Concentration-Based Guarantees for Low-Rank Matrix Reconstruction

TL;DR

This paper investigates low-rank matrix reconstruction from partial observations, comparing trace-norm and max-norm approaches, and provides improved theoretical guarantees based on Rademacher complexity analysis.

Contribution

It introduces novel reconstruction guarantees for both trace-norm and max-norm regularization, demonstrating their advantages over existing specialized analyses.

Findings

Max-norm offers competitive reconstruction guarantees.

Guarantees based on Rademacher complexity are superior to previous bounds.

The analysis applies to approximately low-rank matrices.

Abstract

We consider the problem of approximately reconstructing a partially-observed, approximately low-rank matrix. This problem has received much attention lately, mostly using the trace-norm as a surrogate to the rank. Here we study low-rank matrix reconstruction using both the trace-norm, as well as the less-studied max-norm, and present reconstruction guarantees based on existing analysis on the Rademacher complexity of the unit balls of these norms. We show how these are superior in several ways to recently published guarantees based on specialized analysis.