Computing Fréchet Means on Unknown Submanifolds of Riemannian Manifolds
Abstract
Empirical averages are fundamental statistical primitives. On nonlinear spaces, their generalization, the Fréchet mean, generally lacks a closed-form solution, and its computation requires solving an optimization problem. We study the computation of intrinsic Fréchet means for data supported on an unknown compact d-dimensional Riemannian manifold M1 embedded in an ambient metric space (X,dobs). Such settings arise when observations are better represented in non-Euclidean spaces, like probability distributions in Wasserstein space. We assume access only to a local sampler on M1 and the observation metric dobs, without knowledge of its intrinsic geometry. We propose a graph-based iterative algorithm that adaptively samples the manifold and approximates geodesic interpolation. Under mild assumptions, we establish an O(1/T) expected objective gap guarantee using at most Õ(T4d+3) sampled points. As a byproduct of our analysis we extend the metric equivalence guarantees of ISOMAP from Euclidean to Riemannian ambient spaces, which may be of independent interest. Numerical experiments demonstrate the effectiveness of our method for computing intrinsic averages on unknown data manifolds.
Related
- Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
- Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications
- Four-dimensional pseudo-umbilical biharmonic submanifolds in six-dimensional Riemannian manifolds of constant sectional curvature
- Riemannian Optimization on Unknown Manifolds via Stochastic Proximal Refinement
- Inference for conditional Fréchet means via Riemannian Robust Bias Correction
BibTeX
@misc{mcpherson2026computing,
title = {{Computing Fréchet Means on Unknown Submanifolds of Riemannian Manifolds}},
author = {Ian McPherson and Mauro Maggioni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22592},
eprint = {2610.22592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22592},
}
RIS
TY - UNPB
TI - Computing Fréchet Means on Unknown Submanifolds of Riemannian Manifolds
AU - Ian McPherson
AU - Mauro Maggioni
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.22592
UR - https://arxiv.org/abs/2610.22592
AB - Empirical averages are fundamental statistical primitives. On nonlinear spaces, their generalization, the Fréchet mean, generally lacks a closed-form solution, and its computation requires solving an optimization problem. We study the computation of intrinsic Fréchet means for data supported on an unknown compact d-dimensional Riemannian manifold M_1 embedded in an ambient metric space (X,d_obs). Such settings arise when observations are better represented in non-Euclidean spaces, like probability distributions in Wasserstein space. We assume access only to a local sampler on M_1 and the observation metric d_obs, without knowledge of its intrinsic geometry. We propose a graph-based iterative algorithm that adaptively samples the manifold and approximates geodesic interpolation. Under mild assumptions, we establish an O(1/T) expected objective gap guarantee using at most Õ(T^4d+3) sampled points. As a byproduct of our analysis we extend the metric equivalence guarantees of ISOMAP from Euclidean to Riemannian ambient spaces, which may be of independent interest. Numerical experiments demonstrate the effectiveness of our method for computing intrinsic averages on unknown data manifolds.
ER -