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on a particular signal-to-noise ratio that makes the estimation problem the hardest. As a result, the factor of SNR affecting practical results is masked by the minimax framework. 2. The approximations we obtain for the minimax risk in rate-optimal minimaxity, and even in Theorem 1 are not accurate enough for distingui... | https://arxiv.org/abs/2501.13323v1 |
2≤kτ2o . (6) The new parameter τ, viewed as a constraint on the averaged magnitude of non-zero signal com- ponents, is a measure of signal strength. Compared to the vanilla sparse parameter space in (2), Θ(k, τ) can monitor the changes in SNR. Hence, the minimax framework we will develop with this parameter space can r... | https://arxiv.org/abs/2501.13323v1 |
strength, the minimax framework will set the SNR to a value that falls within Regime III. That is why Theorem 1 is agnostic to SNR. On the other hand, the minimax risk in fact undergoes a phase transition as SNR decreases from Regime III to Regimes I-II. Hence, a minimax analysis that is not considering the SNR will on... | https://arxiv.org/abs/2501.13323v1 |
1 +o(1) . 6 In addition, the ridge estimator ˆβR(λ)with tuning λ=pσ2/(kτ2)is asymptotically minimax optimal up to the second order, i.e. sup β∈Θ(k,τ)Eβ∥ˆβR(λ)−β∥2 2=kτ2 1−kτ2 pσ2 1 +o(1) . The proof of this theorem is presented in Section D. The condition k/n→0 is very mild in the high-dimensional sparse regressi... | https://arxiv.org/abs/2501.13323v1 |
by a larger margin. Remark 7. Compared to the optimal estimator ˆβR(λ)in Regime I, the (nearly) minimax optimal estimator ˆβE(λ, γ)in Regime II employs ℓ1-regularization in addition to the ℓ2-regularization. To shed light on the importance of ℓ1-regularization in Regime II, the following proposition reveals the subopti... | https://arxiv.org/abs/2501.13323v1 |
the elastic-net regularized estimator in our plots is denoted in green with the caption ‘enet’. 4. The ridge regression estimator in (9): ˆβR(λ)∈arg min b∈RpnX i=1(yi−xT ib)2+λ∥b∥2 2, where λ≥0 is the tuning parameter. The implementation is performed by computing the closed form ˆβR(λ) = ( XTX+λIp)−1XTy. The performanc... | https://arxiv.org/abs/2501.13323v1 |
SNR regimes. By leveraging additional convex regularization, [18, 24] developed new variants of best subset selection that perform consistently well in various SNR regimes. [23, 34, 33] adopted the linear asymptotic framework where both the sparsity kand sample sizenscale linearly with the dimension p, to establish con... | https://arxiv.org/abs/2501.13323v1 |
DW Mann. “The discarding of variables in multivariate analysis”. In: Biometrika 54.3-4 (1967), pp. 357–366. [3] Pierre C Bellec, Guillaume Lecu´ e, and Alexandre B Tsybakov. “Slope meets lasso: improved oracle bounds and optimality”. In: The Annals of Statistics 46.6B (2018), pp. 3603–3642. [4] P. J. Bickel, Y. Ritov, ... | https://arxiv.org/abs/2501.13323v1 |
Yu. “Minimax rates of estimation for high- dimensional linear regression over ℓq-balls”. In: IEEE transactions on information theory 57.10 (2011), pp. 6976–6994. [26] Weijie Su and Emmanuel Candes. “SLOPE is adaptive to unknown sparsity and asymptoti- cally minimax”. In: (2016). [27] Robert Tibshirani. “Regression shri... | https://arxiv.org/abs/2501.13323v1 |
. Lemma 5 (Lemma 1.2.1 in [28]) .LetXdenote a non-negative random variable. Then, we have E(X) =Z∞ 0P(X > t )dt. Lemma 6 (Theorems 3 & 4 in [12]) .Suppose Xfollows a noncentral chi-squared distribution with degrees of freedom pand the noncentrality parameter λ, i.e., X∼χ2 p(λ). Then, (i) for c >0,P(X > p +λ+c)≤exph −pc... | https://arxiv.org/abs/2501.13323v1 |
each block 1 ≤j≤k, randomly select an index I∈[m] and set β(j)=λeI; The selection between different blocks are independent. The spike choice λcan depend on the tuple ( n, p, k ). But we drop such a dependence throughout the proof for notational simplicity. From the construction steps, it already implies that πIB(λ;p, k... | https://arxiv.org/abs/2501.13323v1 |
18 By choosing proper values for λ, we can use Proposition 3 to obtain the lower bounds required in Theorem 2. These choices are clarified below. •For Regime I, let λ=µ→0, then λ2=o (2−δ) log( p/k) and hence R(Θ(k, µ),1)≥B(πIB(λ;p, k))≥kµ2 1 +o(1) . •For Regime II, let λ=µ, then λ2=o (2−δ) log( p/k) and hence R(Θ... | https://arxiv.org/abs/2501.13323v1 |
