Samplet limits and multiwavelets

TL;DR

This paper introduces a probabilistic construction of samplets that converge to multiwavelets with flexible vanishing moments, extending classical multiwavelet theory and demonstrated through numerical experiments.

Contribution

It presents a novel probabilistic framework for samplet construction that generalizes multiwavelets beyond tensor product methods.

Findings

Samplet basis converges to signed measures with broken polynomial densities.

Constructs multiwavelets with flexible vanishing moments.

Numerical experiments confirm convergence for random and low-discrepancy data.

Abstract

Samplets are data adapted multiresolution analyses of localized discrete signed measures. They can be constructed on scattered data sites in arbitrary dimension 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, if choosing polynomials as primitives, the resulting samplet basis converges to signed measures with broken polynomial densities in the infinite data limit. 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.