# Federated Learning on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2206.05668 · HTML: https://mlsanity.com/paper/2206.05668/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 32 references that are in the map.
- First posted on arXiv 12 Jun 2022.

## Builds on (most cited first, 19 of 19)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Federated Optimization in Heterogeneous Networks](https://mlsanity.com/paper/1812.06127/) (2018)
- [On the Convergence of FedAvg on Non-IID Data](https://mlsanity.com/paper/1907.02189/) (2019)
- [Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization](https://mlsanity.com/paper/2007.07481/) (2020)
- [Federated Optimization: Distributed Machine Learning for On-Device Intelligence](https://mlsanity.com/paper/1610.02527/) (2016)
- [Linear Convergence in Federated Learning: Tackling Client Heterogeneity and Sparse Gradients](https://mlsanity.com/paper/2102.07053/) (2021)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [FedSplit: An algorithmic framework for fast federated optimization](https://mlsanity.com/paper/2005.05238/) (2020)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [From Local SGD to Local Fixed-Point Methods for Federated Learning](https://mlsanity.com/paper/2004.01442/) (2020)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Convergence and Accuracy Trade-Offs in Federated Learning and Meta-Learning](https://mlsanity.com/paper/2103.05032/) (2021)
- [Decentralized Riemannian Gradient Descent on the Stiefel Manifold](https://mlsanity.com/paper/2102.07091/) (2021)
- [On the Local Linear Rate of Consensus on the Stiefel Manifold](https://mlsanity.com/paper/2101.09346/) (2021)
- [Distributed Principal Component Analysis with Limited Communication](https://mlsanity.com/paper/2110.14391/) (2021)
- [Distributed Optimization on Riemannian Manifolds for multi-agent networks](https://mlsanity.com/paper/1711.11196/) (2017)

## Built on by (most cited first, 13 of 13)

- [Federated Learning on Riemannian Manifolds with Differential Privacy](https://mlsanity.com/paper/2404.10029/) (2024)
- [Nonconvex Federated Learning on Compact Smooth Submanifolds With Heterogeneous Data](https://mlsanity.com/paper/2406.08465/) (2024)
- [Riemannian coordinate descent algorithms on matrix manifolds](https://mlsanity.com/paper/2406.02225/) (2024)
- [Riemannian Federated Learning via Averaging Gradient Streams](https://mlsanity.com/paper/2409.07223/) (2024)
- [Streamlining in the Riemannian Realm: Efficient Riemannian Optimization with Loopless Variance Reduction](https://mlsanity.com/paper/2403.06677/) (2024)
- [FedSPDnet: Geometry-Aware Federated Deep Learning with SPDnet](https://mlsanity.com/paper/2604.22494/) (2026)
- [Aggregation on Learnable Manifolds for Asynchronous Federated Optimization](https://mlsanity.com/paper/2503.14396/) (2025)
- [New Insights on Unfolding and Fine-tuning Quantum Federated Learning](https://mlsanity.com/paper/2506.20016/) (2025)
- [Federated Learning on Riemannian Manifolds: A Gradient-Free Projection-Based Approach](https://mlsanity.com/paper/2507.22855/) (2025)
- [Federated Classification in Hyperbolic Spaces via Secure Aggregation of Convex Hulls](https://mlsanity.com/paper/2308.06895/) (2023)
- [ONG: Orthogonal Natural Gradient Descent](https://mlsanity.com/paper/2508.17169/) (2025)
- [Sheaf-Based Federated Representation Learning](https://mlsanity.com/paper/2608.10016/) (2026)
- [A Stable Aggregation Method for Quantum Federated Learning](https://mlsanity.com/paper/2609.00356/) (2026)

## Abstract (arXiv)

Federated learning (FL) has found many important applications in smart-phone-APP based machine learning applications. Although many algorithms have been studied for FL, to the best of our knowledge, algorithms for FL with nonconvex constraints have not been studied. This paper studies FL over Riemannian manifolds, which finds important applications such as federated PCA and federated kPCA. We propose a Riemannian federated SVRG (RFedSVRG) method to solve federated optimization over Riemannian manifolds. We analyze its convergence rate under different scenarios. Numerical experiments are conducted to compare RFedSVRG with the Riemannian counterparts of FedAvg and FedProx. We observed from the numerical experiments that the advantages of RFedSVRG are significant.