Reconstruction of a Low-rank Matrix in the Presence of Gaussian Noise

TL;DR

This paper introduces a novel matrix reconstruction method called RMT, which effectively reduces noise impact on low-rank matrices observed with Gaussian noise, outperforming existing thresholding techniques.

Contribution

The paper develops a new noise-adjusted reconstruction method based on random matrix theory, improving accuracy over traditional thresholding approaches.

Findings

RMT outperforms oracle soft and hard thresholding methods.

The method closely matches the performance of an ideal oracle scheme.

The approach effectively reverses noise effects on singular values and vectors.

Abstract

In this paper we study the problem of reconstruction of a low-rank matrix observed with additive Gaussian noise. First we show that under mild assumptions (about the prior distribution of the signal matrix) we can restrict our attention to reconstruction methods that are based on the singular value decomposition of the observed matrix and act only on its singular values (preserving the singular vectors). Then we determine the effect of noise on the SVD of low-rank matrices by building a connection between matrix reconstruction problem and spiked population model in random matrix theory. Based on this knowledge, we propose a new reconstruction method, called RMT, that is designed to reverse the effect of the noise on the singular values of the signal matrix and adjust for its effect on the singular vectors. With an extensive simulation study we show that the proposed method outperform even oracle versions of both soft and hard thresholding methods and closely matches the performance of a general oracle scheme.