  In this paper we study the compressed sensing problem of recovering a sparse
signal from a system of underdetermined linear equations when we have prior
information about the probability of each entry of the unknown signal being
nonzero. In particular, we focus on a model where the entries of the unknown
vector fall into two sets, each with a different probability of being nonzero.
We propose a weighted $\ell_1$ minimization recovery algorithm and analyze its
performance using a Grassman 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 non-zero) so
that an iid random Gaussian measurement matrix along with weighted $\ell_1$
minimization recovers almost all such sparse signals with overwhelming
probability as the problem dimension increases. This allows us to compute the
optimal weights. We also provide simulations to demonstrate the advantages of
the method over conventional $\ell_1$ optimization.
