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

arXiv: https://arxiv.org/abs/2502.01995 · HTML: https://mlsanity.com/paper/2502.01995/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2025.

## Abstract (arXiv)

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.