Learning Nonparametric Volterra Kernels with Gaussian Processes

TL;DR

This paper presents a nonparametric Bayesian approach using Gaussian processes to learn nonlinear operators via Volterra series, enabling scalable regression and system identification without Gaussian approximations.

Contribution

It introduces the nonparametric Volterra kernels model (NVKM) that leverages GPs for flexible nonlinear operator learning and scalable inference.

Findings

Effective for multiple output regression

Accurate system identification on benchmarks

Scalable inference with variational methods

Abstract

This paper introduces a method for the nonparametric Bayesian learning of nonlinear operators, through the use of the Volterra series with kernels represented using Gaussian processes (GPs), which we term the nonparametric Volterra kernels model (NVKM). When the input function to the operator is unobserved and has a GP prior, the NVKM constitutes a powerful method for both single and multiple output regression, and can be viewed as a nonlinear and nonparametric latent force model. When the input function is observed, the NVKM can be used to perform Bayesian system identification. We use recent advances in efficient sampling of explicit functions from GPs to map process realisations through the Volterra series without resorting to numerical integration, allowing scalability through doubly stochastic variational inference, and avoiding the need for Gaussian approximations of the output processes. We demonstrate the performance of the model for both multiple output regression and system identification using standard benchmarks.