Universality, Characteristic Kernels and RKHS Embedding of Measures
Universality, Characteristic Kernels and RKHS Embedding of Measures
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DOI：
10.5555/1953048.2021077
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发表时间：
2010-03
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影响因子：
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通讯作者：
Bharath K. Sriperumbudur;K. Fukumizu;Gert R. G. Lanckriet
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作者：
Bharath K. Sriperumbudur;K. Fukumizu;Gert R. G. Lanckriet
在过去的几年里，两种不同的正定（pd）核概念---通用和特征---在机器学习中并行发展：通用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，而特征核是在通过将它们嵌入到再生核希尔伯特空间（RKHS）中来区分概率度量的背景下引入的。然而，这两个概念之间的关系并没有得到很好的理解。本文的主要贡献是澄清通用和特征内核之间的关系，提出了一个统一的研究，将它们与RKHS嵌入的措施，除了澄清它们的关系，严格PD，条件严格PD和整体严格PD内核的其他常见概念。对于径向核上的径向，所有这些概念被证明是等价的。
Over the last few years, two different notions of positive definite (pd) kernels---universal and characteristic---have been developing in parallel in machine learning: universal kernels are proposed in the context of achieving the Bayes risk by kernel-based classification/regression algorithms while characteristic kernels are introduced in the context of distinguishing probability measures by embedding them into a reproducing kernel Hilbert space (RKHS). However, the relation between these two notions is not well understood. The main contribution of this paper is to clarify the relation between universal and characteristic kernels by presenting a unifying study relating them to RKHS embedding of measures, in addition to clarifying their relation to other common notions of strictly pd, conditionally strictly pd and integrally strictly pd kernels. For radial kernels on ℜd, all these notions are shown to be equivalent.