Theoretical and Practical Analysis of Fréchet Regression via Comparison Geometry
This work addresses the challenge of analyzing data relationships on complex structures for researchers and practitioners in fields like machine learning and statistics, offering incremental theoretical and practical insights.
This paper tackles the problem of extending regression to non-Euclidean metric spaces like manifolds and graphs by providing a rigorous theoretical analysis of Fréchet regression using comparison geometry, resulting in statistical guarantees such as exponential concentration bounds and convergence rates, with empirical validation showing effectiveness for data with heteroscedasticity.
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.