Haar-Like Wavelets on Hierarchical Trees

Discrete wavelet methods, originally formulated in the setting of regularly sampled signals, can be adapted to data defined on a point cloud if some multiresolution structure is imposed on the cloud. A wide variety of hierarchical clustering algorithms can be used for this purpose, and the multiresolution structure obtained can be encoded by a hierarchical tree of subsets of the cloud. Prior work introduced the use of Haar-like bases defined with respect to such trees for approximation and learning tasks on unstructured data. This paper builds on that work in two directions. First, we present an algorithm for constructing Haar-like bases on general discrete hierarchical trees. Second, with an eye towards data compression, we present thresholding techniques for data defined on a point cloud with error controlled in the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document} norm and in a Hölder-type norm. In a concluding trio of numerical examples, we apply our methods to compress a point cloud dataset, study the tightness of the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document} error bound, and use thresholding to identify MNIST classifiers with good generalizability.