Low Rank Matrix Completion with Exponential Family Noise

TL;DR

This paper develops new nuclear norm penalized estimators for matrix completion under exponential family noise, providing improved theoretical risk bounds and demonstrating near-minimax optimality.

Contribution

It introduces a general framework for matrix completion with exponential family noise, deriving improved risk bounds and establishing near-minimax optimal rates.

Findings

Proposed estimators achieve better Frobenius risk bounds than previous methods.

Established oracle inequalities for known sampling distributions.

Proved that the rates are minimax optimal up to a logarithmic factor.

Abstract

The matrix completion problem consists in reconstructing a matrix from a sample of entries, possibly observed with noise. A popular class of estimator, known as nuclear norm penalized estimators, are based on minimizing the sum of a data fitting term and a nuclear norm penalization. Here, we investigate the case where the noise distribution belongs to the exponential family and is sub-exponential. Our framework alllows for a general sampling scheme. We first consider an estimator defined as the minimizer of the sum of a log-likelihood term and a nuclear norm penalization and prove an upper bound on the Frobenius prediction risk. The rate obtained improves on previous works on matrix completion for exponential family. When the sampling distribution is known, we propose another estimator and prove an oracle inequality w.r.t. the Kullback-Leibler prediction risk, which translates immediatly into an upper bound on the Frobenius prediction risk. Finally, we show that all the rates obtained are minimax optimal up to a logarithmic factor.