# Bottleneck potentials in Markov Random Fields

## TL;DR

This paper introduces a new class of Markov Random Field models with bottleneck potentials, enabling MAP inference that accounts for maximum local potentials, and demonstrates effective algorithms and results on seismic horizon tracking.

## Contribution

It extends MRFs with bottleneck potentials, providing novel relaxations and algorithms for MAP inference involving maximum-based potentials.

## Key findings

- Effective algorithms for bottleneck MRF inference.
- Successful application to large-scale seismic horizon tracking.
- Improved inference quality with bottleneck potentials.

## Abstract

We consider general discrete Markov Random Fields(MRFs) with additional bottleneck potentials which penalize the maximum (instead of the sum) over local potential value taken by the MRF-assignment. Bottleneck potentials or analogous constructions have been considered in (i) combinatorial optimization (e.g. bottleneck shortest path problem, the minimum bottleneck spanning tree problem, bottleneck function minimization in greedoids), (ii) inverse problems with $L_{\infty}$-norm regularization, and (iii) valued constraint satisfaction on the $(\min,\max)$-pre-semirings. Bottleneck potentials for general discrete MRFs are a natural generalization of the above direction of modeling work to Maximum-A-Posteriori (MAP) inference in MRFs. To this end, we propose MRFs whose objective consists of two parts: terms that factorize according to (i) $(\min,+)$, i.e. potentials as in plain MRFs, and (ii) $(\min,\max)$, i.e. bottleneck potentials. To solve the ensuing inference problem, we propose high-quality relaxations and efficient algorithms for solving them. We empirically show efficacy of our approach on large scale seismic horizon tracking problems.