  We consider the problem of jointly inferring the M-best diverse labelings for
a binary (high-order) submodular energy of a graphical model. Recently, it was
shown that this problem can be solved to a global optimum, for many practically
interesting diversity measures. It was noted that the labelings are, so-called,
nested. This nestedness property also holds for labelings of a class of
parametric submodular minimization problems, where different values of the
global parameter $\gamma$ give rise to different solutions. The popular example
of the parametric submodular minimization is the monotonic parametric max-flow
problem, which is also widely used for computing multiple labelings. As the
main contribution of this work we establish a close relationship between
diversity with submodular energies and the parametric submodular minimization.
In particular, the joint M-best diverse labelings can be obtained by running a
non-parametric submodular minimization (in the special case - max-flow) solver
for M different values of $\gamma$ in parallel, for certain diversity measures.
Importantly, the values for $\gamma$ can be computed in a closed form in
advance, prior to any optimization. These theoretical results suggest two
simple yet efficient algorithms for the joint M-best diverse problem, which
outperform competitors in terms of runtime and quality of results. In
particular, as we show in the paper, the new methods compute the exact M-best
diverse labelings faster than a popular method of Batra et al., which in some
sense only obtains approximate solutions.
