Graph Learning and Optimization for Irregular-Structured Signal Processing

Abstract

Graph Signal Processing (GSP) extends harmonic analysis tools, such as Fourier transforms and wavelets, to discrete signals defined on finite graphs, enabling tasks like signal denoising, prediction, and interpolation on irregular domains. A critical first step in GSP is to learn an appropriate graph that captures pairwise similarities or correlations inherent in the data, ensuring that subsequent graph-based filtering effectively leverages local structure for improved performance.

However, most existing graph learning methods assume static relationships, while real-world interactions often evolve over time. To address this problem, this thesis proposes a slowly time-varying graph learning framework that models the difference between consecutive adjacency matrices as a low-rank matrix. This approach accommodates gradual shifts in node-to-node similarities over time, enabling efficient graph updates with low computational overhead while maintaining alignment with the underlying data.

Beyond graph construction, the challenge of dense or complete graphs often arises, particularly in large-scale applications where representing all possible edges is computationally prohibitive. To address this issue, this thesis introduces a sparsification method guided by the Fiedler number, the second smallest eigenvalue of the Laplacian, which quantifies graph connectivity. By removing edges that minimally affect the Fiedler number, the resulting sparser graph preserves essential connectivity while significantly reducing training and inference costs for deep learning models (e.g, graph convolutional networks (GCNs)). Together, these contributions provide a flexible and computationally efficient approach to GSP in dynamic and large-scale graph settings.