Motivated by mathematical learning from training data, we introduce the notion of refinable kernels. Various characterizations of refinable kernels are presented. The concept of refinable kernels leads to the introduction of wavelet-like reproducing kernels. We also investigate a refinable kernel that forms a Riesz basis. In particular, we characterize refinable translation invariant kernels, a...

We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of ∗-semigroups from the point of view of generation of ∗-representations, linearizations (Kolmogorov decompositions), and reproducing kernel spaces. We obtain a general dilation theorem in both Kolmogorov and reproducing kernel space representations, that unifie...

Almost sure bounds are established on the uniform error of smoothing spline estimators in nonparametric regression with random designs. Some results of Einmahl and Mason (2005) are used to derive uniform error bounds for the approximation of the spline smoother by an “equivalent” reproducing kernel regression estimator, as well as for proving uniform error bounds on the reproducing kernel regre...

Kernel Fisher’s linear discriminant analysis (KFLDA) has been proposed for nonlinear binary classification (Mika, Rätsch, Weston, Schölkopf and Müller, 1999, Baudat and Anouar, 2000). It is a hybrid method of the classical Fisher’s linear discriminant analysis and a kernel machine. Experimental results (e.g., Schölkopf and Smola, 2002) have shown that the KFLDA performs slightly better in terms...

In the analysis of complex phenomena of acoustic systems, the computational model-ing requires special attention for a realistic representation of the physics. As a powerful tool, the nite element method has been widely used in the study of complex systems. In order to capture the important physical phenomena, p-nite elements and/or hp-nite elements are employed. The reproducing kernel particle...

We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn H into a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω. We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for th...

A Gabor based representation for textured images is proposed. Instead of the ordinary filter bank, a reproducing kernel representation is constructed consisting of a sum of several local reproducing kernels. The image representation coefficients are computed by a basis pursuit procedure, and are then considered as the feature vectors. The feature vectors are used to construct a kernel for a sup...

This paper is concerned with a technique for solving Volterra integral equations in the reproducing kernel Hilbert space. In contrast with the conventional reproducing kernel method,the Gram-Schmidt process is omitted here and satisfactory results are obtained. The analytical solution is represented in the form of series. An iterative method is given to obtain the approximate solution. The conv...

We analyze the regularized least square algorithm in learning theory with Reproducing Kernel Hilbert Spaces (RKHS). Explicit convergence rates for the regression and binary classification problems are obtained in particular for the polynomial and Gaussian kernels on the n-dimensional sphere and the hypercube. There are two major ingredients in our approach: (i) a law of large numbers for Hilber...

In this paper we propose a new approach for the modelling of the multi-variable systems (MIMO) on the Reproducing Kernel Hilbert Space (RKHS). The proposed approach considers the MIMO system as a set of MISO processes modelled in RKHS space. We propose also a comparative study of three identification kernel methods of nonlinear systems modelled in Reproducing Kernel Hilbert Space (RKHS), where ...