# Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope

## TL;DR

This paper introduces a novel geometric and probabilistic framework for permutation synchronization using the Riemannian structure of the Birkhoff Polytope, enabling efficient optimization and confidence estimation.

## Contribution

It develops new algorithms based on Riemannian geometry for permutation synchronization, including L-BFGS and Langevin Monte Carlo methods, and introduces a probabilistic model with confidence measures.

## Key findings

- Achieves high-quality multi-graph matching results
- Faster convergence compared to existing methods
- Provides reliable confidence and uncertainty estimates

## Abstract

We present an entirely new geometric and probabilistic approach to synchronization of correspondences across multiple sets of objects or images. In particular, we present two algorithms: (1) Birkhoff-Riemannian L-BFGS for optimizing the relaxed version of the combinatorially intractable cycle consistency loss in a principled manner, (2) Birkhoff-Riemannian Langevin Monte Carlo for generating samples on the Birkhoff Polytope and estimating the confidence of the found solutions. To this end, we first introduce the very recently developed Riemannian geometry of the Birkhoff Polytope. Next, we introduce a new probabilistic synchronization model in the form of a Markov Random Field (MRF). Finally, based on the first order retraction operators, we formulate our problem as simulating a stochastic differential equation and devise new integrators. We show on both synthetic and real datasets that we achieve high quality multi-graph matching results with faster convergence and reliable confidence/uncertainty estimates.