Fr\'echet 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 Fr\'echet 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 Fr\'echet 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.

本研究解决了Fréchet回归在非欧几里得度量空间中的应用空白，提出了一种基于比较几何的严格理论分析方法。研究发现Fréchet均值的存在性、唯一性和稳定性，并为非参数回归提供了统计保障，揭示了流形曲率与回归估计量之间的相互作用，证实了所提超曲映射在异方差数据中的有效性，增强了理论成果的实用性。

通过比较几何对Fréchet回归的理论和实践分析