7,714 research outputs found

    Kernel Bayes' rule

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    A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on representations of probabilities in reproducing kernel Hilbert spaces. Probabilities are uniquely characterized by the mean of the canonical map to the RKHS. The prior and conditional probabilities are expressed in terms of RKHS functions of an empirical sample: no explicit parametric model is needed for these quantities. The posterior is likewise an RKHS mean of a weighted sample. The estimator for the expectation of a function of the posterior is derived, and rates of consistency are shown. Some representative applications of the kernel Bayes' rule are presented, including Baysian computation without likelihood and filtering with a nonparametric state-space model.Comment: 27 pages, 5 figure

    Regularized Regression Problem in hyper-RKHS for Learning Kernels

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    This paper generalizes the two-stage kernel learning framework, illustrates its utility for kernel learning and out-of-sample extensions, and proves {asymptotic} convergence results for the introduced kernel learning model. Algorithmically, we extend target alignment by hyper-kernels in the two-stage kernel learning framework. The associated kernel learning task is formulated as a regression problem in a hyper-reproducing kernel Hilbert space (hyper-RKHS), i.e., learning on the space of kernels itself. To solve this problem, we present two regression models with bivariate forms in this space, including kernel ridge regression (KRR) and support vector regression (SVR) in the hyper-RKHS. By doing so, it provides significant model flexibility for kernel learning with outstanding performance in real-world applications. Specifically, our kernel learning framework is general, that is, the learned underlying kernel can be positive definite or indefinite, which adapts to various requirements in kernel learning. Theoretically, we study the convergence behavior of these learning algorithms in the hyper-RKHS and derive the learning rates. Different from the traditional approximation analysis in RKHS, our analyses need to consider the non-trivial independence of pairwise samples and the characterisation of hyper-RKHS. To the best of our knowledge, this is the first work in learning theory to study the approximation performance of regularized regression problem in hyper-RKHS.Comment: 25 pages, 3 figure
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