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Modern data introduces new challenges to classic signal processingapproaches, leading to a growing interest in the field of graph signalprocessing. A powerful and well established model for real world signals invarious domains is sparse representation over a dictionary, combined with theability to train the dictionary from signal examples. This model has beensuccessfully applied to graph signals as well by integrating the underlyinggraph topology into the learned dictionary. Nonetheless, dictionary learningmethods for graph signals are typically restricted to small dimensions due tothe computational constraints that the dictionary learning problem entails, anddue to the direct use of the graph Laplacian matrix. In this paper, we proposea dictionary learning algorithm that applies to a broader class of graphsignals, and is capable of handling much higher dimensional data. Weincorporate the underlying graph topology both implicitly, by forcing thelearned dictionary atoms to be sparse combinations of graph-wavelet functions,and explicitly, by adding direct graph constraints to promote smoothness inboth the feature and manifold domains. The resulting atoms are thus adapted tothe data of interest while adhering to the underlying graph structure andpossessing a desired multi-scale property. Experimental results on severaldatasets, representing both synthetic and real network data of differentnature, demonstrate the effectiveness of the proposed algorithm for graphsignal processing even in high dimensions.

Finding GEMS: Multi-Scale Dictionaries for High-Dimensional Graph Signals