[Paper Review] A Unified Framework for Structured Graph Learning via Spectral Constraints

Sandeep Kumar, Jiaxi Ying+2 more|arXiv (Cornell University)|Apr 22, 2019

Advanced Graph Neural Networks87 references50 citations

TL;DR

This paper proposes a unified framework that integrates Gaussian graphical models with spectral graph theory to learn graphs with predefined structures via spectral constraints on Laplacian and adjacency matrices.

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.

Motivation & Objective

• Motivate the need to learn graphs with interpretable, predefined structures rather than generic graphs.
• Develop a tractable optimization framework that imposes structural constraints through spectral properties of graph matrices.
• Unify learning and structure inference by transforming combinatorial constraints into spectral constraints on Laplacian and adjacency matrices.
• Provide convergent, efficient algorithms applicable to various structured graphs and real data sets.

Proposed method

• Formulate structured graph learning as maximizing log det(g Theta) minus trace(Theta S) minus alpha h(Theta) with Theta constrained to Laplacian structure.
• Impose spectral constraints on Theta (Laplacian) and Theta_A (adjacency) through eigenvalue sets, enabling k-component, connected sparse, d-regular, and cospectral structures.
• Develop BSUM/block majorization-minimization algorithms to solve the constrained optimization with worst-case O(p^3) complexity, comparable to GLasso.
• Specialize formulations to Laplacian spectral constraints (SGL), adjacency spectral constraints (for bipartite learning), and joint Laplacian-adjacency spectral constraints (SGL-Joint).
• Utilize eigen-decomposition Theta = U Diag(lambda) U^T and Theta_A = V Diag(psi) V^T with appropriate constraint sets on lambda and psi.
• Provide open-source code for reproducing simulations: https://github.com/dppalomar/spectralGraphTopology

Experimental results

Research questions

• RQ1 How can combinatorial graph structure constraints be reformulated as analytical spectral constraints on graph matrices?
• RQ2 Can a single optimization framework learn graphs with multiple structured classes (e.g., multi-component, bipartite, regular) simultaneously?
• RQ3 What convergence properties and computational efficiency can be achieved with BSUM-based algorithms in structured graph learning?
• RQ4 How well do the proposed methods perform on synthetic and real data for tasks like clustering or network inference?
• RQ5 What are the implications of joint Laplacian and adjacency spectral constraints for complex graph structures?

Key findings

• The framework converts structure constraints into spectral constraints on Laplacian and adjacency matrices, enabling tractable optimization.
• Algorithms based on BSUM converge theoretically and offer computational efficiency comparable to GLasso (O(p^3)).
• The approach supports multiple structural classes including k-component, connected sparse, d-regular, and bipartite graphs, and can be extended to cospectral scenarios.
• Extensive experiments on synthetic and real data demonstrate effectiveness of the proposed structured graph learning methods.
• Open-source code is provided for reproducing simulations and experiments.