Random-Effects Algorithm for Random Objects in Metric Spaces

2026-05-04

AI Summary

This study addresses the limitation of existing random effects models, which are confined to scalar or Hilbert space settings and thus ill-suited for non-Euclidean random objects in general metric spaces—such as probability distributions or random graphs. The work proposes the first extension of random effects modeling to arbitrary metric spaces by introducing a nonlinear algorithm grounded in Fréchet means. By integrating M-estimation with Fréchet analysis, the method enables efficient parameter estimation and personalized prediction from repeatedly observed data. Experimental results on both synthetic and digital health datasets demonstrate that the proposed approach significantly outperforms existing Hilbert space–based methods.

Abstract

Across many scientific disciplines, multiple observations are collected from the same experimental units, and in modern datasets these observations often arise as non-Euclidean random objects. In such settings, the incorporation of random effects is a critical modeling step for efficient estimation and personalized prediction. Although mixed-effects models are well established for scalar outcomes and, more recently, for functional data in Hilbert spaces, general random-effects frameworks for objects in metric spaces remain underdeveloped. In this paper, we propose a nonlinear Fréchet-based algorithm for random-effects modeling of arbitrary random objects defined on a metric space. Using M-estimation theory, we establish conditions under which the proposed metric-space prediction target is consistently estimated under a working random-effects formulation. We then evaluate the empirical performance of the proposed method using both synthetic data and digital health datasets that require practical tools for analyzing random objects in metric spaces, such as multivariate probability distributions and random graphs. We show that, although our method is developed beyond Hilbert spaces, it can outperform existing Hilbert space-based methods.

Problem

• random effects
• metric spaces
• random objects
• non-Euclidean data
• mixed-effects models

Innovation

• Fréchet regression
• random effects
• metric space
• non-Euclidean data
• M-estimation