Title: Efficient Recursive Fr\'echet Mean Estimation
Primary area: optimization
Keywords: Riemannian manifold, sphere, Kendall's shape space, Optimization.

Abstract: Estimating the mean is a key aspect of statistical analysis. Doing such an estimation on Riemannian manifolds is complex due to lacking a closed form solution. The gradient descent algorithm is commonly used to approximate the Fr\'echet mean across various applications. Although generally effective,  it can be problematic when the dataset is large as each computation of the gradient can be costly or when the mean is not uniquely defined as in positively curved manifolds. This paper introduces a tree-based, recursive Fr\'echet mean estimator (RFME), designed for data on the hypersphere. We prove the weak consistency of the RFME with the true mean and demonstrate its computational efficiency and accuracy through two simulations and two real-world case studies. We compare our algorithm to the standard gradient descent approach and to the incremental Fr\'echet mean estimator (iFME), a SOTA algorithm that efficiently estimates the mean. Lastly, our algorithm is a generalization of the iFME and thus our algorithm has more flexibility.

TL;DR: We present a novel algorithm for estimating the mean on Riemannian manifolds.