. Then to show Bn,mp→+∞, using the above expression and the continuous mapping theorem, it’s sufficient to see that under the conditions m→ ∞ , λ2≤(2−δ) logm, (1 +o(1))·logm−λ2 2+λOp(1) ≥δ+o(1) 2·logm+Op(p logm)p→+∞. 20 Lemma 11. Assume the same conditions of Lemma 10. Consider the random variable An,mdefined in(24). T... | https://arxiv.org/abs/2501.13323v1 |
P ¯Sn,m−an,m bn,m > ϵ in (27) converges to zero. We first have P ¯Sn,m−an,m bn,m > ϵ ∥λX1+z∥2 ≤ϵ−2b−2 n,m·Var ¯Sn,m| ∥λX1+z∥2 22 ≤ϵ−2b−2 n,m·mX i=2E Y2 m,i 1(Ym,i≤bn,m) ∥λX1+z∥2 , (31) where the first inequality applies Chebychev’s inequality, and the second inequality holds since {¯Ym,i, i= 2, . . . , m }are in... | https://arxiv.org/abs/2501.13323v1 |
Ym,i> b| ∥λX1+z∥2 =P Z−√n λ∥λX1+z∥22 +V < nλ−2 ∥λX1+z∥2 2−2 logb ∥λX1+z∥2 (ii) E Y2 m,i 1(Ym,i≤b)| ∥λX1+z∥2 = 1 +2λ2 n−n 2exp 2∥λX1+z∥2 2 2 +n/λ2! · P Z−n/λ2 p 2 +n/λ2∥λX1+z∥22 +V≥n+ 2λ2 λ2 ∥λX1+z∥2 2−2 logb ∥λX1+z∥2 (iii) E Ym,i 1(Ym,i≤b)| ∥λX1+z∥2 = 1 +λ2 n−n 2exp ∥λX1+z∥2 2 2(1 + n/λ2)! · P Z−... | https://arxiv.org/abs/2501.13323v1 |
that g∼ N(0, Ip), and max Q⊆[p]:|Q|≤k∥XT Q(√nz/∥z∥2)∥2= max u∈SkuTg. We can thus continue from (38) to obtain sup β∈Θ(k,τ)Eβ∥ˆβBS−β∥2 2≥σ2·E1 ¯θ2 k(X)· max u∈SkuTg2 ≥4σ2 9·E 1¯θk(X)≤3/2· max u∈SkuTg2 ≥4σ2 9· E max u∈SkuTg2−r E max u∈SkuTg4·q P(¯θk(X)>3/2) ,(39) where we have used Cauchy–Schwarz inequality... | https://arxiv.org/abs/2501.13323v1 |
Using the independence between Xandz, the second term in (42) can be bounded as E∥(XTX+λI)−1XTz∥2 2≤1 λ2E∥XTz∥2 2=1 λ2·p nE∥z∥2 2=kµ2·kµ2 p. (44) Combining (42), (43) and (44), we conclude sup β∈Θ(k,µ)E∥ˆβR−β∥2 2≤kµ2 1−kµ2 p+okµ2 p . D.2 Lower bound We derive the lower bound (41) based on the independent block prio... | https://arxiv.org/abs/2501.13323v1 |
i=1e−1 2µ2∥Xi∥2 2 eµXT i(µX1+z)+e−µXT i(µX1+z) ≥2mX i=1e−1 2µ2∥Xi∥2 2. (52) Therefore, to show (i), it is sufficient to show EmeµXT 1z e1 2µ2∥X1∥2 2−e−1 2µ2∥X1∥2 2 µ2Pm i=1e−1 2µ2∥Xi∥2 2:=EWn→1. (53) We prove the above result through dominated convergence theorem in two steps: (1) we show Wnp→1; (2) we show EW2 n=O... | https://arxiv.org/abs/2501.13323v1 |
iX1−1)eµXT i(µX1+z)−1 2µ2∥Xi∥2 2 +eµxT 1z−1 2µ2∥X1∥2 2mX i=2(e2µ2XT iX1−1)e−µXT i(µX1+z)−1 2µ2∥Xi∥2 2:= ∆ 21+ ∆ 22. (54) Let us first show E∆1 BA=o µ2 m . We use the same argument as in the proof of Lemma 15 to bound the denominator: A≥2me−µ2 2mPm i=1∥Xi∥2 2, B≥2me−µ2 2mPm i=1∥Xi∥2 2. Furthermore, since ∆1≥0⇔XT 1z≤0,... | https://arxiv.org/abs/2501.13323v1 |
. (56) To simplify the above lower bound, we note that the numerator only depends on X1andv:= µX2+z. We construct a random variable ¯ v:=−n µX2+zsuch that ( X1, v,¯v) are mutually independent. It is clear that the denominator can be written as a function of ( X1, v,¯v). Hence, we can again apply Jensen’s inequality to ... | https://arxiv.org/abs/2501.13323v1 |
σ= 1. Suppose n→ ∞ ,p/k→ ∞ andlog(p/k)/n→0. Ifµ→ ∞ , µ=o p log(p/k) andµ4/n→0, then the Bayes risk satisfies B(π±IB(µ;p, k))≥kµ2 1−k 2p·eµ2 1 +o(1) . Proof. Like in the proof of Proposition 4, let m=p/kand define the symmetric spike prior π±S(µ;m) for β∈Rm: select an index I∈[m] uniformly at random and then set β=... | https://arxiv.org/abs/2501.13323v1 |
+ exp −µXT jy (1 +µ2/n)3/2−µ2 2∥Xj∥2 2 ≥exp −µ4max 2≤j≤m(∥v∥2 2∨ ∥Xj∥2 2) 2n(1 +µ2/n) ·mX j=2 exp µXT jy (1 +µ2/n)3/2−µ2∥Xj∥2 2 2(1 + µ2/n) + exp −µXT jy (1 +µ2/n)3/2−µ2∥Xj∥2 2 2(1 + µ2/n) (67) This result together with Lemma 21 implies E[U 1I∩II]≤Eexpµ2(2 +µ2/n)∥y∥2 2 2n(1 +µ2/n)2+µ4max 2≤j≤m(∥v∥2 2∨ ∥Xj∥2... | https://arxiv.org/abs/2501.13323v1 |
+ µ2/n)∥v∥2 2 + exp −µ (1 +µ2/n)3/2vTy−µ2 2(1 + µ2/n)∥v∥2 2 . Proof. Multiplying both sides of (73) by expµ2(2+µ2/n)∥y∥2 2 2n(1+µ2/n)2+µvTy (1+µ2/n)3/2+µ2∥v∥2 2(1+µ2/n) , we can obtain h expµ2(2 +µ2/n)∥y∥2 2 2n(1 +µ2/n)2+2µvTy (1 +µ2/n)3/2 −1i ·h expµ2(2 +µ2/n)∥y∥2 2 2n(1 +µ2/n)2 −1i ≥1−exp −µ4/n∥y∥2 2 n(1 +µ... | https://arxiv.org/abs/2501.13323v1 |
+ µ2/n) . 43 Also, the other part of the denominator has a simple lower bound: expµ2(2 +µ2/n) 2n(1 +µ2/n)2∥y∥2 2+µ (1 +µ2/n)3/2vTy−µ2 2(1 + µ2/n)∥v∥2 2 + exp −µ2(2 + 3 µ2/n) 2n(1 +µ2/n)2∥y∥2 2−µ(1 + 2 µ2/n) (1 +µ2/n)3/2vTy−µ2 2(1 + µ2/n)∥v∥2 2 ≥exp −µ4max 2≤j≤m(∥v∥2 2∨ ∥Xj∥2 2) 2n(1 +µ2/n) ·expµvTy (1 +µ2/n)3/2... | https://arxiv.org/abs/2501.13323v1 |
proof, we will prove AP→1 and BP→0. Regarding A, using the moment-generating function of noncentral chi-squared distribution, we can obtain that with λ=n∥y∥2 2 µ2(1+µ2/n), EA=E E A y =E"1 + 2 µ2/n 1 +µ2/nn 2exp∥y∥2 2 2(1 + µ2/n)(1 + 2 µ2/n) ·E exp−µ2χ2 n(λ) 2n(1 +µ2/n) y# = 1. Hence, to prove AP→1, it is su... | https://arxiv.org/abs/2501.13323v1 |
