OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage

TL;DR

OptShrink introduces a data-driven singular value shrinkage algorithm that optimally denoises low-rank matrices by leveraging random matrix theory, outperforming traditional convex regularization methods.

Contribution

The paper develops an exact characterization of optimal singular value weights for matrix denoising, enabling a practical algorithm that improves low-rank signal estimation.

Findings

Optimal weights derived from data improve denoising performance.

The method outperforms nuclear norm thresholding in simulations.

Applicable to matrices with missing entries and various noise models.

Abstract

The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable)_denoising_ problem of how to best approximate a low-rank signal matrix buried in noise by optimal (re)weighting of the singular vectors of the measurement matrix. We exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the i.i.d. Gaussian noise case. Our analysis brings into sharp focus the shrinkage-and-thresholding form of the optimal weights, the non-convex nature of the associated shrinkage function (on the singular values) and explains why matrix regularization via singular value thresholding with convex penalty functions (such as the nuclear norm) will always be suboptimal. We validate our theoretical predictions with numerical simulations, develop an implementable algorithm (OptShrink) that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries.