  We consider low-distortion embeddings for subspaces under \emph{entrywise
nonlinear transformations}. In particular we seek embeddings that preserve the
norm of all vectors in a space $S = \{y: y = f(x)\text{ for }x \in Z\}$, where
$Z$ is a $k$-dimensional subspace of $\mathbb{R}^n$ and $f(x)$ is a nonlinear
activation function applied entrywise to $x$. When $f$ is the identity, and so
$S$ is just a $k$-dimensional subspace, it is known that, with high
probability, a random embedding into $O(k/\epsilon^2)$ dimensions preserves the
norm of all $y \in S$ up to $(1\pm \epsilon)$ relative error. Such embeddings
are known as \emph{subspace embeddings}, and have found widespread use in
compressed sensing and approximation algorithms. We give the first
low-distortion embeddings for a wide class of nonlinear functions $f$. In
particular, we give additive $\epsilon$ error embeddings into $O(\frac{k\log
(n/\epsilon)}{\epsilon^2})$ dimensions for a class of nonlinearities that
includes the popular Sigmoid SoftPlus, and Gaussian functions. We strengthen
this result to give relative error embeddings under some further restrictions,
which are satisfied e.g., by the Tanh, SoftSign, Exponential Linear Unit, and
many other `soft' step functions and rectifying units. Understanding embeddings
for subspaces under nonlinear transformations is a key step towards extending
random sketching and compressing sensing techniques for linear problems to
nonlinear ones. We discuss example applications of our results to improved
bounds for compressed sensing via generative neural networks.
