Laplacian matrix learning for smooth graph signal representation

Abstract

Graph learning from signal observations is a crucial problem in modern data-driven applications, particularly in scenarios where meaningful graph structures are not readily available. This dissertation presents a comprehensive study on learning graph topologies, focusing on the GL-SigRep framework, which leverages signal smoothness properties to infer graph structures. The research incorporates theoretical analysis, algorithm design, and practical implementation using MATLAB. The methodology centres on optimizing the graph Laplacian to minimize signal variations while preserving meaningful topological features. The factor analysis model with a Gaussian prior is employed to enhance smoothness, ensuring that the learned graph representation aligns with the underlying data distribution. The implementation is tested using a real-world dataset. Results demonstrate the effectiveness of the GL-SigRep framework in reconstructing meaningful graphs, achieving superior accuracy and robustness compared to existing methods. The findings contribute to the growing field of graph signal processing, with implications for network analysis, biomedical applications, and smart infrastructure systems. Future work aims to enhance scalability and explore alternative probabilistic priors for improved generalization.