Automorphism Groups of Graphical Models and Lifted Variational Inference

TL;DR

This paper introduces the automorphism group of graphical models to formalize symmetry, enabling lifted variational inference that improves efficiency and accuracy in MAP inference tasks.

Contribution

It formalizes the automorphism group concept for graphical models, providing a mathematical framework for lifted inference and demonstrating its application to variational approximations.

Findings

Lifted MAP inference with cycle constraints outperforms local approximations.

The framework reduces inference complexity by exploiting symmetries.

Experimental results show state-of-the-art performance with improved bounds.

Abstract

Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group provides a precise mathematical framework for lifted inference in the general exponential family. Its group action partitions the set of random variables and feature functions into equivalent classes (called orbits) having identical marginals and expectations. Then the inference problem is effectively reduced to that of computing marginals or expectations for each class, thus avoiding the need to deal with each individual variable or feature. We demonstrate the usefulness of this general framework in lifting two classes of variational approximation for MAP inference: local LP relaxation and local LP relaxation with cycle constraints; the latter yields the first lifted inference that operate on a bound tighter than local constraints. Initial experimental results demonstrate that lifted MAP inference with cycle constraints achieved the state of the art performance, obtaining much better objective function values than local approximation while remaining relatively efficient.