Related papers: Kernel-based interpolation at approximate Fekete p…
The kernel-based multi-scale method has been proven to be a powerful approximation method for scattered data approximation problems which is computationally superior to conventional kernel-based interpolation techniques. The multi-scale…
Considering a singularly perturbed convection-diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gau{\ss}-Lobatto type for higher-order streamline diffusion FEM. We show a useful…
We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on $L^2$ spaces weighted by a probability density function.…
We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For…
A fundamental drawback of kernel-based statistical models is their limited scalability to large data sets, which requires resorting to approximations. In this work, we focus on the popular Gaussian kernel and on techniques to linearize…
A kernel based procedure for correcting experimental data for distortions due to the finite resolution and limited detector acceptance is presented. The unfolding problem is known to be an ill-posed problem that can not be solved without…
A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel translates located at the nodes. This is the crux of kernel density estimation, kernel…
Kernel-based methods offer a powerful and flexible mathematical framework for addressing histopolation problems. In histopolation, the available input data does not consist of pointwise function samples but of averages taken over intervals…
Global discrete optimization is notoriously difficult due to the lack of gradient information and the curse of dimensionality, making exhaustive search infeasible. Tensor cross approximation is an efficient technique to approximate…
We discuss the growth of the Lebesgue constants for polynomial interpolation at Fekete points for fixed degree (one) and varying dimension, and underlying set $K\subset \R^d$ a simplex, ball or cube.
We study the worst case error of kernel density estimates via subset approximation. A kernel density estimate of a distribution is the convolution of that distribution with a fixed kernel (e.g. Gaussian kernel). Given a subset (i.e. a point…
Weighted Fekete points are defined as those that maximize the weighted version of the Vandermonde determinant over a fixed set. They can also be viewed as the equilibrium distribution of the unit discrete charges in an external…
Based on direct integrals, a framework allowing to integrate a parametrised family of reproducing kernels with respect to some measure on the parameter space is developed. By pointwise integration, one obtains again a reproducing kernel…
The application of Gaussian processes (GPs) to large data sets is limited due to heavy memory and computational requirements. A variety of methods has been proposed to enable scalability, one of which is to exploit structure in the kernel…
Learning can be seen as approximating an unknown function by interpolating the training data. Kriging offers a solution to this problem based on the prior specification of a kernel. We explore a numerical approximation approach to kernel…
We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and…
We study the generalization error of functions that interpolate prescribed data points and are selected by minimizing a weighted norm. Under natural and general conditions, we prove that both the interpolants and their generalization errors…
The paper addresses the problem to estimate the power spectral density of an ARMA zero mean Gaussian process. We propose a kernel based maximum entropy spectral estimator. The latter searches the optimal spectrum over a class of high order…
Kernel methods are a highly effective and widely used collection of modern machine learning algorithms. A fundamental limitation of virtually all such methods are computations involving the kernel matrix that naively scale quadratically…
We derive a Fast Multipole Method (FMM) where a low-rank approximation of the kernel is obtained using the Empirical Interpolation Method (EIM). Contrary to classical interpolation-based FMM, where the interpolation points and basis are…