Y(e))2] (c) ≤σy+σx√ bσy.(E.23) Here ( a) follows from the property of the operator norm and the definition of β(S); (b) follows from the Cauchy-Schwarz inequality; and ( c) follows from Condition ... | https://arxiv.org/abs/2501.17354v1 |
of two sub-Gaussian variables with parameters σe,S,ℓandσy(1 +σx√ b). Then it follows from the tail bound for exponential random variable that |S| ≤s, ℓ∈[NS],P |Z3(S, ℓ)| ≥C′√ bσ2 xσy√ bu n· |E|+ru n· |E| ≤2e−u,∀u >0. Letting u=t+ log(2 N)≤6 (t+slog(ed/s)), we obtain P" sup |S|≤s,ℓ∈[NS]|Z3(S, k)| ≥6C′σ2 xσyp bζ(s, ... | https://arxiv.org/abs/2501.17354v1 |
≤2e−t, ∀v, v′∈Rd; (E.28) P" |Zv|> Cσ2 x r t en+t en!# ≤2e−t, ∀v∈ B; (E.29) P" Wv> Cd(v,0) r t en+ 1!# ≤2e−t, ∀v∈Rd. (E.30) In the second step, we establish an upper bound on the Talagrand’s γ2functional (Vershynin, 2018) of Θ, which is defined as γ2(Θ,d) := inf {Bk}∞ k=0:|B0|=1,|Bk|≤22ksup v∈Θ∞X k=02k/2d(v,Bk). (E.31) ... | https://arxiv.org/abs/2501.17354v1 |
Σ−1/2x∈ Band for any v∈ B, we have ∥v∥1=∥vS∥1+∥vSc∥1≤(1 +α)∥vS∥1≤(1 +α)√s∥vS∥2≤(1 +α)√s∥v∥2 for some subset |S| ≤sby the definition of Θ; ( b) follows from ∥Σ−1/2x∥2≤κ−1/2∥x∥2=κ−1/2; and ( c) follows from Eg∼N(0,Id)[∥Σ1/2g∥∞]≤Eg∼N(0,Id)[∥g∥∞]≤50√logdby Sudakov-Fernique’s inequality (Conze et al., 1975) and Condition 3.... | https://arxiv.org/abs/2501.17354v1 |
the eventTk0−1 k=k1U2(k), we have sup v0∈Bk0|Zv0−Zπk1(v0)| ≤k0−1X k=k1sup v∈B|Zπk+1(v)−Zπk(v)| ≤40Cσxen−1/2k0−1X k=k12k/2d(πk+1(v), πk(v)) ≤40Cσxen−1/22γ2(B,d).(E.46) For supv1∈Bk1|Zv1|, we define the following event for each 0 ≤k≤k0−1, U3(k) = sup v∈B Zπk(v) ≤Cσ2 x·32q 2k/en (E.47) 66 where the constant Cis the same... | https://arxiv.org/abs/2501.17354v1 |
for a measurement xat a grid with latitude ϕ∈[−π, π], we apply the following transformation: xcos=x∗p cos(ϕ). The cosine transform compensates for the varying areas that grids at different latitudes represent, helping to avoid over-compression or over-amplification of grids at higher latitudes. Next, we estimate the co... | https://arxiv.org/abs/2501.17354v1 |
.0392±0.2569 k= 2 3 .7838±0.3281 2 .0523±0.0883 1 .6077±0.1122 3 .0466±0.1955 k= 3 3 .7652±0.4016 2 .0637±0.0415 1 .6004±0.1042 3 .0379±0.2157 Table 5: The average ±standard deviation of the mean squared error (4.3) of the four tasks air temperature ( air), clear sky upward solar flux ( csulf ), surface pressure ( pres... | https://arxiv.org/abs/2501.17354v1 |
Copula methods in density functional theory Copula methods for modeling pair densities in density functional theory Geneviève Dusson,1Claudia Klüppelberg,2and Gero Friesecke2 1)Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623), F-25000 Besançon, France 2)Department of Mathematics, School of Computation, Informati... | https://arxiv.org/abs/2501.17515v2 |
copula and its marginals on R3, which are given by the single-particle density ρ. Our first achievement in this paper is to generalize the theory of copulas to precisely such factorizations, using optimal transport theory10,11. Roughly, a generalized copula (introduced in detail in section III) is a pair density cwith ... | https://arxiv.org/abs/2501.17515v2 |
AND CONTEXT In electronic structure calculations, we are interested in cal- culating the wavefunction of a system, minimizing the (non- relativistic) electronic energy of Nelectrons in an external po- tential, E[Ψ] =∑ s∈ZN 2Z ΩNh |∇Ψ(r,s)|2+N ∑ i=1Vext(xi)|Ψ(r,s)|2 +∑ 1≤i<j≤Nvee(|ri−rj|)|Ψ(r,s)|2i dr, (2) where Ψis ant... | https://arxiv.org/abs/2501.17515v2 |
and is unique10–12. In copula theory (that is, when d=1 and f0is the uniform density on the unit interval), the Brenier map is given by Ti=Fi, where Fiis the cumulative distribution function (CDF) of fi, Fi(xi) =Zxi −∞fi(t)dt. (8) This is a standard result of optimal transport theory10,11. In higher dimensions, there i... | https://arxiv.org/abs/2501.17515v2 |
