Let κ be an U-invariant reproducing kernel and let H (κ) denote the reproducing kernel Hilbert C[z1, . . . , zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication operators Mz1 , . . . ,Mzd on H (κ). For a positive integer ν and d-tuple T = (T1, . . . , Td), consider the defect operator

A nonparametric kernel-based method for realizing Bayes’ rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes’ ru...

We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

The aim of this paper is to present a unified framework in the setting Hilbert $$C^*$$ -modules for scalar- and vector-valued reproducing kernel spaces -valued spaces. We investigate conditionally negative definite kernels with values -algebra adjointable operators acting on -module. In addition, we show that there exists two-sided connection between positive -modules. Furthermore, explore some...

Let κ be an U-invariant reproducing kernel and let H (κ) denote the reproducing kernel Hilbert C[z1, . . . , zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication operators Mz1 , . . . ,Mzd on H (κ). For a positive integer ν and d-tuple T = (T1, . . . , Td), consider the defect operator

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as kernel probabilistic programming. We illustrate it on synth...

Performance of the linear models, widely used within the framework of adaptive line enhancement (ALE), deteriorates dramatically in the presence of non-Gaussian noises. On the other hand, adaptive implementation of nonlinear models, e.g. the Volterra filters, suffers from the severe problems of large number of parameters and slow convergence. Nonetheless, kernel methods are emerging solutions t...