Low-Rank Matrix Recovery via Heavy-Tailed Quadratic Sampling

2026-07-09

AI Summary

This work addresses the problem of recovering low-rank Hermitian matrices from quadratic measurements under heavy-tailed distributions, assuming only that the sampling vectors possess finite moments of order $4+\delta$. By combining nuclear norm minimization with empirical risk minimization under a semidefinite constraint, and introducing novel techniques for decoupled quadratic form moment estimation and heavy-tailed covariance estimation, the authors establish—for the first time under such weak moment conditions—stable and uniform recovery guarantees with optimal sample complexity $O(rn)$. These results not only apply to general low-rank matrix recovery but also extend to complex projective 4-design sampling and phase retrieval settings, thereby overcoming the limitations of conventional Gaussian or sub-Gaussian assumptions.

Abstract

The problem of recovering an (approximately) low-rank Hermitian matrix $\pmb{M}_0 \in \mathbb{C}^{n \times n}$ of rank $r$ from quadratic sampling matrices of the form $\{\pmb{a}_k \pmb{a}_k^*\}_{k=1}^m$ arises in a variety of applications, including phase retrieval. To obtain rigorous recovery guarantees, the sampling vectors $\{\pmb{a}_k\}_{k=1}^m$ are typically modeled probabilistically. However, most existing theoretical results rely on Gaussian or sub-Gaussian assumptions, which may not accurately capture practical data models. In many applications, sampling vectors exhibit heavier tails, while theoretical understanding in such regimes remains scarce. In this paper, we bridge this gap. We show that two widely used convex approaches, nuclear norm minimization and semidefinite-constrained empirical risk minimization, achieve uniform, stable, and robust recovery under the mild assumption that the entries of the sampling vectors have only finite $4+δ$ moments, with the optimal sample complexity $m = \mathcal{O}(rn)$ up to moment-dependent constants. The two main ingredients of our analysis are moment estimates for quadratic forms established via decoupling, together with recent advances in covariance estimation in heavy-tailed settings. As byproducts, we also establish the optimal sample complexity for low-rank matrix recovery under complex projective $4$-design sampling, thereby improving upon previous results, and obtain stability guarantees for phase retrieval under similarly weak moment assumptions.

Problem

• Low-Rank Matrix Recovery
• Heavy-Tailed Sampling
• Quadratic Sampling
• Phase Retrieval
• Moment Assumptions

Innovation

• low-rank matrix recovery
• heavy-tailed sampling
• nuclear norm minimization
• quadratic sampling
• moment assumptions