by the following classical examples from statistics (for d=1 and the reference density f0≡1 on[0,1]). For Nindependent proba- bility densities, f(x1,..., xN) =f1(x1)·...·fN(xN), C(x1,..., xN) =x1·...·xN,c(x1,..., xN)≡1; (15) and for two fully dependent probability measures, f(x1,x2) = δ(x2−S(x1))for some increasing map... | https://arxiv.org/abs/2501.17515v2 |
-dimensional marginals absence of correlation corresponds to a constant copula . Note that, be- cause the mean field pair density doesn’t quite integrate to the correct number of pairs in the system, the mean field copula doesn’t quite integrate to 1. The mean field interaction energy is obtained by substitut- ing expr... | https://arxiv.org/abs/2501.17515v2 |
with randr′replaced by x andy, and the exchange energy density per unit volume is eLDA x(ρ(x)) =−1 4ρ(x)2η(ρ(x)) (28)with η(a) =Z∞ −∞h(π 2az)vee(|z|)dz. (29) In this derivation the interaction potential v eewas arbitrary; in our numerical results in section VII it will be taken to be the soft Coulomb potential (46) fro... | https://arxiv.org/abs/2501.17515v2 |
uniform density – given by (35). In total we have shown that c(y1,...,yN) =SNδg2(y1)(y2)···δgN(y1)(yN). For- mula (34) now follows by integrating out all but two coordi- nates. V. DISSOCIATING SYSTEMS We now describe the correlation structure of dissociating systems in terms of generalized copulas. This is an important... | https://arxiv.org/abs/2501.17515v2 |
of disjointly supported fewer-particle wave- functions. The pair density of such states is described by an interesting exact expression involving only the subsystem densities and subsystem pair densities. Theorem 2 (Pair density of dissociated systems) Let N A, NBbe positive integers, let ΨAbe an antisymmetric normal- ... | https://arxiv.org/abs/2501.17515v2 |
equals that of subsystem B, translated byNA Nin both the xandydirection and re-scaled to a domain of sizeNB N×NB N. The key advantage of working with copu- las (as compared to pair densities as in Theorem 2) is that the dependence on the density has now disappeared and we have auniversal expression for the copula of th... | https://arxiv.org/abs/2501.17515v2 |
N−1vol(D)−2 N−2 N−1cB ϕ−1 B(˜r),ϕ−1 B(˜r′) where cAand cBare the generalized copulas of ΨA, respec- tively, ΨB, and ϕAandϕBare invertible mappings defined in the proof. Proof of Theorem 4 The proof combines a symmetry ar- gument with a variational argument using the monotonicity of the Brenier map. Starting point is ... | https://arxiv.org/abs/2501.17515v2 |
ΦonDN: Φ(r′ 1,s1,...,r′ N,sN) =Ψ T−1(r′ 1),s1,...,T−1(r′ N),sN ·N ∏ i=11 ρ(T−1(r′ i))/N1/2 ·1 vol(D)N/2. Squaring, summing over spins, and integrating over r′ 3,...,r′ N (using the change-of-variables formula) gives that ρΦ 2(r′ 1,r′ 2) =2N N−1ρΨ 2(T−1(r′ 1),T−1(r′ 2)) ρ(T−1(r′ 1))ρ(T−1(r′ 2))·1 vol(D)2. Hence the ... | https://arxiv.org/abs/2501.17515v2 |
results for the different parameter values a=1,2,3,are shown in Figure 3. The left column represents the external potential for the different values of a, together with the corresponding ground state densities. As expected from the functional form (47)–(48), the potential shows two wells that are separating when the in... | https://arxiv.org/abs/2501.17515v2 |
size 1/3 with constant values 3/2, 3/2 and 0, which corresponds to the one dissociated electron. The last one looks precisely like the copula for the least dis- sociated two-particle system (i.e. Figure 3, (a)), as predicted by theory (Theorem 3). This feature of the copula is even more visible on the ad- ditional exam... | https://arxiv.org/abs/2501.17515v2 |
0.20.40.60.81.00.20.40.60.81.0 0.000.250.500.751.001.25 0.20.40.60.81.00.20.40.60.81.0 0.000.250.500.751.001.25 (d) Copula (17) FIG. 4. Right column: exact copula of ground state for a three-particle system dissociating into three one-electron densities. The other columns show related quantities. Nuclei positions for t... | https://arxiv.org/abs/2501.17515v2 |
