Periodic Kernel Approximation by Index Set Fourier Series Features

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.