µ2/n) 1H3 y# (b)=P Z≤ −µ[(2 + µ2/n)(1 + 2 µ2/n) + 4] 4√n(1 +µ2/n)(1 + 2 µ2/n)1/2∥y∥2 ≤P Z≤ −µ[(2 + µ2/n)(1 + 2 µ2/n) + 4] 4√n(1 +µ2/n)(1 + 2 µ2/n)1/2∥y∥2,∥y∥2≥1 2p n+µ2 +P ∥y∥2<1 2p n+µ2 (c) ≤P Z≤ −µ[(2 + µ2/n)(1 + 2 µ2/n) + 4] 8(1 + µ2/n)1/2(1 + 2 µ2/n)1/2 + expn 2 3 4+ log1 4(d)→0, where H3=n ˜X1,1<−µ(2+... | https://arxiv.org/abs/2501.13323v1 |
find an upper bound for the denominator (inside the square) of (88) on A1∩ A 2∩ A 3. We first have 1 +µ2 n−n 2≤exp −µ2 2 1−µ2 2n ,since log(1 + x)≥x−x2 2,∀x >0. Then, on A1∩ A 2we obtain exp µX1,1∥y∥2−µ2 2∥X1∥2 2 + exp −µX1,1∥y∥2−µ2 2∥X1∥2 2 ≤2 exp3µ2∥y∥2 2 n(1 +µ2/n)−(1−t)µ2 2 . Also, since µ2/n→0, it is ... | https://arxiv.org/abs/2501.13323v1 |
1Ac 2 ≤Eexp 2µX1,1∥y∥2−µ2X2 1,1 ·Eexp −µ2∥X1,−1∥2 2 1∥X1,−1∥2 2<1−t (c)= 1 +2µ2 n−1 2Eexp2µ2∥y∥2 2 n(1 + 2 µ2/n) ·Eexp −µ2∥X1,−1∥2 2 1∥X1,−1∥2 2<1−t (d)= 1 +2µ2 n−1 2 1−4µ2(1 +µ2/n) n(1 + 2 µ2/n)−n 2·Eexp −µ2∥X1,−1∥2 2 1∥X1,−1∥2 2<1−t (e)= 1 +2µ2 n−n 2 1−4µ2(1 +µ2/n) n(1 + 2 µ2/n)−n 2·P χ2 n−1< n ... | https://arxiv.org/abs/2501.13323v1 |
help evaluate the risk of ˆβE. Indeed, as will be shown shortly, the squared distance between ˆβEand˜βE is negligible, and hence their risks are asymptotically the same. We can then focus on ˜βEwhich is more amenable to risk calculation. To this end, (91) allows us to rewrite ˆβEasˆβE=1 1+γη(β+w+∆, λ/2). Combining this... | https://arxiv.org/abs/2501.13323v1 |
n. To evaluate f(δ), we bound the three expectations in (99). Let a2 θ=(kµ2+n)∥θ∥2 2 n, and ϕ(·) and Φ( ·) denote the pdf and cdf of N(0,1), respectively. We obtain ∀δ∈[0, µ], Eη2(ω,2µ) =E 2σ2 θh (1 + 4 µ2σ−2 θ)(1−Φ(2µσ−1 θ))−2µσ−1 θϕ(2µσ−1 θ)i ≤E (σ5 θµ−3/2 + 3 σ7 θµ−5/16)ϕ(2µσ−1 θ) ≤E (a5 θµ−3/2 + 3 a7 θµ−5/16)ϕ... | https://arxiv.org/abs/2501.13323v1 |
< σ−1 θ<∞, Lemma 8 Part (i) implies that σ2 θr(uσ−1 θ;χ1σ−1 θ, χ2), as a function of u, is symmetric, and increasing over u∈[0,+∞). So is its expectation, i.e. g(u;χ1, χ2). Moreover, we have max (x,y):x2+y2=c2 g(x;χ1, χ2) +g(y;χ1, χ2) = max (x,y):x2+y2=c2E σ2 θ· r(xσ−1 θ;χ1σ−1 θ, χ2) +r(yσ−1 θ;χ1σ−1 θ, χ2) ≤E σ2... | https://arxiv.org/abs/2501.13323v1 |
the second term E∥(XTX+λI)−1XTz∥2 2. Then, E∥(XTX+λI)−1XTz∥2 2≥E1 (λ+σ1)2∥XTz∥2 2 =E" 1 (λ+σ1)2∥XTz∥2 2 1 σ1≤ 2+√p n2# +E 1 (λ+σ1)2∥XTz∥2 2 1 σ1> 2+√p n2 ≥1 λ+ 1 +q p n22·E ∥XTz∥2 2 1 σ1≤ 2+√p n2 57 =1 λ+ 1 +q p n22·E E ∥XTz∥2 2 1 σ1≤ 2+√p n2 X =1 λ+ 1 +q p n22·E" ∥X∥... | https://arxiv.org/abs/2501.13323v1 |
Capturing heterogeneous time-variation in covariate effects in non-proportional hazard regression models Niklas Hagemann1,2, Thomas Kneib3and Kathrin M¨ ollenhoff1,2 1Institute of Medical Statistics and Computational Biology, Faculty of Medicine, University of Cologne, Germany 2Division of Mathematics, Department of Ma... | https://arxiv.org/abs/2501.13525v1 |
in practice (Li et al., 2015; Jachno et al., 2019). Frequently, this is caused by non-proportional hazards, i.e. the covariates or their effects are time- dependent. In addition, in many cases the effects might not be linear but of a more complex form. In particular, heterogeneity in terms of treatment-specific, subgro... | https://arxiv.org/abs/2501.13525v1 |
a penalized approach providing a sufficiently good fit while preventing overfitting. Under the assumption of piecewise constant hazard rates, restructuring the data leads to the likelihood of the survival model being proportional to the one of a Poisson regression model (see 2 Section 2 of Bender et al. (2018) for deta... | https://arxiv.org/abs/2501.13525v1 |
based on a Poisson model and, hence, can make use of existing tools for generalized linear models (GLMs) and generalized additive models (GAMs). They require the partition of the time axis into a finite number of intervals and assume the hazard rate to be constant within each interval. The piecewise exponential model i... | https://arxiv.org/abs/2501.13525v1 |
needs to be large enough to provide a sufficiently good fit while overfitting is prevented due to the penalization. Hence, the standard choice of cut-off points for PAMMs is using all unique observed survival times. For sufficiently large datasets with relatively dense and precisely measured survival times (leading to ... | https://arxiv.org/abs/2501.13525v1 |
they are not only heterogeneous and time-varying but the time-variation is potentially heterogeneous, too. We model these effects by functional random coefficients , which have been recently proposed by Hagemann et al. (2024). They are constructed by using functional random effects (FRE; Kneib et al., 2019) as varying ... | https://arxiv.org/abs/2501.13525v1 |
