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: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:2509. 23544v2 Announce Type: replace-cross Abstract: Many modern applications involve predicting structured, non-Euclidean outputs such as probability distributions, networks, and symmetric positive-definite matrices.
By Yidong Zhou, Su I Iao, Hans-Georg M\"uller
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:2606. 05599v1 Announce Type: new Abstract: This paper establishes a theoretical framework for the uniform convergence of smoothly activated deep neural network (DNN) estimators.
By Yizhe Ding, Runze Li, Jia Liu, Lingzhou Xue
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
arXiv:2606. 06772v1 Announce Type: cross Abstract: Understanding the generalization performance of over-parameterized neural networks has become a central topic in deep learning theory.
By Junyu Zhou, Puyu Wang, Yunwen Lei, Marius Kloft, Yiming Ying
arXiv:2502. 05684v5 Announce Type: replace-cross Abstract: How can we effectively remove or ``unlearn'' undesirable information, such as specific features or the influence of individual data points, from a learning outcome while minimizing utility loss and ensuring rigorous guarantees?
By Shizhou Xu, Thomas Strohmer
arXiv:2605. 27991v2 Announce Type: replace-cross Abstract: Gradient-flow optimization is usually viewed as an algorithmic procedure for minimizing empirical loss, with training duration selected by validation or heuristic early-stopping rules.
By Minhao Yao, Ruoyu Wang, Xihong Lin, Lin Liu, Zhonghua Liu
arXiv:2608. 09523v1 Announce Type: new Abstract: Deep neural network (DNN) training with stochastic gradient descent (SGD) and its variants achieves strong empirical performance, yet classical optimization theory does not fully explain this success.
By Binchuan Qi
arXiv:2608.09523v2 Announce Type: replace
Abstract: Stochastic gradient descent (SGD) and its variants train deep neural networks (DNNs) effectively in practice, yet classical optimization theory doe...
By Binchuan Qi
The paper introduces a pointwise generalization theory for fully connected deep neural networks, using a pointwise Riemannian Dimension derived from eigenvalues of learned feature representations across layers. This framework provides hypothesis-dependent, representation-aware generalization bounds that are significantly tighter than traditional size- or norm-based approaches, both theoretically and experimentally. The authors analytically identify structural properties that explain deep networks’ tractability and empirically show that the pointwise Riemannian Dimension captures feature compression, over‑parameterization effects, and optimizer bias.
By Shaojie Li, Yunbei Xu