  We discuss approximation of functions using deep neural nets. Given a
function $f$ on a $d$-dimensional manifold $\Gamma \subset \mathbb{R}^m$, we
construct a sparsely-connected depth-4 neural network and bound its error in
approximating $f$. The size of the network depends on dimension and curvature
of the manifold $\Gamma$, the complexity of $f$, in terms of its wavelet
description, and only weakly on the ambient dimension $m$. Essentially, our
network computes wavelet functions, which are computed from Rectified Linear
Units (ReLU)
