Clusters and Coarse Partitions in LP Relaxations
David Sontag, Amir Globerson and Tommi Jaakkola
Advances in Nerual Information Processing Systems
We propose a new class of consistency constraints for Linear Programming (LP) relaxations for ﬁnding the most probable (MAP) conﬁguration in graphical models. Usual cluster-based LP relaxations enforce joint consistency on the beliefs of a cluster of variables, with computational cost increasing exponentially with the size of the clusters. By partitioning the state space of a cluster and enforcing con-
sistency only across partitions, we obtain a class of constraints which, although less tight, are computationally feasible for large clusters. We show how to solve the cluster selection and partitioning problem monotonically in the dual LP, using the current beliefs to guide these choices. We obtain a dual message passing algorithm and apply it to protein design problems where the variables have large state spaces and the usual cluster-based relaxations are very costly. The result-
ing method solves many of these problems exactly, and signiﬁcantly faster than a method that does not use partitioning.