Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach
This addresses a foundational issue in kernel methods for machine learning on non-Euclidean data, offering a rigorous framework for invariant kernels on symmetric spaces, though it is incremental relative to the broader Bochner-Godement theorem.
The paper tackled the problem of proving that the Gaussian kernel is never positive-definite on non-Euclidean symmetric spaces, achieving this through new geometric and analytical arguments that provide a complete characterization except for some low-dimensional cases handled numerically.
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 arguments. These provide a rigorous characterization of the positive-definiteness of the Gaussian kernel, which is complete but for a limited number of scenarios in low dimensions that are treated by numerical computations. Chief among these results are the L$^{\!\scriptscriptstyle p}$-$\hspace{0.02cm}$Godement theorems (where $p = 1,2$), which provide verifiable necessary and sufficient conditions for a kernel defined on a symmetric space of non-compact type to be positive-definite. A celebrated theorem, sometimes called the Bochner-Godement theorem, already gives such conditions and is far more general in its scope, but is especially hard to apply. Beyond the connection with the Gaussian kernel, the new results in this work lay out a blueprint for the study of invariant kernels on symmetric spaces, bringing forth specific harmonic analysis tools that suggest many future applications.