将回归分析拓展到复杂几何空间，揭示其理论稳定性与实际应用价值
Theoretical and Practical Analysis of Fréchet Regression via Comparison Geometry
- 基于比较几何框架分析非欧空间回归的数学性质
- 证明了弗雷歇均值的存在性、唯一性及估计器的收敛速度
- 适用于带异方差性的数据，特别在双曲映射中表现优异
弗雷歇回归将经典回归方法推广至非欧几里得度量空间，可用于流形和图等复杂结构上的数据分析。本文通过比较几何视角建立严格的理论分析，揭示弗雷歇均值的存在性、唯一性与稳定性，并提供非参数回归的统计保证，包括指数集中界和收敛速率。研究还发现曲率对回归估计器行为的影响，特别是角度稳定性机制。实验验证了理论结果，表明所提出的双曲映射在处理异方差数据时效果显著，凸显其实际应用价值。
原文摘要 · Abstract (English)
Fréchet regression extends classical regression methods to non-Euclidean metric spaces, enabling the analysis of data relationships on complex structures such as manifolds and graphs. This work establishes a rigorous theoretical analysis for Fréchet regression through the lens of comparison geometry which leads to important considerations for its use in practice. The analysis provides key results on the existence, uniqueness, and stability of the Fréchet mean, along with statistical guarantees for nonparametric regression, including exponential concentration bounds and convergence rates. Additionally, insights into angle stability reveal the interplay between curvature of the manifold and the behavior of the regression estimator in these non-Euclidean contexts. Empirical experiments validate the theoretical findings, demonstrating the effectiveness of proposed hyperbolic mappings, particularly for data with heteroscedasticity, and highlighting the practical usefulness of these results.
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