Learning Kronecker-Structured Graphs from Smooth Signals

TL;DR

This paper introduces a novel method for learning Kronecker-structured product graphs from smooth signals, addressing complex dependency modeling with theoretical guarantees and demonstrating superior performance over existing methods.

Contribution

It proposes an alternating optimization algorithm for learning Kronecker and strong product graphs with convergence guarantees, expanding graph learning capabilities.

Findings

Effective learning of Kronecker-structured graphs demonstrated on synthetic data.

Superior performance compared to existing graph learning methods.

Theoretical convergence guarantees provided for the proposed algorithm.

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.