Graph Signal Processing -- Part III: Machine Learning on Graphs, from Graph Topology to Applications

TL;DR

This paper discusses methods for learning graph topology from data, emphasizing correlation-based approaches, sparse graph learning with LASSO variants, and applications in various physical and social systems, with insights into graph neural networks and tensor representations.

Contribution

It introduces a comprehensive framework for data-driven graph topology learning, integrating prior knowledge, structural conditions, and advanced regularization techniques, with practical examples and applications.

Findings

Graph topology can be learned from data using correlation and precision matrices.

LASSO and graphical LASSO effectively identify sparse graph structures.

Tensor representations reveal high-dimensional lattice-structured graph signals.

Abstract

Many modern data analytics applications on graphs operate on domains where graph topology is not known a priori, and hence its determination becomes part of the problem definition, rather than serving as prior knowledge which aids the problem solution. Part III of this monograph starts by addressing ways to learn graph topology, from the case where the physics of the problem already suggest a possible topology, through to most general cases where the graph topology is learned from the data. A particular emphasis is on graph topology definition based on the correlation and precision matrices of the observed data, combined with additional prior knowledge and structural conditions, such as the smoothness or sparsity of graph connections. For learning sparse graphs (with small number of edges), the least absolute shrinkage and selection operator, known as LASSO is employed, along with its graph specific variant, graphical LASSO. For completeness, both variants of LASSO are derived in an intuitive way, and explained. An in-depth elaboration of the graph topology learning paradigm is provided through several examples on physically well defined graphs, such as electric circuits, linear heat transfer, social and computer networks, and spring-mass systems. As many graph neural networks (GNN) and convolutional graph networks (GCN) are emerging, we have also reviewed the main trends in GNNs and GCNs, from the perspective of graph signal filtering. Tensor representation of lattice-structured graphs is next considered, and it is shown that tensors (multidimensional data arrays) are a special class of graph signals, whereby the graph vertices reside on a high-dimensional regular lattice structure. This part of monograph concludes with two emerging applications in financial data processing and underground transportation networks modeling.