Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with a very high number of supports. The m-OT solves several smaller optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, the m-OT does not consider the relationship between mini-batches which leads to undesirable estimation. Moreover, the m-OT does not approximate a proper metric between probability measures since the identity property is not satisfied. To address these problems, we propose a novel mini-batch scheme for optimal transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that finds the optimal coupling between mini-batches and it can be seen as an approximation to a well-defined distance on the space of probability measures. Furthermore, we show that the m-OT is a limit of the entropic regularized version of the BoMb-OT when the regularized parameter goes to infinity. Finally, we carry out experiments on various applications including deep generative models, deep domain adaptation, approximate Bayesian computation, color transfer, and gradient flow to show that the BoMb-OT can be widely applied and performs well in various applications.

翻译：在实际应用中成功地使用了微型和小型最佳运输(m-OT),这涉及到使用大量支持的概率措施。M-OT解决了几个较小的最佳运输问题,然后返回了成本和运输计划的平均值。尽管它具有可扩缩的优势,但M-OT并没有考虑微型小桶之间的关系,从而导致不适当的估计。此外,由于身份属性不满足,M-OT并没有在概率措施之间得出适当的衡量尺度。为了解决这些问题,我们提议了一种新型的小型最佳运输办法,称为小型小桶最佳运输(BoMb-OT),它发现微型小桶之间的最佳结合,并可以被视为接近概率计量空间上一个明确界定的距离。此外,我们表明,当常规参数达到无限时,M-OT是一个定期版本的博姆-OT的限值。最后,我们进行了各种应用的实验,包括深基因化模型、深域适应、近似Bayesian计算、彩色转移和梯流,可以广泛应用BOM-OT和各种应用。