Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems. However, the specific dynamics that DNNs, especially deep residual networks (ResNets), tend to learn during training remain insufficiently characterized. To this end, we model the forward propagation of deep residual networks using continuity equations, in which the measure is conserved and infinite curves in the measure space connect the input distribution to the output one of a ResNet. We find ResNets with $L_2$ regularization attempt to learn the geodesic curve in the Wasserstein space, induced by the optimal transport map. Compared with plain networks, ResNets can better approximate the geodesic curve, which explains why ResNets can be optimized and generalize better. Numerical experiments show that the data tracks of a ResNet tend to be line-shaped in terms of the line-shape score, and the map learned by a ResNet is closer to the optimal transport map in terms of the optimal transport score. In a word, we conclude that ResNets learn the geodesic curve in the Wasserstein space and discretely engineer the data transformation in high-dimensional spaces.
Accelerated Multiple Wasserstein Gradient Flows for Multi-objective Distributional Optimization
Graduate School of Information Science and Technology, Hokkaido University, Japan（信息科学与技术研究生学校，北海道大学，日本）；Bioinformatics Center, Kyoto University, Japan（生物信息学中心，京都大学，日本）；Institute of Information Technology, Vietnam Academy of Science and Technology, Vietnam（信息技术研究所，越南科学与技术学院，越南）
Wasserstein Parallel Transport for Predicting the Dynamics of Statistical Systems
Department of Statistics and Data Science, Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）；Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Department of Economics, Emory University（埃默里大学经济系）
Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence
Department of Statistics University of Illinois Urbana-Champaign（统计学系伊利诺伊大学厄巴纳-香槟分校）；Department of ACMS University of Notre Dame（ACMS系诺丁汉大学）；Department of Mathematics University of British Columbia（数学系不列颠哥伦比亚大学）；Department of Mathematics Thomas Lord Department of Computer Science University of Southern California（数学系托马斯·劳德计算机科学系南加州大学）
The Wasserstein-Fourier Distance for Stationary Time Series
IRIT, Université de Toulouse, CNRS（图卢兹大学法国国家科学研究中心图卢兹信息学研究所）；Imperial College London（帝国理工学院）；Center for Mathematical Modeling, Universidad de Chile（智利大学数学建模中心）
Interpolation for Robust Learning: Data Augmentation on Wasserstein Geodesics
Carnegie Mellon University（卡内基梅隆大学）；AT&T Chief Data Office（AT&T首席数据办公室）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Michigan（密歇根大学）
On the Wasserstein Geodesic Principal Component Analysis of probability measures
CREST-ENSAE, IP Paris（CREST-ENSAE，IP巴黎）；CNRS, IRIT, Université de Toulouse（CNRS，IRIT，图卢兹大学）；Université Paris 1 Panthéon Sorbonne（巴黎第一大学巴黎政治学院）；ENAC, IMT, Université de Toulouse（ENAC，IMT，图卢兹大学）
GeoECG: Data Augmentation via Wasserstein Geodesic Perturbation for Robust Electrocardiogram Prediction
Carnegie Mellon University（卡内基梅隆大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Colorado Denver - Anschutz Medical Campus（科罗拉多大学丹佛分校安舒茨医学校区）；Allegheny Health Network（阿勒格尼健康网络）
Contact Wasserstein Geodesics for Non-Conservative Schrödinger Bridges
Bosch Center for Artificial Intelligence (BCAI)（博世人工智能中心）；Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）；Technical University of Denmark (DTU)（丹麦技术大学）；Italian Institute of Artificial Intelligence (AI4I)（意大利人工智能研究所）
Encoded Prior Sliced Wasserstein AutoEncoder for learning latent manifold representations
Gatsby Computational Neuroscience Unit, University College London（伦敦大学学院盖茨比计算神经科学单元）；Department of Mathematics and Center for Scientific Computation and Mathematical Modeling, University of Maryland College Park（马里兰大学帕克分校数学系与科学计算与数学建模中心）
Linear convergence of proximal descent schemes on the Wasserstein space
Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan（日本东京RIKEN先进智能研究中心）；School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK（赫瑞-瓦特大学数学与计算机科学学院）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（爱丁堡数学科学麦克斯韦研究所）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；School of Mathematics, University of Edinburgh, UK, The Alan Turing Institute, UK（爱丁堡大学数学学院、英国艾伦·图灵研究所）
