Zero Probability of the Cut Locus of a Fréchet Mean on a Riemannian Manifold

2025/08/01 by Alexander Lytchak, Lytchak, Alexander, Stephan Huckemann +1
Computer Science · Mathematics · #53C20 #60D05 #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Morphological variations and asymmetry #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2508.00747

openalex publication_date 2025/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases and conjectured in general. In application, we rule out stickiness, while providing examples of nowhere smooth Fréchet functions and we discuss extensions of the statement to Fréchet p-means, for p≠ 2, as well as to noncomplete manifolds and more general metric spaces.

Citations

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Related

• Stability of the Cut Locus and a Central Limit Theorem for Fréchet Means of Riemannian Manifolds
• Central limit theorem for intrinsic Frechet means in smooth compact Riemannian manifolds
• Some properties of Fréchet medians in Riemannian manifolds
• A genericity property of Fréchet sample means on Riemannian manifolds
• Generalized Fréchet means with random minimizing domains and its strong consistency