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 on graphs: localization, decomposition, and dictionary learning of piecewise-constant signals. These graph-based problems are related to many real-world applications, such as localizing virus attacks in cyber-physical systems, localizing stimulus in brain connectivity networks, and mining traffic events in city street networks, where the key issue is to separate localized activated patterns and background noise; in other words, we aim to find the supports of localized activated patterns. Counterparts of these problems in classical signal/image processing, such as impulse detection, foreground detection, and wavelet construction, have been intensely studied over the past few decades. We use piecewise-constant graph signals to model localized patterns, where each piece indicates a localized pattern that exhibits homogeneous internal behavior and the number of pieces indicates the number of localized patterns. For such signals, we show that decomposition and dictionary learning are natural extensions of localization, the goal of which is not only to efficiently approximate graph signals, but also to accurately find supports of localized patterns. For each of the three problems, i.e., localization, decomposition, and dictionary learning, we propose a specific graph signal model, an optimization problem, and a computationally efficient solver. The proposed solvers directly find the supports of arbitrary localized activated patterns without tuning any thresholds, which is a notorious challenge in many localization problems. 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. From taxi-pickup activities and using our methods, we are able to detect both everyday as well as special events and distinguish weekdays from weekends. Our findings validate the effectiveness of the approach and suggest that graph signal processing tools may aid in urban planning and traffic forecasting.