Unbiased Risk Estimates for Singular Value Thresholding and Spectral Estimators

TL;DR

This paper introduces an unbiased risk estimation method for spectral estimators like singular value thresholding, enabling automatic regularization parameter selection and improving low-rank matrix recovery from noisy data.

Contribution

It develops a general unbiased risk estimate formula for spectral estimators under Gaussian noise, including new results on matrix function differentiability.

Findings

Provides a formula for unbiased risk estimation of SVT.

Demonstrates improved parameter selection in MRI data denoising.

Offers theoretical insights into matrix function differentiability.

Abstract

In an increasing number of applications, it is of interest to recover an approximately low-rank data matrix from noisy observations. This paper develops an unbiased risk estimate---holding in a Gaussian model---for any spectral estimator obeying some mild regularity assumptions. In particular, we give an unbiased risk estimate formula for singular value thresholding (SVT), a popular estimation strategy which applies a soft-thresholding rule to the singular values of the noisy observations. Among other things, our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems. In particular, we demonstrate the utility of the unbiased risk estimation for SVT-based denoising of real clinical cardiac MRI series data. We also give new results concerning the differentiability of certain matrix-valued functions.