Noisy low-rank matrix completion with general sampling distribution

TL;DR

This paper introduces new nuclear-norm penalized estimators for noisy low-rank matrix completion under general sampling distributions, providing near-optimal error bounds without noise variance estimation.

Contribution

It proposes two novel estimators, including a square-root type, and analyzes their performance with non-asymptotic bounds under broad sampling conditions.

Findings

Performance guarantees are minimax optimal up to a logarithmic factor.

The estimators do not require noise variance knowledge.

Results hold under high-dimensional scaling with general sampling distributions.

Abstract

In the present paper, we consider the problem of matrix completion with noise. Unlike previous works, we consider quite general sampling distribution and we do not need to know or to estimate the variance of the noise. Two new nuclear-norm penalized estimators are proposed, one of them of "square-root" type. We analyse their performance under high-dimensional scaling and provide non-asymptotic bounds on the Frobenius norm error. Up to a logarithmic factor, these performance guarantees are minimax optimal in a number of circumstances.