# Consensus on Lie groups for the Riemannian Center of Mass

arXiv: https://arxiv.org/abs/2308.08054 · HTML: https://mlsanity.com/paper/2308.08054/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Aug 2023.

## Built on by (most cited first, 2 of 2)

- [The Asymptotic Behavior of Attention in Transformers](https://mlsanity.com/paper/2412.02682/) (2024)
- [Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds](https://mlsanity.com/paper/2509.07779/) (2025)

## Abstract (arXiv)

In this paper, we develop a consensus algorithm for distributed computation of the Riemannian center of mass (RCM) on Lie Groups. The algorithm is built upon a distributed optimization reformulation that allows developing an intrinsic, distributed (without relying on a consensus subroutine), and a computationally efficient protocol for the RCM computation. The novel idea for developing this fast distributed algorithm is to utilize a Riemannian version of distributed gradient flow combined with a gradient tracking technique. We first guarantee that, under certain conditions, the limit point of our algorithm is the RCM point of interest. We then provide a proof of global convergence in the Euclidean setting, that can be viewed as a "geometric" dynamic consensus that converges to the average from arbitrary initial points. Finally, we proceed to showcase the superior convergence properties of the proposed approach as compared with other classes of consensus optimization-based algorithms for the RCM computation.