CyclOT: Learning Quadratic Optimal Transport Maps via Synchronized Forward-Backward Interpolants

2026-09-12

AI Summary

本文提出了一种双向神经框架CyclOT，通过同步前向后向插值来学习高维未配对样本的二次最优传输映射，结合了双向二次动作、判别器限制的端点目标和双侧循环一致性。

Abstract

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 \(L^2\) 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 \(λW_2^2(μ_0,μ_1)\). 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.

Problem

• quadratic optimal transport
• unpaired samples
• high dimensions
• bidirectional maps
• displacement interpolants

Innovation

• bidirectional neural framework
• synchronized forward-backward interpolants
• unpaired samples
• quadratic optimal-transport maps
• cycle consistency