Universal kernels via harmonic analysis on Riemannian symmetric spaces

2025-06-23

AI Summary

Existing kernel theory is largely confined to Euclidean spaces, limiting the application of kernel methods to non-Euclidean manifold learning. Method: This work systematically extends the notion of kernel universality to Riemannian symmetric spaces—leveraging Lie group representation theory, harmonic analysis, and the reproducing kernel Hilbert space (RKHS) framework—to establish rigorous criteria for the approximation capability of kernels on non-Euclidean domains. Contribution/Results: We prove the universality of several classical positive-definite kernels—including the heat kernel and Gaussian-type kernels—on canonical symmetric spaces such as the sphere, hyperbolic space, and Grassmannian manifolds. To our knowledge, this is the first unified analytical framework for designing and analyzing kernels on manifold-valued data. The theory provides foundational guarantees for non-Euclidean kernel methods, enabling principled generalization of kernel-based learning beyond flat geometries.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Knowledge Acquisition
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Study universality of kernels in Riemannian symmetric spaces
• Extend kernel methods to non-Euclidean domains
• Prove universality of positive definite kernels on manifolds

Innovation

• Harmonic analysis on Riemannian symmetric spaces
• Universality properties of non-Euclidean kernels
• Theoretical justification for manifold-valued data