Multi-dimensional Parametric Mincuts for Constrained MAP Inference

TL;DR

This paper introduces novel algorithms for solving constrained MAP inference in discrete models by formulating it as a multi-dimensional parametric mincut problem, enabling exact solutions and handling soft constraints efficiently.

Contribution

The paper presents a new multi-dimensional parametric mincut approach for constrained MAP inference, including variants for hard constraints and applications to various discrete optimization problems.

Findings

Faster solutions compared to continuous relaxation methods.

Closer to ground truth in image segmentation tasks.

Effectively handles soft and hard constraints.

Abstract

In this paper, we propose novel algorithms for inferring the Maximum a Posteriori (MAP) solution of discrete pairwise random field models under multiple constraints. We show how this constrained discrete optimization problem can be formulated as a multi-dimensional parametric mincut problem via its Lagrangian dual, and prove that our algorithm isolates all constraint instances for which the problem can be solved exactly. These multiple solutions enable us to even deal with `soft constraints' (higher order penalty functions). Moreover, we propose two practical variants of our algorithm to solve problems with hard constraints. We also show how our method can be applied to solve various constrained discrete optimization problems such as submodular minimization and shortest path computation. Experimental evaluation using the foreground-background image segmentation problem with statistic constraints reveals that our method is faster and its results are closer to the ground truth labellings compared with the popular continuous relaxation based methods.