related quantities. Parameter afor top row: a=1, second row: a=2, bottom row: a=3. of the two-particle system of (b) in Figure 3, as predicted by theory. VIII. FITTING THE COPULA FOR TWO-PARTICLE SYSTEMS In this section, we turn to the approximation of the copula and the pair density for two-particle systems. More prec... | https://arxiv.org/abs/2501.17515v2 |
the copulas c1andc3. These barycenters are computed using a Sinkhorn algorithm from the Python Optimal Transport library24with a regularization parameter of 10−3and 1000 iterations. The results are shown in the third row of Figure 8. We observe that the copulas for a=2 and a=2.5 are reasonably well approximated, while ... | https://arxiv.org/abs/2501.17515v2 |
a=2.5 Linear barycenter copula 2.04e-2 1.74e-2 5.26e-3 Wasserstein barycenter copula 2.70e-2 1.85e-2 6.21e-3 Sigmoid copula 1.01e-2 8.57e-3 4.79e-3 TABLE III. L2error on the pair densitya = 1.5 a=2 a=2.5 Linear barycenter copula 2.85e-2 4.03e-2 1.07e-2 Wasserstein barycenter copula 2.33e-2 9.49e-3 2.27e-2 one-parameter... | https://arxiv.org/abs/2501.17515v2 |
copula (second row), Wasserstein barycenter copula (third row), one-parameter neural net copula (fourth row), LDA-0 (bottom row). The color limits are (0,0.1). The one-parameter neural net copula is seen to provide an excellent fit at all values of the bond length parameter a. In Figure 11, we plot the exchange-correla... | https://arxiv.org/abs/2501.17515v2 |
N}) =σ({k+ 1,..., N}, so as before summation over τ∈SNcan be re- placed by summation over τ= (τ1×τ2)σwhere τ1∈Sk, τ2∈SN−k, and we obtain Copula methods in density functional theory 18 Σ2(x1,x2) =k ∑ i=1N ∑ j=k+1∑ σ∈SN σ(i)=1,σ(j)=2∑ τ1∈Sk∑ τ2∈SN−kε(τ1)ε(τ2)Zn ΨA(xσ(1),...,xσ(k))ΨA(xτ1σ(1),...,xτ1σ(k))∗ ·ΨB(xσ(k+1),...,... | https://arxiv.org/abs/2501.17515v2 |
and optimal transportation with coulomb cost,” Communications on Pure and Applied Mathematics 66, 548–599 (2013). 19M. Colombo, L. De Pascale, and S. Di Marino, “Multimarginal optimal transport maps for one–dimensional repulsive costs,” Canad. J. Math. 67, 350–368 (2015). 20P. Hohenberg and W. Kohn, “Inhomogeneous elec... | https://arxiv.org/abs/2501.17515v2 |
On the double robustness of Conditional Feature Importance Angel Reyero Lobo Institut de Math ´ematiques de Toulouse Universit ´e de Toulouse angel.reyero-lobo@inria.fr Pierre Neuvial Institut de Math ´ematiques de Toulouse Universit ´e de Toulouse pierre.neuvial@math.univ-toulouse.frBertrand Thirion Inria Universit ´e... | https://arxiv.org/abs/2501.17520v3 |
has been a shift towards conditional approaches. These include Conditional Variable Importance—also called Conditional Feature Importance (CFI) in Ewald et al. (2024)—and Conditional Model Reliance (CMR), dened as a ratio rather than a difference in Fisher et al. (2019). However, the conditional sampling step is often... | https://arxiv.org/abs/2501.17520v3 |
about the relationship between inputs and outputs, as it can be complex. However, we assume that the relationship among the input covariates is simple, typically because they originate from the same generative process (e.g. a measurement device). For this reason, it can be much more efcient and accurate to study the r... | https://arxiv.org/abs/2501.17520v3 |
model, m−j, for each coordinate j. On the one hand, this is computationally intensive, and on the other hand, it introduces optimization errors that do not compensate as desired, as will be discussed in Section 3.2. Arst naivepermutation-basedapproach consists in comparing the performance of the estimate on a test se... | https://arxiv.org/abs/2501.17520v3 |
address the fact that, under the conditional null hypothesis, similarly to what happens with LOCO, the inuence function vanishes. Consequently, there is a need to correct the estimated variance to ensure type-I error control (see Williamson et al. (2023); Dai et al. (2024); Verdinelli & Wasserman (2024)). In Section 4... | https://arxiv.org/abs/2501.17520v3 |
we prove adouble robustnessproperty for detecting conditionally null covariates: to identify a null covariate, it is sufcient that one of the two estimates is consistent. This contrasts with LOCO, where errors in both estimates must compensate. This property explains the good empirical results obtained by CPI for vari... | https://arxiv.org/abs/2501.17520v3 |
