Theoretical and Practical Analysis of Fr\'echet Regression via Comparison Geometry

TL;DR

This paper provides a comprehensive theoretical and practical analysis of Fréchet regression on complex metric spaces using comparison geometry, offering new insights into its stability, existence, and statistical guarantees, supported by empirical validation.

Contribution

It introduces a rigorous theoretical framework for Fréchet regression via comparison geometry, including stability, existence, and convergence results, and demonstrates practical effectiveness through experiments.

Findings

Established conditions for existence and uniqueness of Fréchet means.

Provided statistical guarantees including concentration bounds and convergence rates.

Validated hyperbolic mappings for heteroscedastic data in empirical experiments.

Abstract

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.