Compact Relaxations for MAP Inference in Pairwise MRFs with Piecewise Linear Priors

TL;DR

This paper introduces a compact linear programming relaxation for MAP inference in pairwise Markov Random Fields with piecewise linear priors, significantly reducing complexity while maintaining solution quality.

Contribution

It presents a novel, compact LP relaxation method that scales linearly with label segments, improving efficiency over traditional quadratic-sized relaxations.

Findings

The compact LP matches the standard LP in label assignment quality.

Complexity is reduced from O(L^2) to O(LK) per clique.

The method is applicable to general piecewise linear priors.

Abstract

Label assignment problems with large state spaces are important tasks especially in computer vision. Often the pairwise interaction (or smoothness prior) between labels assigned at adjacent nodes (or pixels) can be described as a function of the label difference. Exact inference in such labeling tasks is still difficult, and therefore approximate inference methods based on a linear programming (LP) relaxation are commonly used in practice. In this work we study how compact linear programs can be constructed for general piecwise linear smoothness priors. The number of unknowns is O(LK) per pairwise clique in terms of the state space size $L$ and the number of linear segments K. This compares to an O(L^2) size complexity of the standard LP relaxation if the piecewise linear structure is ignored. Our compact construction and the standard LP relaxation are equivalent and lead to the same (approximate) label assignment.