AI and the World's Hard Problems · Theoretical computer science · arXiv · 2026-10-23
Deep Learning for Random Objects in Metric Spaces: Beyond Manifold Assumptions
Deep learning has improved prediction across many tasks, but principled methods for random objects in non-Euclidean metric spaces, such as distributions and graphs, are scarce. The authors propose a deep learning framework for regression with random objects that works directly on the metric structure of the response space, avoiding manifold or Euclidean embedding assumptions, and establish statistical consistency and convergence rates. It shows competitive predictive performance in simulations and was applied to two digital health case studies with distribution-valued outcomes in Wasserstein spaces, using accelerometry and continuous glucose monitoring data.