Recursive Fréchet Mean Estimation

Cheng Wang, Carlos J Soto
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:7103-7120, 2026.

Abstract

Estimating the mean of manifold-valued data is a central problem in modern statistics, yet it remains challenging due to the lack of a closed-form expression for the {Fréchet} mean. The gradient descent algorithm is widely used to approximate this quantity across various applications. Although generally effective, it can be computationally intensive for large datasets, as each iteration requires evaluating gradients with respect to the entire dataset. To address these limitations, we propose a tree-based, Recursive {Fréchet} Mean Estimator ({RFME}), tailored to data on manifolds. The proposed method leverages a hierarchical aggregation strategy to reduce computational complexity while preserving statistical accuracy. We establish the weak consistency of {RFME} with respect to the population {Fréchet} mean and discuss its computational properties. Through simulation studies and real-world applications, we demonstrate that {RFME} achieves competitive estimation accuracy with substantially improved efficiency. Moreover, as a generalization of the incremental {Fréchet} mean estimator, {RFME} also offers enhanced flexibility while maintaining practical advantages.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-wang26d, title = {Recursive {Fréchet} Mean Estimation}, author = {Wang, Cheng and Soto, Carlos J}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {7103--7120}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/wang26d/wang26d.pdf}, url = {https://proceedings.mlr.press/v337/wang26d.html}, abstract = {Estimating the mean of manifold-valued data is a central problem in modern statistics, yet it remains challenging due to the lack of a closed-form expression for the {Fréchet} mean. The gradient descent algorithm is widely used to approximate this quantity across various applications. Although generally effective, it can be computationally intensive for large datasets, as each iteration requires evaluating gradients with respect to the entire dataset. To address these limitations, we propose a tree-based, Recursive {Fréchet} Mean Estimator ({RFME}), tailored to data on manifolds. The proposed method leverages a hierarchical aggregation strategy to reduce computational complexity while preserving statistical accuracy. We establish the weak consistency of {RFME} with respect to the population {Fréchet} mean and discuss its computational properties. Through simulation studies and real-world applications, we demonstrate that {RFME} achieves competitive estimation accuracy with substantially improved efficiency. Moreover, as a generalization of the incremental {Fréchet} mean estimator, {RFME} also offers enhanced flexibility while maintaining practical advantages.} }
Endnote
%0 Conference Paper %T Recursive Fréchet Mean Estimation %A Cheng Wang %A Carlos J Soto %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-wang26d %I PMLR %P 7103--7120 %U https://proceedings.mlr.press/v337/wang26d.html %V 337 %X Estimating the mean of manifold-valued data is a central problem in modern statistics, yet it remains challenging due to the lack of a closed-form expression for the {Fréchet} mean. The gradient descent algorithm is widely used to approximate this quantity across various applications. Although generally effective, it can be computationally intensive for large datasets, as each iteration requires evaluating gradients with respect to the entire dataset. To address these limitations, we propose a tree-based, Recursive {Fréchet} Mean Estimator ({RFME}), tailored to data on manifolds. The proposed method leverages a hierarchical aggregation strategy to reduce computational complexity while preserving statistical accuracy. We establish the weak consistency of {RFME} with respect to the population {Fréchet} mean and discuss its computational properties. Through simulation studies and real-world applications, we demonstrate that {RFME} achieves competitive estimation accuracy with substantially improved efficiency. Moreover, as a generalization of the incremental {Fréchet} mean estimator, {RFME} also offers enhanced flexibility while maintaining practical advantages.
APA
Wang, C. & Soto, C.J.. (2026). Recursive Fréchet Mean Estimation. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:7103-7120 Available from https://proceedings.mlr.press/v337/wang26d.html.

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