in each component being subject to at most one penalty which makes estimation numerically stable. For the detailed step-wise construction procedure see section 3 of Wood et al. (2013). 6 4 Simulations In this simulation study, we investigate the performance of the proposed approach with regard to two critical aspects: ... | https://arxiv.org/abs/2501.13525v1 |
Effect of x2in the DGPs of the four scenarios. The first subplot (from the left) shows scenario (I), i.e. heterogeneous time-variation, the second one scenario (II), i.e. the combination of heterogeneity and time-variation (without interaction), the third one scenario (III), i.e. heterogeneity only, and the last one sc... | https://arxiv.org/abs/2501.13525v1 |
- (IV) there is one block of consisting of four boxplots, one for each model (i) - (iv). 9 equal for all three fit measures. This strongly indicates that model (i) is penalized towards the true (less complex) model, as desired. In contrast, in scenario (III) we can observe model (i) to fit slightly better than model (i... | https://arxiv.org/abs/2501.13525v1 |
choose cubic P-splines with first order penalty and 9 inner knots, such that each of the intervals corresponds to one year. The estimated regression coefficients are shown in Table 2. A higher age significantly increases the hazard rate, which is an expected result as age usually increases the risk of death. In contras... | https://arxiv.org/abs/2501.13525v1 |
only slightly increases the model fit compared to the increase that is achieved by using the functional random coefficient. This coincides with the impression from Figure 3, indicating that the time-variation might cancel out if it is not modeled diagnoses specifically. The fact that this also applies to the AIC indica... | https://arxiv.org/abs/2501.13525v1 |
for High- Dimensional Data in Toxicology“ (RTG 2624, P7) funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, Project Number 427806116). Competing interests The authors declare no competing interests. References Andersen, P. K. and Gill, R. D. (1982). Cox’s regression model for counting proce... | https://arxiv.org/abs/2501.13525v1 |
methods. PLoS One , 10(1):e0116774. Mehta, K. R., Nakao, K., Zuraek, M. B., et al. (2005). Fractional genomic alteration detected by array- based comparative genomic hybridization independently predicts survival after hepatic resection for metastatic colorectal cancer. Clinical Cancer Research , 11(5):1791–1797. Murphy... | https://arxiv.org/abs/2501.13525v1 |
1Information-theoretic limits and vector approximate message-passing for high-dimensional time series Daria Tieplova, Samriddha Lahiry, Jean Barbier Abstract High-dimensional time series appear in many scientific setups, demanding a nuanced approach to model and analyze the underlying dependence structure. Theoretical ... | https://arxiv.org/abs/2501.13625v2 |
this article, we make an important step in that direction by rigorously establishing single-letter formulas for information-theoretic quantities in a stochastic regression model, where the rows of the measurement matrix come from an AR(1) process [11]. D. Tieplova is with The Abdus Salam International Centre for Theore... | https://arxiv.org/abs/2501.13625v2 |
. , λ p)with|λi|<1for all i. Under this assumption, equation (I.1) breaks down into pindependent equations xµ+1,i=λixµ,i+ξµ,i, i= 1, . . . , p , which allows us to easily calculate the variance of xµ,iand the correlations between xµ,iandxµ+s,i: E(xµ+1,i)2=λ2 iE(xµ,i)2+ 1. (I.4) Since xµis a stationary process, its vari... | https://arxiv.org/abs/2501.13625v2 |
every i= 1, . . . , k ,G(i) Nis an N× |Ii|independent matrix with i.i.d. standard Gaussian elements. With this, (I.3) can be rewritten as YN=1√p(Λ1/2 1,NG(1) N, . . . ,Λ1/2 k,NG(k) N)β0+ZN. (I.8) The block structure of the design matrix naturally propagates to the signal. In what follows we define β(i) 0:= (β0,j)j∈Ii∈R... | https://arxiv.org/abs/2501.13625v2 |
the dependent (or structured) design setup less is known. Yet, a single-letter formula for a large class of correlated matrices forming a subclass of the right-rotationally invariant ensemble has been proved in [8] and later extended in [35]. From the algorithmic perspective, for the measurement matrices with i.i.d. Ga... | https://arxiv.org/abs/2501.13625v2 |
