Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, several important machine learning applications call for the computation of exact Wasserstein distance, and regularized variations with small regularization parameter will fail due to numerical stability issues or degradate the performance. We address this challenge by developing an Inexact Proximal point method for Optimal Transport (IPOT) with the proximal operator approximately evaluated at each iteration using projections to the probability simplex. We also simplify the architecture for learning generative models based on optimal transport solution, and generalize the idea of IPOT to a new method for computing Wasserstein barycenter. We provide convergence analysis of IPOT and experiments showing our new methods outperform the state-of-the-art methods in terms of both effectiveness and efficiency.

开发了一种基于Inexact Proximal point方法的算法(IPOT)，该算法通过将投影用于概率单纯形来近似评估近端算子，以解决精确最优输运问题，并具有具有理论保证和强健的正则化参数选择，同时缓解了数值稳定性问题并避免了应用于生成模型时的收缩问题。此外，基于IPOT提出了一种新的算法，用于获得更锐利的Wasserstein重心。

计算精确瓦瑟斯坦距离的快速近端点方法