Local law and Tracy–Widom limit for sparse stochastic block models

Hwang, Jong Yun; Lee, Ji Oon; Yang, Wooseok

doi:10.3150/20-bej1201

Cited by 10 publications

(3 citation statements)

References 38 publications

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“…Methods based on likelihood or cross-validation are actively researched, yet the majority of them are commonly restrained by the scale of networks. Spectral methods are highly scalable for estimating k in large networks, although their rigorous analyses require delicate, highly technical random matrix arguments [Ajanki et al, 2017, Benaych-Georges et al, 2019, Chakrabarty et al, 2020, Dumitriu and Zhu, 2019, Benaych-Georges et al, 2020, Hwang et al, 2020.…”

Section: Prior Literaturementioning

confidence: 99%

Estimating Graph Dimension with Cross-validated Eigenvalues

Chen¹,

Roch²,

Rohe³

et al. 2021

Preprint

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In applied multivariate statistics, estimating the number of latent dimensions or the number of clusters is a fundamental and recurring problem. One common diagnostic is the scree plot, which shows the largest eigenvalues of the data matrix in decreasing order; the user searches for a "gap" or "elbow" in the decaying eigenvalues; unfortunately, these patterns can hide beneath the bias of the sample eigenvalues. This methodological problem is conceptually difficult because, in many situations, there is only enough signal to detect a subset of the k population dimensions/eigenvectors. In this situation, one could argue that the correct choice of k is the number of detectable dimensions. We alleviate these problems with cross-validated eigenvalues. Under a large class of random graph models, without any parametric assumptions, we provide a p-value for each sample eigenvector. It tests the null hypothesis that this sample eigenvector is orthogonal to (i.e., uncorrelated with) the true latent dimensions. This approach naturally adapts to problems where some dimensions are not statistically detectable. In scenarios where all k dimensions can be estimated, we prove that our procedure consistently estimates k. In simulations and a data example, the proposed estimator compares favorably to alternative approaches in both computational and statistical performance.

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Section: Prior Literaturementioning

confidence: 99%

Estimating Graph Dimension with Cross-validated Eigenvalues

Chen¹,

Roch²,

Rohe³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…On the methodological side, we aim to lift the second moment matching restriction for comparisons in previous works, e.g., [17,22,40], to prove edge universality. In applications generalized Wigner matrices arise as centered adjacency matrices of balanced stochastic block models [6,29,41].…”

Section: Introductionmentioning

confidence: 99%

Quantitative Tracy-Widom laws for the largest eigenvalue of generalized Wigner matrices

Schnelli¹,

Xu²

2022

Preprint

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We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix H converge to the Tracy-Widom laws at a rate nearly O(N −1/3 ), as the matrix dimension N tends to infinity. We allow the variances of the entries of H to have distinct values but of comparable sizes such that i E|h ij | 2 = 1. Our result improves the previous rate O(N −2/9 ) by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.

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“…for i = 1, 2, it is often useful to let ϕ approximate an indicator function. In particular, allowing ϕ to depend on n, one can use Theorem 1.2 (along with the explicit formula for C 1.2 given in (2.15)) to obtain local laws which describe the mesoscopic behavior of the eigenvalues; such local laws have been established in the random matrix theory literature for a variety of ensembles, see [1,2,3,6,11,13,19,20,26,27,28,30,36,37,40,43,45,47,49,51,67,69,75,76,77] and references therein for a partial list of such results. We will use Theorem 1.2 to establish a rate of convergence for the empirical spectral measure of Toeplitz matrices subject to small random perturbations in Section 1.2.…”

Section: Introductionmentioning

confidence: 99%

Quantitative results for non-normal matrices subject to random and deterministic perturbations

O’Rourke¹,

Wood²

2021

Preprint

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We consider the eigenvalue distribution of a fixed matrix subject to a small perturbation. In particular, we prove quantitative comparison results which show that the logarithmic potential is stable under perturbations with small norm or low rank, provided the smallest and largest singular values are not too extreme. We also establish a quantitative version of the Tao-Vu replacement principle. As an application of our results, we study the spectral distribution of banded Toeplitz matrices subject to small random perturbations, including the case when the bandwidth grows with the dimension. For this model, we obtain a rate of convergence in Wasserstein distance of the empirical spectral measure to its limiting distribution. We give a number of examples of our methods including random multiplicative perturbations and large classes of deterministic perturbations that behave similar to random perturbations.

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Local law and Tracy–Widom limit for sparse stochastic block models

Cited by 10 publications

References 38 publications

Estimating Graph Dimension with Cross-validated Eigenvalues

Estimating Graph Dimension with Cross-validated Eigenvalues

Quantitative Tracy-Widom laws for the largest eigenvalue of generalized Wigner matrices

Quantitative results for non-normal matrices subject to random and deterministic perturbations

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