We first introduce some related definitions of the bounded linear operator L in the reproducing kernel space W(2)(m)(D). Then we show spectral analysis of L and derive several property theorems.

We review recent methods for learning with positive definite kernels. All these methods formulate learning and estimation problems as linear tasks in a reproducing kernel Hilbert space (RKHS) associated with a kernel. We cover a wide range of methods, ranging from simple classifiers to sophisticated methods for estimation with structured data. (AMS 2000 subject classifications: primary 30C40 Ke...

Early stopping is a form of regularization based on choosing when to stop running an iterative algorithm. Focusing on non-parametric regression in a reproducing kernel Hilbert space, we analyze the early stopping strategy for a form of gradient-descent applied to the least-squares loss function. We propose a data-dependent stopping rule that does not involve hold-out or cross-validation data, a...

In the present contribution we tackle the problem of nonlinear independent component analysis by non-Euclidean Hebbian-like learning. Independent component analysis (ICA) and blind source separation originally were introduced as tools for the linear unmixing of the signals to detect the underlying sources. Hebbian methods became very popular and succesfully in this context. Many nonlinear ICA e...

A reproducing kernel Hilbert space (RKHS) has four well-known easily derived properties. Since these properties are usually not emphasized as a simple means of gaining insight into RKHS structure, they are singled out and proved here.

In this paper, we apply the new implementation of reproducing kernel Hilbert space method to give the approximate solution to some functional integral equations of the second kind. To show its effectiveness and convenience, some examples are given.

Problems in R are addressed where the scalar potential of an associated vector field satisfies Laplace’s equation in some unbounded external region and is to be approximated by unknown (point) sources contained in the complimentary subregion. Two specific field geometries are considered: R half-space and the exterior of an R sphere, which are the two standard settings for geophysical and geoexp...