Bach (2020)). Therefore, CPI leverages this regularization to achieve lower variability, especially in the main setting of interest in this paper, where estimating ygiven Xis assumed to be challenging, but the relationship between covariates is relatively simple. Linear model.In the remainder of this section, we focus ... | https://arxiv.org/abs/2501.17520v3 |
dened as j SCPI=1 ntestntest i=1ℓ 1 ncalncal k=1m(x′(j) i,k), yi −ℓ(m(x i), yi), where thej-th coordinate ofx′(j) i,kis conditionally sampled, and the restxed tox−j i(see (1)). 4.1 Asymptotic efciency Under the same assumptions as those given in Williamson et al. (2023), along with an additional assumption... | https://arxiv.org/abs/2501.17520v3 |
empirical variance exactly. Nevertheless, under the null hypothesis, the inuence function vanishes, preventing a Gaussian asymptotic distribution. This poses a challenge, as it hinders variable selection with direct statistical guarantees—an essential component of reliable scientic discovery. This issue is the same a... | https://arxiv.org/abs/2501.17520v3 |
Boosting. We applied Sobol-CPI with ncal= 1 andn cal= 100. In Figure 1, we observe that even if we use a complex model to estimate m−j, but a simple one to estimate ν−j, we not only achieve a more computationally efcient estimate but also obtain better 8 results with Sobol-CPI than with LOCO. First, for an important... | https://arxiv.org/abs/2501.17520v3 |
Lobo, A. R., Linhart, J., Thirion, B., and Neuvial, P. When knockoffs fail: diagnosing and xing non-exchangeability of knockoffs, 2025. URLhttps://arxiv.org/abs/2407.06892. B´enard, C., Da Veiga, S., and Scornet, E. Mean decrease accuracy for random forests: inconsistency, and a practical solution via the sobol-mda.Bi... | https://arxiv.org/abs/2501.17520v3 |
Kolodyazhniy, V., Thirion, B., and Engemann, D. A. Measuring variable importance in heterogeneous treatment effects with condence. InProceedings of the 42nd International Conference on Machine Learning (ICML). PMLR, 2025. URL https:// arxiv.org/abs/2408.13002. arXiv:2408.13002. Papamakarios, G., Nalisnick, E., Rezende... | https://arxiv.org/abs/2501.17520v3 |
. First, note that they are both Gaussian as E XjX−j =µ j+Σj,−jΣ−1 −j,−j(X−j−µ −j), therefore they are linear combinations of coordinates of a Gaussian vector. 12 Then, note that ϵjis centered. Finally, to see that they are independent, as they are both Gaussian variables, we just need to prove that their covariance... | https://arxiv.org/abs/2501.17520v3 |
=E yX−j =m −j(X−j)∈F −j, whereF −j:=f∈F:f(u) =f(v)for allu, v∈Rpsuch thatu −j=v −j In particular, we have that m(X) =m(X′(j))asX−j=X′(j)−j. Therefore, using the continuity of ℓ, we conclude that CPI(j) =1 ntestntest i=1ℓ m(x′(j) i), yi −ℓ(m(x i), yi) ntrain→∞− −−−−− →1 ntestntest i=1ℓ m(x′(j) i), yi... | https://arxiv.org/abs/2501.17520v3 |
linear model mandm−j. Then, in Lemma 3.10, we use the underlying distribution to compute the expectation and prove the linear decay. Proposition D.1(LM LOCO bias).Under Assumption 3.8 withXGaussian, we have that E LOCO (j, P 0)Dtrain =TSI(j, P 0) +∆β′⊤Σ−j∆β′−∆β⊤Σ∆β =TSI(j, P 0) +∥∆β′∥2 Σ−j−∥∆β∥2 Σ Proof.Fir... | https://arxiv.org/abs/2501.17520v3 |
expectation isId(ntrain−p−1). Therefore, we have that E ∥∆β∥2 Σ =σ4 1 ntrain−p−1tr(I p) =σ4 p ntrain−p−1 Similarly forE ∥∆β′∥2 Σ−j , we have that E ∥∆β′∥2 Σ−j =σ4 p−1 ntrain−p−2 17 Therefore, we conclude that E ∥∆β′∥2 Σ−j −E ∥∆β∥2 Σ =σ4 p−1 ntrain−p−2−σ4 p ntrain−p−1 =σ4 −ntrain+ 2p+ 1 (ntrain−p−1)(n train... | https://arxiv.org/abs/2501.17520v3 |
To do so, werst introduce the same notations as in this paper. Denote R:=c(P 1−P 2) :c∈[0,∞), P 1, P2the linear space ofnite signed measures generated by the class of distributions M. In R, the supremum norm ∥·∥∞is the supremum difference of their distribution functions. The G ˆateaux derivative of P→E P[ℓ(f(X), y... | https://arxiv.org/abs/2501.17520v3 |
