  Submodular extensions of an energy function can be used to efficiently
compute approximate marginals via variational inference. The accuracy of the
marginals depends crucially on the quality of the submodular extension. To
identify the best possible extension, we show an equivalence between the
submodular extensions of the energy and the objective functions of linear
programming (LP) relaxations for the corresponding MAP estimation problem. This
allows us to (i) establish the worst-case optimality of the submodular
extension for Potts model used in the literature; (ii) identify the worst-case
optimal submodular extension for the more general class of metric labeling; and
(iii) efficiently compute the marginals for the widely used dense CRF model
with the help of a recently proposed Gaussian filtering method. Using synthetic
and real data, we show that our approach provides comparable upper bounds on
the log-partition function to those obtained using tree-reweighted message
passing (TRW) in cases where the latter is computationally feasible.
Importantly, unlike TRW, our approach provides the first practical algorithm to
compute an upper bound on the dense CRF model.
