Related papers: Positive definite kernels and boundary spaces
Motivated by applications, we introduce a general and new framework for operator valued positive definite kernels. We further give applications both to operator theory and to stochastic processes. The first one yields several dilation…
The paper introduces new sufficient conditions of strict positive definiteness for kernels on d-dimensional spheres which are not radially symmetric but possess specific coefficient structures. The results use the series expansion of the…
This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and…
We study positive definiteness of kernels $K(x,y)$ on two-point homogeneous spaces. As opposed to the classical case, which has been developed and studied in the existing literature, we allow the kernel to have an (integrable) singularity…
We study how iterated and composed completely positive maps act on operator-valued kernels. Each kernel is realized inside a single Hilbert space where composition corresponds to applying bounded creation operators to feature vectors. This…
In the paper we investigate a method of quantization based on the concept of positive definite kernel on a principal $G$-bundle with compact structural group G. For G=U(1) our approach leads to Kostant-Souriau geometric quantization as well…
We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes…
We provide a characterization for the continuous positive definite kernels on $\mathbb R^d$ that are invariant to linear isometries, i.e. invariant under the orthogonal group $O(d)$. Furthermore, we provide necessary and sufficient…
We present a characterization for a positive definite operator valued kernel to be universal or $C_{0}$-universal, and apply these characterizations to a family of operator valued kernels that are shown to be well behaved. Later, we obtain…
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…
The generalization properties of Gaussian processes depend heavily on the choice of kernel, and this choice remains a dark art. We present the Neural Kernel Network (NKN), a flexible family of kernels represented by a neural network. 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…
We review machine learning methods employing positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain, expanded in terms of a…
The aim of the paper is to give a full characterization of functions f from I into the real line R (where I is an interval in R that satisfies certain natural conditions) such that for any I-valued positive definite kernel K defined on an…
We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical…
We investigate iterated compositions of weighted sums of Gaussian kernels and provide an interpretation of the construction that shows some similarities with the architectures of deep neural networks. On the theoretical side, we show that…
We consider the problem of positive-semidefinite continuation: extending a partially specified covariance kernel from a subdomain $\Omega$ of a rectangular domain $I\times I$ to a covariance kernel on the entire domain $I\times I$. For a…
We present sufficient condition for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their representation as a series of spherical harmonics. The family analyzed is a…
In this paper, we develop an approach to exploiting kernel methods with manifold-valued data. In many computer vision problems, the data can be naturally represented as points on a Riemannian manifold. Due to the non-Euclidean geometry of…
We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our…