Sharp Oracle Inequalities in Low Rank Estimation

TL;DR

This paper establishes sharp oracle inequalities for low rank matrix estimation using nuclear norm penalization, providing precise bounds on excess risk and error dependence on rank.

Contribution

It proves sharp low rank oracle inequalities with optimal constants and rank-dependent error bounds in the context of nuclear norm penalized empirical risk minimization.

Findings

Sharp oracle inequalities with constant one for excess risk

Error bounds explicitly depend on the rank of the target matrix

Results applicable to low rank matrix recovery problems

Abstract

The paper deals with the problem of penalized empirical risk minimization over a convex set of linear functionals on the space of Hermitian matrices with convex loss and nuclear norm penalty. Such penalization is often used in low rank matrix recovery in the cases when the target function can be well approximated by a linear functional generated by a Hermitian matrix of relatively small rank (comparing with the size of the matrix). Our goal is to prove sharp low rank oracle inequalities that involve the excess risk (the approximation error) with constant equal to one and the random error term with correct dependence on the rank of the oracle.