标题： 泛化性，特征核和测度的RKHS嵌入 显示英文标题

标题： Universality, Characteristic Kernels and RKHS Embedding of Measures

摘要： 一种用于概率测度的希尔伯特空间嵌入最近被提出，其中任何概率测度都被表示为再生核希尔伯特空间（RKHS）中的一个均值元素。 这种嵌入在同质性检验、独立性检验、降维等应用中得到了应用，要求再生核是特征性的，即嵌入是单射的。 在本文中，我们将这种嵌入推广到有限带符号博雷尔测度，其中任何有限带符号博雷尔测度都被表示为一个RKHS中的均值元素。 我们证明，所提出的嵌入是单射的当且仅当核是通用的。 因此，这提供了一种通用核的新表征，这些核是在通过基于核的分类/回归算法实现贝叶斯风险的背景下提出的。 通过利用通用性和将有限带符号博雷尔测度嵌入到RKHS之间的关系，我们建立了通用核和特征核之间的关系。 显示英文摘要

摘要： A Hilbert space embedding for probability measures has recently been proposed, wherein any probability measure is represented as a mean element in a reproducing kernel Hilbert space (RKHS). Such an embedding has found applications in homogeneity testing, independence testing, dimensionality reduction, etc., with the requirement that the reproducing kernel is characteristic, i.e., the embedding is injective. In this paper, we generalize this embedding to finite signed Borel measures, wherein any finite signed Borel measure is represented as a mean element in an RKHS. We show that the proposed embedding is injective if and only if the kernel is universal. This therefore, provides a novel characterization of universal kernels, which are proposed in the context of achieving the Bayes risk by kernel-based classification/regression algorithms. By exploiting this relation between universality and the embedding of finite signed Borel measures into an RKHS, we establish the relation between universal and characteristic kernels.