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

2025-01-20

AI Summary

Computing data averages on Stiefel and Grassmann manifolds faces fundamental bottlenecks: the Fréchet mean lacks closed-form solutions, while Riemannian barycenter methods suffer from high computational complexity and limited applicability. Method: This paper introduces RL-barycenter—a novel framework that reformulates manifold averaging as Euclidean arithmetic averaging followed by orthogonal projection onto the manifold, thereby bypassing iterative optimization entirely. Grounded in differential geometry and manifold optimization theory, the method ensures theoretical elegance and computational efficiency. Contribution/Results: On synthetic benchmarks, RL-barycenter achieves accuracy comparable to state-of-the-art Riemannian barycenter algorithms, yet reduces computational cost by one to two orders of magnitude. It is the first method enabling real-time manifold mean estimation, offering a scalable, principled tool for Riemannian statistics and high-dimensional manifold learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• Stiefel and Grassmann manifolds
• Frechet mean infeasibility
• R-centroid complexity

Innovation

• RL-Center Method
• Stiefel and Grassmann Manifolds
• Efficient Computation