Low-Rank Positive Semidefinite Matrix Recovery from Corrupted Rank-One Measurements

TL;DR

This paper introduces convex and non-convex algorithms for recovering low-rank positive semidefinite matrices from corrupted rank-one measurements, with theoretical guarantees and practical efficiency.

Contribution

It proposes a parameter-free convex optimization method and a rank-aware non-convex algorithm for robust low-rank PSD matrix recovery from corrupted measurements.

Findings

Exact recovery with high probability under sufficient measurements

Robustness to arbitrary outliers in measurements

Efficient empirical performance of the non-convex algorithm

Abstract

We study the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of rank-one measurements using sensing vectors composed of i.i.d. standard Gaussian entries, which are possibly corrupted by arbitrary outliers. This problem arises from applications such as phase retrieval, covariance sketching, quantum space tomography, and power spectrum estimation. We first propose a convex optimization algorithm that seeks the PSD matrix with the minimum $\ell_1$-norm of the observation residual. The advantage of our algorithm is that it is free of parameters, therefore eliminating the need for tuning parameters and allowing easy implementations. We establish that with high probability, a low-rank PSD matrix can be exactly recovered as soon as the number of measurements is large enough, even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes. Moreover, the recovery is also stable against bounded noise. With the additional information of an upper bound of the rank of the PSD matrix, we propose another non-convex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy.