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Construction of generalized samplets in Banach spaces

Peter Balazs, Michael Multerer

TL;DR

This work extends samplets from point evaluations to general functionals in Banach spaces by leveraging Banach frames or Riesz bases with square-summable coefficients, enabling stable representations through an isometry on the analysis-operator image. A multilevel hierarchy is built via spectral clustering of functionals’ supports, which permits generalized samplets with vanishing moments against a chosen primitive set and yields an abstract localization result for samplet coefficients. The construction hinges on a QR-based moment-decomposition to enforce vanishing moments, and the framework is demonstrated through three illustrative examples, including RKHS, Tausch–White wavelets, and operator-adapted wavelets. The approach generalizes wavelet-like multiresolution analysis to unstructured data and functionals beyond point evaluations, providing flexible, localized representations in Banach spaces with potential impact on unstructured data processing and numerical analysis of PDEs.

Abstract

Recently, samplets have been introduced as localized discrete signed measures which are tailored to an underlying data set. Samplets exhibit vanishing moments, i.e., their measure integrals vanish for all polynomials up to a certain degree, which allows for feature detection and data compression. In the present article, we extend the different construction steps of samplets to functionals in Banach spaces more general than point evaluations. To obtain stable representations, we assume that these functionals form frames with square-summable coefficients or even Riesz bases with square-summable coefficients. In either case, the corresponding analysis operator is injective and we obtain samplet bases with the desired properties by means of constructing an isometry of the analysis operator's image. Making the assumption that the dual of the Banach space under consideration is imbedded into the space of compactly supported distributions, the multilevel hierarchy for the generalized samplet construction is obtained by spectral clustering of a similarity graph for the functionals' supports. Based on this multilevel hierarchy, generalized samplets exhibit vanishing moments with respect to a given set of primitives within the Banach space. We derive an abstract localization result for the generalized samplet coefficients with respect to the samplets' support sizes and the approximability of the Banach space elements by the chosen primitives. Finally, we present three examples showcasing the generalized samplet framework.

Construction of generalized samplets in Banach spaces

TL;DR

This work extends samplets from point evaluations to general functionals in Banach spaces by leveraging Banach frames or Riesz bases with square-summable coefficients, enabling stable representations through an isometry on the analysis-operator image. A multilevel hierarchy is built via spectral clustering of functionals’ supports, which permits generalized samplets with vanishing moments against a chosen primitive set and yields an abstract localization result for samplet coefficients. The construction hinges on a QR-based moment-decomposition to enforce vanishing moments, and the framework is demonstrated through three illustrative examples, including RKHS, Tausch–White wavelets, and operator-adapted wavelets. The approach generalizes wavelet-like multiresolution analysis to unstructured data and functionals beyond point evaluations, providing flexible, localized representations in Banach spaces with potential impact on unstructured data processing and numerical analysis of PDEs.

Abstract

Recently, samplets have been introduced as localized discrete signed measures which are tailored to an underlying data set. Samplets exhibit vanishing moments, i.e., their measure integrals vanish for all polynomials up to a certain degree, which allows for feature detection and data compression. In the present article, we extend the different construction steps of samplets to functionals in Banach spaces more general than point evaluations. To obtain stable representations, we assume that these functionals form frames with square-summable coefficients or even Riesz bases with square-summable coefficients. In either case, the corresponding analysis operator is injective and we obtain samplet bases with the desired properties by means of constructing an isometry of the analysis operator's image. Making the assumption that the dual of the Banach space under consideration is imbedded into the space of compactly supported distributions, the multilevel hierarchy for the generalized samplet construction is obtained by spectral clustering of a similarity graph for the functionals' supports. Based on this multilevel hierarchy, generalized samplets exhibit vanishing moments with respect to a given set of primitives within the Banach space. We derive an abstract localization result for the generalized samplet coefficients with respect to the samplets' support sizes and the approximability of the Banach space elements by the chosen primitives. Finally, we present three examples showcasing the generalized samplet framework.

Paper Structure

This paper contains 11 sections, 10 theorems, 86 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Banach frames with square-summable coefficients
  4. Banach Riesz bases with square-summable coefficients
  5. Matrices acting on sequences in Banach spaces
  6. Finite sets of functionals
  7. Construction of generalized samplet bases
  8. Multiresolution analysis of functionals
  9. Computation of the basis transformation
  10. Properties of generalized samplets
  11. Examples

Key Result

Lemma 1

Let $\{f_i\}_{i\in I}\subset \mathcal{B}'$ be a frame. Then, the minimum norm pseudo-inverse $T^\dagger \colon \mathcal{B}'\to\ell^2(I)$ is given by $T^\dag=T^\star S^{-1}= \widetilde{T}^\star$ with $\|T^\dagger u'\|_{\ell^2}\leq A^{-1}\|u'\|_{ \mathcal{B}'}$ for any $u'\in \mathcal{B}'$. Parti for any sequence ${\boldsymbol\beta}\in\ell^2(I)$ with $u'=T{\boldsymbol\beta}$.

Figures (1)

  • Figure : Spectral bisection algorithm

Theorems & Definitions (25)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Remark 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 15 more