Efficient Exact Inference in Planar Ising Models

TL;DR

This paper introduces polynomial-time algorithms for exact inference in planar Ising models, enabling efficient computation of various probabilistic and energy-based quantities without submodularity constraints.

Contribution

It presents a novel approach that leverages planarity and perfect matchings for exact inference, extending beyond traditional graph cut methods.

Findings

Efficient exact computation of ground states and partition functions.

Applicable to penalized maximum-likelihood and maximum-margin estimation.

Effective in image denoising and segmentation tasks.

Abstract

We give polynomial-time algorithms for the exact computation of lowest-energy (ground) states, worst margin violators, log partition functions, and marginal edge probabilities in certain binary undirected graphical models. Our approach provides an interesting alternative to the well-known graph cut paradigm in that it does not impose any submodularity constraints; instead we require planarity to establish a correspondence with perfect matchings (dimer coverings) in an expanded dual graph. We implement a unified framework while delegating complex but well-understood subproblems (planar embedding, maximum-weight perfect matching) to established algorithms for which efficient implementations are freely available. Unlike graph cut methods, we can perform penalized maximum-likelihood as well as maximum-margin parameter estimation in the associated conditional random fields (CRFs), and employ marginal posterior probabilities as well as maximum a posteriori (MAP) states for prediction. Maximum-margin CRF parameter estimation on image denoising and segmentation problems shows our approach to be efficient and effective. A C++ implementation is available from http://nic.schraudolph.org/isinf/