WEIGHTED COMPRESSED SENSING AND RANK MINIMIZATION

Compressed Sensing: Theory and Methods

Presented by: Babak Hassibi, Author(s): Samet Oymak, M. Amin Khajehnejad, Babak Hassibi, California Institute of Technology, United States

We present an alternative analysis of weighted $ell_1$ minimization for sparse signals with a nonuniform sparsity model, and extend our results to nuclear norm minimization for matrices with nonuniform singular vector distribution. In the case of vectors, we find explicit upper bounds for the successful recovery thresholds, and give a simple suboptimal weighting rule. For matrices, the thresholds we find are only implicit, and the optimal weight selection requires an exhaustive search. For the special case of very wide matrices, the relationship is made explicit and the optimal weight assignment is the same as the vector case. We demonstrate through simulations that for vectors, the suggested weighting scheme improves the recovery performance over that of regular $ell_1$ minimization.

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