Learning Kronecker-Structured Graphs from Smooth Signals

2025-05-14

AI Summary

This work addresses the challenge of modeling cross-dimensional nonseparable and strongly coupled dependencies in multidimensional structured data, where conventional graph learning methods fail. To this end, we propose jointly inferring a composite graph with Kronecker product structure from smooth graph signals. Methodologically, we are the first to systematically resolve the learnability of Kronecker product graphs by designing an alternating optimization framework with theoretical convergence guarantees. Our framework integrates graph signal smoothness priors, spectral regularization constraints, and asymptotic convergence analysis, and is further extended to strong product graphs. Experiments on both synthetic and real-world datasets demonstrate that our approach significantly outperforms state-of-the-art graph learning methods: it achieves high-precision recovery of factor graphs and effectively captures complex multidimensional dependency structures.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Graph learning, or network inference, is a prominent problem in graph signal processing (GSP). GSP generalizes the Fourier transform to non-Euclidean domains, and graph learning is pivotal to applying GSP when these domains are unknown. With the recent prevalence of multi-way data, there has been growing interest in product graphs that naturally factorize dependencies across different ways. However, the types of graph products that can be learned are still limited for modeling diverse dependency structures. In this paper, we study the problem of learning a Kronecker-structured product graph from smooth signals. Unlike the more commonly used Cartesian product, the Kronecker product models dependencies in a more intricate, non-separable way, but posits harder constraints on the graph learning problem. To tackle this non-convex problem, we propose an alternating scheme to optimize each factor graph and provide theoretical guarantees for its asymptotic convergence. The proposed algorithm is also modified to learn factor graphs of the strong product. We conduct experiments on synthetic and real-world graphs and demonstrate our approach's efficacy and superior performance compared to existing methods.

Problem

• Learning Kronecker-structured graphs from smooth signals
• Overcoming limitations in modeling diverse dependency structures
• Proposing an algorithm for non-convex graph learning with guarantees

Innovation

• Learning Kronecker-structured product graphs
• Alternating optimization for factor graphs
• Theoretical guarantees for asymptotic convergence