Theoretical and Practical Analysis of Fréchet Regression via Comparison Geometry

TL;DR

This work extends Fréchet regression to non-Euclidean settings by leveraging comparison geometry in CAT$(K)$ spaces. It establishes foundational results on the existence and uniqueness of Fréchet means, their stability under measure perturbations, and nonparametric regression guarantees with exponential concentration bounds and convergence rates. The authors develop angle-stability analyses and local jet expansions of Fréchet functionals, providing geometric insight into estimator behavior under curvature, and validate the theory with experiments that demonstrate superior performance of hyperbolic mappings in heteroscedastic regimes. Together, these results offer a rigorous, curvature-aware framework for regression with non-Euclidean responses and guide practical applications in manifolds, graphs, and spherical-to-hyperbolic mappings.

Abstract

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.

Figures (9)

• Figure 1: Mapping from spherical data into hyperbolic space.
• Figure 2: Visualization of the HYG Stellar database.
• Figure 3: Heteroscedasticity in the HYG Stellar dataset.
• Figure 4: Illustrative example of transformed responses. Under the heteroscedastic errors assumption, the appropriate transformations of response variable yield stabilized variance. In this figure, $Y$ is the original response variables, $\log(Y)$ is the log-transformed variables and $W$ is the hyperbolic mapped variables.
• Figure 5: Python code for the Fréchet regression.

Theorems & Definitions (58)

• Definition 2.1: $\mathrm{CAT}(K)$ space
• Definition 2.2: Geodesic convexity
• Definition 2.3: $\lambda$-strong geodesic convexity
• Definition 2.4: Lower semicontinuity
• Definition 2.5: Weak convergence in metric space
• Definition 2.6: Alexandrov angle
• Definition 2.7: Riemannian exponential map
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3