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In this paper we introduce a nonuniform sparsity model and analyze theperformance of an optimized weighted $\ell_1$ minimization over that sparsitymodel. In particular, we focus on a model where the entries of the unknownvector fall into two sets, with entries of each set having a specificprobability of being nonzero. We propose a weighted $\ell_1$ minimizationrecovery algorithm and analyze its performance using a Grassmann angleapproach. We compute explicitly the relationship between the systemparameters-the weights, the number of measurements, the size of the two sets,the probabilities of being nonzero- so that when i.i.d. random Gaussianmeasurement matrices are used, the weighted $\ell_1$ minimization recovers arandomly selected signal drawn from the considered sparsity model withoverwhelming probability as the problem dimension increases. This allows us tocompute the optimal weights. We demonstrate through rigorous analysis andsimulations that for the case when the support of the signal can be dividedinto two different subclasses with unequal sparsity fractions, the optimalweighted $\ell_1$ minimization outperforms the regular $\ell_1$ minimizationsubstantially. We also generalize the results to an arbitrary number ofclasses.

Analyzing Weighted $\ell_1$ Minimization for Sparse Recovery with  Nonuniform Sparse Models\footnote{The results of this paper were presented in  part at the International Symposium on Information Theory, ISIT 2009}