  We consider the problem of approximating partition functions for Ising
models. We make use of recent tools in combinatorial optimization: the
Sherali-Adams and Lasserre convex programming hierarchies, in combination with
variational methods to get algorithms for calculating partition functions in
these families. These techniques give new, non-trivial approximation guarantees
for the partition function beyond the regime of correlation decay. They also
generalize some classical results from statistical physics about the
Curie-Weiss ferromagnetic Ising model, as well as provide a partition function
counterpart of classical results about max-cut on dense graphs
\cite{arora1995polynomial}. With this, we connect techniques from two
apparently disparate research areas -- optimization and counting/partition
function approximations. (i.e. \#-P type of problems).
  Furthermore, we design to the best of our knowledge the first provable,
convex variational methods. Though in the literature there are a host of convex
versions of variational methods \cite{wainwright2003tree, wainwright2005new,
heskes2006convexity, meshi2009convexifying}, they come with no guarantees
(apart from some extremely special cases, like e.g. the graph has a single
cycle \cite{weiss2000correctness}). We consider dense and low threshold rank
graphs, and interestingly, the reason our approach works on these types of
graphs is because local correlations propagate to global correlations --
completely the opposite of algorithms based on correlation decay. In the
process we design novel entropy approximations based on the low-order moments
of a distribution.
  Our proof techniques are very simple and generic, and likely to be applicable
to many other settings other than Ising models.
