无界特征核与通用核

中文摘要 AI 辅助

核方法是机器学习和统计学中最强大的工具之一，具有大量成功的应用。其巨大成功源于每个核所关联的灵活函数类——即其再生核希尔伯特空间（RKHS）——这促进了统计分析，同时也源于其计算可行性和对许多领域的适用性。多个概念（如特征核、$L_p$-通用核和积分严格正定核）刻画了核及其RKHS的表达能力，并在理解核方法的统计性质中发挥关键作用；这些概念及其关系在有界核情形下已被充分理解。尽管在过去十年中，无界核已受到显著关注（例如，在构建基于核的差异和依赖度量中，如最大均值差异、希尔伯特-施密特独立性准则和核斯坦因差异），但令人惊讶的是，关于这些概念在无界情形下的关系知之甚少。在本文中，我们解决了这一严重瓶颈，在温和假设下建立了它们之间的关系。

英文摘要

Kernel methods are among the most powerful tools in machine learning and statistics, with a large number of successful applications. Their immense success stems from the flexible function class associated to each kernel---its reproducing kernel Hilbert space (RKHS)---which facilitates statistical analysis, as well as from their computational tractability and applicability to many domains. Multiple notions (such as characteristic, $L_p$-universal, and integrally strictly positive definite) capture the expressivity of kernels and their RKHSs and play a key role in understanding the statistical properties of kernel methods; these concepts and their relations are well-understood for bounded kernels. Even though unbounded kernels have received significant attention over the past decade (for instance, in the construction of kernel-based discrepancy and dependence measures such as the maximum mean discrepancy, the Hilbert-Schmidt independence criterion, and the kernel Stein discrepancy), surprisingly little is known about the relations of these notions in the unbounded case. In the present paper we tackle this severe bottleneck, establishing their relations under mild assumptions.

发表机构

• Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
• London School of Economics（伦敦政治经济学院）

机构由 AI 辅助整理，请以论文原文为准。