  We consider the problem of estimating the parameters of a $d$-dimensional
rectified Gaussian distribution from i.i.d. samples. A rectified Gaussian
distribution is defined by passing a standard Gaussian distribution through a
one-layer ReLU neural network. We give a simple algorithm to estimate the
parameters (i.e., the weight matrix and bias vector of the ReLU neural network)
up to an error $\epsilon||W||_F$ using $\tilde{O}(1/\epsilon^2)$ samples and
$\tilde{O}(d^2/\epsilon^2)$ time (log factors are ignored for simplicity). This
implies that we can estimate the distribution up to $\epsilon$ in total
variation distance using $\tilde{O}(\kappa^2d^2/\epsilon^2)$ samples, where
$\kappa$ is the condition number of the covariance matrix. Our only assumption
is that the bias vector is non-negative. Without this non-negativity
assumption, we show that estimating the bias vector within any error requires
the number of samples at least exponential in the infinity norm of the bias
vector. Our algorithm is based on the key observation that vector norms and
pairwise angles can be estimated separately. We use a recent result on learning
from truncated samples. We also prove two sample complexity lower bounds:
$\Omega(1/\epsilon^2)$ samples are required to estimate the parameters up to
error $\epsilon$, while $\Omega(d/\epsilon^2)$ samples are necessary to
estimate the distribution up to $\epsilon$ in total variation distance. The
first lower bound implies that our algorithm is optimal for parameter
estimation. Finally, we show an interesting connection between learning a
two-layer generative model and non-negative matrix factorization. Experimental
results are provided to support our analysis.
