Representation of functions on big data associated with directed graphs

Abstract

This paper is an extension of the previous work of Chui, Filbir, and Mhaskar (Appl. Comput. Harm. Anal. 38 (3) 2015:489-509), not only from numeric data to include non-numeric data as in that paper, but also from undirected graphs to directed graphs (called digraphs, for simplicity). Besides theoretical development, this paper introduces effective mathematical tools in terms of certain data-dependent orthogonal systems for function representation and analysis directly on the digraphs. In addition, this paper also includes algorithmic development and discussion of various experimental results on such data-sets as CORA, Proposition, and Wikivotes.

Adaptive Directional Haar Tight Framelets on Bounded Domains for Digraph Signal Representations

“…Typical harmonic analysis is on theory and applications related to functions defined on regular Euclidean domains [5,11,12,23,25,27,30,37]. In recent years, driven by the rapid progress of deep learning and their successful applications in solving AI (artificial intelligence) related tasks, such as natural language processing, autonomous systems, robotics, medical diagnostics, and so on, there has been a great interest in developing harmonic analysis for data defined on non-Euclidean domains such as manifold data or graph data, e.g., see [1,4,7,8,14,16,20,29,40,41,49] and many references therein. For example, data in machine/statistical learning, are typically from social networks, biology, physics, finance, etc., and can be naturally obtained or organized as graphs or graph data.…”

“…The interested reader can refer to [38,42,46] and many references therein. Similar to the wavelets and framelets for signal/image processing, multiscale representation systems based on various approaches such as spectral theory [9], diffusion wavelets [6], non-spectral construction [7,8], etc., have also been developed for graph signal representation and processing.…”

“…In this paper, motivated by the recent development of directional Haar framelet systems on R d [13,22,32,35] as well as wavelet-like systems for graph signal processing [1,8,33,50,51], we focus on the development of directional multiscale representation systems for signals defined on digraphs. We are going to investigate the following two main problems: 1) How to construct directional Haar tight framelets on bounded domains with adaptivity?…”

“…Haar wavelet system [19] is the first ever constructed orthonormal wavelet system on the interval [0, 1]. It is a very simple yet elegant system that even nowadays there are many literatures on Haar-type systems, e.g., see [7,8,22,33,50]. Starting from a scaling (refinable) function φ := χ [0,1] , which is a characteristic function defined on the unit interval I := [0, 1], and the mother wavelet function…”

“…In fact, one typical approach is to define a counterpart graph Laplacian on digraph, such as the Hodge Laplacian in [36], the weighted adjacency matrix in [10], the so-called dilaplacian in [34], and so on. In this paper, we use the idea developed in [8]: to lift the dimension from one to two by using a pair of undirected graphs to represent a given digraph. In a nutshell, a digraph signal is identified as a signal defined on I 2 = [0, 1] × [0, 1] through the following steps.…”