Learning graphs with monotone topology properties

Learning graphs with topology properties is a non-convex optimization problem. We propose a tractable algorithm that finds the generalized Laplacian matrix of a graph with the desired type of topology. Given the covariance/similarity matrix, our algorithm first solves a combinatorial optimization problem to find a graph topology that satisfies the desired structural property, and second, it estimates a generalized Laplacian matrix by solving a sparsity constrained Log-Determinant divergence minimization problem. The proposed method is guaranteed to find a near optimal generalized Laplacian matrix if the graph family is monotone, i.e. it is closed under edge removal operations. We derive an specific instance of our algorithm to learn bipartite graphs and we evaluate the performance of our graph learning method via numerical experiments with synthetic and image data.