Worst-case Optimal Submodular Extensions for Marginal Estimation

TL;DR

This paper introduces a method to identify worst-case optimal submodular extensions for energy functions, enabling more accurate marginal estimation in graphical models, including dense CRFs, with practical algorithms and theoretical guarantees.

Contribution

It establishes an equivalence between submodular extensions and LP relaxations, leading to worst-case optimal extensions and efficient marginal computation methods.

Findings

Proves worst-case optimality of certain submodular extensions for Potts and metric labeling models.

Provides a practical algorithm for upper bounds on dense CRF models.

Achieves comparable bounds to TRW in synthetic and real data experiments.

Abstract

Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of the energy and the objective functions of linear programming (LP) relaxations for the corresponding MAP estimation problem. This allows us to (i) establish the worst-case optimality of the submodular extension for Potts model used in the literature; (ii) identify the worst-case optimal submodular extension for the more general class of metric labeling; and (iii) efficiently compute the marginals for the widely used dense CRF model with the help of a recently proposed Gaussian filtering method. Using synthetic and real data, we show that our approach provides comparable upper bounds on the log-partition function to those obtained using tree-reweighted message passing (TRW) in cases where the latter is computationally feasible. Importantly, unlike TRW, our approach provides the first practical algorithm to compute an upper bound on the dense CRF model.