Learning Graphs with Monotone Topology Properties and Multiple Connected Components

Learning graphs with structural properties is in general a non-convex optimization problem. We consider graph families closed under edge removal operations, and graphs with multiple connected components. We propose a tractable algorithm that finds the generalized Laplacian matrix of a graph with the desired type of structure. Our algorithm has two steps, first it solves a combinatorial optimization problem to find a graph topology that satisfies the desired structural property. Second, it estimates a generalized Laplacian matrix by solving a sparsity constrained log-determinant divergence minimization problem. Our results are based on the analysis of a convex relaxation via weighted $\ell_1$-regularization. We derive specific instances of our algorithm to learn tree structured graphs, sparse connected graphs and bipartite graphs. We evaluate the performance of our graph learning method via numerical experiments with synthetic and image data.