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

Jérémie Bigot (1) , Charles-Alban Deledalle (1) , Delphine Féral (1)

1 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: matrix denoising; singular value decomposition; low-rank model; spectral estimator; Stein's unbiased risk estimate; random matrix theory; exponential family; optimal shrinkage rule; degrees of freedom

Domaines: Statistiques [stat]; Mathématiques [math]

Jérémie Bigot, Charles-Alban Deledalle, Delphine Féral. Generalized SURE for optimal shrinkage of singular values in low-rank matrix denoising. Journal of Machine Learning Research, 2017. ⟨hal-01323285⟩