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February 2021 Singular vector and singular subspace distribution for the matrix denoising model
Zhigang Bao, Xiucai Ding, and Ke Wang
Ann. Statist. 49(1): 370-392 (February 2021). DOI: 10.1214/20-AOS1960


In this paper, we study the matrix denoising model $Y=S+X$, where $S$ is a low rank deterministic signal matrix and $X$ is a random noise matrix, and both are $M\times n$. In the scenario that $M$ and $n$ are comparably large and the signals are supercritical, we study the fluctuation of the outlier singular vectors of $Y$, under fully general assumptions on the structure of $S$ and the distribution of $X$. More specifically, we derive the limiting distribution of angles between the principal singular vectors of $Y$ and their deterministic counterparts, the singular vectors of $S$. Further, we also derive the distribution of the distance between the subspace spanned by the principal singular vectors of $Y$ and that spanned by the singular vectors of $S$. It turns out that the limiting distributions depend on the structure of the singular vectors of $S$ and the distribution of $X$, and thus they are nonuniversal. Statistical applications of our results to singular vector and singular subspace inferences are also discussed.


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Zhigang Bao. Xiucai Ding. and Ke Wang. "Singular vector and singular subspace distribution for the matrix denoising model." Ann. Statist. 49 (1) 370 - 392, February 2021.


Received: 1 December 2018; Revised: 1 October 2019; Published: February 2021
First available in Project Euclid: 29 January 2021

Digital Object Identifier: 10.1214/20-AOS1960

Primary: 60B20, 62G10
Secondary: 15B52, 62H10, 62H25

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 1 • February 2021
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