将Bregman散度拓展至函数空间，用核方法实现高效计算与应用。
Generalising maximum mean discrepancy: kernelised functional Bregman divergences
- 在希尔伯特空间中定义函数型Bregman散度，利用核均值嵌入简化估计。
- 支持聚类、通用估计、鲁棒估计和生成建模等任务，性能优于传统方法。
- 适合从事机器学习理论、函数数据分析与核方法研究者参考。
Bregman散度在统计学、机器学习和计算信息几何中具有核心作用，尤其在聚类、指数族、参数估计和优化中广泛应用。然而，希尔伯特空间，特别是再生核希尔伯特空间（RKHS）的工具尚未系统应用于函数型Bregman散度（即点为函数而非有限维参数向量的情形）。尽管已有研究涉及函数型散度，但多在巴拿赫空间中进行，与主流机器学习中的核方法和希尔伯特空间几何不一致。本文在希尔伯特空间中研究函数型Bregman散度，利用自对偶配对和里斯表示子带来便捷的微分运算；进一步将散度生成器设为核均值嵌入的复合形式，使散度可高效估计。我们探讨了其在聚类、通用估计、鲁棒估计和生成建模中的应用，并与其它类型的Bregman散度进行了对比。
原文摘要 · Abstract (English)
Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to clustering, exponential families, parameter estimation and optimisation, among other things. Despite this, the full toolkit of Hilbert spaces and in particular reproducing kernel Hilbert spaces have not been systematically developed and applied to functional Bregman divergences, where points are functions rather than finite-dimensional parameter vectors. While other types of functional Bregman divergences have been studied, these are typically in a Banach space rather than more directly aligned with kernel methods and Hilbert-space geometry commonly used in machine learning. We consider functional Bregman divergences on a Hilbert space, where the self-dual pairing and Riesz representer afford us particularly convenient calculus. Further specialising Bregman generators as a composition involving a kernel mean embedding makes such divergences easy to estimate. We discuss applications in clustering, universal estimation, robust estimation and generative modelling, and contrast our approach with other types of Bregman divergences.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。