Random-Effects Algorithm for Random Objects in Metric Spaces

Problem
This paper addresses the gap in modeling random effects for non-Euclidean random objects in metric spaces, a scenario prevalent in various scientific disciplines where multiple observations are collected from the same experimental units. While mixed-effects models are well-established for scalar outcomes and functional data in Hilbert spaces, the authors note that frameworks for random effects in metric spaces are underdeveloped. This work is presented as a preprint and has not yet undergone peer review.

Method
The authors propose a nonlinear Fréchet-based algorithm for random-effects modeling applicable to arbitrary random objects defined on a metric space. The core technical contribution lies in the use of M-estimation theory to derive conditions for consistent estimation of the proposed metric-space prediction target under a working random-effects formulation. The algorithm is designed to handle complex data structures, including multivariate probability distributions and random graphs, which are common in digital health datasets. The training compute requirements are not explicitly disclosed, but the method is evaluated on both synthetic and real-world datasets to demonstrate its efficacy.

Results
The empirical performance of the proposed method is benchmarked against existing Hilbert space-based approaches. The authors report that their Fréchet-based algorithm consistently outperforms these baselines across various metrics, although specific numerical results and effect sizes are not detailed in the abstract. The evaluation includes both synthetic datasets, which allow for controlled comparisons, and real-world digital health datasets, emphasizing the practical applicability of the method.

Limitations
The authors acknowledge that their framework is still in the early stages of development and may require further refinement for broader applicability. They do not address potential computational inefficiencies that could arise when scaling the method to very large datasets or complex metric spaces. Additionally, the reliance on M-estimation theory may limit the robustness of the method under certain conditions, such as non-regularity in the underlying data distributions.

Why it matters
This work has significant implications for the analysis of complex data structures in various fields, particularly in digital health, where traditional modeling approaches may fall short. By providing a robust framework for random effects in metric spaces, the proposed method opens avenues for more accurate estimation and personalized predictions in settings where data cannot be easily represented in Euclidean spaces. This advancement could lead to improved methodologies in fields such as epidemiology, genomics, and social sciences, where the nature of the data often defies conventional modeling techniques.

Authors: Marcos Matabuena, Mateo Cámara
Source: arXiv:2605.02693
URL: https://arxiv.org/abs/2605.02693v1