Adaptive Estimation of Noise Variance and Matrix Estimation via USVT Algorithm

TL;DR

This paper introduces an adaptive method for estimating noise variance and the underlying matrix in large noisy matrices without assuming low rank, using an improved variance estimator and USVT algorithm.

Contribution

It presents a novel approach combining an existing variance estimator with a modified USVT algorithm for improved matrix estimation when noise variance is unknown.

Findings

Provides an upper bound on the mean squared error of the variance estimator.

Establishes an upper bound on the mean squared error of the matrix estimate.

Demonstrates the effectiveness of the adaptive method in noisy matrix estimation.

Abstract

We propose a method for estimating the entries of a large noisy matrix when the variance of the noise, $\sigma^2$, is unknown without putting any assumption on the rank of the matrix. We consider the estimator for $\sigma$ introduced by Gavish and Donoho \cite{Gavish} and give an upper bound on its mean squared error. Then with the estimate of the variance, we use a modified version of the Universal Singular Value Thresholding (USVT) algorithm introduced by Chatterjee \cite{Chatterjee} to estimate the noisy matrix. Finally, we give an upper bound on the mean squared error of the estimated matrix.