  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, independently. 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.
