🤖 AI Summary
This paper addresses exact recovery of low-rank matrices from an extremely small number of noisy observations. To overcome the limitations of existing methods—which rely on strong spectral assumptions such as small condition numbers and large singular value gaps—we propose the first polynomial-time algorithm that guarantees high-probability exact recovery under only three fundamental assumptions: low rank, delocalization of singular vectors, and sufficiently random sampling. Our key innovation is a contour integral analysis framework, synergistically integrating ∞-norm error bounds with random sampling theory to break the theoretical barrier of zero-error reconstruction under noise. Under bounded-precision noise, we establish, for the first time without imposing additional spectral structural assumptions, rigorous guarantees of perfect matrix recovery. This significantly extends both the applicability boundary and robustness of low-rank matrix completion.
📝 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 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 three basic assumptions. This is the first such algorithm. To analyse the algorithm, we introduce a contour integration argument which is totally different from all previous methods and may be of independent interest.