Sep 12, 2026J/K move · Enter open · S save
Shizhou Xu, Jiachen Liu, Shih-Hsin Wang, Stefan Broecker+3
SLAC National Accelerator Laboratory, Stanford University, Menlo Park, CA 94025, USA · Department of Mathematics, University of California, Davis, Davis, CA 95616, USA · Department of Computer Science and Information Engineering, National Taiwan University, Taipei 10617, Taiwan+2
We study the recovery of forward and reverse quadratic optimal-transport maps from unpaired samples in high dimensions. We introduce a bidirectional neural framework in which the learned maps induce forward and backward displacement interpolants, while the training objective combines bidirectional quadratic action, discriminator-restricted Jensen-Shannon endpoint objectives, and two-sided cycle consistency. The construction requires neither precomputed sample pairings nor an explicit convex-potential parameterization. For absolutely continuous probability measures supported on a compact convex set, and under the stated generator-approximation, discriminator-richness, and minimizer-attainment conditions, we prove a population recovery theorem: for every prescribed accuracy, the sum of the corresponding
L2 errors between any global minimizer and the forward and reverse quadratic Brenier maps is below that accuracy, provided the discriminator level is sufficiently large and the annealing action weight becomes sufficiently small. Moreover, the cycle loss is bounded above by
λW22(μ0,μ1). Complementary results quantify approximate invertibility and show that exact endpoint Jensen-Shannon divergence and cycle consistency control missing target mass and many-to-one map collapse, respectively. Experiments on Swiss roll, MNIST, CelebA, single-cell perturbation data, and chest X-ray images evaluate endpoint fidelity, transport cost, inverse consistency, and the geometry of the induced interpolations.