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

Abstract

We consider the problem of recovering an unknown low-rank matrix X with (possibly) non-orthogonal, effectively sparse rank-1 decomposition from incomplete and inaccurate 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 from y up to noise level. Working with a variant of robust injectivity, we derive reconstruction guarantees for various choices of A including subgaussian, Gaussian rank-1, and heavy-tailed measurements. Numerical experiments support the validity of our theoretical considerations.

Completion of Structured Low-Rank Matrices via Iteratively Reweighted Least Squares

“…In this section, we explore the empirical performance of IRLS in view of the theoretical results of Theorems 2.4 and 2.5, and compare its ability to recover simultaneous lowrank and row-sparse data matrices with the state-of-the-art methods Sparse Power Factorization (SPF) [LWB18] and Riemannian adaptive iterative hard thresholding (RiemAdaIHT) [EKPU22], which are among the methods with the best empirical performance reported in the literature. The method ATLAS [FMN21] and its successor [Mal21] are not used in our empirical studies since they are tailored to robust recovery and yield suboptimal performance when seeking high-precision reconstruction in low noise scenarios. We use spectral initialization for SPF and RiemAdaIHT.…”

“…A third line of work, approaches the problem from a variational point of view. In [FMN21,Mal21] the authors aim at enhancing robustness of recovery by alternating minimization of an ℓ 1 -norm based multi-penalty functional. In essence, the theoretical results bound the reconstruction error of global minimizers of the proposed functional depending on the number of observations.…”