The unsupervised task of aligning two or more distributions in a shared latent space has many applications including fair representations, batch effect mitigation, and unsupervised domain adaptation. Existing flow-based approaches estimate multiple flows independently, which is equivalent to learning multiple full generative models. Other approaches require adversarial learning, which can be computationally expensive and challenging to optimize. Thus, we aim to jointly align multiple distributions while avoiding adversarial learning. Inspired by efficient alignment algorithms from optimal transport (OT) theory for univariate distributions, we develop a simple iterative method to build deep and expressive flows. Our method decouples each iteration into two subproblems: 1) form a variational approximation of a distribution divergence and 2) minimize this variational approximation via closed-form invertible alignment maps based on known OT results. Our empirical results give evidence that this iterative algorithm achieves competitive distribution alignment at low computational cost while being able to naturally handle more than two distributions.

翻译：在共享的潜层空间对两种或两种以上分布进行匹配这一未经监督的任务有许多应用,包括公平表述、批量效应减缓和不受监督的域适应。现有的基于流动的方法独立地估计多种流动,这相当于学习多种完整的基因模型。其他方法需要对抗性学习,这种学习在计算上成本很高,而且具有优化的挑战性。因此,我们的目标是在避免对抗性学习的同时,联合对多个分布进行匹配。我们受来自最佳运输理论(OT)的高效匹配算法的启发,我们开发了一个简单的迭接法,以构建深度和表达性流动。我们的方法将每个循环分为两个子问题:1)形成分布差异的变异近似值,2)根据已知的OT结果,通过封闭式的不可忽略的调整图将这种变异近值最小化。我们的经验结果证明,这种迭代算法以低计算成本实现了竞争性分配一致性,同时能够自然处理两个以上的分布。