  Computing maximum a posteriori (MAP) estimation in graphical models is an
important inference problem with many applications. We present message-passing
algorithms for quadratic programming (QP) formulations of MAP estimation for
pairwise Markov random fields. In particular, we use the concave-convex
procedure (CCCP) to obtain a locally optimal algorithm for the non-convex QP
formulation. A similar technique is used to derive a globally convergent
algorithm for the convex QP relaxation of MAP. We also show that a recently
developed expectation-maximization (EM) algorithm for the QP formulation of MAP
can be derived from the CCCP perspective. Experiments on synthetic and
real-world problems confirm that our new approach is competitive with
max-product and its variations. Compared with CPLEX, we achieve more than an
order-of-magnitude speedup in solving optimally the convex QP relaxation.
