Global universal approximation of functional input maps on weighted spaces

TL;DR

This paper develops a universal approximation theory for functional input neural networks on weighted infinite-dimensional spaces, extending classical results to non-compact settings and applications in path space functionals and signature kernels.

Contribution

It introduces a universal approximation theorem for neural networks on weighted spaces, applicable to path functionals and signature kernel methods, with implications for uncertainty quantification.

Findings

Proves universal approximation on weighted spaces beyond compact sets.

Applies results to path space functionals and signature kernel linear functions.

Links Gaussian process regression with signature kernels for uncertainty quantification.

Abstract

We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family to map the input weighted space to the hidden layer, on which a non-linear scalar activation function is applied to each neuron, and finally return the output via some linear readouts. Relying on Stone-Weierstrass theorems on weighted spaces, we can prove a global universal approximation result on weighted spaces for continuous functions going beyond the usual approximation on compact sets. This then applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks. As a further application of the weighted Stone-Weierstrass theorem we prove a global universal approximation result for linear functions of the signature. We also introduce the viewpoint of Gaussian process regression in this setting and emphasize that the reproducing kernel Hilbert space of the signature kernels are Cameron-Martin spaces of certain Gaussian processes. This paves a way towards uncertainty quantification for signature kernel regression.