  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 $\ell_1$-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 i.i.d
Gaussian entries, weighted $\ell_1$-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.
