arXiv subjects

Franziskus Steinert

Publications and source records attributed to Franziskus Steinert.

3 recordsLinked to original sources

arXiv2026

Universality of kernels on Riemannian symmetric spaces

We investigate universality properties of continuous, positive-definite invariant kernels on Riemannian symmetric spaces, providing a unified harmonic-analytic characterization across compact and non-compact settings. In the compact symmetric case, we prove that a continuous, positive-definite invariant kernel is $C$-universal if and only if all of its spherical coefficients are strictly positive, a sharpening of classical Bochner-type results. This characterization is extended to compact homogeneous spaces, where universality is shown to be equivalent to the strict positive-definiteness of coefficient matrices arising from a representation-theoretic expansion. In contrast, for non-compact symmetric spaces, we establish that any continuous, positive-definite invariant kernel that is $C_0$ and integrable is automatically $C_0$-universal. Our analysis essentially relies on spectral decompositions using spherical functions and group Fourier transforms, in order to provide alternative, harmonic-analytic formulations of universality conditions. Several examples (including kernels on spheres and compact Lie groups, as well as integrable kernels on hyperbolic spaces and symmetric cones) both illustrate the theory and demonstrate practical criteria for constructing universal kernels.

math.FA↗

arXiv2025

Universal kernels via harmonic analysis on Riemannian symmetric spaces

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 machine learning. In this work, we establish fundamental tools for investigating universality properties of kernels in Riemannian symmetric spaces, thereby extending the study of this important topic to kernels in non-Euclidean domains. Moreover, we use the developed tools to prove the universality of several recent examples from the literature on positive definite kernels defined on Riemannian symmetric spaces, thus providing theoretical justification for their use in applications involving manifold-valued data.

cs.LG↗

arXiv2025

Invariant kernels on the space of complex covariance matrices

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 kernel, which is (in a certain natural sense) also integrable, can be computed by taking the inverse spherical transform of a positive function. General expressions for inverse spherical transforms are then provided, which can be used to explore new families of invariant kernels, at a rather moderate computational cost. A further, alternative approach for constructing new invariant kernels is also introduced, based on Ramanujan's master theorem for symmetric cones. This leads to a novel closed-form invariant kernel, called the Beta-prime kernel. Numerical experiments highlight the computational and performance advantages of this kernel, especially in the context of two-sample hypothesis testing.

math.FA↗