Index Set Fourier Series Features for Approximating Multi-dimensional Periodic Kernels
This work addresses a computational bottleneck for researchers and practitioners in fields like textures and quantum mechanics, offering an incremental improvement over existing kernel approximation methods.
The paper tackles the problem of scaling Gaussian Processes to large datasets with periodic patterns in high dimensions by introducing Index Set Fourier Series Features, which reduces predictive error and improves generalization compared to random Fourier features.
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.