Random-Effects Algorithm for Random Objects in Metric Spaces

TL;DR

This paper introduces a nonlinear Fréchet-based algorithm for modeling random effects of objects in metric spaces, enabling efficient estimation and prediction for complex non-Euclidean data.

Contribution

It develops a general random-effects framework for metric space objects, extending beyond Hilbert spaces, with theoretical consistency and practical evaluation.

Findings

Method outperforms existing Hilbert space-based approaches.

Consistent estimation established under certain conditions.

Effective on synthetic and digital health datasets.

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 Fr\'echet-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.