Learning Graphs With Monotone Topology Properties and Multiple Connected Components

Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation problem with graph Laplacian constraints. While such problems are convex, existing methods cannot guarantee that solutions will have specific graph topology properties (e.g., being a tree), which are desirable for some applications. The problem of learning a graph with topology properties is in general non-convex. In this paper, we propose an approach to solve these problems by decomposing them into two sub-problems for which efficient solutions are known. Specifically, a graph topology inference (GTI) step is employed to select a feasible graph topology. Then, a graph weight estimation (GWE) step is performed by solving a generalized graph Laplacian estimation problem, where edges are constrained by the topology found in the GTI step. Our main result is a bound on the error of the GWE step as a function of the error in the GTI step. This error bound indicates that the GTI step should be solved using an algorithm that approximates the data similarity matrix by another matrix whose entries have been thresholded to zero to have the desired type of graph topology. The GTI stage can leverage existing methods, which are typically based on minimizing the total weight of removed edges. Since the GWE stage is an inverse covariance estimation problem with linear constraints, it can be solved using existing convex optimization methods. We demonstrate that our approach can achieve good results for both synthetic and texture image data.