  The goal of compressed sensing is to learn a structured signal $x$
  from a limited number of noisy linear measurements $y \approx Ax$.  In
  traditional compressed sensing, ``structure'' is represented by
  sparsity in some known basis.   Inspired by the success of deep
  learning in modeling images, recent work starting with~\cite{BDJP17}
  has instead considered structure to come from a generative model
  $G: \R^k \to \R^n$.  We present two results establishing the
  difficulty of this latter task, showing that existing bounds are
  tight.

   First, we provide a lower bound matching the~\cite{BDJP17} upper
  bound for compressed sensing from $L$-Lipschitz generative models
  $G$.  In particular, there exists such a function that requires
  roughly $\Omega(k \log L)$ linear measurements for sparse recovery
  to be possible.   This holds even for the more relaxed goal of
  \emph{nonuniform} recovery.

   Second, we show that generative models generalize sparsity as a
  representation of structure.   In particular, we construct a
  ReLU-based neural network $G: \R^{2k} \to \R^n$ with $O(1)$ layers
  and $O(kn)$ activations per layer, such that the range of $G$
  contains all $k$-sparse vectors.

Lower bound for compressed sensing w/ generative models that matches known upper bounds