Signal Localization, Decomposition and Dictionary Learning on Graphs

Motivated by the need to extract meaning from large amounts of complex data that can be best modeled by graphs, we consider three critical problems in graph signal processing: localization, decomposition and dictionary learning. We are particularly interested in piecewise-constant graph signals that efficiently model local information in the vertex-domain; for such signals, we show that decomposition and dictionary learning are natural extensions of localization. For each of the three problems, we propose a specific graph signal model, an optimization problem and a computationally efficient solver. We then conduct an extensive empirical study to validate the proposed methods on both simulated and real data including the analysis of a large volume of spatio-temporal Manhattan urban data. Using our methods, we are able to detect both everyday and special events and distinguish weekdays from weekends from taxi-pickup activities. The findings validate the effectiveness of our approach and suggest that graph signal processing tools may aid in urban planning and traffic forecasting.