Singular vector distribution of sample covariance matrices

Ding, Xiucai

doi:10.1017/apr.2019.10

Cited by 10 publications

(5 citation statements)

References 37 publications

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“…Remark 2.6. The above assumption has previously appeared in [6,10,11,21]. It guarantees a regular square-root behavior of the spectral densities ρ 1,2c near λ r (see Lemma 3.6 below), which is used to prove the local deformed MP law at the soft edge.…”

Section: )mentioning

confidence: 89%

“…For this reason, the bad event {|x ij | ≥ q for some i, j} is negligible, and we will not consider the case it happens throughout the proof. Next we introduce a convenient self-adjoint linearization trick, which has been proved to be useful in studying the local laws of deformed sample random matrices [10,21,38]. We define the following (N + M ) × (N + M ) block matrix, which is a linear function of X.…”

Section: )mentioning

confidence: 99%

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A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

Ding¹,

Yang²

2018

Ann. Appl. Probab.

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In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form Q = T X(T X) * , where the sample X is an M2 × N random matrix with i.i.d. entries with mean zero and variance N −1 , and T is an M1 × M2 deterministic matrix satisfying T * T is diagonal. We study the asymptotic behavior of the largest eigenvalues of Q when M := min{M1, M2} and N tends to infinity with limN→∞ N/M = d ∈ (0, ∞). Under mild assumptions of T , we prove that the Tracy-Widom law holds for the largest eigenvalue of Q if and only if lims→∞ s 4 P(| √ N xij| ≥ s) = 0. This condition was first proposed for Wigner matrices by Lee and Yin [24]. *

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Section: )mentioning

confidence: 89%

Section: )mentioning

confidence: 99%

A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

Ding¹,

Yang²

2018

Ann. Appl. Probab.

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“…holds with high probability. Furthermore, denote ξ i , ζ i as the singular vectors of X, for some large constant C > 0, with high probability, we have [8] max…”

Section: Introductionmentioning

confidence: 99%

“…To illustrate our results and ideas, we give an overview of the present paper. As we have seen from [8,10], the self-adjoint linearization technique is quite useful in dealing with rectangular matrices. Hence, in a first step, we denote by…”

Section: Introductionmentioning

confidence: 99%

High dimensional deformed rectangular matrices with applications in matrix denoising

Ding¹

2020

Bernoulli

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We consider the recovery of a low rank M × N matrix S from its noisy observationS in the high dimensional framework when M is comparable to N . We propose two efficient estimators for S under two different regimes. Our analysis relies on the local asymptotics of the eigenstructure of large dimensional rectangular matrices with finite rank perturbation. We derive the convergent limits and rates for the singular values and vectors for such matrices.

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“…Gaussian X in [17,40]; the edge universality was later proved under various moment assumptions on the entries x ij [5,14,30,34]. Finally, for the (non-outlier) sample eigenvectors, the completely delocalization [30,43], quantum unique ergodicity [8], distribution of the eigenvector components [13] and convergence of eigenvector empirical spectral distribution [56] have been constructed.…”

Section: Introductionmentioning

confidence: 99%

Spiked separable covariance matrices and principal components

Ding

Yang

2019

Preprint

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We study a class of separable sample covariance matrices of the form r Q1 :" r A 1{2 X r BX ˚r A 1{2 . Here r A and r B are positive definite matrices whose spectrums consist of bulk spectrums plus several spikes, i.e. larger eigenvalues that are separated from the bulks. Conceptually, we call r Q1 a spiked separable covariance matrix model. On the one hand, this model includes the spiked covariance matrix as a special case with r B " I. On the other hand, it allows for more general correlations of datasets. In particular, for spatio-temporal dataset, r A and r B represent the spatial and temporal correlations, respectively. In this paper, we study the outlier eigenvalues and eigenvectors, i.e. the principal components, of the spiked separable covariance model r Q1. We prove the convergence of the outlier eigenvalues r λi and the generalized components (i.e. xv, r ξiy for any deterministic vector v) of the outlier eigenvectors r ξi with optimal convergence rates. Moreover, we also prove the delocalization of the non-outlier eigenvectors. We state our results in full generality, in the sense that they also hold near the so-called BBP transition and for degenerate outliers. Our results highlight both the similarity and difference between the spiked separable covariance matrix model and the spiked covariance model matrix in [8]. In particular, we show that the spikes of both r A and r B will cause outliers of the eigenvalue spectrum, and the eigenvectors can help to select the outliers that correspond to the spikes of r A (or r B). Contents 1 Introduction 2 Definitions and main results 3 Statistical estimation for spiked separable covariance matrices 4 Basic tools

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Singular vector distribution of sample covariance matrices

Cited by 10 publications

References 37 publications

A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

High dimensional deformed rectangular matrices with applications in matrix denoising

Spiked separable covariance matrices and principal components

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