The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from numerical analysis standpoint. basic problem of finding an algorithm for efficient integration functions reproduced by kernels not been fully solved. In this article we construct two classes algorithms that use <inline-formu...

We give several properties of the reproducing kernel Hilbert spaces induced by the Gaussian kernel and their implications for recent results in the complexity of the regularized least square algorithm in learning theory.

Gaussian kernels with flexible variances provide a rich family of Mercer kernels for learning algorithms. We show that the union of the unit balls of reproducing kernel Hilbert spaces generated by Gaussian kernels with flexible variances is a uniform Glivenko-Cantelli (uGC) class. This result confirms a conjecture concerning learnability of Gaussian kernels and verifies the uniform convergence ...

We describe a data complexity approach to kernel selection based on the behavior of polynomial and Gaussian kernels. Our results show how the use of a Gaussian kernel produces a gram matrix with useful local information that has no equivalent counterpart in polynomial kernels. By exploiting neighborhood information embedded by data complexity measures, we are able to carry out a form of meta-ge...

Gaussian processes provide a principled Bayesian framework, but direct implementations are restricted to small data sets due to the cubic time cost in the data size. In case the kernel function is expressible as a tensor product kernel and input data lies on a multidimensional grid it has been shown that the computational cost for Gaussian process regression can be reduced considerably. Tensor ...

We compare a kernel-based collocation method (meshfree approximation method) with a Galerkin finite element method for solving elliptic stochastic partial differential equations driven by Gaussian noise. The kernel-based collocation solution is a linear combination of reproducing kernels obtained from related differential and boundary operators centered at chosen collocation points. Its random ...

There has been an increasing interest in kernel-based techniques, such as support vector techniques, regularization networks, and gaussian processes. There are inner relationships among those techniques, with the kernel function playing a central role. This article discusses a new class of kernel functions derived from the so-called frames in a function Hilbert space.