Asymptotic Estimates for Spectral Estimators of Rotationally Invariant Matrices
Zhuohang He, Xiaojun Yuan, Junjie Ma · 2024
In this paper, we consider the recovery of low-rank matrices from noisy observations using spectral denoisers, where the singular values are denoised through an identical scalar smoothing function. We explore the asymptotic mean squared error (AMSE) of these denoisers within a framework where the rank of the matrix to be recovered grows linearly with the matrix size. We demonstrate that, under arbitrary i.i.d. noise and some mild regularity assumptions, the AMSE converges in probability to a deterministic function of the noise power. Our results are applicable to commonly used denoisers, including the best-rank-r denoiser, the singular-value soft-threshold denoiser, and the singular-value hard-threshold denoiser. To the best of our knowledge, this is the first study to establish an analytical expression for the asymptotic MSE under arbitrary i.i.d. noise. The derived analytical expression depends solely on the empirical distribution of the singular values of the low-rank matrix and the specific form of the spectral denoiser employed.