some auxiliary results used in Section IV. II. M AIN RESULTS In this section we state our main theorem on the asymptotic expression for the normalized mutual information. Consider the stochastic regression model given by equations (I.2) and (I.1). We assume that we are in the Bayesian optimal setting, meaning that the ... | https://arxiv.org/abs/2501.13625v2 |
setup follows along similar lines. 5 a) General Ap:Theorem II.1 can be generalized to a broader class of matrices Ap. Let us consider the sequence of diagonal matrices {Ap}∞ p=1with eigenvalues (λ1,p, . . . , λ p,p)and their respective empirical eigenvalue measure ηp=1 pPp i=1δλi,p. We assume the following natural hypo... | https://arxiv.org/abs/2501.13625v2 |
:=1 NE∥1√pΦNβ0− ⟨1√pΦNβ⟩∥2. (II.18) It is easy to find an expression for ymmse through the I-MMSE formula [28] dip dσ−2=cN 2ymmse . (II.19) From Theorem II.1 we obtain the expression for the limit of ymmse lim p→∞ymmse =2 clim p→∞dip dσ−2=2 cdiRS(˜r) dσ−2, (II.20) where ˜r= (˜r1,˜r2)is the global minimum of iRSonΓ. The... | https://arxiv.org/abs/2501.13625v2 |
of two models with Gaussian signal βand equal SNR. The red curve corresponds to the model with Ap= 0and the blue one to the model where Aphas eigenvalues {0.9,0.7,0.5,0.1} with equal multiplicity each. We can observe that MSE obtained with V AMP (marked with "+") matches replica prediction (continuous lines). As was me... | https://arxiv.org/abs/2501.13625v2 |
AMP algorithm. The plot on the right shows two models with right rotationally invariant design matrices ( Ap= 0andAp= 0.9Ipand on the left we see two models with block right rotationally invariant design matrices ( Ap={0.9,0.1}andAp={0.9,0.8,0.7}). highly unstable. Furthermore, the greater the dispersion of the eigenva... | https://arxiv.org/abs/2501.13625v2 |
normalized mutual information for the interpolating model is given by ip,ϵ(t) :=−1 pEΦ,V,β0lnZ dP0(β)Dvexp − H(t,β,v,Yt,˜Yt,Φ) −c+ 1 2. (IV .8) First note that ip=ip,ϵ(0) + o(1). Indeed, denote the interpolating free energy, i.e. the first term on the RHS of (IV .8)) by fp,ϵ(t). Define the Hamiltonian of the original ... | https://arxiv.org/abs/2501.13625v2 |
expectation w.r.t. all quenched variables (i.e. fixed by the realization of the problem), namely (β0,V,Yt,˜Yt,Φ), or equivalently w.r.t. (β0,V,Z,˜Z,Φ). Here H′means a time-derivative, given by H′(t,β0,V,Yt,˜Yt,Φ) =1 2σ2NX µ=1Zµs 1 p(1−t)(Φβ0)µ−kX i=1r2,i(t)√lip R2,i(t)(Λ(i)V(i))µ −1 2kX i=1|Ii|X j=1˜Z(i) jr1,i(t)p R1... | https://arxiv.org/abs/2501.13625v2 |
moments of prior distribution P0andcfor which: 1 s2kpZ [sp,2sp]2kdϵZ1 0dtE⟨(Qi−E⟨Qi⟩t,ϵ)2⟩t,ϵ≤Cp−γ,fori= 1, . . . , k. (IV .15) Proof. The proof follows in the same vein as the concentration of overlaps proved [4], [5], hence we outline the steps and omit the details. Indeed it is enough to show that for all ϵ E⟨(Qi−E⟨... | https://arxiv.org/abs/2501.13625v2 |
the proof of Lemma 2.2 in [8]. For this we fix R= (R1,1, . . . , R 2,k), where for each i= 1. . . , k R 1,i(t,ϵ) =ϵ1,i+Rt 0r1,i(s,ϵ)dsandR2,i(t,ϵ) =ϵ2,i+Rt 0r2,i(s,ϵ)dsare the solutions to the Cauchy problem ∂R1,i=Ai(ρ−E⟨Q1⟩t,ϵ, . . . , ρ −E⟨Qk⟩t,ϵ) =:F1,i(t,R(t)) ∂R2,i=ρ−E⟨Qi⟩t,ϵ=:F2,i(t,R(t)) R(0) =ϵ(IV .25) w... | https://arxiv.org/abs/2501.13625v2 |
integrate by parts the RHS with respect to noise, we obtain −1 N√pNX µ=1E[Zµ⟨(E2Φ(β0−β))µ⟩] =1 N√pNX µ=1E[⟨(E2Φ(β0−β))µ(1√pΦ(β0−β) +Z)µ⟩] −1 N√pNX µ=1E[⟨(E2Φ(β0−β))µ⟩⟨(1√pΦ(β0−β) +Z)µ⟩] =1 NE[⟨∥1√pEΦ(β0−β)∥2⟩]−1 NE[∥⟨1√pEΦ(β0−β)⟩∥2] (IV .32) Due to Nishimori identity 1 NE[⟨∥1√pEΦ(β0−β)∥2⟩] = 21 NE[∥⟨1√pEΦ(β0−β)⟩∥2] (IV... | https://arxiv.org/abs/2501.13625v2 |
=Sk i=1[αi−1, αi], withαi=a+i(b−a)/k, and define a sequence of p×pdiagonal matrices ¯Ap(k) = diag( ¯λ1,p(k), . . . , ¯λp,p(k)), where each matrix has exactly kdifferent eigenvalues: {α0, . . . , α k−1}. We remind that the initial sequence of {Ap}∞ p=1is of the form Ap= diag( λ1,p, . . . , λ p,p). We desire that ¯Ap(k)b... | https://arxiv.org/abs/2501.13625v2 |
the random part is of order one. To bound limp→∞max i≤k∥Λi,N−¯Λi(k)∥we recall that asymptotically the eigenspaces of Λiand¯Λi(k)are the same (see Appendix B) so lim p→∞∥Λi,N−¯Λi(k)∥ ≤ 1 (1−λi)2−1 (1−¯λi(k))2 ≤|λi−¯λi(k)|(2−λi−¯λi(k)) (1−¯λi(k))2(1−λi)2≤C k. (IV .49) This gives us limp→∞|ip−¯ip(k)|=O(1/k). Now, to deal ... | https://arxiv.org/abs/2501.13625v2 |
2019. [6] J. Barbier and N. Macris. The adaptive interpolation method for proving replica formulas. applications to the curie–weiss and wigner spike models. Journal of Physics A: Mathematical and Theoretical , 52(29):294002, 2019. [7] J. Barbier, N. Macris, M. Dia, and F. Krzakala. Mutual information and optimality of ... | https://arxiv.org/abs/2501.13625v2 |
