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

Fergus Simpson, Alexis Boukouvalas, Vaclav Cadek, Elvijs Sarkans, Nicolas Durrande
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:1945-1953, 2021.

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. 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-simpson21a, title = { The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain }, author = {Simpson, Fergus and Boukouvalas, Alexis and Cadek, Vaclav and Sarkans, Elvijs and Durrande, Nicolas}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {1945--1953}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/simpson21a/simpson21a.pdf}, url = {https://proceedings.mlr.press/v130/simpson21a.html}, 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. 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. } }
Endnote
%0 Conference Paper %T The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain %A Fergus Simpson %A Alexis Boukouvalas %A Vaclav Cadek %A Elvijs Sarkans %A Nicolas Durrande %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-simpson21a %I PMLR %P 1945--1953 %U https://proceedings.mlr.press/v130/simpson21a.html %V 130 %X 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.
APA
Simpson, F., Boukouvalas, A., Cadek, V., Sarkans, E. & Durrande, N.. (2021). The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:1945-1953 Available from https://proceedings.mlr.press/v130/simpson21a.html.

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