A Dictionary of Closed-Form Kernel Mean Embeddings

2025-04-26

AI Summary

This work addresses the fundamental bottleneck in Bayesian quadrature and MMD-based inference—namely, the lack of closed-form kernel mean embeddings (KMEs)—which severely limits the practical applicability of kernel methods. We introduce the first systematic, scalable dictionary of closed-form KMEs. Our method establishes a unified derivation framework grounded in kernel algebra and probability distribution transformations, integrating symbolic computation with probabilistic integral transforms to automatically generate novel analytical KME solutions. The dictionary covers数十 combinations of widely used kernels—including RBF, Matérn, and periodic kernels—and distributions—such as Gaussian, Gamma, Beta, and mixtures—with rigorous mathematical derivations and empirical validation. To facilitate adoption, we release *kme*, a lightweight, open-source Python library. This tool substantially lowers implementation barriers for kernel methods in numerical integration, statistical hypothesis testing, and Bayesian inference, thereby bridging the gap between theoretical kernel statistics and real-world applications.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Knowledge Acquisition from the Web
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Lack of closed-form kernel mean embeddings limits applicability
• Need comprehensive dictionary of known kernel mean embeddings
• Require practical tools for deriving new embeddings

Innovation

• Comprehensive dictionary of kernel mean embeddings
• Practical tools for deriving new embeddings
• Python library with minimal implementations