The equivalent kernel [1] is a way of understanding how Gaussian process regression works for large sample sizes based on a continuum limit. In this paper we show how to approximate the equivalent kernel of the widely-used squared exponential (or Gaussian) kernel and related kernels. This is easiest for uniform input densities, but we also discuss the generalization to the non-uniform case. We ...

We propose a bootstrap sampling method jackboot sampling This provides more accu rate inferences than ordinary bootstrap sampling better con dence interval coverage and less biased or unbiased standard errors The method is simple to implement We also prescribe a smoothing parameter for use in smoothed bootstrapping using or dinary kernel smoothing The e ect is similar to that of jackboot sampling

Let f be a density on the real line and let f,~ be the kernel estimate of f in which the smoothing factor is obtained by maximizing the cross-validated likelihood product according to the method of Duin and Habbema, Hermans and Vandenbroek . Under mild regularity conditions on the kernel and f, we show, among other things that f Jf,~ f ( --~ 0 almost surely if and only if the sample extremes of...

A foveated image is a non-uniform resolution image whose resolution is highest at a point (fovea) but falling off away from the fovea. It can be obtained from a uniform image through a space-variant smoothing process, where the width of the smoothing function is small near the fovea and gradually expanding as the distance from the fovea increases. We treat this process as an integral operator a...

This paper provides a comparison study among a set of robust diffusion algorithms for processing optical flows. The proposed algorithms combine the smoothing ability of the heat kernel, modelled by the local Hessian, and the outlier rejection mechanisms of robust statistics algorithms. Smooth optical flow variation can be modelled very well using heat kernels. The diffusion kernel is considered...

In data analytic applications of density estimation one is usually interested in estimating the density over its support. However, common estimators such as the basic kernel estimator use a single smoothing parameter over the whole of the support. While this will be adequate for some densities there will be other densities that will be very difficult to estimate using this approach. The purpose...

Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and nume...

We propose and analyze a novel framework for learning sparse representations, based on two statistical techniques: kernel smoothing and marginal regression. The proposed approach provides a flexible framework for incorporating feature similarity or temporal information present in data sets, via non-parametric kernel smoothing. We provide generalization bounds for dictionary learning using smoot...

In brain imaging, cortical data such as the cortical thickness, cortical surface curvatures and surface coordinates have been mapped to a unit sphere for the purpose of visualization, surface registration and statistical analysis. Since the unit sphere provides a readily available parametrization and basis functions, cortical data can be easily quantified with respect to the spherical parametri...

Bayesian Gaussian processes and Support Vector machines are powerful kernel-based methods to attack the pattern recognition problem. Probably due to the very different philosophies of the fields they have been originally proposed in, techniques for these two models have been developed somewhat in isolation from each other. This tutorial paper reviews relationships between Bayesian Gaussian proc...