Computing Fr\'echet Means on Unknown Submanifolds of Riemannian Manifolds

Empirical averages are fundamental statistical primitives. On nonlinear spaces, their generalization, the Fr\'echet mean, generally lacks a closed-form solution, and its computation requires solving an optimization problem. We study the computation of intrinsic Fr\'echet means for data supported on an unknown compact $d$-dimensional Riemannian manifold $M_{1}$ embedded in an ambient metric space $(X,d_{\operatorname{obs}})$. 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 $d_{\operatorname{obs}}$, 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 $\widetilde{O}(T^{4d+3})$ 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.