Finite-Sample Bounds for Fréchet Means of Dependent Data in Metric Spaces with Curvature Bounded Above
Abstract
We prove finite-sample risk bounds for empirical quadratic and power Fréchet means in geodesic metric spaces with curvature bounded above for observations with arbitrary joint law. We compare the sample with exchangeable copies in which one observation is redrawn independently of the original sample and the others move to preserve the joint law. These displacements control the error, with no entropy or dimension assumptions. For quadratic Fréchet means, this yields a general ``coupling bound'' in any separable complete CAT(κ) space, where for κ>0 the observations must lie in a closed convex set of circumradius below π/(2√κ). This bound is exact on the real line for nonnegatively correlated Gaussian observations and for comonotone functions of a common factor. We apply it in two settings. Under local dependence, the resulting risk bound grows linearly with the average size of the dependency neighborhoods. Under transport sensitivity and for bounded metric variances, summable sensitivities of the conditional laws to a change in one observation preserve the parametric mean squared error rate, while nonsummable polynomial decay gives explicit slower rates. We verify transport sensitivity for stationary reversible Markov chains with Wasserstein-contractive kernels and for bi-Lipschitz images of Gaussian sequences. In the more challenging setting of heavy-tailed, dependent data with possibly infinite variance, we consider power Fréchet means of order α∈(1,2). Under nonpositive curvature, we bound their Lα risk assuming only finite α-th moments.
Related
- Weighted Power Fréchet Means in Metric Spaces with Curvature Bounded Above
- Central Limit Theorems for Sample Fréchet Means of Manifold-Valued Markov Chains
- Gaussian Fluctuations of the Gradient of the Fréchet Function and Nonstandard Asymptotics of Sample Fréchet Means
- Sample-Optimal Estimation of the Fréchet Inception Distance
- Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below
BibTeX
@misc{rico2026finitesample,
title = {{Finite-Sample Bounds for Fréchet Means of Dependent Data in Metric Spaces with Curvature Bounded Above}},
author = {Zoraida F. Rico and Christof Schötz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13362},
eprint = {2610.13362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13362},
}
RIS
TY - UNPB
TI - Finite-Sample Bounds for Fréchet Means of Dependent Data in Metric Spaces with Curvature Bounded Above
AU - Zoraida F. Rico
AU - Christof Schötz
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13362
UR - https://arxiv.org/abs/2610.13362
AB - We prove finite-sample risk bounds for empirical quadratic and power Fréchet means in geodesic metric spaces with curvature bounded above for observations with arbitrary joint law. We compare the sample with exchangeable copies in which one observation is redrawn independently of the original sample and the others move to preserve the joint law. These displacements control the error, with no entropy or dimension assumptions. For quadratic Fréchet means, this yields a general ``coupling bound'' in any separable complete CAT(κ) space, where for κ>0 the observations must lie in a closed convex set of circumradius below π/(2√κ). This bound is exact on the real line for nonnegatively correlated Gaussian observations and for comonotone functions of a common factor. We apply it in two settings. Under local dependence, the resulting risk bound grows linearly with the average size of the dependency neighborhoods. Under transport sensitivity and for bounded metric variances, summable sensitivities of the conditional laws to a change in one observation preserve the parametric mean squared error rate, while nonsummable polynomial decay gives explicit slower rates. We verify transport sensitivity for stationary reversible Markov chains with Wasserstein-contractive kernels and for bi-Lipschitz images of Gaussian sequences. In the more challenging setting of heavy-tailed, dependent data with possibly infinite variance, we consider power Fréchet means of order α∈(1,2). Under nonpositive curvature, we bound their L^α risk assuming only finite α-th moments.
ER -