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Samplets: Wavelet concepts for scattered data

Helmut Harbrecht, Michael Multerer

TL;DR

This chapter develops a wavelet-inspired framework for scattered data by introducing samplets, signed measures with vanishing moments that form an orthonormal multiresolution basis on irregular data sites. It covers construction via a balanced cluster tree, a fast linear-cost transform, and data-compression techniques, while extending the approach to reproducing kernel Hilbert spaces for efficient kernel-matrix representations and arithmetic. The work demonstrates adaptive subsampling, RKHS embedding of samplets, and a sparse, L1-regularized pursuit (basis pursuit) in the samplet coordinates, including space–time kernel decompositions. Collectively, the results enable scalable, high-accuracy interpolation, compression, and operator manipulation for nonlocal kernels on scattered data, with practical impact in high-dimensional settings and irregular sampling scenarios.

Abstract

This chapter is dedicated to recent developments in the field of wavelet analysis for scattered data. We introduce the concept of samplets, which are signed measures of wavelet type and may be defined on sets of arbitrarily distributed data sites in possibly high dimension. By employing samplets, we transfer well-known concepts known from wavelet analysis, namely the fast basis transform, data compression, operator compression and operator arithmetics to scattered data problems. Especially, samplet matrix compression facilitates the rapid solution of scattered data interpolation problems, even for kernel functions with nonlocal support. Finally, we demonstrate that sparsity constraints for scattered data approximation problems become meaningful and can efficiently be solved in samplet coordinates.

Samplets: Wavelet concepts for scattered data

TL;DR

This chapter develops a wavelet-inspired framework for scattered data by introducing samplets, signed measures with vanishing moments that form an orthonormal multiresolution basis on irregular data sites. It covers construction via a balanced cluster tree, a fast linear-cost transform, and data-compression techniques, while extending the approach to reproducing kernel Hilbert spaces for efficient kernel-matrix representations and arithmetic. The work demonstrates adaptive subsampling, RKHS embedding of samplets, and a sparse, L1-regularized pursuit (basis pursuit) in the samplet coordinates, including space–time kernel decompositions. Collectively, the results enable scalable, high-accuracy interpolation, compression, and operator manipulation for nonlocal kernels on scattered data, with practical impact in high-dimensional settings and irregular sampling scenarios.

Abstract

This chapter is dedicated to recent developments in the field of wavelet analysis for scattered data. We introduce the concept of samplets, which are signed measures of wavelet type and may be defined on sets of arbitrarily distributed data sites in possibly high dimension. By employing samplets, we transfer well-known concepts known from wavelet analysis, namely the fast basis transform, data compression, operator compression and operator arithmetics to scattered data problems. Especially, samplet matrix compression facilitates the rapid solution of scattered data interpolation problems, even for kernel functions with nonlocal support. Finally, we demonstrate that sparsity constraints for scattered data approximation problems become meaningful and can efficiently be solved in samplet coordinates.

Paper Structure

This paper contains 18 sections, 5 theorems, 77 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Multiresolution analysis of scattered data
  3. Samplet bases
  4. Cluster tree
  5. Construction of the samplets
  6. Properties of the samplets
  7. Scattered data analysis
  8. Fast samplet transform
  9. Data compression
  10. Adaptive subsampling
  11. Scattered data approximation
  12. Reproducing kernel Hilbert spaces
  13. Samplets in reproducing kernel Hilbert spaces
  14. Asymptotically smooth kernels
  15. Matrix compression
  16. ...and 3 more sections

Key Result

theorem 1

The spaces $\mathcal{X}_{j}$ defined in equation HM_eg:Vspaces form a multiresolution analysis where the respective complement spaces $\mathcal{S}_{j}$ from HM_eg:Wspaces satisfy The associated samplet basis ${{\boldsymbol\Sigma}}$ defined in HM_eg:Wbasis is an orthonormal basis in $\mathcal{X}$. In particular, there holds:

Figures (10)

  • Figure 1: Scaling distribution (left), samplet on level $j=1$ (middle) and samplet on level $j=2$ (right) for $N=200$ uniformly distributed data sites and $q+1=3$.
  • Figure 2: Fishbone scheme of the fast samplet transform.
  • Figure 3: Reconstruction of the temperature based on the hard-thresholded coefficient vector for the relative thresholds $10^{-k}$, $k=2,3,4,5$ (from top left to bottom right). Notice that in case of $k=2$ only the coarse level samplets remain after thresholding.
  • Figure 4: Reconstruction error of the temperature with respect to the hard-thresholded coefficient vector for the relative thresholds $10^{-k}$, $k=2,3,4,5$ (from top left to bottom right).
  • Figure 5: Average temperature in October 2022 (left) and leafs of the corresponding adaptive tree with relative threshold $\varepsilon=10^{-4}$ (right).
  • ...and 5 more figures

Theorems & Definitions (11)

  • definition 1
  • remark 1
  • definition 2
  • remark 2
  • theorem 1
  • lemma 1
  • definition 3
  • lemma 2
  • theorem 2
  • remark 3
  • ...and 1 more