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We derived the first sparse recovery guarantees for weighted $\ell_1$minimization with sparse random matrices and the class of weighted sparsesignals, using a weighted versions of the null space property to derive theseguarantees. These sparse matrices from expender graphs can be applied very fastand have other better computational complexities than their dense counterparts.In addition we show that, using such sparse matrices, weighted sparse recoverywith weighted $\ell_1$ minimization leads to sample complexities that arelinear in the weighted sparsity of the signal and these sampling rates can besmaller than those of standard sparse recovery. Moreover, these results reduceto known results in standard sparse recovery and sparse recovery with priorinformation and the results are supported by numerical experiments.

Sparse matrices for weighted sparse recovery