Approximation and non-parametric estimation of functions over high-dimensional spheres via deep ReLU networks

Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a d * -dimensional manifold that is embedded into a space with potentially much larger ambient dimension. It is shown that sparsely connected deep ReLU networks can approximate a Hölder function with smoothness index β up to error ε using of the order of ε -d * /β log(1/ε) many non-zero network parameters. As an application, we derive statistical convergence rates for the estimator minimizing the empirical risk over all possible choices of bounded network parameters.