Extreme value theory for singular subspace estimation in the matrix denoising model

Music City Center 

This paper studies fine-grained singular subspace inference in the matrix denoising model where a deterministic low-rank signal matrix is additively perturbed by a stochastic matrix of independent Gaussian noise. We establish that the maximum Euclidean row norm of the aligned difference between the top-$r$ sample and population singular vector matrices approaches the Gumbel distribution in the large-matrix limit under suitable signal-to-noise conditions after appropriate centering and scaling. Our main results are obtained by a novel synthesis of entrywise matrix perturbation theory and saddle point approximation methods in statistics. The theoretical developments in this paper lead to methodology for hypothesis testing low-rank signal structure encoded in the singular subspaces spanned by the top-$r$ singular vectors. To develop a data-driven inference procedure, shrinkage-type de-biased estimators are derived for the signal singular values. The features of our test include an asymptotic control over the size, and a power phase transition analysis under simple alternative structures.

Singular subspace inference

Two-to-infinity norm

Gumbel convergence

Saddle point approximation

Singular value shrinkage 

Section on Statistical Learning and Data Science