# Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds

arXiv: https://arxiv.org/abs/2501.11555 · HTML: https://mlsanity.com/paper/2501.11555/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 41 references that are in the map.
- First posted on arXiv 20 Jan 2025.

## Builds on (most cited first, 5 of 5)

- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [Normalization Techniques in Training DNNs: Methodology, Analysis and Application](https://mlsanity.com/paper/2009.12836/) (2020)
- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [Huber means on Riemannian manifolds](https://mlsanity.com/paper/2407.15764/) (2024)
- [Linear pooling of sample covariance matrices](https://mlsanity.com/paper/2008.05854/) (2020)

## Built on by (most cited first, 1 of 1)

- [FedSPDnet: Geometry-Aware Federated Deep Learning with SPDnet](https://mlsanity.com/paper/2604.22494/) (2026)

## Abstract (arXiv)

In this paper, the issue of averaging data on a manifold is addressed. While the Fr'echet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this, R-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However, R-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call RL-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.