A Dictionary of Closed-Form Kernel Mean Embeddings

Francois-Xavier Briol, Toni Karvonen, Alexandra Gessner, Maren Mahsereci
Proceedings of the First International Conference on Probabilistic Numerics, PMLR 271:84-94, 2025.

Abstract

Kernel mean embeddings – integrals of a kernel with respect to a probability distribution – are essential in Bayesian quadrature, but also widely used in other computational tools for numerical integration or methods for statistical inference. These methods often require, or are enhanced by, the availability of a closed-form expression for the kernel mean embedding. However, deriving such expressions can be challenging, limiting the applicability of kernel-based techniques when practitioners do not have access to a closed-form embedding. This paper addresses this limitation by providing a comprehensive dictionary of known kernel mean embeddings, along with practical tools for deriving new embeddings from known ones. We also provide a Python library that includes minimal implementations of the embeddings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v271-briol25a, title = {A Dictionary of Closed-Form Kernel Mean Embeddings}, author = {Briol, Francois-Xavier and Karvonen, Toni and Gessner, Alexandra and Mahsereci, Maren}, booktitle = {Proceedings of the First International Conference on Probabilistic Numerics}, pages = {84--94}, year = {2025}, editor = {Kanagawa, Motonobu and Cockayne, Jon and Gessner, Alexandra and Hennig, Philipp}, volume = {271}, series = {Proceedings of Machine Learning Research}, month = {01--03 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v271/main/assets/briol25a/briol25a.pdf}, url = {https://proceedings.mlr.press/v271/briol25a.html}, abstract = {Kernel mean embeddings – integrals of a kernel with respect to a probability distribution – are essential in Bayesian quadrature, but also widely used in other computational tools for numerical integration or methods for statistical inference. These methods often require, or are enhanced by, the availability of a closed-form expression for the kernel mean embedding. However, deriving such expressions can be challenging, limiting the applicability of kernel-based techniques when practitioners do not have access to a closed-form embedding. This paper addresses this limitation by providing a comprehensive dictionary of known kernel mean embeddings, along with practical tools for deriving new embeddings from known ones. We also provide a Python library that includes minimal implementations of the embeddings.} }
Endnote
%0 Conference Paper %T A Dictionary of Closed-Form Kernel Mean Embeddings %A Francois-Xavier Briol %A Toni Karvonen %A Alexandra Gessner %A Maren Mahsereci %B Proceedings of the First International Conference on Probabilistic Numerics %C Proceedings of Machine Learning Research %D 2025 %E Motonobu Kanagawa %E Jon Cockayne %E Alexandra Gessner %E Philipp Hennig %F pmlr-v271-briol25a %I PMLR %P 84--94 %U https://proceedings.mlr.press/v271/briol25a.html %V 271 %X Kernel mean embeddings – integrals of a kernel with respect to a probability distribution – are essential in Bayesian quadrature, but also widely used in other computational tools for numerical integration or methods for statistical inference. These methods often require, or are enhanced by, the availability of a closed-form expression for the kernel mean embedding. However, deriving such expressions can be challenging, limiting the applicability of kernel-based techniques when practitioners do not have access to a closed-form embedding. This paper addresses this limitation by providing a comprehensive dictionary of known kernel mean embeddings, along with practical tools for deriving new embeddings from known ones. We also provide a Python library that includes minimal implementations of the embeddings.
APA
Briol, F., Karvonen, T., Gessner, A. & Mahsereci, M.. (2025). A Dictionary of Closed-Form Kernel Mean Embeddings. Proceedings of the First International Conference on Probabilistic Numerics, in Proceedings of Machine Learning Research 271:84-94 Available from https://proceedings.mlr.press/v271/briol25a.html.

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