With the discovery of Wasserstein GANs, Optimal Transport (OT) has become a
powerful tool for large-scale generative modeling tasks. In these tasks, OT
cost is typically used as the loss for training GANs. In contrast to this
approach, we show that the OT map itself can be used as a generative model,
providing comparable performance. Previous analogous approaches consider OT
maps as generative models only in the latent spaces due to their poor
performance in the original high-dimensional ambient space. In contrast, we
apply OT maps directly in the ambient space, e.g., a space of high-dimensional
images. First, we derive a min-max optimization algorithm to efficiently
compute OT maps for the quadratic cost (Wasserstein-2 distance). Next, we
extend the approach to the case when the input and output distributions are
located in the spaces of different dimensions and derive error bounds for the
computed OT map. We evaluate the algorithm on image generation and unpaired
image restoration tasks. In particular, we consider denoising, colorization,
and inpainting, where the optimality of the restoration map is a desired
attribute, since the output (restored) image is expected to be close to the
input (degraded) one.

该研究提出了一种使用 Optimal Transport map 直接在高维图片空间中进行生成建模的算法，并在图像恢复等任务上验证了其有效性。