Theory and Related Fields , 70:95–175, 2018. [26] A. Fletcher, M. Sahraee-Ardakan, S. Rangan, and P. Schniter. Expectation consistent approximate inference: Generalizations and convergence. In 2016 IEEE International Symposium on Information Theory (ISIT) , pages 190–194. IEEE, 2016. [27] U. Grenander and G. Szegö. Toe... | https://arxiv.org/abs/2501.13625v2 |
Rangan. Generalized approximate message passing for estimation with random linear mixing. In 2011 IEEE International Symposium on Information Theory Proceedings , pages 2168–2172. IEEE, 2011. [48] S. Rangan, P. Schniter, and A. K. Fletcher. Vector approximate message passing. IEEE Transactions on Information Theory , 6... | https://arxiv.org/abs/2501.13625v2 |
Bernoulli , 28(3):1835 – 1861, 2022. APPENDIX A REPLICA CALCULATION In this appendix we derive the replica symmetric potential (A.46) of the free energy using the so called replica method [40] which was first introduced in the context of the spin glasses and became a powerful method of statistical physics. It is based ... | https://arxiv.org/abs/2501.13625v2 |
λN−2 ............ λN−1λN−2. . . 1 , (A.11) then the eigenvalues of matrix INu+Rv/σ2are, for each i= 1, . . . N : (δi/σ2+ 1/Q12)((u−1)Q12+Q11) + 2−Q11/Q12=δi((u−1)Q12+Q11)/σ2+u+ 1with multiplicity 1 (δi/σ2+ 1/Q12)(Q11−Q12) + 2−Q11/Q12=δi(Q11−Q12)/σ2+ 1with multiplicity u−1. This allows us to find (A.10) explicitly:... | https://arxiv.org/abs/2501.13625v2 |
In other words, Q12=mmse (β|q ˆQ12β+Z). 25 This leaves us with only two simple equation ˆQ12=Z Rdµ(δ)cδ Q12δ+σ2, (A.28) Q12=mmse (β|q ˆQ12β+Z). To simplify (A.28) we use properties of Kac-Murdock-Szegö (A.11) matrix which are described in details in the literature, we need only to know that for any nice function Fwe ha... | https://arxiv.org/abs/2501.13625v2 |
that KMS matrices asymptotically share the same eigenvector space (for more details we again refer to [27]), so in the limit we have lim p,u1 puln(det( INu+Rv/σ2))−1/2 =−c 2πhZπ 0ln(kX i=1liδi(θ)(Q(i) 11−Q(i) 12)/σ2+ 1)dθ+Zπ 0Pk i=1liδi(θ)Q(i) 12/σ2+ 1 Pk i=1liδi(θ)(Q(i) 11−Q(i) 12)/σ2+ 1dθi ,(A.37) where we denote δi(... | https://arxiv.org/abs/2501.13625v2 |
come from a Fourier coefficients of the function Fρ(θ) =1−ρ2 1−2ρcosθ+ρ2=+∞X n=−∞ρ|n|einθ, (B.2) and because of this such function also sometimes called generating function of AN,ρ. From [27], the eigenvalues ofλ1(AN,ρ)< λ2(AN,ρ)< . . . < λ N(AN,ρ)are λi(AN,ρ) =Fρ(θi,N), (B.3) 28 where(i−1)π N+1< θi,N<iπ N+1, fori= 1, ... | https://arxiv.org/abs/2501.13625v2 |
. . . , u′ i, . . . , u p)| ≤ci,1≤i≤p. IfX= (X1, . . . , X p)be a random vector with i.i.d. N(0,1)entries. Then the following holds: Var(g(X))≤1 4pX i=1c2 i. LetW1= (Z,˜Z). As a first step, we prove the following lemma 29 Lemma D.3. Eh 1 plogZ −EW1h1 plogZ 2i ≤Cp−1. Proof. Letg(W1) = log Z. The partial derivatives wit... | https://arxiv.org/abs/2501.13625v2 |
=1 pβ(i)⊺β′(i),˜Qi(β′,β) =1 Np(Φβ′)⊺E2Λi,NΦβ. (E.1) Note that we have −1 Np2E[⟨(Φβ′)⊺E2Λi,N(Φ(β0−β))(β(i) 0−β(i))⊺β(i)⟩ =E[⟨Qi(β,β)˜Qi(β′,β0)⟩]−E[⟨Qi(β,β)˜Qi(β′,β)⟩] +E[⟨Qi(β0,β)˜Qi(β′,β)⟩]−E[⟨Qi(β0,β)˜Qi(β′,β0)⟩] The third and fourth terms of the RHS cancel each other due to the Nishimori identity and the first two te... | https://arxiv.org/abs/2501.13625v2 |
q:=1 N(˜Φβ)⊺˜Φ(β0− β)concentrates around its expected value. The proof now follows in the same vein as in [7]. One defines Mas M:=1 pNX µ=1(˜Φβ)2 µ 2−(˜Φβ)µ(˜Φβ0)µ−σ 2(˜Φβ)µZµ. The following equations hold: Za ϵdσ−1E[⟨(M − ⟨M⟩ )2⟩] =o(1), (E.9) Za ϵdσ−1E[ (E⟨M⟩ − ⟨M⟩ )2] =o(1), (E.10) Za ϵdσ−1E[⟨(q−E⟨q⟩)2⟩] =o(1). (E.1... | https://arxiv.org/abs/2501.13625v2 |
Consistent spectral clustering in sparse tensor block models Ian Välimaa and Lasse Leskelä 24 January 2025 Abstract High-order clustering aims to classify objects in multiway datasets that are preva- lent in various fields such as bioinformatics, social network analysis, and recommen- dation systems. These tasks often ... | https://arxiv.org/abs/2501.13820v1 |
question of interest is to find how small ρcan be for the recovery to be possible and computationally feasible. As the main motivation comes from hypergraphs, the focus is mostly on binary arrays. However, the developed theory includes also other more general integer-valued distributions such as Poisson distributions. ... | https://arxiv.org/abs/2501.13820v1 |
