Related papers: Positive definite kernels and lattice paths
The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this work, we present a novel analysis tool to handle discrete…
Over the last decade several positive definite kernels have been proposed to treat spike trains as objects in Hilbert space. However, for the most part, such attempts still remain a mere curiosity for both computational neuroscientists and…
It is shown that a positive (bounded linear) operator on a Hilbert space with trivial kernel is unitarily equivalent to a Hankel operator that satisfies double positivity condition if and only if it is non-invertible and has simple spectrum…
The main purpose of our paper is a new approach to design of algorithms of Kaczmarz type in the framework of operators in Hilbert space. Our applications include a diverse list of optimization problems, new Karhunen-Lo\`eve transforms, and…
We tackle the problem of optimizing over all possible positive definite radial kernels on Riemannian manifolds for classification. Kernel methods on Riemannian manifolds have recently become increasingly popular in computer vision. However,…
In this note we refine the notion of conditionally negative definite kernels to the notion of conditionally strictly negative definite kernels and study its properties. We show that the class of these kernels carries some surprising…
Motivated by practical applications, I present a novel and comprehensive framework for operator-valued positive definite kernels. This framework is applied to both operator theory and stochastic processes. The first application focuses on…
We study representations of positive definite kernels $K$ in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for…
We supply a Fourier characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of circles.
For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a…
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…
We introduce a method to construct general multivariate positive definite kernels on a nonempty set $X$ that employs a prescribed bounded completely monotone function and special multivariate functions on $X$.\ The method is consistent with…
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 article, we use the knowledge of positive definite tensors to develop a concept of positive definite multi-kernels to construct the kernel-based interpolants of scattered data. By the techniques of reproducing kernel Banach spaces,…
We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, in the cases in which at least one of the spaces is a sphere of…
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown…
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…
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…
Positive-definite kernel functions are fundamental elements of kernel methods and Gaussian processes. A well-known construction of such functions comes from Bochner's characterization, which connects a positive-definite function with a…
We study both canonical reproducing kernels and constructive reproducing kernels for holomorphic functions in $\CC^1$ and $\CC^n$. We compare and contrast the two, and also develop important relations between the two types of kernels. We…