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

Modern data introduces new challenges to classic signal processing approaches, leading to a growing interest in the field of graph signal processing. A powerful and well established model for real world signals in various domains is sparse representation over a dictionary, combined with the ability to train the dictionary from signal examples. This model has been successfully applied to graph signals as well by integrating the underlying graph topology into the learned dictionary. Nonetheless, dictionary learning methods for graph signals are typically restricted to small dimensions due to the computational constraints that the dictionary learning problem entails, and due to the direct use of the graph Laplacian matrix. In this paper, we propose a dictionary learning algorithm that applies to a broader class of graph signals, and is capable of handling much higher dimensional data. We incorporate the underlying graph topology both implicitly, by forcing the learned dictionary atoms to be sparse combinations of graph-wavelet functions, and explicitly, by adding direct graph constraints to promote smoothness in both the feature and manifold domains. The resulting atoms are thus adapted to the data of interest while adhering to the underlying graph structure and possessing a desired multi-scale property. Experimental results on several datasets, representing both synthetic and real network data of different nature, demonstrate the effectiveness of the proposed algorithm for graph signal processing even in high dimensions.