Node-based Multiple Graph Learning with Theoretical Guarantees

Mohammad Alwardat, Bisakh Banerjee, Tapabrata Maiti, Selin Aviyente

Keywords: graph learning, multiple graphs, perturbation, graph signal smoothness

TL;DR: This paper introduces a node-based framework for learning two closely related graphs from observed node data.

Abstract: In many applications, inferring graph topology, i.e., learning the graph structure from a given set of nodal observations, is a significant task. Existing approaches are mostly limited to learning a single graph assuming that the observed data are homogeneous. In many applications, data sets are heterogeneous and involve multiple related graphs, i.e., multiview graphs. Recent work on learning multiview graphs ensures the similarity of learned view graphs through edge-based similarity between the graphs. In this paper, we take a node-based approach instead of assuming that similarities and differences between networks are driven by individual edges, providing a more intuitive interpretation of network differences. Moreover, unlike existing methods that employ Gaussian Graphical Models (GGM), which learn precision matrices rather than the actual graph structures, we characterize the graph using a Laplacian matrix. Thus, the approach is expected to work broadly beyond Gaussian graphical learning. We develop an optimization framework to learn the individual graphical structures, assuming that the differences are due to individual nodes that are perturbed across views. The proposed optimization framework is presented for the special case of two views. Furthermore, we derive the upper bound on the estimation error of the proposed graph estimator and characterize the impact of the sample size, number of nodes, and the spectrum of the graph Laplacians on estimation errors. The approach is evaluated on synthetic graph data for robustness against noise, graph density, and sample size. Finally, the proposed framework is applied to two-view real-world graph data for graph learning and clustering.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 7827