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Orthonormal Expansions for Translation-Invariant Kernels

Filip Tronarp, Toni Karvonen

TL;DR

This paper introduces a general Fourier-analytic framework for constructing orthonormal bases of translation-invariant kernels on $\mathbb{R}$ from bases of $\mathscr{L}_2(\mathbb{R})$, yielding pointwise-convergent RKHS expansions $r(t,u)=\sum_m\psi_m^*(t)\psi_m(u)$. It applies the method to three canonical kernels: Matérn kernels with half-integer order via Laguerre-based bases (including a finite null-space portion), the Cauchy kernel via complex/real Cauchy–Laguerre bases, and the Gaussian kernel via Hermite functions; each yields explicit, computable expansions and tractable truncation errors. The Gaussian expansion recovers a Mercer expansion consistent with Mehler’s formula, while the Matérn expansions are nearly Mercer, up to a finite null-space. The results provide practical, scalable representations of these kernels for numerical methods and theoretical analysis, with clear pathways for extending to other translations-invariant kernels and scaling regimes.

Abstract

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of $\mathscr{L}_2(\mathbb{R})$. This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.

Orthonormal Expansions for Translation-Invariant Kernels

TL;DR

This paper introduces a general Fourier-analytic framework for constructing orthonormal bases of translation-invariant kernels on from bases of , yielding pointwise-convergent RKHS expansions . It applies the method to three canonical kernels: Matérn kernels with half-integer order via Laguerre-based bases (including a finite null-space portion), the Cauchy kernel via complex/real Cauchy–Laguerre bases, and the Gaussian kernel via Hermite functions; each yields explicit, computable expansions and tractable truncation errors. The Gaussian expansion recovers a Mercer expansion consistent with Mehler’s formula, while the Matérn expansions are nearly Mercer, up to a finite null-space. The results provide practical, scalable representations of these kernels for numerical methods and theoretical analysis, with clear pathways for extending to other translations-invariant kernels and scaling regimes.

Abstract

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of . This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.
Paper Structure (21 sections, 7 theorems, 120 equations, 8 figures)

This paper contains 21 sections, 7 theorems, 120 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Construction of orthonormal bases
  3. On Mercer expansions
  4. Summary of expansions
  5. Matérn kernels
  6. Cauchy kernel
  7. Gaussian kernel
  8. Expansions of Matérn kernels
  9. Laguerre functions
  10. Matérn--Laguerre functions
  11. Classification of Matérn--Laguerre functions
  12. Truncation error
  13. The null-space $\mathscr{M}_{\nu}^0$
  14. Expansions of the Cauchy kernel
  15. Expansion in complex-valued Cauchy--Laguerre functions
  16. ...and 6 more sections

Key Result

theorem 1.1

Suppose that $r(t, u) = \Phi(t - u)$ is a translation-invariant symmetric positive-definite kernel with $\Phi \in C(\mathbb{R}) \cap \mathscr{L}_1(\mathbb{R})$. Let $\{\varphi_m\}_{m \in I}$ be an orthonormal basis of $\mathscr{L}_2(\mathbb{R})$ and $h$ a function such that $\abs[0]{\hat{h}(\omega)} for $m \in I$ form an orthonormal basis of $\mathscr{H}_r(\mathbb{R})$ and the kernel $r$ has the p

Figures (8)

  • Figure 1: The Matérn--Laguerre functions $\psi_{m, \nu}^{+}$ in \ref{['eq:psi-positive']} for $m = 0, \ldots, 6$. Observe that the functions vanish on the negative real line.
  • Figure 2: Translates $\rho_{\nu + 1/2}(\cdot, u)$ of the kernel in \ref{['eq:rho-kernel']} for $u \in \{-2, -1.2, -0.4, 0.4, 1.2, 2\}$. Observe that each translate is supported on the axis that $u$ lies on.
  • Figure 3: The truncation in \ref{['eq:matern-truncation']} for two Matérn kernels. Because the second kernel argument has been fixed to a positive value, the truncations are exact on the negative real line by \ref{['eq:matern-simplification-sign']}.
  • Figure 4: The null-space Matérn--Laguerre functions $\psi_{m, \nu}^{0}$ in \ref{['eq:psi-null']} for $\nu = 2$ and $\nu = 9$.
  • Figure 5: The real-valued Cauchy--Laguerre functions$\alpha_{m}$ and $\beta_{m}$ in \ref{['eq:trig-cauchy-laguerre']}.
  • ...and 3 more figures

Theorems & Definitions (15)

  • theorem 1.1: Construction of orthonormal bases
  • proof
  • Proposition 3.1: Matérn--Laguerre upper bound
  • proof
  • Proposition 3.2: Matérn--Laguerre orthogonality
  • proof
  • Proposition 3.3: Matérn truncation
  • proof
  • Proposition 3.4: Null-space symmetry
  • proof
  • ...and 5 more