A Unified Framework for Structured Graph Learning via Spectral Constraints

TL;DR

This paper introduces a unified spectral constraint-based framework for structured graph learning, enabling efficient and interpretable graph models with specific structures from data.

Contribution

It proposes a novel optimization framework integrating spectral graph theory to impose various structures on learned graphs, addressing NP-hard combinatorial challenges.

Findings

Framework effectively learns structured graphs from synthetic data.

Algorithms demonstrate efficiency and convergence on real datasets.

Structured graphs improve interpretability and data relationship understanding.

Abstract

Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying graphical models from data. Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. Useful structured graphs include the multi-component graph, bipartite graph, connected graph, sparse graph, and regular graph. In general, structured graph learning is an NP-hard combinatorial problem, therefore, designing a general tractable optimization method is extremely challenging. In this paper, we introduce a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory. To impose a particular structure on a graph, we first show how to formulate the combinatorial constraints as an analytical property of the graph matrix. Then we develop an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices. The proposed algorithms are provably convergent, computationally efficient, and practically amenable for numerous graph-based tasks. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. The code for all the simulations is made available as an open source repository.