整理了常用核函数的闭式均值嵌入，方便直接使用。
A Dictionary of Closed-Form Kernel Mean Embeddings
- 系统整理已知核函数的闭式均值嵌入公式
- 提供从已有嵌入推导新嵌入的方法工具
- 适合需要高效积分或统计推断的研究者
核均值嵌入——即核函数关于概率分布的积分——在贝叶斯求积中至关重要，也被广泛用于数值积分或基于最大均值差异的统计推断。这些方法通常需要或受益于核均值嵌入的闭式表达式。然而，推导这类表达式往往困难，限制了核方法在缺乏闭式解时的应用。本文通过构建一个全面的已知核均值嵌入词典，并提供从已知嵌入推导新嵌入的实用工具，解决了这一限制。此外，还提供了包含最小实现的Python库。
原文摘要 · Abstract (English)
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.
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