Open Access

Cheng Wang, Carlos J Soto

• Estimating the mean of manifold-valued data is a central problem in modern statistics, yet it remains challenging due to the lack of a closed-form expression for the Fréchet mean. The gradient descent algorithm is widely used to approximate this quantity across various applications. Although generally effective, it can be computationally intensive for large datasets, as each iteration requires evaluating gradients with respect to the entire dataset. To address these limitations, we propose a tree-based, Recursive Fréchet Mean Estimator (RFME), tailored to data on manifolds. The proposed method leverages a hierarchical aggregation strategy to reduce computational complexity while preserving statistical accuracy. We establish the weak consistency of RFME with respect to the population Fréchet mean and discuss its computational properties. Through simulation studies and real-world applications, we demonstrate that RFME achieves competitive estimation accuracy with substantially improved efficiency. Moreover, as a generalization of the incremental Fréchet mean estimator, RFME also offers enhanced flexibility while maintaining practical advantages.