et al. (2023), the asymptotic efciency will be established by decomposing each predictiveness measure. Indeed, we are going torst prove that 1 ntestntest i=1ℓ(m(x i), yi)→E[ℓ(m(X), y)] This comes directly from Theorem 2 of Williamson et al. (2023). Then, we need to proof this efcient asymptotic convergence 1 ntes... | https://arxiv.org/abs/2501.17520v3 |
hypothesis, using Chebyshev’s inequality, we have that PH0 ≥z αsen+c√n ≤V() (zαsen+c√n)2→0 Improving the corrective term in linear settings:We observe that using Markov’s inequality we have that PH0 ≥z αsen+c√n ≤E() (zαsen+c√n) From Lemmas 3.9 and 3.10, we observe that in the linear setting, it is pos... | https://arxiv.org/abs/2501.17520v3 |
independent from Xand that m−j(X−j)is constant conditionally onX−j, theny⊥ ⊥XjX−j. To prove the other way, werst observe that E y2X−j =E (m(X) +ϵ)2X−j =E m(X)2X−j +σ2, using that ϵis centered and independent of X. On the other hand, we observe that using the conditional independence and also thatϵis centered... | https://arxiv.org/abs/2501.17520v3 |
Proposition 4.3, with ℓ=ℓ2, it is possible tox the calibration set size and correct the bias generated. Moreover, when ncalisxed to 1, as it is just a correction of CPI, it benets from its double robustness, making it easier to separate null covariates from important ones. In this way, this hyperparameterrepresents ... | https://arxiv.org/abs/2501.17520v3 |
Sobol-CPI achieves better discrimination of important covariates, assigns no importance to null covariates, and is signicantly more computationally efcient. covariates, while the thirdgure represents an unimportant covariate. They are theoretically obtained in Example K.2. On the bottom, the leftgure shows the AUC,... | https://arxiv.org/abs/2501.17520v3 |
powerful method, beneting from its double robustness property. In Figures 8 and 9, we observe a detailed comparison of the type-I error and power, respectively. Among the methods, the most powerful tests involve the quadratic correction. Nevertheless, these do not control the type-I error. Additionally, the uncorrecte... | https://arxiv.org/abs/2501.17520v3 |
no clear distinction between important and null covariates. In Figure 11, we observe, similarly to the linear case, that the quadratic correction is not sufcient to control the type-I error. However, we also see that the linear correction with the variance estimated via bootstrap is not sufcient either. This contrast... | https://arxiv.org/abs/2501.17520v3 |
is exactly Σ0,0−Σ0,−0Σ−1 −0,−0 Σ−0,0. We also observe that as it is a Toeplitz matrix, we have the property that Σ−0,0=ρΣ−0,1=ρΣ−0,−0 (1,0,,0)⊤. Thus, we can develop the last term as E V(X0X−0) =Σ0,0−Σ0,−0Σ−1 −0,−0 Σ−0,0 = 1−ρΣ 0,−0Σ−1 −0,−0 Σ−0,−0 (1,0,,0)⊤ = 1−ρΣ 0,−0(1,0,,0)⊤ = 1−ρ2 31 Figure 10:General... | https://arxiv.org/abs/2501.17520v3 |
the paper. • The authors are encouraged to create a separate ”Limitations” section in their paper. •The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specication, asymptotic approximations ... | https://arxiv.org/abs/2501.17520v3 |
the simulations and models used are carefully described, allowing for full reproducibility of all datasets and models. Additionally, the code is publicly available. Guidelines: • The answer NA means that the paper does not include experiments. •If the paper includes experiments, a No answer to this question will not be... | https://arxiv.org/abs/2501.17520v3 |
for a new open-source benchmark). •The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy) for more details. •The authors should provide instructions on data acces... | https://arxiv.org/abs/2501.17520v3 |
the text. 8.Experiments compute resources Question: For each experiment, does the paper provide sufcient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] Justication: A comparison of computation time is provided in the supp... | https://arxiv.org/abs/2501.17520v3 |
