Learning Graphs with Monotone Topology Properties and Multiple Connected Components
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Learning graphs with topology properties is a non-convex optimization problem. We propose an algorithm that finds the generalized Laplacian matrix of a graph with the desired type of graph topology. The algorithm has three steps, first, given data it computes a similarity/covariance matrix. Second, using the similarity matrix, it finds a feasible graph topology. Second, it estimates a generalized Laplacian matrix by solving a sparsity constrained log-determinant divergence minimization problem. The algorithm works when the graph family is closed under edge removal operations, or corresponds to graphs with multiple connected components. By analyzing the non-convex problem via a convex relaxation based on weighted ℓ_1-regularization, we derive an error bound between the solution of the non-convex problem and the output of our algorithm. We use this bound to design algorithms for the graph topology inference step. 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 data and texture images.
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