Learning Kronecker-Structured Graphs from Smooth Signals

TL;DR

This work addresses the challenge of learning Kronecker-structured graphs from smooth multi-way signals in GSP. It formulates a penalized maximum likelihood estimator under Kronecker constraints and solves it via an alternating optimization scheme, with extensions to strong product graphs and rigorous guarantees on existence, subproblem uniqueness, and asymptotic consistency. Empirically, KSGL outperforms competing GSP and GM methods on synthetic Kronecker and strong-product graphs and yields meaningful brain connectivity patterns in EEG data, illustrating the practical value of product-structure priors. Overall, the paper provides a principled, provably convergent approach for Kronecker-structured graph learning that scales to multi-way data and real-world applications.

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.

Figures (9)

• Figure 1: An example that compares the Cartesian, Kronecker, and strong graph products.
• Figure 2: Comparison of different methods on various synthetic Kronecker product graphs and signals. Each sub-figure shows the trend of Rel-Err of the product (top row) or factor (middle and bottom rows) Laplacian matrices as $n$ increases. Black dash lines fit the theory in \ref{['ieq:convergence_rate']} to the KSGL results.
• Figure 3: Comparison of different methods on various synthetic strong product graphs and signals. Each sub-figure shows the trend of Rel-Err as $n$ increases. Black dash lines fit the theory in \ref{['ieq:convergence_rate']} to our results.
• Figure 4: The brain connectivity inferred by MWGL and KSGL. Nodes reflect the actual electrode positions in the 10-20 system. The background color shows the mean EEG activity of each status.
• Figure 5: Comparison of different methods on various synthetic Kronecker product graphs and signals. Each sub-figure shows the trend of PR-AUC of the product or factor edge prediction as $n$ increases.

Theorems & Definitions (12)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 4.1: Existence of MLE
• Theorem 4.2: Uniqueness of MLE
• Corollary 4.3
• Theorem 4.4: High-dimensional consistency
• proof
• proof