The minecraft kernel: Modelling correlated Gaussian Processes in the fourier domain

Elvijs Sarkans, Fergus Simpson, Alexis Boukouvalas, Nicolas Durrande

Abstract:

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. For a given pair
of processes, the cross-covariance is not reproducible across the full range of permitted
correlations, aside from the special case where
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.