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Explicit Universal and Approximate-Universal Kernels on Compact Metric Spaces

Eloi Tanguy

TL;DR

This work tackles the problem of constructing universal kernels on compact metric spaces by embedding the space into a separable Hilbert space. The authors present an explicit injection $\varphi: \mathcal{X} \to \ell^2$ based on a countable basis and show that composing a universal kernel on the image with this injection yields a universal kernel on $\mathcal{X}$. To address practicality, they develop approximate-universal, tractable variants $\hat{k}$ and $k_t$ using finite-dimensional surrogates $\hat{\varphi}$ and truncations, with explicit error bounds that depend on discretization and kernel constants. The results provide a principled route to explicit, computable universal kernels for non-Euclidean spaces, along with a quantitative understanding of the tradeoffs between accuracy and tractability in kernel-based learning tasks.

Abstract

Universal kernels, whose Reproducing Kernel Hilbert Space is dense in the space of continuous functions are of great practical and theoretical interest. In this paper, we introduce an explicit construction of universal kernels on compact metric spaces. We also introduce a notion of approximate universality, and construct tractable kernels that are approximately universal.

Explicit Universal and Approximate-Universal Kernels on Compact Metric Spaces

TL;DR

This work tackles the problem of constructing universal kernels on compact metric spaces by embedding the space into a separable Hilbert space. The authors present an explicit injection based on a countable basis and show that composing a universal kernel on the image with this injection yields a universal kernel on . To address practicality, they develop approximate-universal, tractable variants and using finite-dimensional surrogates and truncations, with explicit error bounds that depend on discretization and kernel constants. The results provide a principled route to explicit, computable universal kernels for non-Euclidean spaces, along with a quantitative understanding of the tradeoffs between accuracy and tractability in kernel-based learning tasks.

Abstract

Universal kernels, whose Reproducing Kernel Hilbert Space is dense in the space of continuous functions are of great practical and theoretical interest. In this paper, we introduce an explicit construction of universal kernels on compact metric spaces. We also introduce a notion of approximate universality, and construct tractable kernels that are approximately universal.

Paper Structure

This paper contains 14 sections, 13 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Kernels in Practice and Related Works
  3. Elements of RKHS Theory
  4. Paper Outline and Contributions
  5. Construction of universal kernels in \ref{['sec:univ_kernel']}
  6. Approximate universal kernels in \ref{['sec:approx_univ_kernels']}
  7. Explicit Universal Taylor and Radial Kernels on a Compact Metric Space
  8. Injection of S into l2
  9. Universal Kernels on X
  10. A strictly convex functional on $\mathcal{P}(\mathcal{X})$
  11. Approximate Universal Kernels
  12. Constructing a Smaller RKHS hatH
  13. Showing that hatH is Approximately Universal
  14. An Approximate Universal Truncated Kernel

Key Result

Lemma 4

Figures (4)

  • Figure 1: Given a basis $(x_n)_{n\in \mathbb{N}}$ of $\mathcal{X}$, the mapping $\varphi: \mathcal{X} \longrightarrow \ell^2$ maps a point $x \in \mathcal{X}$ to the sequence of its distances to the points of the basis.
  • Figure 2: Discretisation of the space $\mathcal{X}$ into a cover of $J$ balls of radius $\eta>0$ centred at each $(y_j)_{j\in \llbracket 1, J \rrbracket}$.
  • Figure 3: The mapping $\hat{\varphi}: \mathcal{X} \longrightarrow \mathbb{R}^J$ maps a point $x\in \mathcal{X}$ to the vector of normalised distances between $x$ and the centres $y_j$ of the covering.
  • Figure 4: The basis $(x_n)_{n\in \mathbb{N}}$ is such that there equally as many $(x_n)$ in each region $\mathcal{X}_j$ of points closest to $y_j$. In the figure, we observe a zoom on the region $\mathcal{X}_4$, where the example point $x_n$ is closest to $y_4$. In mathematical terms, we write this property as $\beta(n) = y_j$, and in \ref{['prop:adapted_basis']} we will construct $(x_n)$ such that the sum $\sum_nq^{-2n}$ is split evenly between the sets $\beta^{-1}(\{j\})$.

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Definition 7
  • ...and 20 more