Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows

We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter.

Kernel Methods

During the past decade, a major revolution has taken place in pattern-recognition technology with the introduction of rigorous and powerful mathematical approaches in problem domains previously treated with heuristic and less efficient techniques. The use of convex optimisation and statistical learning theory has been combined with ideas from functional analysis and classical statistics to produce a class of algorithms called kernel methods (KMs), which have rapidly become commonplace in applications. This book, and others, provides evidence of the practical applications that have made kernel methods a fundamental part of the toolbox for machine learning, statistics, and signal processing practitioners. The field of kernel methods has not only provided new insights and therefore new algorithms, but it has also created much discussion on well-established techniques such as Parzen windows and Gaussian processes, which use essentially the same technique but in different frameworks.