a sufficiently large constant Cand computationally feasible for ρ≥Cn−d/2with a sufficiently large constant C. 1.2 Organization This paper is structured as follows. Section 2 discusses the used tensor formalism and statis- tical model. Section 3 presents a spectral clustering algorithm and states the main theorem descri... | https://arxiv.org/abs/2501.13820v1 |
community or the block) of ik∈[nk]along mode kislkifzk(ik) =lk. In this work, we are interested in large and sparse TBMs. A large TBM is a sequence of TBMsindexedby ν= 1,2,...sothattheorder disfixedandthedatatensorgrowsaccording ton1,ν···nd,ν→∞asν→∞. A sequence of integer-valued TBMs (i.e., Yν∈Zn1,ν×···×nd,ν) is consid... | https://arxiv.org/abs/2501.13820v1 |
To estimate the cluster membership vectors of all modes, it may be run d times, once for each mode. Algorithm 1. Hollow spectral clustering input:Data tensorY∈Rn1×···×nd, modek, number of clusters rk, trimming thresholdCtrimρ2, relaxation constant Q> 1 output: Cluster membership vector ˆzk: [nk]→[rk] SetAto be the Gram... | https://arxiv.org/abs/2501.13820v1 |
coefficientα(Assumption 2.3). Consider parameter regime r3/2 k√n1···nd∨nklognk n1···nd≪ρ≪n−1 k(n1···nd/nk)−ε(3.2) for some constant ε > 0andrk≪n2/3 k. Then there exists a constant Csuch that if Ctrim≥C∨ε−1, then the estimated cluster membership vector ˆzkgiven by Algorithm 1 on modekis weakly consistent. Furthermore, t... | https://arxiv.org/abs/2501.13820v1 |
on the left depicts the core tensor with a noninformative aggregate matrix and the cube on the right with an informative aggregate matrix. is independent of z(i1). This implies that aggregating the data into an order- 1tensor loses information. Similartohypergraphs,thedatatensorisassumedtobebinary Y∈{ 0,1}n×n×n so that... | https://arxiv.org/abs/2501.13820v1 |
will slightly speed up the computations. Then the algorithm clusters the first mode of the low-rank tensor with k-means++. Since HSC is initialized with a simple rank-approximation (via eigenvalue decomposition of (mat 1Y)(mat 1Y)T), it is expected not to improve γ. That is, we expect γ≈1.33. (iv)Aggregate SVD. Spectra... | https://arxiv.org/abs/2501.13820v1 |
180 Number of nodes (n)102 Density ( ) HSC 0.50.60.70.80.91.0 Clustering accuracy 30 60 100 180 Number of nodes (n)102 Density ( ) Aggregate SVD 0.50.60.70.80.91.0 Clustering accuracyFigure 3: Comparison of the four algorithms when the aggregate matrix is not informative. The number of nodes varies logarithmically betw... | https://arxiv.org/abs/2501.13820v1 |
from left to right. Accuracy ( 1−ℓ) ofk-means clustering is also recorded. hypergraph), Ais a symmetric matrix with upper-diagonal entries Ai1i2=/summationdisplay i3,...,idYi1...id= (d−2)!/summationdisplay i3<···<idYi1...id= (d−2)!/summationdisplay hyperedgeeI{{i1,i2}⊂e}, i 1<i2. Thatis, uptoamultiplicativeconstant(whi... | https://arxiv.org/abs/2501.13820v1 |
states that the detection task is impossible ifρ≤cn−d 2D−d−2 2for a sufficiently small constant candD≤cn. Since polynomial-time algorithms correspond to D≍lnn, the threshold corresponds roughly to ρ≤cn−d 2log−d−2 2n. Moreover, Kunisky showed that the detection task becomes easier when an aggregated tensor remains infor... | https://arxiv.org/abs/2501.13820v1 |
suboptimal. In the case of subgaussian TBM, Han, Luo, Wang, and Zhang [21] approach by analyzing concentrationofasingularsubspaceofapossiblywiderandommatrix(i.e., avectorsubspace spanned by the first rright or left singular vectors corresponding to the rlargest singular values of a random matrix). For this purpose, the... | https://arxiv.org/abs/2501.13820v1 |
anyt>0, Markov’s inequality (with respect to the counting measure) gives |{j|∥ˆθˆzj−Xj∥2≥t2}|≤/summationtext j∥ˆθˆzj−Xj∥2 t2≤32Qr∥E∥2 t2. Consider the set A={j∈[n]|∥ˆθˆzj−Xj∥<t}. Ifzi̸=zjfori,j∈A, then ∥ˆθˆzi−Szj∥≥∥Szi−Szj∥−∥Szi−ˆθˆzi∥>∆−t. Ifzi=zjfor somei,j∈A, then∥ˆθˆzi−Szj∥< t. Consider tsatisfying ∆−t=t, or equiva... | https://arxiv.org/abs/2501.13820v1 |
0) satisfying the MGF bound with variance proxy σ2(Assumption 2.1). Define a trimmed version of the matrix as X′ ij= Xij,∥Xi:∥1∨∥X:j∥1≤Ctrim(n∨m)σ2, 0,otherwise. Then for all t≥1, P/parenleftbigg ∥X′∥op≥(28t+ 8.09Ctrim+ 417)/radicalig (n∨m)σ2/parenrightbigg ≤2(e(n∨m))−2t. Proof.See Appendix A. ■ Theorem 6.4. Let... | https://arxiv.org/abs/2501.13820v1 |
