Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion

Abstract

This paper deals with the trace regression model where n entries or linear combinations of entries of an unknown m₁ × m₂ matrix A₀ corrupted by noise are observed. We propose a new nuclear-norm penalized estimator of A₀ and establish a general sharp oracle inequality for this estimator for arbitrary values of n, m₁, m₂ under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form, and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the high-dimensional setting m₁m₂ ≫ n. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix A₀, a nonminimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of A₀ with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by A₀, and the aim is to find the best trace regression model approximating the data. As a by-product, we show that, under the restricted eigenvalue condition, the usual vector Lasso estimator satisfies a sharp oracle inequality (i.e., an oracle inequality with leading constant 1).