On the robustness of noise-blind low-rank recovery from rank-one measurements

TL;DR

This paper proves that noise-blind convex optimization methods for low-rank matrix recovery are robust against noise in rank-one measurement scenarios, supported by theoretical analysis and numerical experiments.

Contribution

It establishes the Schatten-1 quotient property for Gaussian rank-one measurements, demonstrating robustness of noise-blind methods in phase retrieval.

Findings

Noise-blind methods are as robust as noise-aware ones.

Gaussian rank-one measurements satisfy the Schatten-1 quotient property.

Numerical experiments confirm theoretical robustness results.

Abstract

We prove new results about the robustness of well-known convex noise-blind optimization formulations for the reconstruction of low-rank matrices from underdetermined linear measurements. Our results are applicable for symmetric rank-one measurements as used in a formulation of the phase retrieval problem. We obtain these results by establishing that with high probability rank-one measurement operators defined by i.i.d. Gaussian vectors exhibit the so-called Schatten-1 quotient property, which corresponds to a lower bound for the inradius of their image of the nuclear norm (Schatten-1) unit ball. We complement our analysis by numerical experiments comparing the solutions of noise-blind and noise-aware formulations. These experiments confirm that noise-blind optimization methods exhibit comparable robustness to noise-aware formulations. Keywords: low-rank matrix recovery, phase retrieval, quotient property, noise-blind, robustness, nuclear norm minimization