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

2025-02-04

AI Summary

Existing regression frameworks lack theoretical foundations for non-Euclidean domains—such as manifolds and graphs—where classical Euclidean assumptions fail. Method: We establish a unified Fréchet regression framework grounded in comparison geometry, the first to systematically incorporate tools from this field to rigorously characterize how curvature influences existence, uniqueness, stability, and statistical convergence of Fréchet regression estimators. We further propose a theoretically guaranteed hyperbolic mapping strategy to robustly handle heteroscedasticity, circumventing restrictive Euclidean hypotheses. Contribution/Results: Our analysis yields optimal convergence rates and exponential concentration bounds for Fréchet regression estimators under curvature constraints. Empirical evaluation demonstrates that the proposed method significantly outperforms Euclidean baselines in both predictive robustness and interpretability across diverse non-Euclidean settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Extends regression to non-Euclidean metric spaces
• Analyzes existence and stability of Fréchet mean
• Validates theory with empirical experiments

Innovation

• Extends regression to non-Euclidean metric spaces
• Uses comparison geometry for theoretical analysis
• Proposes hyperbolic mappings for heteroscedastic data