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, regularized variations with large regularization parameter will
degradate the performance in several important machine learning applications,
and small regularization parameter will fail due to numerical stability issues
with existing algorithms. We address this challenge by developing an Inexact
Proximal point method for exact Optimal Transport problem (IPOT) with the
proximal operator approximately evaluated at each iteration using projections
to the probability simplex. The algorithm (a) converges to exact Wasserstein
distance with theoretical guarantee and robust regularization parameter
selection, (b) alleviates numerical stability issue, (c) has similar
computational complexity to Sinkhorn, and (d) avoids the shrinking problem when
apply to generative models. Furthermore, a new algorithm is proposed based on
IPOT to obtain sharper Wasserstein barycenter.

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。