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

*Jérémie Bigot, Charles Deledalle, Delphine Féral*; 18(137):1−50, 2017.

### Abstract

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 Stein's unbiased
risk estimation (SURE) formulas that hold for any spectral
estimators which shrink or threshold the singular values of the
data matrix. This leads to new data-driven spectral estimators,
whose optimality is discussed using tools from random matrix
theory and through numerical experiments. Under the spiked
population model and in the asymptotic setting where the
dimensions of the data matrix are let going to infinity, some
theoretical properties of our approach are compared to recent
results on asymptotically optimal shrinking rules for Gaussian
noise. It also leads to new procedures for singular values
shrinkage in finite-dimensional matrix denoising for Gamma-
distributed and Poisson-distributed measurements.

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