Worst-Case Polynomial-Time Exact MAP Inference on Discrete Models with Global Dependencies
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Considering the worst-case scenario, junction tree algorithm remains the most efficient and general solution for exact MAP inference on discrete graphical models. Unfortunately, its main tractability assumption requires the treewidth of a corresponding MRF to be bounded strongly limiting the range of admissible applications. In fact, many practical problems in the area of structured prediction require modelling of global dependencies by either directly introducing global factors or enforcing global constraints on the prediction variables. This, however, always results in a fully-connected graph making exact inference by means of this algorithm intractable. Nevertheless, depending on the structure of the global factors, we can further relax the conditions for an efficient inference. In this paper we reformulate the work in [1] and present a better way to establish the theory also extending the set of handleable problem instances for free - since it requires only a simple modification of the originally presented algorithm. To demonstrate that this extension is not of a purely theoretical interest we identify one further use case in the context of generalisation bounds for structured learning which cannot be handled by the previous formulation. Finally, we accordingly adjust the theoretical guarantees that the modified algorithm always finds an optimal solution in polynomial time.
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