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arXiv · 2106.05183

Shrinkage Estimation of Functions of Large Noisy Symmetric Matrices

Panagiotis Lolas·Lexing Ying
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Abstract

We study the problem of estimating functions of a large symmetric matrix $A_n$ when we only have access to a noisy estimate $\hat{A}_n=A_n+σZ_n/\sqrt{n}.$ We are interested in the case that $Z_n$ is a Wigner ensemble and suggest an algorithm based on nonlinear shrinkage of the eigenvalues of $\hat{A}_n.$ As an intermediate step we explain how recovery of the spectrum of $A_n$ is possible using only the spectrum of $\hat{A}_n$. Our algorithm has important applications, for example, in solving high-dimensional noisy systems of equations or symmetric matrix denoising. Throughout our analysis we rely on tools from random matrix theory.

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Panagiotis Lolas, Lexing Ying. 2021-06-09. Shrinkage Estimation of Functions of Large Noisy Symmetric Matrices. https://arxiv.org/abs/2106.05183

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