Deep learning predicts complex random objects without manifold limits
Deep Learning for Random Objects in Metric Spaces: Beyond Manifold Assumptions
Machine Learning
Summary
Many data types in areas like healthcare don’t fit neatly into regular shapes or spaces, which makes using standard deep learning tricky. The authors designed a way for deep learning to work directly with complex objects measured by distances, not assuming any simple shape or embedding. Their method has strong mathematical backing and beats other approaches in tests, especially when data have complicated structures. They also showed it works well on health data like body movement and blood sugar readings.
What this means in practice
- •For health data scientists: Predict distribution-valued health outcomes like glucose levels directly without simplifying assumptions about data shape.
- •For wearable device engineers: Improve analysis of complex sensor data such as high-resolution accelerometry by modeling responses as random objects in metric spaces.
Authors
Marcos Matabuena, Xiaokai Luo, Oscar Hernan Madrid Padilla
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.