Periodic Kernel Approximation by Index Set Fourier Series Features

Abstract

Periodicity is often studied in timeseries modelling with autoregressive methods but is less popular in the kernel literature, particularly for multi-dimensional problems such as in textures, crystallography, quantum mechanics, and robotics. 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 cost, while approximate feature methods are impeded by their approximate accuracy. We introduce a method that efficiently decomposes multi-dimensional periodic kernels into a set of basis functions by exploiting multivariate Fourier series. Termed \emph{Index Set Fourier Series Features}, we show that our approximation produces significantly less predictive generalisation error than alternative approximations such as those based on random and deterministic Fourier features on regression problems with periodic data.