models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efciency and accessibility of ML). 11.Safeguards Question: Does the paper describe safeguards that have been put in place for responsible release of data... | https://arxiv.org/abs/2501.17520v3 |
paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [NA] Justication: There is no human subjects. Guidelines: •The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. •Inclu... | https://arxiv.org/abs/2501.17520v3 |
Wireless Network Topology Inference: A Markov Chains Approach James Martin∗Tristan Pryer∗†Luca Zanetti∗†‡ Abstract In this work, we address the problem of inferring the topology of a wireless network using limited observational data. Specifically, we assume that we can detect when a node is transmitting, but no further... | https://arxiv.org/abs/2501.17532v1 |
assumption regarding the network’s dynamics: we assume that the future state of the network depends only on its current state and is independent of its past states. While this assumption is not strictly valid for communication networks, we argue that it simplifies the model sufficiently to allow rigorous analysis while... | https://arxiv.org/abs/2501.17532v1 |
step in a new line of research on wireless network inference. Further investigation is required to fully assess the strengths and limitations of the methodology. We discuss potential directions for future research in the conclusion section. 2 Problem setup LetG= (V, E) be a graph representing the wireless network we wo... | https://arxiv.org/abs/2501.17532v1 |
u’s transmission being immediately followed by v’s, while N(u) counts the to- tal number of times uis transmitting. Notice that N(u) =P v∈VN(u, v). Moreover, these quantities can be computed having access only to St,vfor 0≤t≤T, v∈V. Let also M∈Rn×nbe defined as M(u, v) =N(u, v) N(u). Essentially, M(u, v) represents the... | https://arxiv.org/abs/2501.17532v1 |
Then, with high probability, ∥P−(M−(k−1)bΠ)∥ ≤ϵ. The theorem essentially tells us that ˆP=M−(k−1)bΠ is a good estimator forP. In particular, by observing the network for a large enough time, we can approximate Parbitrarily well in the operator norm. This approximation preserves important quantities of the network. For ... | https://arxiv.org/abs/2501.17532v1 |
size of the network suffices to well approximate P. The proof of Theorem 1, deferred to Section 7, is inspired by the techniques of [19]. In particular, it uses matrix concentration inequalities for Markov chains [10] and martingales [18]. We end this section by briefly discussing the validity of our Markovian model fo... | https://arxiv.org/abs/2501.17532v1 |
some simulations with TCP obtaining very similar results. We sample 500 pairs of sender/receiver nodes ( i, j) uniformly at random (with replacement) and transmit 3 packets of size 100 bytes from itojat time 30 + t, where tis uniformly distributed in [0 , T] for some transmission window length T >0. The time series mat... | https://arxiv.org/abs/2501.17532v1 |
14 16 18 20 Transmission window length, T0.50.60.70.80.91.0Proportion of correctly identified links Transfer entropy Lsym (d) Varying transmis- sion length Figure 1: Experimental results for a cyclic network topology of six nodes. Fig- ure (a) displays the adjacency matrix of the network. Figure (b) the estimated matri... | https://arxiv.org/abs/2501.17532v1 |
0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.02 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.03 0.00(c) Transfer entropy matrix 2 4 6 810 12 14 16 18 20 Transmission window length, T0.50.60.70.80.91.0Proportion of correctly identified links Transfer entropy Lsym (d) Varying transmis- sion length Figure 3: Experimental result... | https://arxiv.org/abs/2501.17532v1 |
0.15 0.06 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.07 0.06 0.15 0.00 0.14 0.06 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.01 0.02 0.03 0.04 0.05 0.06 0.06 0.07 0.08 0.06 0.14 0.00 0.15 0.05 0.07 0.05 0.04 0.03 0.02 0.01 0.01 0.02 0.03 0.04 0.05 0.06 0.06 0.06 0.07 0.08 0.06 0.15 0.00 0.15 ... | https://arxiv.org/abs/2501.17532v1 |
