  The goal of compressed sensing is to estimate a vector from an
underdetermined system of noisy linear measurements, by making use of prior
knowledge on the structure of vectors in the relevant domain. For almost all
results in this literature, the structure is represented by sparsity in a
well-chosen basis. We show how to achieve guarantees similar to standard
compressed sensing but without employing sparsity at all. Instead, we suppose
that vectors lie near the range of a generative model $G: \mathbb{R}^k \to
\mathbb{R}^n$. Our main theorem is that, if $G$ is $L$-Lipschitz, then roughly
$O(k \log L)$ random Gaussian measurements suffice for an $\ell_2/\ell_2$
recovery guarantee. We demonstrate our results using generative models from
published variational autoencoder and generative adversarial networks. Our
method can use $5$-$10$x fewer measurements than Lasso for the same accuracy.
