# Random-Effects Algorithm for Random Objects in Metric Spaces

arXiv: https://arxiv.org/abs/2605.02693 · HTML: https://mlsanity.com/paper/2605.02693/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 46 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 17 of 17)

- [Why do tree-based models still outperform deep learning on tabular data?](https://mlsanity.com/paper/2207.08815/) (2022)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Universal Bayes consistency in metric spaces](https://mlsanity.com/paper/1906.09855/) (2019)
- [Gaussian Process Boosting](https://mlsanity.com/paper/2004.02653/) (2020)
- [Network Regression with Graph Laplacians](https://mlsanity.com/paper/2109.02981/) (2021)
- [Single Index Fr'echet Regression](https://mlsanity.com/paper/2108.05437/) (2021)
- [Wasserstein F-tests and Confidence Bands for the Fr`echet Regression of Density Response Curves](https://mlsanity.com/paper/1910.13418/) (2019)
- [Dimension Reduction for Fr'echet Regression](https://mlsanity.com/paper/2110.00467/) (2021)
- [Sliced Wasserstein Regression](https://mlsanity.com/paper/2306.10601/) (2023)
- [Nonlinear global Fr'echet regression for random objects via weak conditional expectation](https://mlsanity.com/paper/2310.07817/) (2023)
- [Modeling Time-Varying Random Objects and Dynamic Networks](https://mlsanity.com/paper/2104.04628/) (2021)
- [Medoid splits for efficient random forests in metric spaces](https://mlsanity.com/paper/2306.17031/) (2023)
- [Multilevel functional distributional models with application to continuous glucose monitoring in diabetes clinical trials](https://mlsanity.com/paper/2403.10514/) (2024)
- [Fr'echet Covariance and MANOVA Tests for Random Objects in Multiple Metric Spaces](https://mlsanity.com/paper/2306.12066/) (2023)
- [Geodesic Mixed Effects Models for Repeatedly Observed/Longitudinal Random Objects](https://mlsanity.com/paper/2307.05726/) (2023)
- [Personalized Imputation in metric spaces via conformal prediction: Applications in Predicting Diabetes Development with Continuous Glucose Monitoring Information](https://mlsanity.com/paper/2403.18069/) (2024)
- [Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces](https://mlsanity.com/paper/2507.15741/) (2025)

## Abstract (arXiv)

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.