High dimensional deformed rectangular matrices with applications in matrix denoising

Ding, Xiucai

doi:10.3150/19-bej1129

Cited by 42 publications

(37 citation statements)

References 34 publications

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“…, whereλ 2 k :=λ 2 k − 2m n ·σ 2 ξ . The shrinkage estimators {λ k } r k≥1 are inspired by random matrix theory ( [16]). Similarly, we define the estimator of Λ −2 2 F aŝ…”

Section: Data-dependent Confidence Regions Of Singular Subspacesmentioning

confidence: 99%

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Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

Dong

2019

IEEE Trans. Inform. Theory

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Low-rank matrix regression refers to the instances of recovering a low-rank matrix based on specially designed measurements and the corresponding noisy outcomes. In the last decade, numerous statistical methodologies have been developed for efficiently recovering the unknown low-rank matrices. However, in some applications, the unknown singular subspace is scientifically more important than the low-rank matrix itself. In this article, we revisit the low-rank matrix regression model and introduce a two-step procedure to construct confidence regions of the singular subspace. The procedure involves the de-biasing for the typical low-rank estimators after which we calculate the empirical singular vectors. We investigate the distribution of the joint projection distance between the empirical singular subspace and the unknown true singular subspace. We specifically prove the asymptotical normality of the joint projection distance with data-dependent centering and normalization when r 3/2 (m 1 +m 2 ) 3/2 = o(n/ log n) where m 1 , m 2 denote the matrix row and column sizes, r is the rank and n is the number of independent random measurements. Consequently, we propose data-dependent confidence regions of the true singular subspace which attains any pre-determined confidence level asymptotically. In addition, non-asymptotical convergence rates are also established. Numerical results are presented to demonstrate the merits of our methods.

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“…, whereλ 2 k :=λ 2 k − 2m n ·σ 2 ξ . The shrinkage estimators {λ k } r k≥1 are inspired by random matrix theory ( [16]). Similarly, we define the estimator of Λ −2 2 F aŝ…”

Section: Data-dependent Confidence Regions Of Singular Subspacesmentioning

confidence: 99%

“…for some absolute constant C 1 > 0, where we also used the fact ∆ 2 F = O P σ ξ · rm n and n rm. To prove the the concentration bound forB n andV n , we apply the results from random matrix theory [16]. Then, we can immediate show that the following bounds hold with probability at least 1 − 1 m 2 for all 1 ≤ j ≤ r,…”

Section: Proof Of Lemma 14mentioning

confidence: 99%

Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

Dong

2019

IEEE Trans. Inform. Theory

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“…It was shown in [13], [16], [26], and [39] that the linearizing block matrix is quite useful in dealing with rectangular matrices.…”

Section: Notation and Toolsmentioning

confidence: 99%

“…Another important piece of information from our result is that the singular vectors are completely delocalized. This property can be applied to the problem of low rank matrix denoising [13], i.e. \[\hat S = TX + S,\] where S is a deterministic low rank matrix.…”

Section: Introductionmentioning

confidence: 99%

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Singular vector distribution of sample covariance matrices

Ding

2019

Adv. Appl. Probab.

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We consider a class of sample covariance matrices of the form Q = T XX * T * , where X = (xij) is an M × N rectangular matrix consisting of i.i.d entries and T is a deterministic matrix satisfying T * T is diagonal. Assuming M is comparable to N , we prove that the distribution of the components of the singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of xij coincide with the Gaussian random variables. For the singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of xij match with those of Gaussian random variables. Similar results have been proved for Wigner matrices by Knowles and Yin in [19].

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Limiting Canonical Distribution of Two Large-Dimensional Random Vectors

Bai

Hou

et al. 2021

Methodology and Applications of Statistics

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High dimensional deformed rectangular matrices with applications in matrix denoising

Cited by 42 publications

References 34 publications

Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

Singular vector distribution of sample covariance matrices

Limiting Canonical Distribution of Two Large-Dimensional Random Vectors

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