Approximation by neural networks with a bounded number of nodes at each level (Q1395810)

The paper deals with approximation by neural networks in the space \(C\) of continuous functions \(f: {\mathbb R}^d \to {\mathbb R}^{d'}\), in the topology of uniform convergence on compacta. It is well-known that the set of all neural networks with one hidden layer is dense in \(C\) if and only if the activation function \(\sigma\) is not a polynomial; the number of nodes in the network is allowed to be arbitrarily large. In the paper under review the density is established for the set of all multilayer networks with at most \(d+d'+2\) nodes in each layer. The restriction on \(\sigma\) is even weaker - it should not be linear - but the number of layers is allowed to be arbitrarily large.