Approximately low-rank recovery from noisy and local measurements by convex program

TL;DR

This paper introduces a tensor-norm-constrained convex estimator for recovering approximately low-rank matrices from noisy, local measurements, with theoretical guarantees and practical algorithms.

Contribution

It proposes a novel tensor-norm regularizer for low-rank matrix recovery that adapts to local structure and provides near-optimal error bounds.

Findings

Achieves near-minimax optimal error bounds.

Provides a polynomial-time algorithm for matrix completion and sketching.

Demonstrates effectiveness through statistical analysis.

Abstract

Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the nuclear norm for low rankness has an excellent analogy with the $\ell_1$ norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the nuclear norm. We provide a tensor-norm-constrained estimator for the recovery of approximately low-rank matrices from local measurements corrupted with noise. A tensor-norm regularizer is designed to adapt to the local structure. We derive statistical analysis of the estimator over matrix completion and decentralized sketching by applying Maurey's empirical method to tensor products of Banach spaces. The estimator provides a near-optimal error bound in a minimax sense and admits a polynomial-time algorithm for these applications.