Tightening LP Relaxations for MAP using message passing
David Sontag, Talya Meltzer, Amir Globerson, Tommi Jaakkola and Yair Weiss
Proceedings of Uncertainty in Artificial Intelligence (UAI)
Linear Programming (LP) relaxations have become powerful tools for
finding the most probable (MAP) configuration in graphical models.
These relaxations can be solved efficiently using
message-passing algorithms such as belief propagation and,
when the relaxation is tight, provably find the MAP configuration.
The standard LP relaxation is not tight
enough in many real-world problems, however, and this has lead to the use of higher order cluster-based LP relaxations. The computational
cost increases exponentially with the size of the clusters and
limits the number and type of clusters we can use. We propose to
solve the cluster selection problem monotonically in the dual LP, iteratively
selecting clusters with guaranteed improvement, and quickly re-solving
with the added clusters by reusing the existing solution. Our dual message-passing algorithm
finds the MAP configuration in protein side-chain
placement, protein design, and stereo problems,
in cases where the standard LP relaxation fails.