  This paper is concerned with the problem of exact MAP inference in general
higher-order graphical models by means of a traditional linear programming
relaxation approach. In fact, the proof that we have developed in this paper is
a rather simple algebraic proof being made straightforward, above all, by the
introduction of two novel algebraic tools. Indeed, on the one hand, we
introduce the notion of delta-distribution which merely stands for the
difference of two arbitrary probability distributions, and which mainly serves
to alleviate the sign constraint inherent to a traditional probability
distribution. On the other hand, we develop an approximation framework of
general discrete functions by means of an orthogonal projection expressing in
terms of linear combinations of function margins with respect to a given
collection of point subsets, though, we rather exploit the latter approach for
the purpose of modeling locally consistent sets of discrete functions from a
global perspective. After that, as a first step, we develop from scratch the
expectation optimization framework which is nothing else than a reformulation,
on stochastic grounds, of the convex-hull approach, as a second step, we
develop the traditional LP relaxation of such an expectation optimization
approach, and we show that it enables to solve the MAP inference problem in
graphical models under rather general assumptions. Last but not least, we
describe an algorithm which allows to compute an exact MAP solution from a
perhaps fractional optimal (probability) solution of the proposed LP
relaxation.
