A comment on Stein's unbiased risk estimate for reduced rank estimators

TL;DR

This paper examines the applicability of Stein's unbiased risk estimate (SURE) to reduced rank estimators, providing conditions under which SURE can be used for risk estimation despite estimator discontinuities.

Contribution

It establishes a sufficient condition for Stein's lemma to hold for discontinuous spectral estimators, including reduced rank estimators, clarifying when SURE is applicable.

Findings

SURE applies to certain spectral estimators with discontinuities

Reduced rank estimators satisfy the new Stein's lemma condition

Hard thresholding estimators do not satisfy the condition and are not covered

Abstract

In the framework of matrix valued observables with low rank means, Stein's unbiased risk estimate (SURE) can be useful for risk estimation and for tuning the amount of shrinkage towards low rank matrices. This was demonstrated by Cand\`es et al. (2013) for singular value soft thresholding, which is a Lipschitz continuous estimator. SURE provides an unbiased risk estimate for an estimator whenever the differentiability requirements for Stein's lemma are satisfied. Lipschitz continuity of the estimator is sufficient, but it is emphasized that differentiability Lebesgue almost everywhere isn't. The reduced rank estimator, which gives the best approximation of the observation with a fixed rank, is an example of a discontinuous estimator for which Stein's lemma actually applies. This was observed by Mukherjee et al. (2015), but the proof was incomplete. This brief note gives a sufficient condition for Stein's lemma to hold for estimators with discontinuities, which is then shown to be fulfilled for a class of spectral function estimators including the reduced rank estimator. Singular value hard thresholding does, however, not satisfy the condition, and Stein's lemma does not apply to this estimator.