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Invariant kernels on the space of complex covariance matrices

Salem Said, Franziskus Steinert, Cyrus Mostajeran

TL;DR

The paper develops a rigorous and computational framework for invariant kernels on the space $M$ of complex covariance matrices by leveraging the $L^1$-Godement theorem and the spherical transform. It provides explicit transform/inverse formulas, a practical Lang-trick approach, and a Ramanujan-based spherical power-series construction, enabling closed-form kernels such as the Beta-prime kernel with favorable computational properties. The Beta-prime kernel demonstrates substantial speedups over traditional kernels (e.g., heat or Matérn) in two-sample testing on SPD matrices, while remaining broadly applicable to other matrix spaces admitting the same symmetry. Overall, the work advances both theory and computation for invariant kernel methods on non-Euclidean spaces relevant in statistics and machine learning.

Abstract

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.

Invariant kernels on the space of complex covariance matrices

TL;DR

The paper develops a rigorous and computational framework for invariant kernels on the space of complex covariance matrices by leveraging the -Godement theorem and the spherical transform. It provides explicit transform/inverse formulas, a practical Lang-trick approach, and a Ramanujan-based spherical power-series construction, enabling closed-form kernels such as the Beta-prime kernel with favorable computational properties. The Beta-prime kernel demonstrates substantial speedups over traditional kernels (e.g., heat or Matérn) in two-sample testing on SPD matrices, while remaining broadly applicable to other matrix spaces admitting the same symmetry. Overall, the work advances both theory and computation for invariant kernel methods on non-Euclidean spaces relevant in statistics and machine learning.

Abstract

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 --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.
Paper Structure (15 sections, 7 theorems, 89 equations, 3 figures)

This paper contains 15 sections, 7 theorems, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General background
  3. Positive definite functions
  4. Riemannian geometry
  5. The spherical transform
  6. Constructing invariant kernels
  7. Applying Ramanujan's theorem
  8. Numerical Experiments
  9. Computational considerations
  10. Application: the two-sample test
  11. Proof of Proposition \ref{['prop:gn']}
  12. Proof of Proposition \ref{['prop:gauss_spherical_trans']}
  13. Proof of Theorem \ref{['th:L1G']}
  14. Proof of Proposition \ref{['prop:langtrick']}
  15. Proof of Proposition \ref{['prop:ramanujan']}

Key Result

Proposition 1

If $\mathcal{K}$ is an invariant kernel, then the function $f(x) = \mathcal{K}(x,\mathrm{id})$ is $U$-invariant and positive definite. Conversely, if $f$ is a $U$-invariant and positive definite function, then $\mathcal{K}(x,y) = f(y^{ -1/2}xy^{ -1/2})$ defines an invariant kernel.

Figures (3)

  • Figure 1: Color plot of $f(x)/\Gamma_M(2\alpha)$, given by (\ref{['eq:betaprime']}) with $\alpha = 2$, restricted to real $2\times 2$ SPD matrices, here visualized as points in a cone viewed from two perspectives. The domain of the depicted plot is restricted to $0<\operatorname{tr}(x)<2$.
  • Figure 2: Computation runtime for $100 \times 100$ kernel Gram matrices, plotted against matrix dimension $N$. Beta-prime kernel (left) and 1-Matérn kernel (right). The $1$-Matérn kernel is the Matérn kernel with smoothness parameter $= 1$.
  • Figure 3: Rejection rate of the null hypothesis "same distribution" plotted against the spectral scaling factor $r_k\space$. Beta-prime kernel (yellow) and 1-Matérn kernel (blue).

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Proposition 6
  • Proposition 7