This paper concerns solving the sparse deconvolution and demixing problem using l1,2-minimization. We show that under a certain structured random model, robust and stable recovery is possible. The results extend results of Ling and Strohmer (Inverse Probl. 31, 115002 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.
However, as was pointed out in (Ling and Strohmer 2015 @cite_1 ), numerical experiments show that both @math - and @math -minimization perform significantly better than nuclear norm minimization when it comes to recovering matrices with the structure described above. This is not hard to argue heuristically: Assuming @math , the above approach tries to recover a @math -matrix of rank @math , which needs an order of @math measurements. On the other hand @math (or @math ) tries to recover an @math -column sparse (or @math -sparse) @math matrix , which only needs @math . For really small sparsities and moderate @math , @math can be smaller than @math . Therefore, the authors of the mentioned paper concentrate their efforts, as will we, on analysing the @math -minimization (and @math -minimization, respectively).
@cite_1
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The design of high-precision sensing devises becomes ever more difficult and expensive. At the same time, the need for precise calibration of these devices (ranging from tiny sensors to space telescopes) manifests itself as a major roadblock in many scientific and technological endeavors. To achieve optimal performance of advanced high-performance sensors one must carefully calibrate them, which is often difficult or even impossible to do in practice. In this work we bring together three seemingly unrelated concepts, namely self-calibration, compressive sensing, and biconvex optimization. The idea behind self-calibration is to equip a hardware device with a smart algorithm that can compensate automatically for the lack of calibration. We show how several self-calibration problems can be treated efficiently within the framework of biconvex compressive sensing via a new method called SparseLift. More specifically, we consider a linear system of equations where both and the diagonal matrix (which models the calibration error) are unknown. By 'lifting' this biconvex inverse problem we arrive at a convex optimization problem. By exploiting sparsity in the signal model, we derive explicit theoretical guarantees under which both and can be recovered exactly, robustly, and numerically efficiently via linear programming. Applications in array calibration and wireless communications are discussed and numerical simulations are presented, confirming and complementing our theoretical analysis.