The task of mapping two or more distributions to a shared representation has many applications including fair representations, batch effect mitigation, and unsupervised domain adaptation. However, most existing formulations only consider the setting of two distributions, and moreover, do not have an identifiable, unique shared latent representation. We use optimal transport theory to consider a natural multiple distribution extension of the Monge assignment problem we call the symmetric Monge map problem and show that it is equivalent to the Wasserstein barycenter problem. Yet, the maps to the barycenter are challenging to estimate. Prior methods often ignore transportation cost, rely on adversarial methods, or only work for discrete distributions. Therefore, our goal is to estimate invertible maps between two or more distributions and their corresponding barycenter via a simple iterative flow method. Our method decouples each iteration into two subproblems: 1) estimate simple distributions and 2) estimate the invertible maps to the barycenter via known closed-form OT results. Our empirical results give evidence that this iterative algorithm approximates the maps to the barycenter.

翻译：绘制两个或更多分布到共同代表处的任务有许多应用,包括公平代表、分批效果减缓和不受监督的域适应。然而,大多数现有配方只考虑设置两种分布,而且没有可识别的、独特的共同的潜在代表。我们使用最佳运输理论来考虑蒙古人分配问题的自然多重分布延伸,我们称之为对称蒙古人地图问题,并表明它相当于瓦塞斯坦巴利中心的问题。然而,向巴利中心绘制的地图很难估算。以前的方法往往忽略运输成本,依赖对抗性方法,或仅用于分散分布。因此,我们的目标是通过简单的迭接流方法估计两个或两个以上分布处及其相应的中间点之间的不可忽略的地图。我们的方法将每个分解成两个子问题:1)估计简单的分布,2)通过已知的封闭式OT结果估计向巴利中心绘制的不可倒置的地图。我们的经验结果证明,这个迭代算算法接近了巴利中心地图。