Optimal tuning-free convex relaxation for noisy matrix completion

TL;DR

This paper introduces a tuning-free square-root convex relaxation method for noisy matrix completion, achieving optimal statistical performance without needing noise level knowledge, and connects it to nonconvex estimators.

Contribution

It proposes a novel tuning-free convex estimator for noisy matrix completion with proven optimality and links it to nonconvex rank-constrained approaches.

Findings

Achieves optimal statistical performance under standard assumptions.

Does not require prior noise level knowledge.

Establishes connections between convex and nonconvex estimators.

Abstract

This paper is concerned with noisy matrix completion--the problem of recovering a low-rank matrix from partial and noisy entries. Under uniform sampling and incoherence assumptions, we prove that a tuning-free square-root matrix completion estimator (square-root MC) achieves optimal statistical performance for solving the noisy matrix completion problem. Similar to the square-root Lasso estimator in high-dimensional linear regression, square-root MC does not rely on the knowledge of the size of the noise. While solving square-root MC is a convex program, our statistical analysis of square-root MC hinges on its intimate connections to a nonconvex rank-constrained estimator.