Random matrices: Overcrowding estimates for the spectrum

Nguyen, Hoi H.

doi:10.1016/j.jfa.2018.06.010

Cited by 16 publications

(28 citation statements)

References 29 publications

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“…The estimate (1.14) follows from the bulk universality result in [31]. Further, both estimates, for both symmetry classes, also follow from the main results of [47], which appeared later. At the edge, [14,Theorem 2.7] showed (1.14)…”

Section: Introductionsupporting

confidence: 54%

Optimal Delocalization for Generalized Wigner Matrices

Benigni¹,

Lopatto²

2020

Preprint

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We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove high probability delocalization bounds with sharp constants. Our proof uses an analysis of the eigenvector moment flow introduced by Bourgade and Yau (2017) to bound logarithmic moments of eigenvector entries for random matrices with small Gaussian components. We then extend this control to all generalized Wigner matrices by comparison arguments based on a framework of regularized eigenvectors, level repulsion, and the observable employed by Landon, Lopatto, and Marcinek (2018) to compare extremal eigenvalue statistics. Additionally, we prove level repulsion and eigenvalue overcrowding estimates for the entire spectrum, which may be of independent interest.

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Section: Introductionsupporting

confidence: 54%

Optimal Delocalization for Generalized Wigner Matrices

Benigni¹,

Lopatto²

2020

Preprint

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“…This will be achieved by applying the restricted invertibility principle originating in the classical work of Bourgain and Tzafriri [7]. The argument based on the restricted invertibility was used in the recent papers of Cook [11] and Nguyen [23], but our method will be different.…”

Section: Randomized Restricted Invertibilitymentioning

confidence: 99%

The sparse circular law under minimal assumptions

Rudelson

Tikhomirov

2019

Geom. Funct. Anal.

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“…We first state a special case of Corollary 1.17 from [Nguyen, 2018]. Here, N I (X) denotes the number of eigenvalues of X ∈ S n×n in the real interval I.…”

Section: B Lower-bound For Smallest Singular Valuesmentioning

confidence: 99%

Smoothed analysis of the low-rank approach for smooth semidefinite programs

Pumir¹,

Jelassi²,

Boumal³

2018

Preprint

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We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size n × k such that X = Y Y * is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced, and positive semidefiniteness is naturally enforced. However, optimization in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To answer it, under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem. We particularize our results to an SDP relaxation of phase retrieval.

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Random matrices: Overcrowding estimates for the spectrum

Cited by 16 publications

References 29 publications

Optimal Delocalization for Generalized Wigner Matrices

Optimal Delocalization for Generalized Wigner Matrices

The sparse circular law under minimal assumptions

Smoothed analysis of the low-rank approach for smooth semidefinite programs

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