Product Graph Learning from Multi-domain Data with Sparsity and Rank Constraints

TL;DR

This paper introduces a method for learning product graphs from multi-domain data by estimating sparse graph factors with rank constraints, enabling efficient clustering and approximation of complex graph structures.

Contribution

It proposes an iterative solver for sparse product graph learning and extends it to infer multi-component graph factors with rank constraints, including approximation algorithms for non-exact Cartesian products.

Findings

Effective in synthetic data experiments

Successful application to real data clustering

Improves computational efficiency for large graphs

Abstract

In this paper, we focus on learning product graphs from multi-domain data. We assume that the product graph is formed by the Cartesian product of two smaller graphs, which we refer to as graph factors. We pose the product graph learning problem as the problem of estimating the graph factor Laplacian matrices. To capture local interactions in data, we seek sparse graph factors and assume a smoothness model for data. We propose an efficient iterative solver for learning sparse product graphs from data. We then extend this solver to infer multi-component graph factors with applications to product graph clustering by imposing rank constraints on the graph Laplacian matrices. Although working with smaller graph factors is computationally more attractive, not all graphs may readily admit an exact Cartesian product factorization. To this end, we propose efficient algorithms to approximate a graph by a nearest Cartesian product of two smaller graphs. The efficacy of the developed framework is demonstrated using several numerical experiments on synthetic data and real data.