Universality of Shallow and Deep Neural Networks on Non-Euclidean Spaces
Vugar Ismailov
TL;DR
A central focus of the paper is the deep narrow framework, in which the width of each hidden layer is uniformly bounded while the depth is allowed to grow, and conditions under which such width constrained deep networks retain universal approximation power.
Abstract
We develop a framework for shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and a fixed scalar activation function, and it reduces to multilayer feedforward networks in the Euclidean case. We focus on the universal approximation property and establish general conditions under which such networks are dense in spaces of continuous vector-valued functions on arbitrary and locally convex topological spaces. In the absence of width constraints, we obtain universality results that extend classical approximation theorems to non-Euclidean settings. A central focus of the paper is the deep narrow framework, in which the width of each hidden layer is uniformly bounded while the depth is allowed to grow. We identify conditions under which such width constrained deep networks retain universal approximation power. As a concrete example, we employ Ostrand's extension of the Kolmogorov superposition theorem to derive an explicit universality result for products of compact metric spaces, with width bounds expressed in terms of topological dimension.