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Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach

Nathael Da Costa, Cyrus Mostajeran, Juan-Pablo Ortega, Salem Said

TL;DR

This work proves that the Gaussian kernel $k(x,y)=\exp(-\lambda d^2(x,y))$ is never positive-definite on non-Euclidean Riemannian symmetric spaces, except for a few low-dimensional scenarios treated numerically. It develops a rigorous harmonic-analytic framework based on Godement's theorem and the L$^{\!2}$- and L$^{\!1}$-Godement theorems, tying positive-definiteness to the nonnegativity of the spherical transform $\hat{f}$ and its integrability with respect to the Harish-Chandra measure. The authors systematically analyze both compact and non-compact spaces: they show non-PD for the Gaussian kernel on all compact symmetric spaces via embedding into circles, and for non-compact spaces by embedding hyperbolic planes and using explicit transforms in hyperbolic plane/space settings. They also introduce the hyperbolic secant kernel $k(x,y)=(\cosh(d(x,y)))^{-a}$, which is PD on hyperbolic spaces (and related 3D cases for integer $a$), and connect these results to non-Euclidean Herschel-Maxwell distributions, providing probabilistic interpretations and PD conditions for related kernels. The work offers a blueprint for studying invariant kernels on symmetric spaces and paves the way for kernel methods in geometric settings with strong harmonic-analytic structure.

Abstract

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.

Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach

TL;DR

This work proves that the Gaussian kernel is never positive-definite on non-Euclidean Riemannian symmetric spaces, except for a few low-dimensional scenarios treated numerically. It develops a rigorous harmonic-analytic framework based on Godement's theorem and the L- and L-Godement theorems, tying positive-definiteness to the nonnegativity of the spherical transform and its integrability with respect to the Harish-Chandra measure. The authors systematically analyze both compact and non-compact spaces: they show non-PD for the Gaussian kernel on all compact symmetric spaces via embedding into circles, and for non-compact spaces by embedding hyperbolic planes and using explicit transforms in hyperbolic plane/space settings. They also introduce the hyperbolic secant kernel , which is PD on hyperbolic spaces (and related 3D cases for integer ), and connect these results to non-Euclidean Herschel-Maxwell distributions, providing probabilistic interpretations and PD conditions for related kernels. The work offers a blueprint for studying invariant kernels on symmetric spaces and paves the way for kernel methods in geometric settings with strong harmonic-analytic structure.

Abstract

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-Godement theorems (where ), 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.
Paper Structure (28 sections, 11 theorems, 91 equations, 2 figures)

This paper contains 28 sections, 11 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Godement's theorem
  3. L$^\textbf{\!2}$-Godement theorem
  4. Basic examples
  5. Hyperbolic plane
  6. Hyperbolic space
  7. Distance kernels
  8. Wishart kernel
  9. Hyperbolic secant kernel
  10. Compact symmetric spaces
  11. Non-compact symmetric spaces
  12. Hyperbolic plane
  13. Hyperbolic space
  14. The general case
  15. The hyperbolic secant revisited: non-Euclidean Herschel-Maxwell distributions
  16. ...and 13 more sections

Key Result

Theorem 3.1

\newlabelprop:l2god0 A continuous function $f \in L^2_H(M)$ is positive-definite if and only if $\hat{f}(t) \geq 0$ for almost all $t \in \mathfrak{a}^*$ and $\hat{f}$ is integrable with respect to the measure $|c(t)|^{-2}dt$.

Figures (2)

  • Figure 1: Spherical transform for the Gaussian kernel on the hyperbolic plane
  • Figure 1: Plots of the normalized hyperbolic Herschel-Maxwell (solid curve, (\ref{['2D hyperbolc HM']})) and Gaussian (dashed curve, (\ref{['2D hyperbolic Gauss']})) densities as functions of $r=d(o,x)$ for parameters $a=4$ and $\lambda = 1.66$.

Theorems & Definitions (16)

  • Theorem 3.1: L$^{\! 2}$-$\space$Godement Theorem
  • Proposition 5.1
  • Lemma 5.2: Embedding Lemma
  • Lemma 5.3
  • Proof 1
  • Proposition 6.1
  • Remark 6.2
  • Proposition 6.3
  • Lemma 6.4
  • Lemma 6.5
  • ...and 6 more