提出简单高效的流形平均方法，计算更快且精度不降。
Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds
- 用投影算术平均替代复杂迭代算法
- 在模拟数据上性能接近现有方法，计算成本更低
- 适合需要快速流形平均的工程应用
本文研究流形上的数据平均问题。虽然基于黎曼几何的弗雷歇均值理论上理想，但往往不可行且计算代价高昂。为克服此问题，已有R-重心被提出并成功应用于施蒂费尔和格拉斯曼流形。然而，R-重心仍依赖迭代算法和复杂算子，存在显著局限。本文提出更简洁、高效的重心方法，称为RL-重心。我们证明，在大多数实际应用场景下，该框架产生的重心极为简单：即算术平均投影到流形上。该方法被应用于施蒂费尔和格拉斯曼流形。在模拟数据上，其性能与现有方法相当，但计算开销显著降低。
原文摘要 · Abstract (English)
In this paper, the issue of averaging data on a manifold is addressed. While the Fréchet 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.
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