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The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain

Fergus Simpson · Alexis Boukouvalas · Vaclav Cadek · Elvijs Sarkans · Nicolas Durrande

Keywords: [ Models and Methods ] [ Gaussian Processes ]


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.

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