A function space analysis of finite neural networks with insights from sampling theory

Abstract: This work suggests using sampling theory to analyze the function space represented by neural networks. First, it shows, under the assumption of a finite input domain, which is the common case in training neural networks, that the function space generated by multi-layer networks with non-expansive activation functions is smooth. This extends over previous works that show results for the case of infinite width ReLU networks. Then, under the assumption that the input is band-limited, we provide novel error bounds for univariate neural networks. We analyze both deterministic uniform and random sampling showing the advantage of the former.