Varying-Coefficient Fréchet Regression

2025-09-13

AI Summary

To address regression modeling with random object responses—such as distributions, shapes, or networks—this paper introduces the first varying-coefficient extension of Fréchet regression to non-Euclidean spaces. The proposed varying-coefficient Fréchet regression framework unifies treatment of both Euclidean and non-Euclidean responses and accommodates diverse effect modifiers, including scalars, functional covariates, and manifold-valued variables. It constructs interpretable, smooth estimators via local linear Fréchet regression. Theoretically, we establish the asymptotic convergence rate of the estimator, separately characterizing the bias and stochastic error orders. Empirically, comprehensive simulations and real-data analyses—including neuroimaging and meteorological applications—demonstrate its superior performance and robustness. This work fills a critical theoretical gap in non-Euclidean varying-coefficient modeling and provides a unified, scalable analytical framework for complex random object responses.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

As a growing number of problems involve variables that are random objects, the development of models for such data has become increasingly important. This paper introduces a novel varying-coefficient Fréchet regression model that extends the classical varying-coefficient framework to accommodate random objects as responses. The proposed model provides a unified methodology for analyzing both Euclidean and non-Euclidean response variables. We develop a comprehensive estimation procedure that accommodates diverse predictor settings. Specifically, the model allows the effect-modifier variable U to be either Euclidean or non-Euclidean, while the predictors X are assumed to be Euclidean. Tailored estimation methods are provided for each scenario. To examine the asymptotic properties of the estimators, we introduce a smoothed version of the model and establish convergence rates through separate theoretical analyses of the bias and stochastic terms. The effectiveness and practical utility of the proposed methodology are demonstrated through extensive simulation studies and a real-data application.

Problem

• Extends varying-coefficient regression to handle random object responses
• Provides unified methodology for Euclidean and non-Euclidean response variables
• Develops estimation procedures for diverse predictor settings and scenarios

Innovation

• Varying-coefficient Fréchet regression for random objects
• Unified methodology for Euclidean and non-Euclidean responses
• Tailored estimation methods for diverse predictor settings