The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:1945-1953, 2021.
In the univariate setting, using the kernel spectral representation is an appealing approach for generating stationary covariance functions. However, performing the same task for multiple-output Gaussian processes is substantially more challenging. We demonstrate that current approaches to modelling cross-covariances with a spectral mixture kernel possess a critical blind spot. Pairs of highly correlated (or highly anti-correlated) processes are not reproducible, aside from the special case when their spectral densities are of identical shape. We present a solution to this issue by replacing the conventional Gaussian components of a spectral mixture with block components of finite bandwidth (i.e. rectangular step functions). The proposed family of kernel represents the first multi-output generalisation of the spectral mixture kernel that can approximate any stationary multi-output kernel to arbitrary precision.