Generative Modeling with Optimal Transport Maps

TL;DR

This paper demonstrates that optimal transport maps can be directly used as generative models in high-dimensional spaces, providing a new approach for image generation and restoration tasks with competitive performance.

Contribution

The authors develop a min-max algorithm for computing OT maps in high-dimensional spaces and extend it to different dimensions, enabling direct application in ambient spaces.

Findings

Effective OT map computation for high-dimensional data

Successful application to image generation tasks

Improved unpaired image restoration results

Abstract

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.