Matrix Denoising with Partial Noise Statistics: Optimal Singular Value Shrinkage of Spiked F-Matrices

TL;DR

This paper introduces an asymptotically optimal singular value shrinkage method for denoising large low-rank matrices corrupted by unknown noise, leveraging side noise-only measurements to improve performance.

Contribution

It develops a novel noise covariance whitening and re-coloring workflow with a closed-form optimal shrinkage nonlinearity, enhancing matrix denoising with side information.

Findings

Outperforms traditional methods without side information in high-dimensional settings

Derives a closed-form optimal shrinkage function under mean square error loss

Provides new insights into eigenvector rotation of spiked F-matrices

Abstract

We study the problem of estimating a large, low-rank matrix corrupted by additive noise of unknown covariance, assuming one has access to additional side information in the form of noise-only measurements. We study the Whiten-Shrink-reColor (WSC) workflow, where a "noise covariance whitening" transformation is applied to the observations, followed by appropriate singular value shrinkage and a "noise covariance re-coloring" transformation. We show that under the mean square error loss, a unique, asymptotically optimal shrinkage nonlinearity exists for the WSC denoising workflow, and calculate it in closed form. To this end, we calculate the asymptotic eigenvector rotation of the random spiked F-matrix ensemble, a result which may be of independent interest. With sufficiently many pure-noise measurements, our optimally-tuned WSC denoising workflow outperforms, in mean square error, matrix denoising algorithms based on optimal singular value shrinkage which do not make similar use of noise-only side information; numerical experiments show that our procedure's relative performance is particularly strong in challenging statistical settings with high dimensionality and large degree of heteroscedasticity.