Equitable Partitions of Concave Free Energies

Abstract

Recently, exploiting symmetries within variational inference has been algebraically formalized. With the exception of TRW for marginal inference, however, the framework resulted in approximate MAP algorithms only, based on equitable and orbit partitions of the graphical model. Here, we deepen our understandnig of it for marginal inference. Specifically, we show that a large class of concave free energies admits equitable partitions, of which orbit partitions are a special case, that can be exploited for lifting. Although already interesting on its own, we go one step further. We demonstrate that concave free energies can be reparameterized so that existing convergent algorithms can be used for lifted variational marginal inference without modification.