Signal processing and optimization on graphs
Learning time-varying structures and generalizing convolution principles
Alberto Natali (TU Delft - Signal Processing Systems)
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Abstract
Graph signal processing is a field that focuses on extracting valuable information from data collected in networks, such as social, transportation, and brain networks. This doctoral thesis makes significant contributions to two important aspects of graph signal processing: network topology identification and the convolution theorem. The thesis begins by introducing the fundamental concepts of signal processing, such as shifting, convolution, and filtering, in the discrete-time domain. It then extends these concepts to a graph-based context, where classical discrete-time signal processing can be seen as a special case of graph signal processing. The thesis then presents novel theories and algorithms in three main areas. Specifically:
(1) Graph Topology and Filter Estimation: It proposes an algorithmic approach to jointly learn the graph structure and filter coefficients from input-output graph-based data. The method addresses the non-convexity of the problem using an alternating-minimization scheme, ensuring global convergence.
(2) Time-Varying Graph Topology Inference: A new framework based on time-varying convex optimization tools is introduced for inferring time-varying network structures from graph-based data. This framework offers flexibility to users in balancing execution speed and algorithm accuracy through tunable parameters.
(3) Generalizing the Convolution Theorem: A generalization of the convolution theorem that encompasses both the graph convolution theorem and the one related to time-varying filters is introduced. This generalization has implications for (non-) stationarity and spectral analysis of signals and enables the casting of a graph learning problem to infer a potential graph structure for the frequency domain.
In summary, this thesis significantly contributes to advancing graph signal processing by addressing fundamental challenges and introducing innovative methodologies. It holds the potential to inspire further innovation in the field and deepen our understanding of complex network dynamics.