Robust Local Fréchet Regression: Nonasymptotic Theory and Minimax Optimality
Abstract
We develop robust local Fréchet regression for heavy-tailed responses in a general metric space and propose a distance-based median-of-means estimator. We derive nonasymptotic pointwise high-probability bounds separately under local second-order smoothness and local $1$-Wasserstein Hölder continuity, and show that the corresponding optimized rates match the classical pointwise rates for nonparametric regression. We also establish matching pointwise minimax lower bounds over embedded Gaussian regression subclasses. Remarkably, our results do not require an entropy condition on the response space. Simulation studies with positive-definite matrix-valued responses, together with a real-data application involving Wasserstein-valued responses, illustrate the robustness--efficiency tradeoff and practical performance of the proposed estimator.
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BibTeX
@misc{im2026robust,
title = {{Robust Local Fréchet Regression: Nonasymptotic Theory and Minimax Optimality}},
author = {Chang Jun Im and Daisuke Kurisu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23757},
eprint = {2610.23757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23757},
}
RIS
TY - UNPB
TI - Robust Local Fréchet Regression: Nonasymptotic Theory and Minimax Optimality
AU - Chang Jun Im
AU - Daisuke Kurisu
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23757
UR - https://arxiv.org/abs/2610.23757
AB - We develop robust local Fréchet regression for heavy-tailed responses in a general metric space and propose a distance-based median-of-means estimator. We derive nonasymptotic pointwise high-probability bounds separately under local second-order smoothness and local $1$-Wasserstein Hölder continuity, and show that the corresponding optimized rates match the classical pointwise rates for nonparametric regression. We also establish matching pointwise minimax lower bounds over embedded Gaussian regression subclasses. Remarkably, our results do not require an entropy condition on the response space. Simulation studies with positive-definite matrix-valued responses, together with a real-data application involving Wasserstein-valued responses, illustrate the robustness--efficiency tradeoff and practical performance of the proposed estimator.
ER -