Index Set Fourier Series Features for Approximating Multi-dimensional Periodic Kernels

TL;DR

This paper introduces Index Set Fourier Series Features, a method to efficiently approximate multivariate periodic kernels in Gaussian Processes, reducing computational costs and improving predictive accuracy for high-dimensional periodic data.

Contribution

The paper presents a novel approach to decompose multivariate periodic kernels using Fourier series, enabling scalable Gaussian Process modeling for high-dimensional periodic datasets.

Findings

Significantly less predictive error than random Fourier features

Better generalization on periodic regression tasks

Efficient decomposition of high-dimensional periodic kernels

Abstract

Periodicity is often studied in timeseries modelling with autoregressive methods but is less popular in the kernel literature, particularly for higher dimensional problems such as in textures, crystallography, and quantum mechanics. Large datasets often make modelling periodicity untenable for otherwise powerful non-parametric methods like Gaussian Processes (GPs) which typically incur an $\mathcal{O}(N^3)$ computational burden and, consequently, are unable to scale to larger datasets. To this end we introduce a method termed \emph{Index Set Fourier Series Features} to tractably exploit multivariate Fourier series and efficiently decompose periodic kernels on higher-dimensional data into a series of basis functions. We show that our approximation produces significantly less predictive error than alternative approaches such as those based on random Fourier features and achieves better generalisation on regression problems with periodic data.