Adaptive Shrinkage of singular values

TL;DR

This paper introduces a new adaptive singular value shrinkage estimator that improves low-rank matrix recovery from noisy data by using a data-driven parameter selection method, outperforming existing techniques.

Contribution

It proposes a continuum of thresholding functions and a fast, variance-free risk estimation criterion for improved low-rank matrix denoising.

Findings

Outperforms existing methods in mean squared error across various noise levels.

Accurately estimates the rank of the underlying signal.

Provides a computationally efficient parameter selection strategy.

Abstract

To recover a low rank structure from a noisy matrix, truncated singular value decomposition has been extensively used and studied. Recent studies suggested that the signal can be better estimated by shrinking the singular values. We pursue this line of research and propose a new estimator offering a continuum of thresholding and shrinking functions. To avoid an unstable and costly cross-validation search, we propose new rules to select two thresholding and shrinking parameters from the data. In particular we propose a generalized Stein unbiased risk estimation criterion that does not require knowledge of the variance of the noise and that is computationally fast. A Monte Carlo simulation reveals that our estimator outperforms the tested methods in terms of mean squared error on both low-rank and general signal matrices across different signal to noise ratio regimes. In addition, it accurately estimates the rank of the signal when it is detectable.