# Max-Sliced Wasserstein Distance and its use for GANs

## TL;DR

This paper introduces the max-sliced Wasserstein distance, which improves sample complexity and reduces projection complexity, enabling efficient training of GANs on high-resolution images.

## Contribution

It proposes the max-sliced Wasserstein distance, enhancing sliced Wasserstein with better sample and projection complexity for high-dimensional GAN training.

## Key findings

- Max-sliced Wasserstein has superior sample complexity compared to Wasserstein.
- The method enables GAN training on 256x256 images efficiently.
- It reduces projection complexity in sliced Wasserstein computations.

## Abstract

Generative adversarial nets (GANs) and variational auto-encoders have significantly improved our distribution modeling capabilities, showing promise for dataset augmentation, image-to-image translation and feature learning. However, to model high-dimensional distributions, sequential training and stacked architectures are common, increasing the number of tunable hyper-parameters as well as the training time. Nonetheless, the sample complexity of the distance metrics remains one of the factors affecting GAN training. We first show that the recently proposed sliced Wasserstein distance has compelling sample complexity properties when compared to the Wasserstein distance. To further improve the sliced Wasserstein distance we then analyze its `projection complexity' and develop the max-sliced Wasserstein distance which enjoys compelling sample complexity while reducing projection complexity, albeit necessitating a max estimation. We finally illustrate that the proposed distance trains GANs on high-dimensional images up to a resolution of 256x256 easily.