Recovery of row-sparse low-rank matrices with application to SAR/ISAR autofocus

In this paper we consider a blind de-factorization problem for linear measurements of an unknown sparse vector that are distorted by a multiplicative interference signal that is also unknown. This scenario applies, for example, to SAR measurements in the slow time domain that are corrupted by uncompensated motion errors. We will show that lifting into a matrix space leads to the question of how to reconstruct a row-sparse low-rank matrix from linear measurements, and to a solution based on the minimization of a cost function containing both the nuclear norm and a mixed $\ell_{2} / \ell_{1}$ term.