Unbiased risk estimates for matrix estimation in the elliptical case

This paper is concerned with additive models of the form Y = M + epsilon, where Y is an observed n x m matrix with m < n, M is an unknown n x m matrix of interest with low rank, and epsilon is a random noise whose distribution is elliptically symmetric. For general estimators <(M)over cap> of M, we develop unbiased risk estimates, including in the special case where epsilon is Gaussian with covariance matrix proportional to the identity matrix. To this end, we develop a new Stein-Haff type identity. We apply the theory to a model selection framework with estimators defined through a soft-thresholding function. We establish the robustness of our approach within a large subclass of elliptical distributions. (C) 2017 Elsevier Inc. All rights reserved.