Related papers: Representation by Integrating Reproducing Kernels

The one-sided and full Hilbert transforms are evaluated exactly by means of the method of finite-part integration [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. In general, the result consists of two terms -- the…

The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in…

In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data…

Important information on the structure of complex systems, consisting of more than one component, can be obtained by measuring to which extent the individual components exchange information among each other. Such knowledge is needed to…

We introduce a vector differential operator $\mathbf{P}$ and a vector boundary operator $\mathbf{B}$ to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This…

We present an embedding of stochastic optimal control problems, of the so called path integral form, into reproducing kernel Hilbert spaces. Using consistent, sample based estimates of the embedding leads to a model free, non-parametric…

A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing…

This paper is devoted to the study of reproducing kernel Hilbert spaces. We focus on multipliers of reproducing kernel Banach and Hilbert spaces. In particular we tried to extend this concept and prove some theorems.

We investigate the question of studying spectral clustering in a Hilbert space where the set of points to cluster are drawn i.i.d. according to an unknown probability distribution whose support is a union of compact connected components. We…

The classical Mercer's theorem claims that a continuous positive definite kernel $K({\mathbf x}, {\mathbf y})$ on a compact set can be represented as $\sum_{i=1}^\infty \lambda_i\phi_i({\mathbf x})\phi_i({\mathbf y})$ where…

In this paper we analyze a greedy procedure to approximate a linear functional defined in a Reproducing Kernel Hilbert Space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals, including…

This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace.…

It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories,…

We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas…

We introduce a divergence measure between data distributions based on operators in reproducing kernel Hilbert spaces defined by kernels. The empirical estimator of the divergence is computed using the eigenvalues of positive definite Gram…

We extend the diffusion-map formalism to data sets that are induced by asymmetric kernels. Analytical convergence results of the resulting expansion are proved, and an algorithm is proposed to perform the dimensional reduction. In this work…

In this paper, a spectral theorem is proved for self-adjoint cyclically compact partial integral operators in the space of functions with mixed norm, which is a Kaplansky--Hilbert module. The decomposition through eigenfunctions, integral…

Reproducing kernel Hilbert spaces (RKHSs) are special Hilbert spaces in one-to-one correspondence with positive definite maps called kernels. They are widely employed in machine learning to reconstruct unknown functions from sparse and…

We analyse the convergence of sampling algorithms for functions in reproducing kernel Hilbert spaces (RKHS). To this end, we discuss approximation properties of kernel regression under minimalistic assumptions on both the kernel and the…

Kernel theorems, in general, provide a convenient representation of bounded linear operators. For the operator acting on a concrete function space, this means that its action on any element of the space can be expressed as a generalised…