Robust Sensing of Low-Rank Matrices with Non-Orthogonal Sparse Decomposition

TL;DR

This paper introduces a variational approach for robustly recovering low-rank matrices with sparse, non-orthogonal decompositions from linear measurements, with provable guarantees and numerical validation.

Contribution

It proposes a new variational formulation and provides theoretical guarantees for recovering low-rank matrices with sparse decompositions under various measurement models.

Findings

Reconstruction guarantees for sub-gaussian, Gaussian, and heavy-tailed measurements.

Numerical experiments confirm theoretical predictions.

Method effectively handles non-orthogonal sparse decompositions.

Abstract

We consider the problem of recovering an unknown low-rank matrix X with (possibly) non-orthogonal, effectively sparse rank-1 decomposition from measurements y gathered in a linear measurement process A. We propose a variational formulation that lends itself to alternating minimization and whose global minimizers provably approximate X up to noise level. Working with a variant of robust injectivity, we derive reconstruction guarantees for various choices of A including sub-gaussian, Gaussian rank-1, and heavy-tailed measurements. Numerical experiments support the validity of our theoretical considerations.