Title: A comment on Stein's unbiased risk estimate for reduced rank estimators,
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
Candès 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.