Universal kernels via harmonic analysis on Riemannian symmetric spaces

2025/06/24 by Franziskus Steinert, Steinert, Franziskus, Salem Said +3
Mathematics · Physics and Astronomy · #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.2506.19245

Abstract

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.

Citations

• Invariant kernels on the space of complex covariance matrices
• Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach
• Geometric Learning with Positively Decomposable Kernels
• Functional inequalities on symmetric spaces of noncompact type and applications
• Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices
• Hyperbolic Graph Neural Networks
• Hyperbolic Graph Convolutional Neural Networks
• A Riemannian Network for SPD Matrix Learning
• Geodesic Exponential Kernels: When Curvature and Linearity Conflict
• A Discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces
• Universality, Characteristic Kernels and RKHS Embedding of Measures
• Quaternions and matrices of quaternions

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