A Dictionary of Closed-Form Kernel Mean Embeddings
This work addresses a practical bottleneck for practitioners in computational statistics and machine learning by making kernel-based techniques more accessible, though it is incremental as it compiles and extends existing knowledge.
The paper tackles the challenge of deriving closed-form expressions for kernel mean embeddings, which are essential for Bayesian quadrature and other kernel-based methods, by providing a comprehensive dictionary of known embeddings and tools for deriving new ones, along with a Python library for implementation.
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.