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.00 0.03 0.00 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.04 0.00 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.00 0.05 0.00 0.00 0.00 0.00 0.00 ... | https://arxiv.org/abs/2501.17532v1 |
bound on the following quantities ( u, v∈V): eN(u, v) =T−1X t=0X i̸=j1{X(i) t=u, X(j) t=v}, (2) Q(u, v) =T−1X t=0X i,i′1{X(i) t=u, X(i′) t=v}. (3) We will use the following recent matrix Bernstein inequality for Markov chains [10]. Theorem 2. Let{Xi}∞ i=1be a Markov chain with finite state space V, sta- tionary distrib... | https://arxiv.org/abs/2501.17532v1 |
which implies {Xk}kis a martingale difference sequence for {Yk}k, i.e., E k−1[Xk] = 0 for all k= 1,2, . . . . Assume the operator norm of Xkis bounded by Ralmost surely: ∥Xk∥ ≤Ra.s. for all k= 1,2, . . . Define W(1) k=Pk j=1Ej−1[X∗ jXj]andW(2) k=Pk j=1Ej−1[XjX∗ j]fork= 1,2, . . .. Further assume their operator norm is ... | https://arxiv.org/abs/2501.17532v1 |
t=Pt j=1Ej−1[YT jYj] andW(2) k=Pt j=1Ej−1[YjYT j]. Then, with high probability, it holds that ∥W(1) T∥ ≤8k3Tπ⋆. and ∥W(2) T∥ ≤2k3Tπ⋆. 14 Proof. We first obtain a bound on the operator norm of W(1) tLetu, v∈V. Then, YT jYj(u, v) =X w∈VYj(w, u)Yj(w, v) =X w∈VkX i=11{X(i) j−1=w} 1{X(i) j=u} −P(w, u) +X w∈VX 1≤i̸=ℓ≤k1{X(... | https://arxiv.org/abs/2501.17532v1 |
≤4k(k−1)Tmax u∈VX w∈Vπ(w)P(w, u) ≤4k(k−1)Tπ⋆ and TX t=1Bt ∞≤4k(k−1)Tπ⋆, which implies that TX t=1Bt ≤4k(k−1)Tπ⋆. Analogously, we can show that TX t=1Ct ≤4k(k−1)Tπ⋆. In a similar way we bound the contribution given by Dt. With high proba- bility, by Lemma 3 it holds that, TX t=1Dt ≤2k2max u∈VX w∈VN(w)P(w, u) ≤4k3Tπ⋆. By... | https://arxiv.org/abs/2501.17532v1 |
(k−1)Π)∥ ≤ϵ. 8 Acknowledgements All authors were partially supported by the Defence Science and Technology Laboratory. References [1] Ivan Brugere, Brian Gallagher, and Tanya Y Berger-Wolf. “Network struc- ture inference, a survey: Motivations, methods, and applications”. In: ACM Computing Surveys (CSUR) 51.2 (2018), p... | https://arxiv.org/abs/2501.17532v1 |
LIKELIHOOD LANDSCAPE OF BINARY LATENT MODEL ON A TREE DA VID CLANCY , JR., HANBAEK LYU, AND SEBASTIEN ROCH Abstract . We study the optimization landscape of maximum likelihood estimation for a binary la- tent tree model with hidden variables at internal nodes and observed variables at the leaves. This model, known as t... | https://arxiv.org/abs/2501.17622v1 |
independent observations of the states at the leaves of the tree. Formally, let σ(1),...,σ(m)be in- dependent samples from the model described above with true parameter Pθ∗. We only observe the leaf states, denoted by σ(j)|L=(σ(j) v;v∈L) for j=1,..., m. Our goal is to estimate θ∗from these partial observations. This es... | https://arxiv.org/abs/2501.17622v1 |
as F81 [11], when restricted to a single edge parameter ˆθe, the log-likelihood ℓis strictly concave and attains a unique maximizer. For more general discrete models, Dinh and Matsen [9] provide conditions under which the one-dimensional likelihood is guaranteed to have at most one stationary point, a condition satisfi... | https://arxiv.org/abs/2501.17622v1 |
above (Definition 3.1). In Section 4, we bound the diagonal entries of the expected Hessian. In Section 5, we bound the off-diagonal entries of the expected Hessian (41), which involves a di fficult-to-analyze product of dependent random variables. Roughly speaking, we handle this di fficulty by grouping consecutive te... | https://arxiv.org/abs/2501.17622v1 |
are of order O(δ−1) while the o ff-diagonal entries are exponentially small in the shortest path distance between the two corresponding edges. For its statement, we let H(ˆθ)∈R|E|×|E|denote be the Hessian of the expected log-likelihood, whose ( e,f) entry for e,f∈Eis defined as H(ˆθ)e,f=∂2 ∂ˆθe∂ˆθfEθ∗h ℓ(ˆθ;σ|L)i =Eθ∗... | https://arxiv.org/abs/2501.17622v1 |
