Related papers: General superconvergence for kernel-based approxim…
We design a quasi-interpolation operator from the Sobolev space $H^1_0(\Omega)$ to its finite-dimensional finite element subspace formed by piecewise polynomials on a simplicial mesh with a computable approximation constant. The operator 1)…
Singular and oscillatory functions feature in numerous applications. The high-accuracy approximation of such functions shall greatly help us develop high-order methods for solving applied mathematics problems. This paper demonstrates that…
Indefinite inner product spaces of entire functions and functions analytic inside a disk are considered and their completeness studied. Spaces induced by the rotation invariant reproducing kernels in the form of the generalized…
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a…
In this paper we first identify a basic limitation in gradient descent-based optimization methods when used in conjunctions with smooth kernels. An analysis based on the spectral properties of the kernel demonstrates that only a vanishingly…
The support vector machine (SVM) is a supervised learning algorithm that finds a maximum-margin linear classifier, often after mapping the data to a high-dimensional feature space via the kernel trick. Recent work has demonstrated that in…
A spectral mixture (SM) kernel is a flexible kernel used to model any stationary covariance function. Although it is useful in modeling data, the learning of the SM kernel is generally difficult because optimizing a large number of…
Given the joint distribution of two random variables $X,Y$ on some second countable locally compact Hausdorff space, we investigate the statistical approximation of the $L^2$-operator defined by $[Pf](x) := \mathbb{E}[ f(Y) \mid X = x ]$…
We consider conditions on a given system $\mathcal{F}$ of vectors in Hilbert space $\mathcal{H}$, forming a frame, which turn $\mathcal{H}$ into a reproducing kernel Hilbert space. It is assumed that the vectors in $\mathcal{F}$ are…
Representing images by compact codes has proven beneficial for many visual recognition tasks. Most existing techniques, however, perform this coding step directly in image feature space, where the distributions of the different classes are…
Stable Diffusion (SD) has evolved DDPM (Denoising Diffusion Probabilistic Model) based image generation significantly by denoising in latent space instead of feature space. This popularized DDPM-based image generation as the cost and…
A Bayes point machine is a single classifier that approximates the majority decision of an ensemble of classifiers. This paper observes that kernel interpolation is a Bayes point machine for Gaussian process classification. This observation…
This paper studies the eigenvalue problem $K \psi = \lambda \psi$ associated with a Fredholm integral operator $K$ defined by a smooth kernel. The focus is on analyzing the convergence behaviour of numerical approximations to eigenvalues…
This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: \Omega \times \Omega \rightarrow \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity…
These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…
We extend the herding algorithm to continuous spaces by using the kernel trick. The resulting "kernel herding" algorithm is an infinite memory deterministic process that learns to approximate a PDF with a collection of samples. We show that…
We study strictly proper scoring rules in the Reproducing Kernel Hilbert Space. We propose a general Kernel Scoring rule and associated Kernel Divergence. We consider conditions under which the Kernel Score is strictly proper. We then…
In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are…
Interpolation of sparse pixel information towards a dense target resolution finds its application across multiple disciplines in computer vision. State-of-the-art interpolation of motion fields applies model-based interpolation that makes…
This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a…