we consider the problem of recovering a structured signal x that lies close to a subset of interest t in $r^n$, from its random noisy linear measurements y = b a x + w, i.e., a generalization of the classic compressed sensing problem. above, b is a fixed matrix and a has independent sub-gaussian rows. by varying b, and the sub-gaussian distribution of a, this gives a family of measurement matrices which may have heavy tails, dependent rows and columns, and singular values with large dynamic range. typically, generalized compressed sensing assumes a random measurement matrix with nearly uniform singular values (with high probability), and asks: how many measurements are needed to recover x? in our setting, this question becomes: what properties of b allow good recovery? we show that the “effective rank'' of b may be used as a surrogate for the number of measurements, and if this exceeds the squared gaussian complexity of t-t then accurate recovery is guaranteed. we also derive the optimal dependence on the sub-gaussian norm of the rows of a, to our knowledge this was not known previously even in the case when b is the identity. we allow model mismatch and inexact optimization with the aim of creating an easily accessible theory that specializes well to the case when t is the range of an expansive neural net.