Joint Network Topology Inference via Structural Fusion Regularization

Abstract

Joint network topology inference represents a canonical problem of jointly learning multiple graph Laplacian matrices from heterogeneous graph signals. In such a problem, a widely employed assumption is that of a simple common component shared among multiple graphs. However, in practice, a more intricate topological pattern, comprising simultaneously of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">homogeneous</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">heterogeneous</i> components, would exhibit in multiple graphs. In this paper, we propose a general graph estimator based on a novel structural fusion regularization that enables us to jointly learn multiple graphs with such complex topological patterns, and enjoys rigorous theoretical guarantees. Specifically, in the proposed regularization term, the structural similarity among graphs is characterized by a Gram matrix, which enables us to flexibly model different types of network structural similarities through different Gram matrix choices. Algorithmically, the regularization term, coupling the parameters together, makes the formulated optimization problem intractable, and thus, we develop an implementable algorithm based on the alternating direction method of multipliers (ADMM) to solve it. Theoretically, non-asymptotic statistical analysis is provided, which precisely characterizes the minimum sample size required for the consistency of the graph estimator. This analysis also provides high-probability bounds on the estimation error as a function of graph structural similarities and other key problem parameters. Finally, the superior performance of the proposed method is demonstrated through simulated and real data examples.