On the Links between Probabilistic Graphical Models and Submodular Optimisation. (Liens entre modèles graphiques probabilistes et optimisation sous-modulaire)

A probabilistic graphical model encodes conditional independences among random variables, which is related to factorisable distributions. Moreover, the entropy of a probability distribution on a set of discrete random variables is always bounded by the entropy of its factorisable counterpart. This is due to the submodularity of entropy on the set of discrete random variables. Submodular functions are also generalisation of matroid rank function; therefore, linear functions may be optimised on the associated polytopes exactly using a greedy algorithm. In this manuscript, we exploit these links between the structures of graphical models and submodular functions: we use greedy algorithms to optimise linear functions on the polytopes related to graphic and hypergraphic matroids for learning the structures of graphical models, while we use inference algorithms on graphs to optimise submodular functions. The irst main contribution of the thesis aims at approximating a probabilistic distribution with a factorisable tractable distribution under the maximum likelihood framework. Since the tractability of exact inference is exponential in the treewidth of the decomposable graph, our goal is to learn bounded treewidth decomposable graphs, which is known to be NP-hard. We pose this as a combinatorial optimisation problem and provide convex relaxations based on graphic and hypergraphic matroids. This leads to an approximate solution with good empirical performance. In the second main contribution, we use the fact that the entropy of a probability distribution is always bounded by the entropy of its factorisable counterpart mainly as a consequence of submodularity. This property of entropy is generalised to all submodular functions and bounds based on graphical models are proposed. We refer to them as graph-based bounds. An algorithm is developped to maximise submodular functions, which is NP-hard, by maximising the graph-based bound using variational inference algorithms on graphs. The third main contribution of the thesis deals with minimising submodular functions that can be written as sum of “simple” submodular functions. It is broadly subdivided into two parts. The irst part deals with reviewing algorithms that minimise sum of “simple” submodular functions using minimisation oracles of the “simple” functions. Here, we speciically deal with cut functions in large scale problems and the minimisation oracles of the “simple” functions are graph inference algorithms. The second part proposes algorithms to minimise sum of general submodular functions using the structure of the polytopes related to individual “simple” submodular functions.