# Distributed methods for synchronization of orthogonal matrices over graphs

## TL;DR

This paper introduces two novel distributed algorithms for synchronizing orthogonal matrices over directed graphs, with guarantees of convergence and stability under various graph conditions, applicable to 3D localization and image registration.

## Contribution

The paper presents two new algorithms for orthogonal matrix synchronization over directed graphs, one for symmetric and one for asymmetric graphs, with proven convergence and stability properties.

## Key findings

- Algorithms converge under general conditions.
- Methods are stable for small step sizes.
- Numerical simulations verify effectiveness.

## Abstract

This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We formulate the synchronization problem as a least-squares optimization problem with nonlinear constraints. The synchronization problem appears as one of the key components in applications ranging from 3D-localization to image registration. The main contributions of this work can be summarized as the introduction of two novel algorithms; one for symmetric graphs and one for graphs that are possibly asymmetric. Under general conditions, the former has guaranteed convergence to the solution of a spectral relaxation to the synchronization problem. The latter is stable for small step sizes when the graph is quasi-strongly connected. The proposed methods are verified in numerical simulations.