Provable approximation properties for deep neural networks

We discuss approximation of functions using deep neural nets. Given a function f on a d-dimensional manifold Γ⊂Rm, 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 Γ, 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).