用神经网络直接学习最优传输映射，提升图像变形与高维传输效率。
GradNetOT: Learning Optimal Transport Maps with GradNets
- 基于单调梯度网络结构，直接参数化最优传输映射。
- 在图像变形和高维传输任务中均实现高效准确的映射学习。
- 适合需要精确传输路径的生成模型与控制问题研究者。
单调梯度函数在求解最优传输（OT）问题的Monge形式中起核心作用，该问题广泛应用于流体动力学到机器人集群控制等领域。当传输成本为平方欧几里得距离时，Brenier定理保证唯一最优传输映射满足Monge-Ampère方程，且是某个凸函数的梯度。我们此前在[arXiv:2301.10862]与[arXiv:2404.07361]中提出单调梯度网络（mGradNets），即直接参数化单调梯度映射空间的神经网络。本文利用mGradNets，通过最小化基于Monge-Ampère方程定义的训练损失函数，直接学习最优传输映射。实验表明，mGradNets的结构先验有助于在图像形态变换任务和高维OT问题中高效学习最优传输映射。
原文摘要 · Abstract (English)
Monotone gradient functions play a central role in solving the Monge formulation of the optimal transport (OT) problem, which arises in modern applications ranging from fluid dynamics to robot swarm control. When the transport cost is the squared Euclidean distance, Brenier's theorem guarantees that the unique optimal transport map satisfies a Monge-Ampère equation and is the gradient of a convex function. In [arXiv:2301.10862] [arXiv:2404.07361], we proposed Monotone Gradient Networks (mGradNets), neural networks that directly parameterize the space of monotone gradient maps. In this work, we leverage mGradNets to directly learn the optimal transport mapping by minimizing a training loss function defined using the Monge-Ampère equation. We empirically show that the structural bias of mGradNets facilitates the learning of optimal transport maps across both image morphing tasks and high-dimensional OT problems.
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