Fast exact recovery of noisy matrix from few entries: the infinity norm approach

TL;DR

This paper introduces a novel, simple algorithm for exactly recovering noisy low-rank matrices from few entries using the infinity norm, removing previous spectral assumptions and employing a new contour integration analysis.

Contribution

It presents the first algorithm capable of exact matrix recovery in noisy settings under only basic low-rank, incoherence, and sample size assumptions, without additional spectral conditions.

Findings

Achieves exact recovery in noisy conditions with minimal assumptions.

Introduces a new contour integration method for analysis.

Removes spectral assumptions like small condition number or large singular value gaps.

Abstract

The matrix recovery (completion) problem, a central problem in data science and theoretical computer science, is to recover a matrix $A$ from a relatively small sample of entries. While such a task is impossible in general, it has been shown that one can recover $A$ exactly in polynomial time, with high probability, from a random subset of entries, under three (basic and necessary) assumptions: (1) the rank of $A$ is very small compared to its dimensions (low rank), (2) $A$ has delocalized singular vectors (incoherence), and (3) the sample size is sufficiently large. There are many different algorithms for the task, including convex optimization by Candes, Tao and Recht (2009), alternating projection by Hardt and Wooters (2014) and low rank approximation with gradient descent by Keshavan, Montanari and Oh (2009, 2010). In applications, it is more realistic to assume that data is noisy. In this case, these approaches provide an approximate recovery with small root mean square error. However, it is hard to transform such an approximate recovery to an exact one. Recently, results by Abbe et al. (2017) and Bhardwaj et al. (2023) concerning approximation in the infinity norm showed that we can achieve exact recovery even in the noisy case, given that the ground matrix has bounded precision. Beyond the three basic assumptions above, they required either the condition number of $A$ is small (Abbe et al.) or the gap between consecutive singular values is large (Bhardwaj et al.). In this paper, we remove these extra spectral assumptions. As a result, we obtain a simple algorithm for exact recovery in the noisy case, under only the three basic assumptions. This is the first such algorithm. To analyse this algorithm, we introduce a contour integration argument which is totally different from all previous methods and may be of independent interest.