Analyzing Weighted $\ell _1$ Minimization for Sparse Recovery With Nonuniform Sparse Models

In this paper, we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted ℓ₁ minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into two sets, with entries of each set having a specific probability of being nonzero. We propose a weighted ℓ₁ minimization recovery algorithm and analyze its performance using a Grassmann angle approach. We compute explicitly the relationship between the system parameters-the weights, the number of measurements, the size of the two sets, the probabilities of being nonzero-so that when i.i.d. random Gaussian measurement matrices are used, the weighted ℓ₁ minimization recovers a randomly selected signal drawn from the considered sparsity model with overwhelming probability as the problem dimension increases. This allows us to compute the optimal weights. We demonstrate through rigorous analysis and simulations that for the case when the support of the signal can be divided into two different subclasses with unequal sparsity fractions, the weighted ℓ₁ minimization outperforms the regular ℓ₁ minimization substantially. We also generalize our results to signal vectors with an arbitrary number of subclasses for sparsity.