Weighted $\ell_1$-minimization for generalized non-uniform sparse model

Sidhant Misra; Pablo A. Parrilo

doi:10.1109/tit.2015.2442922

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arXiv · 1301.1327

Weighted $\ell_1$-minimization for generalized non-uniform sparse model

Sidhant Misra·Pablo A. Parrilo

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Abstract

Model-based compressed sensing refers to compressed sensing with extra structure about the underlying sparse signal known a priori. Recent work has demonstrated that both for deterministic and probabilistic models imposed on the signal, this extra information can be successfully exploited to enhance recovery performance. In particular, weighted $\ell_1$-minimization with suitable choice of weights has been shown to improve performance in the so called non-uniform sparse model of signals. In this paper, we consider a full generalization of the non-uniform sparse model with very mild assumptions. We prove that when the measurements are obtained using a matrix with i.i.d Gaussian entries, weighted $\ell_1$-minimization successfully recovers the sparse signal from its measurements with overwhelming probability. We also provide a method to choose these weights for any general signal model from the non-uniform sparse class of signal models.

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Sidhant Misra, Pablo A. Parrilo. 2014-09-16. Weighted $\ell_1$-minimization for generalized non-uniform sparse model. https://doi.org/10.1109/tit.2015.2442922

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