标题： 闭式核均值嵌入词典 显示英文标题

标题： A Dictionary of Closed-Form Kernel Mean Embeddings

摘要： 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，同时也广泛用于其他数值积分或基于最大均值差异的统计推断的计算工具中。这些方法通常需要，或者因闭式表达式的可用性而得到增强。然而，推导此类表达式可能具有挑战性，限制了核方法的应用，当从业者无法获得闭式嵌入时尤其如此。本文通过提供已知核均值嵌入的综合字典以及从已知嵌入推导新嵌入的实用工具来解决这一局限性。我们还提供了一个Python库，其中包含嵌入的最小实现。 显示英文摘要

摘要： 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.