Riemannian thresholding methods for row-sparse and low-rank matrix recovery

TL;DR

This paper introduces Riemannian thresholding techniques for efficient recovery of matrices that are both row-sparse and low-rank, with applications in blind deconvolution, demonstrating near-optimal results and adaptive step size improvements.

Contribution

It develops a Riemannian iterative hard thresholding method that reduces computational costs and extends to a proximal gradient approach for unknown sparsity.

Findings

Near-optimal recovery with Gaussian and rank-one measurements

Significant computational cost reduction for rank-one measurement operators

Adaptive stepsizes improve recovery performance

Abstract

In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show near-optimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the special case of unknown sparsity.