A Dictionary of Closed-Form Kernel Mean Embeddings

TL;DR

This work tackles the bottleneck of obtaining closed-form kernel mean embeddings $K_P$ and $K_{PP}$ for kernel-based methods such as Bayesian quadrature and MMD. It offers a comprehensive dictionary of known kernel–distribution pairs and practical methods to derive new embeddings via transformations, mixtures, changes of measure or variable, and Stein constructions. A Python library kernel_embedding_dictionary accompanies the dictionary, providing reusable, testable embeddings as code to lower the barrier for practitioners. By centralizing and systematizing embeddings, the paper enables direct, tractable implementation of kernel-based numerical integration and inference in a wide range of applications.

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 for statistical inference based on the maximum mean discrepancy. 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.