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Self-calibrating compressive sensing — a bilinear inverse problem

Student-Run Applied & Math Seminar

Speaker: Shuyang Ling, UC Davis
Location: 2112 MSB
Start time: Wed, May 14 2014, 12:10PM

In signal processing, problems such as blind deconvolution and blind source deconvolution are all bilinear inverse problems. The observable data is a bilinear mapping of two vectors. In the recent few years, a unifying approach is proposed to solve this class of problems by relating them to low rank matrix recovery problems or heuristically nuclear norm minimization problems. A crucial mathematical problem here is under what suitable conditions the solutions of primal and dual problems are equivalent. Several worked examples include phase retrieval via matrix completion and blind deconvolution via convex programming. The problem we are interested in is called SCCS, short for self-calibrating compressive sensing. It is a problem of recovering sparse signal with measurement matrices which depend on another linear parameter. It could be naturally reformulated as a low-rank matrix recovery problem as well. Here we want to solve the similar mathematical question : what conditions could guarantee that the recovered low-rank matrix via nuclear norm minimization is the solution of primal problem if sparsity is imposed on signal?