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arXiv · 2103.06950

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

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

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Fergus Simpson, Alexis Boukouvalas, Vaclav Cadek, Elvijs Sarkans, Nicolas Durrande. 2021-03-11. The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain. https://arxiv.org/abs/2103.06950

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