  Probabilistic inference in pairwise Markov Random Fields (MRFs), i.e.
computing the partition function or computing a MAP estimate of the variables,
is a foundational problem in probabilistic graphical models. Semidefinite
programming relaxations have long been a theoretically powerful tool for
analyzing properties of probabilistic inference, but have not been practical
owing to the high computational cost of typical solvers for solving the
resulting SDPs. In this paper, we propose an efficient method for computing the
partition function or MAP estimate in a pairwise MRF by instead exploiting a
recently proposed coordinate-descent-based fast semidefinite solver. We also
extend semidefinite relaxations from the typical binary MRF to the full
multi-class setting, and develop a compact semidefinite relaxation that can
again be solved efficiently using the solver. We show that the method
substantially outperforms (both in terms of solution quality and speed) the
existing state of the art in approximate inference, on benchmark problems drawn
from previous work. We also show that our approach can scale to large MRF
domains such as fully-connected pairwise CRF models used in computer vision.

Machine Learning;Machine Learning