[2] D. Arthur and S. Vassilvitskii. “k-means++: the advantages of careful seeding”. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’07) . New Orleans, Louisiana, 2007, pp. 1027–1035. [3] K. Avrachenkov, M. Dreveton, and L. Leskelä. “Community recovery in non-binary and temporal... | https://arxiv.org/abs/2501.13820v1 |
M. Wang, and A. R. Zhang. “Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit”. In: Journal of the Royal Statistical Society Series B: Statistical Methodology 84.5 (2022), pp. 1666–1698. [22] C. J. Hillar and L.-H. Lim. “Most tensor problems are NP-hard”. In: Journal of the ACM60.6 (... | https://arxiv.org/abs/2501.13820v1 |
1103–1127. [41] M. Wang and Y. Zeng. “Multiway clustering via tensor block models”. In: Advances in Neural Information Processing Systems (NeurIPS) 32 (2019). [42] H. Weyl. “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Dif- ferentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraum... | https://arxiv.org/abs/2501.13820v1 |
P/parenleftiggn/summationdisplay i=1I/braceleftigg Si≥(1 +t)/parenleftigg/radicalbigg 8mσ2logen k∨4 logen k 3/parenrightigg/bracerightigg ≥k/parenrightigg ≤(en)−t for allk≥1. Choosek= 1∨en emσ2/8so that ifmσ2≤8 logen, thenk=en emσ2/8and /radicalbigg 8mσ2logen k∨4 logen k 3=mσ2∨mσ2 6=mσ2, P/parenleftiggn/summatio... | https://arxiv.org/abs/2501.13820v1 |
(4ε−1+2)nandwhenever (u1,...,un)∈Sε,also (u1,...,−ui,...,un)∈ Sε. Define a constant Cε= 2 log(4ε−1+ 2)(so that|Sε×Sε|≤eCεn). By Exercise 4.4.3 in [39], we have ∥X′∥≤1 1−2εmax u,v∈SεuTX′v=1 1−2εmax u,v∈Sε/summationdisplay i,juivjX′ ij. ApplyingaChernoffbounddirectlyonsums/summationtext i,juivjX′ ijwillworksufficientlywe... | https://arxiv.org/abs/2501.13820v1 |
jσ2 τ2(e|λ|−1−|λ|) ≤exp/parenleftiggσ2 τ2(e|λ|−1−|λ|)/parenrightigg . For anyt≥3σ2/τ, Bernstein’s inequality (D.3) gives P /uniondisplay u,v∈Sε /summationdisplay (k,l)∈L/summationdisplay i∈Sk,j∈TluivjXij≥t ≤/summationdisplay u,v∈SεP /summationdisplay (k,l)∈L/summationdisplay i∈Sk,j∈Tluivj τXij≥t τ ... | https://arxiv.org/abs/2501.13820v1 |
Without loss of generality, since (k,l)/∈H3, we may assume that log2en sk≥ CH3logX(Sk,Tl) esktlσ2and CH3CLe 3a2 kn≤/parenleftbigg2en sk/parenrightbigg1 CH3log2en sk =/parenleftbigg2en sk/parenrightbigg1 CH3/parenleftigg CH4log/parenleftbigg2en sk/parenrightbigg1 CH4/parenrightigg ≤CH4/parenleftbigg2en sk/parenrightbi... | https://arxiv.org/abs/2501.13820v1 |
4ms2σ6log2en s∨4σ2slog2en s 3satisfiesslog2en s=t2 s 4msσ6∧3ts 4σ2so that P n/uniondisplay i=1/uniondisplay ∅̸=S⊂[n]/braceleftig (AXT)i:(S)≥(1 +t)ts/bracerightig ≤nn/summationdisplay s=1exp/parenleftigg slog2en s−(1 +t)/parenleftiggt2 s 4msσ6∧3ts 4σ2/parenrightigg/parenrightigg =nn/summationdisplay s=1exp/par... | https://arxiv.org/abs/2501.13820v1 |
ijEX2 i′j = (n−1)mp2(1−p)2. ■ This suggests that XXT−EXXTconsists of two parts, a diagonal and an off-diagonal part. For small p≪1, we expect the norm of the diagonal to be of order√mpand the norm of the off-diagonal to be of order√nmp. If this holds, then the diagonal dominates the off-diagonal, when√mp≫√nmp, or equiv... | https://arxiv.org/abs/2501.13820v1 |
matrices as each entry of X2is finite almost surely. Let Kmaxdenote the number of nonzero components (which is random as X2 is random). By Lemma D.1:(v), we have Eeλ(|X2(i,j)−EX2(i,j)|−E|X2(i,j)−EX2(i,j)|)≤e2σ2(eλ−1−λ)≤e3σ2(eλ−1−λ)for allλ≥0 and consequently Eeλ/summationtext i(|X2(i,j)−EX2(i,j)|−E|X2(i,j)−EX2(i,j)|)≤E... | https://arxiv.org/abs/2501.13820v1 |
t 4log1 3enσ2−√ t 4 ≤3√ 5n−2e−√ t/5+m−1e−√ t/4 ≤3√ 5 + 1 ne−√ t/4 ≤8 ne−(t3/2)1/3/4. 53 The claim follows by writing the assumption on tast3/2≥53/2∨83log3m log3(1/3enσ2). ■ Lemma C.3 (Decoupling) .LetX∈Zn×mbe a random matrix with independent integer- valued entries that satisfy Assumption 2.1 with variance proxy σ2andE... | https://arxiv.org/abs/2501.13820v1 |
Theorem 6.4. Lett≥0. Since E|Xij−EXij|≤2E|Xij|≤2σ2andmσ2≥8 logen, 56 applying Lemma A.1 gives P/parenleftiggn/uniondisplay i=1/braceleftbigg ∥(X−EX)i:∥1≥2mσ2+ (1 +t)/radicalig 8mσ2logen/bracerightbigg/parenrightigg ≤(en)−t. Underahigh-probabilityevent/intersectiontextn i=1/braceleftig ∥(X−EX)i:∥1<2mσ2+ (1 +t)√8mσ2l... | https://arxiv.org/abs/2501.13820v1 |
IfXis a centered random variable satisfying Condition (2.1) with variance proxy σ2 andξis a random variable independent of Xsatisfying|ξ|≤1almost surely, then their productξXsatisfies EξX=EξEX= 0and EeλξX=EE(eλξX|ξ)≤Eeσ2(e|λξ|−1−|λξ|)≤eσ2(e|λ|−1−|λ|). (v) IfXis a centered random variable satisfying Condition (2.1) with... | https://arxiv.org/abs/2501.13820v1 |
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