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On Quasi-Localized Dual Pairs in Reproducing Kernel Hilbert Spaces

Helmut Harbrecht, Rüdiger Kempf, Michael Multerer

TL;DR

The paper addresses scalable scattered data approximation in RKHS by developing and comparing dual representations of the projection operator, including localized Lagrange functions, orthogonal Newton-type bases, and multiresolution samplets. It introduces symmetric preconditioners built from footprint-based localizations and matrix square roots or Cholesky factorizations, and demonstrates how samplets yield sparse kernel representations and direct solvers in compressed coordinates. The authors provide decay and compressibility insights for kernel matrices, particularly for Matérn kernels, and validate the approaches through extensive 2D and 3D numerical experiments, including implicit surface reconstruction. The work offers practical, scalable techniques for fast interpolation and surface processing in scattered data settings, with clear guidance on when to use local Lagrange versus samplet-based methods.

Abstract

In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice is by no means mandatory and different choices, like, for example, the Lagrange basis, are possible and might offer additional features. In this article, we discuss different alternatives together with their canonical duals. We study a localized version of the Lagrange basis, localized orthogonal bases, such as the Newton basis, and multiresolution versions thereof, constructed by means of samplets. We argue that the choice of orthogonal bases is particularly useful as they lead to symmetric preconditioners. All bases under consideration are compared numerically to illustrate their feasibility for scattered data approximation. We provide benchmark experiments in two spatial dimensions and consider the reconstruction of an implicit surface as a relevant application from computer graphics.

On Quasi-Localized Dual Pairs in Reproducing Kernel Hilbert Spaces

TL;DR

The paper addresses scalable scattered data approximation in RKHS by developing and comparing dual representations of the projection operator, including localized Lagrange functions, orthogonal Newton-type bases, and multiresolution samplets. It introduces symmetric preconditioners built from footprint-based localizations and matrix square roots or Cholesky factorizations, and demonstrates how samplets yield sparse kernel representations and direct solvers in compressed coordinates. The authors provide decay and compressibility insights for kernel matrices, particularly for Matérn kernels, and validate the approaches through extensive 2D and 3D numerical experiments, including implicit surface reconstruction. The work offers practical, scalable techniques for fast interpolation and surface processing in scattered data settings, with clear guidance on when to use local Lagrange versus samplet-based methods.

Abstract

In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice is by no means mandatory and different choices, like, for example, the Lagrange basis, are possible and might offer additional features. In this article, we discuss different alternatives together with their canonical duals. We study a localized version of the Lagrange basis, localized orthogonal bases, such as the Newton basis, and multiresolution versions thereof, constructed by means of samplets. We argue that the choice of orthogonal bases is particularly useful as they lead to symmetric preconditioners. All bases under consideration are compared numerically to illustrate their feasibility for scattered data approximation. We provide benchmark experiments in two spatial dimensions and consider the reconstruction of an implicit surface as a relevant application from computer graphics.
Paper Structure (19 sections, 3 theorems, 59 equations, 6 figures, 2 tables)

This paper contains 19 sections, 3 theorems, 59 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Scattered Data Approximation
  3. Other Bases for $\mathcal{H}_X$
  4. Dual Pairs
  5. About the Lagrange Basis
  6. Alternative Point of View: Pseudo-differential Operators
  7. Efficient Computation of the Lagrange Basis
  8. Cut-off and Localized Lagrange Functions
  9. Symmetrized Version for Preconditioning
  10. Dual Pairs in Samplet Coordinates
  11. Samplet Basis Construction
  12. Samplet Representation of the Lagrange Basis
  13. Kernel Matrix Compression and Associated Algebra
  14. Numerical Tests
  15. Setup
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\Omega \subseteq \mathbb{R}^d$ be a domain. Let $K$ be the reproducing kernel of an RKHS $\mathcal{H}$ on $\Omega$ and $X = \{ \boldsymbol{x}_1, \dots, \boldsymbol{x}_N\} \subseteq \Omega$ be a set of sites. Then, for all $\lambda \geq 0$, there exists a unique solution $s^*_{\lambda}$ of $\min i.e., $s^*_{\lambda} \in\mathcal{H}_X$. Furthermore, $\boldsymbol{\alpha}$ is the unique solution o

Figures (6)

  • Figure 1: Basis of kernel translates, localized Lagrange basis for $H^1( \mathbb{R})$ (top row). The bottom row shows the corresponding orthogonal functions obtained with the square root of the inverse Gramian ${{\boldsymbol A}}^{-1/2}$.
  • Figure 2: $H^1( \mathbb{R})$-embedded primal and dual scaling distribution (top left), samplet on level $j=1$ (top middle) and samplet on level $j=2$ (top right) for $N=200$ equidistant data sites and $q+1=3$ vanishing moments. The bottom row shows the corresponding orthogonal samplets obtained with the square root of the inverse kernel matrix ${{\boldsymbol A}}_\Sigma^{-1/2}$.
  • Figure 3: Typical matrix pattern of the samplet compressed kernel matrix for $d=2$ (left), its reordering by means of nested dissection (middle), and the associated Cholesky factor (right).
  • Figure 4: Localized Lagrange approach: Errors as a function of the compression rate in case of varying smoothness parameters $\nu = 0.5, 1.0, 1.5$ and sample sizes $N = 10^3, 10^4, 10^5$.
  • Figure 5: Samplet matrix compression: Errors as a function of the compression rate in case of varying smoothness parameters $\nu = 0.5, 1.0, 1.5$ and sample sizes $N = 10^3, 10^4, 10^5$.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • Corollary 3.1
  • Proposition 4.1