arXiv Machine Learning
Oct 23

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

The paper introduces a deep learning framework for regression with random objects in metric spaces, bypassing restrictive manifold or Euclidean embedding assumptions. It provides statistical consistency, convergence rates, and conditions that mitigate the curse of dimensionality for hierarchically compositional regression functions. Extensive simulations and two digital health case studies with distribution-valued outcomes demonstrate competitive predictive performance over existing methods.

By Marcos Matabuena, Xiaokai Luo, Oscar Hernan Madrid Padilla
arXiv Machine Learning
Jun 8

Deep Single-Index Fr\'echet Regression

arXiv:2606. 06957v1 Announce Type: cross Abstract: Predicting outputs that are located in non-Euclidean spaces, such as probability distributions, networks, and symmetric positive-definite matrices, is becoming increasingly important in modern data analysis, particularly when inputs are high-dimensional.

By Muqing Cui, Yidong Zhou, Su I Iao, Hans-Georg M\"uller
arXiv AI
Sep 10

Deep Barycentric Regression for Optimal Transport Map Estimation and its Statistical Optimality

The paper introduces BROT, a two‑step approach for estimating optimal transport maps. First, it computes the unregularized OT plan, then fits a deep neural network to the resulting barycentric targets using least‑squares regression. The authors prove that, under standard regularity conditions, BROT achieves the minimax convergence rate when the true OT map is Lipschitz, and demonstrate its effectiveness on synthetic data, images, and downstream tasks such as single‑cell perturbation prediction and unsupervised domain adaptation.

By Kunwoong Kim, Insung Kong, Yongdai Kim
arXiv Machine Learning
Jul 30

Minimax-Optimal Generalization Bounds for Smooth Deep Neural Networks Trained by (Stochastic) Gradient Descent

arXiv:2606. 06772v2 Announce Type: replace-cross Abstract: Characterizing the optimization dynamics and statistical performance of over-parameterized deep neural networks (DNNs) remains a central challenge in understanding the remarkable success of deep learning.

By Junyu Zhou, Puyu Wang, Dennis Wagner, Yunwen Lei, Marius Kloft, Yiming Ying