Computing high-dimensional optimal transport by flow neural networks

The problem of finding a transport map between two general distributions P and Q in high dimension is essential in statistics, optimization, and machine learning. When both distributions are only accessible via finite samples, the transport map needs to be learned from data. In spite of the modeling and computational challenges, this setting has applications in many fields. For example, transfer learning in domain adaption aims to obtain a model on the target domain at a lower cost by making use of an existing pre-trained model on the source domain [Courty et al., 2014, 2017], and this can be achieved by transporting the source domain samples to the target domain using the transport map. The (optimal) transport has also been applied to achieve model fairness [Silvia et al., 2020]. By transporting distributions corresponding to different sensitive attributes to a common distribution, an unfair model is calibrated to match certain desired fairness criteria (e.g., demographic parity [Jiang et al., 2020]). The transport map can also be used to provide intermediate interpolating distributions between P and Q. In density ratio estimation (DRE), this bridging facilitates the so-called "telescopic" DRE [Rhodes et al., 2020] which has been shown to be more accurate when P and Q significantly differ. Furthermore, learning such a transport map between two sets of images can facilitate solving problems in computer vision, such as image restoration and image-to-image translation [Isola et al., 2017].