from the context. IfTuconsists of a single node u, then Zu=Xuas we get to observe the spin at u. In general, Zuis a random variable determined by the spin configuration XLuon the leaves of Tuand takes values in [−1,1]. In fact, there is a recursive structure of magnetization, first established in Borgs, Chayes, Mossel,... | https://arxiv.org/abs/2501.17622v1 |
For edges e ={x,y}and f ={u,v}with dist(e,f)=N as above, we have ∂ ∂ˆθfZy=ZvNY j=1ˆθ{yj,yj−1}NY j=01−(ˆθ{yj,wj}Zwj)2 1+ˆθ{yj,wj}ˆθ{yj,yj−1}ZwjZyj−12. (16) 8 DA VID CLANCY , JR., HANBAEK LYU, AND SEBASTIEN ROCH In particular, ∂2 ∂ˆθe∂ˆθfℓ(ˆθ;σ|L)=ˆθeZxZv (1+ˆθeZxZy)2NY j=1ˆθ{yj,yj−1}NY j=0 1−(ˆθ{yj,w... | https://arxiv.org/abs/2501.17622v1 |
suppose we have two node-disjoint descendant subtrees TuandTv. Then for any node walong the shortest path between uandv, ZuyZv|σw. (25) which follows from the Markov property of the CFN model and the fact that Zuis determined byσLu. In fact, the ‘unsigned magnetizations’ σuZuare independent as long as the supporting de... | https://arxiv.org/abs/2501.17622v1 |
independence properties of unsigned magnetizations in Claim 3.4. Define events F :={flip on the edge e={x,y}}={σx,σy} A :={good reconstruction at both xandy} M:={one moderate failure and one good reconstruction among xandy} E :=A∪(Fc∩M).(28) Upper bound. Observe that the function t7→t2 (1+tˆθe)2is maxi... | https://arxiv.org/abs/2501.17622v1 |
Hessian yields the following upper bound ∂2 ∂ˆθe∂ˆθfℓ(ˆθ;σ|L) ≤(1−2c6δ)N (1+ˆθeZxZy)2NY j=01−(ˆθ{yj,wj}Zwj)2 (1+ˆθ{yj,wj}Zwjˆθ{yj,yj−1}Zyj−1)2, (33) recalling that N=dist(e,f) denotes the shortest-path distance between the ends of eand f(see Figure 2 b). We next introduce some notation to further simplify the bound in ... | https://arxiv.org/abs/2501.17622v1 |
(1+ξjηj)2for 4≤i≤N. (37) The above is a function of random variables ξi−3,ηi−3,ηi−2,ηi−1,ηisince they determine the value of ξi−2,ξi−1,ξithrough the recursion ξj+1:=ˆθjq(ηj,ξj). Since|ξi−3|≤1−2c6δ, if we fix a deterministic value xwith absolute value ≤1−2c6δand define random variables ξ◦ i−2,ξ◦ i−1,ξ◦ iusing the recurs... | https://arxiv.org/abs/2501.17622v1 |
the random variables eWN,WN−4,..., Wr+3,Rrdepend on disjoint sets of independent random variables, which yields the desired independence between them. □ It is important to note that the random variables that appear in the expression on the right-hand side of (41) above are independent by Lemma 5.2 so that we can easily... | https://arxiv.org/abs/2501.17622v1 |
Assumption A1 holds and let e ,f∈E(T)be any two edges such that N=1+dist(e,f)≥3and r∈{0,1,2,3}. Then for eWNand R rdefined in (39) and(40) (resp.) Eθ∗heWNi ≤C44 and Eθ∗[Rr]≤C44. (44) Furthermore, Eθ∗heW2 Ni ≤C45 δ4and Eθ∗h R2 ri ≤C45 δ3. (45) As we will see below, Lemma 5.3 and Lemma 5.4 give us good control on the fir... | https://arxiv.org/abs/2501.17622v1 |
will prove Lemmas 5.3, 5.4, and 5.5, which were used to prove Theorem 2.2 (ii)-(iii)in Section 5. For notational simplicity, we will prove Lemmas 5.3 and 5.4 for the case of W4andeW4, respec- tively. By shifting the index, we can conclude Lemmas 5.3 and 5.4 for the general case. We will also writeξjinstead ofξ◦ jso tha... | https://arxiv.org/abs/2501.17622v1 |
in (35), let us consider the function F(ˆθ1,η1,η2,ξ1) :=(1−η2 1)(1−η2 2) (1+ξ2η2)2(1+ξ1η1)2, whereξ2=ˆθ1η1+ξ1 1+η1ξ1. Recall that Za∈[−1,1] a.s. so under Assumption A1 (in particular (6)), our new random variables ξj,ηjsatisfy (36) a.s. and ˆθi∈[1−2C6δ,1−2c6δ]. The term Fappears when grouping two consecutive terms in t... | https://arxiv.org/abs/2501.17622v1 |
x,y∈(−1,1). □ We will also need a similar representation when dealing with eW4. We include this as a lemma. It follows from repeated applications of Lemma 6.6 (i). Lemma 6.10. Letηj,ξjsatisfy the recursions in (34). Then 1 (1+ξ5η5)24Y j=1(1−η2 j) (1+ξjηj)2=1 (1+ξ1˜η1)24Y j=1(1−η2 j) (1+ˆθjηj˜ηj+1)2 where ˜η5=η5and for ... | https://arxiv.org/abs/2501.17622v1 |
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