  Rectified linear units, or ReLUs, have become the preferred activation
function for artificial neural networks. In this paper we consider two basic
learning problems assuming that the underlying data follow a generative model
based on a ReLU-network -- a neural network with ReLU activations. As a
primarily theoretical study, we limit ourselves to a single-layer network. The
first problem we study corresponds to dictionary-learning in the presence of
nonlinearity (modeled by the ReLU functions). Given a set of observation
vectors $\mathbf{y}^i \in \mathbb{R}^d, i =1, 2, \dots , n$, we aim to recover
$d\times k$ matrix $A$ and the latent vectors $\{\mathbf{c}^i\} \subset
\mathbb{R}^k$ under the model $\mathbf{y}^i = \mathrm{ReLU}(A\mathbf{c}^i
+\mathbf{b})$, where $\mathbf{b}\in \mathbb{R}^d$ is a random bias. We show
that it is possible to recover the column space of $A$ within an error of
$O(d)$ (in Frobenius norm) under certain conditions on the probability
distribution of $\mathbf{b}$.
  The second problem we consider is that of robust recovery of the signal in
the presence of outliers, i.e., large but sparse noise. In this setting we are
interested in recovering the latent vector $\mathbf{c}$ from its noisy
nonlinear sketches of the form $\mathbf{v} = \mathrm{ReLU}(A\mathbf{c}) +
\mathbf{e}+\mathbf{w}$, where $\mathbf{e} \in \mathbb{R}^d$ denotes the
outliers with sparsity $s$ and $\mathbf{w} \in \mathbb{R}^d$ denote the dense
but small noise. This line of work has recently been studied (Soltanolkotabi,
2017) without the presence of outliers. For this problem, we show that a
generalized LASSO algorithm is able to recover the signal $\mathbf{c} \in
\mathbb{R}^k$ within an $\ell_2$ error of $O(\sqrt{\frac{(k+s)\log d}{d}})$
when $A$ is a random Gaussian matrix.
