Sparse Blind Deconvolution and Demixing Through $\ell_{1,2}$-Minimization

TL;DR

This paper demonstrates that under a structured random model, robust and stable recovery of sparse deconvolution and demixing can be achieved using $\,ell_{1,2}$-minimization, extending prior theoretical results and explaining experimental findings.

Contribution

It provides a theoretical framework for sparse deconvolution and demixing via $\,ell_{1,2}$-minimization under structured randomness, generalizing previous work.

Findings

Robust recovery is possible under certain structured random models.

Theoretical results extend prior work by Ling and Strohmer.

Applicable to recovery of column-sparse matrices in general.

Abstract

This paper concerns solving the sparse deconvolution and demixing problem using $\ell_{1,2}$-minimization. We show that under a certain structured random model, robust and stable recovery is possible. The results extend results of Ling and Strohmer [Self Calibration and Biconvex Compressive Sensing, Inverse Problems, 2015], and in particular theoretically explain certain experimental findings from that paper. Our results do not only apply to the deconvolution and demixing problem, but to recovery of column-sparse matrices in general.