Related papers: On the completely positive kernels for nonuniform …
This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as…
We survey results concerning behavior of positivity of line bundles and possible vanishing theorems in positive characteristic. We also try to describe variation of positivity in mixed characteristic. These problems are very much related to…
The present work develops certain analytical tools required to construct and compute invariant kernels on the space of complex covariance matrices. The main result is the $\mathrm{L}^1$--Godement theorem, which states that any invariant…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
A rectangular matrix is called totally positive, if all its minors are positive. A point of a real Grassmanian manifold $G_{l,m}$ of $l$-dimensional subspaces in $\mathbb R^m$ is called strictly totally positive, if one can normalize its…
We establish a domination principle for positive operators, which provides an upper bound on the essential spectral radius and yields quasi-compactness criteria on weighted supremum spaces with Lyapunov type functions and local domination.…
This work aims to prove that the classical Gaussian kernel, when defined on a non-Euclidean symmetric space, is never positive-definite for any choice of parameter. To achieve this goal, the paper develops new geometric and analytical…
We study the positivity and asymptotic behaviour of nonnegative solutions of a general nonlocal fast diffusion equation, \[\partial_t u + \mathcal{L}\varphi(u) = 0,\] and the interplay between these two properties. Here $\mathcal{L}$ is a…
Let G be a simple and simply-connected complex algebraic group, and let X \subset G^\vee be the centralizer subgroup of a principal nilpotent element. Ginzburg and Peterson independently related the ring of functions on X with the homology…
We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules…
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 continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.
Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define…
Fractional powers and polynomial maps preserving structured totally positive matrices, one-sided Polya frequency functions, or totally positive kernels are treated from a unifying perspective. Besides the stark rigidity of the polynomial…
We consider positive semidefinite kernels valued in the $*$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $*$-semigroups. A rather general dilation theorem is stated and…
In this article we study the field of Hilbertian metrics and positive definit (pd) kernels on probability measures, they have a real interest in kernel methods. Firstly we will make a study based on the Alpha-Beta-divergence to have a…
$C^*$-algebra-valued kernels could pave the way for the next generation of kernel machines. To further our fundamental understanding of learning with $C^*$-algebraic kernels, we propose a new class of positive definite kernels based on the…
It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global…
This paper studies random operator-valued positive definite (p.d.) kernels and their connection to moment dilations. A class of random p.d. kernels is introduced in which the positivity requirement is imposed only in expectation, extending…
We develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a…