stat.MLOct 22, 2026

Deep Learning for Random Objects in Metric Spaces: Beyond Manifold Assumptions

Authors: Marcos Matabuena, Xiaokai Luo, Oscar Hernan Madrid Padilla

Organizations: Mohamed bin Zayed University of Artificial Intelligence · University of Notre Dame · Department of Statistics at University of California, Los Angeles

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Abstract

Deep learning (DL) methods have substantially improved predictive performance across a wide range of downstream prediction tasks, enabling new discoveries in multiple domains. However, principled DL methods for non-Euclidean random objects remain relatively scarce, despite the growing importance of modern biomedical outcomes taking the form of probability distributions, random graphs, functions, and other random objects in metric spaces. Motivated by a new generation of digital and clinical biomarkers, we propose a deep learning framework for regression with random objects in metric spaces. Our approach operates directly on the metric structure of the response space, avoiding restrictive manifold or Euclidean embedding assumptions imposed by many existing methods. At the same time, it retains modern statistical guarantees under the stated statistical assumptions. We establish statistical consistency and convergence rates for the proposed estimator, and characterize conditions under which it can mitigate the curse of dimensionality for hierarchically compositional regression functions. Extensive simulation studies demonstrate competitive predictive performance, with gains over several competing approaches in multiple Euclidean and non-Euclidean configurations. Finally, we demonstrate the practical utility of the proposed framework in two digital health case studies with distribution-valued outcomes in Wasserstein spaces, using high-resolution accelerometry and continuous glucose monitoring data. Embedding-based deep-learning methods may be less suitable when no informative low-dimensional representation exists or when the learned embedding does not adequately preserve the intrinsic geometry of the response space.

Figures & tables

Figure 1: Bivariate temporal distributions derived from continuous glucose monitoring data for three participants. The horizontal axis represents time of day, and the vertical axis represents glucose concentration.
Figure 2: Boxplots of prediction error for the competing methods in Scenario 1. Each panel corresponds to a different predictor dimension pp .
Figure 3: Boxplots of prediction error for the competing methods in Scenario 2. Each panel corresponds to a different predictor dimension pp .
Figure 4: Boxplots of prediction error for the competing methods in Scenario 3 with q=1q=1 . Each panel corresponds to a different predictor dimension pp .
Figure 5: Boxplots of prediction error for the competing methods in Scenario 4 with q=2q=2 . Each panel corresponds to a different predictor dimension pp .
Figure 6: Boxplots of prediction error for the competing methods in Scenario 4 with q=4q=4 . Each panel corresponds to a different predictor dimension pp .
Figure 7: 3D distributions for Participants 115, 368, and 500.
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