Uniform Convergence of Generalized Conditional Fréchet Means with Applications to Weighted Fréchet Aggregation and Exceedance Set Estimation

2026-07-29

AI Summary

This study addresses the challenge of establishing uniform convergence for generalized conditional Fréchet means in non-Euclidean spaces, where the absence of linear structure impedes conventional analytical approaches. The authors propose a novel theoretical framework that circumvents the need to verify uniform equicontinuity—a standard requirement in existing theory—by leveraging structural conditions on the empirical loss function, thereby providing the first uniform convergence guarantees for this estimator. The approach naturally extends to weighted Fréchet aggregation and exceedance set estimation, and holds in general metric spaces. The validity and practical utility of the theory are demonstrated through Monte Carlo simulations and an empirical analysis of dynamic traffic networks in New York City.

Abstract

The statistical analysis of object oriented data in non-Euclidean spaces heavily relies on generalized conditional Fréchet means, notably in the context of Fréchet regression. However, establishing the uniform convergence of these estimators presents several theoretical challenges. The difficulties are caused primarily by the absence of linear structures in general metric spaces, rendering standard techniques for verifying the asymptotic uniform equicontinuity of the estimator largely intractable. To overcome this limitation, this paper introduces an alternative theoretical framework for establishing uniform convergence that bypasses the need to verify uniform equicontinuity, under a novel structural condition on the empirical cost function of the generalized conditional Fréchet means. We demonstrate that this analytical condition is satisfied by various prominent Fréchet regression models across broad classes of metric spaces. Leveraging these foundational uniform convergence guarantees, we subsequently extend two widely used frameworks from Euclidean to non-Euclidean spaces: (i) a weighted Fréchet aggregation framework that facilitates both distributed regression and robust median-of-means regression; and (ii) an exceedance set estimation framework to identify critical covariate regions where the conditional generalized Fréchet mean surpasses a prescribed threshold, alongside a metric to quantify the aggregate magnitude of the exceedance. The theoretical properties of these proposed methods are empirically validated through Monte Carlo simulations and an application to dynamic transportation networks in New York City.

Problem

• uniform convergence
• Fréchet means
• non-Euclidean spaces
• metric spaces
• conditional Fréchet regression

Innovation

• Uniform convergence
• Generalized conditional Fréchet means
• Non-Euclidean spaces
• Weighted Fréchet aggregation
• Exceedance set estimation