Weighted $\ell _{1}$ -Minimization for Generalized Non-Uniform Sparse Model

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 ℓ
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 independent identically distributed Gaussian entries, weighted ℓ
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.