# Convex Relaxations for Learning Bounded-Treewidth Decomposable Graphs

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K. S. Sesh Kumar,
Francis Bach
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Proceedings of the 30th International Conference on Machine Learning, PMLR 28(1):525-533, 2013.

#### Abstract

We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NP-hard problem and most approaches consider local search techniques. In this paper, we pose it as a combinatorial optimization problem, which is then relaxed to a convex optimization problem that involves searching over the forest and hyperforest polytopes with special structures. A supergradient method is used to solve the dual problem, with a run-time complexity of O(k^3 n^k+2 \log n) for each iteration, where n is the number of variables and k is a bound on the treewidth. We compare our approach to state-of-the-art methods on synthetic datasets and classical benchmarks, showing the gains of the novel convex approach.

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