Learning Kronecker-Structured Graphs from Smooth Signals (2505.09822v1)

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.

Insights into Kronecker-Structured Graph Learning from Smooth Signals

The paper under discussion focuses on an advanced topic in the field of Graph Signal Processing (GSP), specifically addressing the challenge of learning graphs structured through Kronecker products from smooth signals. This work builds upon the fundamental principles of GSP, which extends classical signal processing to non-Euclidean domains and finds applications in various disciplines like network analysis, neuroscience, and beyond.

Key Contributions

The authors tackle the inherent complexity of learning Kronecker-structured graphs, which offer a robust modeling framework that can accommodate intricate dependencies inherent in multi-way data. Unlike traditional Cartesian product graphs, Kronecker-structured graphs introduce non-separability and recursive self-similarity, posing significant challenges in graph learning due to the non-convex constraints.

The paper demonstrates the following primary contributions:

1. Formulation of Penalized MLE: The authors propose a novel penalized Maximum Likelihood Estimation (MLE) approach to learn Kronecker product graph Laplacians. This formulation considers the probabilistic framework of Graphical Models and GSP, focusing on decomposing the graph Laplacian leveraging smooth signals.
2. Alternating Optimization Scheme: An alternating scheme is introduced to handle the non-convex nature of the problem, where individual optimization on each factor graph is executed sequentially. This method assures asymptotic convergence, theoretically improving on existing methods by recognizing product structures.
3. Algorithm Variant for Strong Product Learning: Extending the methodology, the authors also propose modifications to accommodate strong product graphs within their framework, broadening the applicability of their algorithm beyond Kronecker products.
4. Theoretical Guarantees: The paper provides rigorous theoretical guarantees for the proposed algorithm, establishing its statistical consistency and improved convergence rates compared to non-structured methods. This affords significant advantages when efficient and accurate graph learning from smooth signals is required.

Numerical Results and Implications

The empirical evaluation on both synthetic and real-world datasets, such as EEG data, showcases the superior performance of the proposed approach over existing methods, including conventional and spectral graph learning techniques. For instance, KSGL (Kronecker Structured Graph Learning) demonstrated marked improvement in reconstructing both the factor and product graph Laplacians, aligning well with the theoretical predictions.

These improvements have valuable implications for fields relying on the analysis of multi-way signals or high-dimensional data, suggesting improvements in understanding dependencies in complex systems such as neural activities and communication networks.

Future Directions

The research opens pathways for further exploration, particularly in enhancing the scalability of the algorithm to handle very large datasets, which is critical as the volume of multi-way data continues to grow rapidly. Additionally, exploring extensions or variations of the proposed method to accommodate a broader class of graph products or incorporating more extensive prior information about the graph structures could be of considerable interest.

The methodological innovations presented in this paper contribute a critical advancement in the field of graph learning from smooth signals, establishing a foundation upon which future research in GSP and related fields can build.