Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Samplet limits and multiwavelets

Gianluca Giacchi, Michael Multerer, Jacopo Quizi

Abstract

Samplets are data adapted multiresolution analyses of localized discrete signed measures. They can be constructed on scattered data sites in arbitrary dimension and such that they exhibit vanishing moments with respect to any prescribed set of primitives. We consider the samplet construction in a probabilistic framework and show that, when choosing polynomials as primitives, the resulting samplet basis converges in the infinite data limit to signed measures with broken polynomial densities. These densities amount to multiwavelets with respect to a hierarchical partition of the region containing the data sites. As a byproduct, we therefore obtain a construction of general multiwavelets that allows for a flexible prescription of vanishing moments going beyond tensor product constructions. For congruent partitions we particularly recover classical multiwavelets with scale- and partition- independent filter coefficients. The theoretical findings are complemented by numerical experiments that illustrate the convergence results in case of random as well as low-discrepancy data sites.

Samplet limits and multiwavelets

Abstract

Samplets are data adapted multiresolution analyses of localized discrete signed measures. They can be constructed on scattered data sites in arbitrary dimension and such that they exhibit vanishing moments with respect to any prescribed set of primitives. We consider the samplet construction in a probabilistic framework and show that, when choosing polynomials as primitives, the resulting samplet basis converges in the infinite data limit to signed measures with broken polynomial densities. These densities amount to multiwavelets with respect to a hierarchical partition of the region containing the data sites. As a byproduct, we therefore obtain a construction of general multiwavelets that allows for a flexible prescription of vanishing moments going beyond tensor product constructions. For congruent partitions we particularly recover classical multiwavelets with scale- and partition- independent filter coefficients. The theoretical findings are complemented by numerical experiments that illustrate the convergence results in case of random as well as low-discrepancy data sites.

Paper Structure

This paper contains 23 sections, 20 theorems, 209 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Probabilistic setting
  4. Orthogonal polynomials
  5. Convergence of empirical Gram matrices
  6. Convergence of discrete orthogonal polynomials
  7. Rates and discrepancy effects
  8. Samplets
  9. Banach frame setting
  10. Samplet construction
  11. Asymptotics of samplets and broken polynomials
  12. Convergence of local Gram matrices
  13. Samplets and broken orthonormal polynomials
  14. Coarsening of broken orthonormal polynomials
  15. Extension to downward index sets
  16. ...and 8 more sections

Key Result

Theorem 2.1

The empirical measure $\widehat{ \mathbb{P}}_N$, defined in eq:emp_meas, converges $\mathbb{P}$-almost surely to $\mathbb{P}$, i.e., $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Number of samples $N$ and minimum number of samples per tree leaf for random sampling (left, average over 10 runs) and Halton points (right).
  • Figure 2: Visualization of the samplet convergence in $d=1$ for increasing number of samples. A scaling distribution is shown on the left and a samplet on the right.
  • Figure 3: Visualization of the samplet convergence in $d=2$ for increasing number of samples. A scaling distribution is shown on the left and a samplet on the right.
  • Figure 4: Average projection error of filter coefficients of scaling distribution (left) and samplets (right) taken over all non-leaf clusters for increasing $N$ and uniformly random points. The error bars denote the standard deviation from 10 runs.
  • Figure 5: Average projection error of filter coefficients of scaling distribution (left) and samplets (right) taken over all non-leaf clusters for increasing $N$ for Halton points

Theorems & Definitions (41)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • ...and 31 more