Lipschitz最优迁移映射的GAN估计
GAN Estimation of Lipschitz Optimal Transport Maps
摘要：
本文介绍了基于神经网络的两个概率分布之间的最优传输映射的第一个统计相合估计量。基于Lipschitz神经网络领域的理论和实践进展，我们定义了一个受二次运输成本惩罚的Lipschitz约束生成对抗网络。然后，我们证明了，在正则性假设下，所得到的生成器一致收敛到最优的运输映射的样本量增加到无穷大。此外，我们通过大量的数值实验表明，学习映射具有良好的性能。与以前的工作处理统计保证或实用性相比，我们提供了一个表达和可行的估计铺平了道路，为最佳运输应用的渐近行为必须得到认证。
英文摘要：
This paper introduces the first statistically consistent estimator of the optimal transport map between two probability distributions, based on neural networks. Building on theoretical and practical advances in the field of Lipschitz neural networks, we define a Lipschitz-constrained generative adversarial network penalized by the quadratic transportation cost. Then, we demonstrate that, under regularity assumptions, the obtained generator converges uniformly to the optimal transport map as the sample size increases to infinity. Furthermore, we show through a number of numerical experiments that the learnt mapping has promising performances. In contrast to previous work tackling either statistical guarantees or practicality, we provide an expressive and feasible estimator which paves way for optimal transport applications where the asymptotic behaviour must be certified.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
