Tractable Evaluation of Stein’s Unbiased Risk Estimator with Convex Regularizers

P. Nobel, E. Candès, and S. Boyd

IEEE Transactions on Signal Processing, 71:4330–4341, 2023.

Stein's unbiased risk estimate (SURE) gives an unbiased estimate of the ell_2 risk of any estimator of the mean of a Gaussian random vector. We focus here on the case when the estimator minimizes a quadratic loss term plus a convex regularizer. For these estimators SURE can be evaluated analytically for a few special cases, and generically using recently developed general purpose methods for differentiating through convex optimization problems; these generic methods however do not scale to large problems. In this paper we describe methods for evaluating SURE that handle a wide class of estimators, and also scale to large problem sizes.