Generalized SURE for optimal shrinkage of singular values in low-rank matrix denoising

We consider the problem of estimating a low-rank signal matrix from noisy measurements under the assumption that the distribution of the data matrix belongs to an exponential family. In this setting, we derive generalized SURE formulas that hold for any smooth spectral estimators which shrink or threshold the singular values of the data matrix. This allows to obtain new data-driven shrinkage rules, whose optimality is discussed using tools from random matrix theory and through numerical experiments.