(q,p)-Wasserstein GANs: Comparing Ground Metrics for Wasserstein GANs

Abstract

Generative Adversial Networks (GANs) have made a major impact in computer vision and machine learning as generative models. Wasserstein GANs (WGANs) brought Optimal Transport (OT) theory into GANs, by minimizing the $1$-Wasserstein distance between model and data distributions as their objective function. Since then, WGANs have gained considerable interest due to their stability and theoretical framework. We contribute to the WGAN literature by introducing the family of $(q,p)$-Wasserstein GANs, which allow the use of more general $p$-Wasserstein metrics for $p\ge 1$ in the GAN learning procedure. While the method is able to incorporate any cost function as the ground metric, we focus on studying the $l^q$ metrics for $q\ge 1$. This is a notable generalization as in the WGAN literature the OT distances are commonly based on the $l^2$ ground metric. We demonstrate the effect of different $p$-Wasserstein distances in two toy examples. Furthermore, we show that the ground metric does make a difference, by comparing different $(q,p)$ pairs on the MNIST and CIFAR-10 datasets. Our experiments demonstrate that changing the ground metric and $p$ can notably improve on the common $(q,p)=(2,1)$ case.

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• Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks
• GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium
• Improved Techniques for Training GANs
• Wasserstein Auto-Encoders
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• Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark
• Generative Modeling with Optimal Transport Maps
• Learning normalizing flows from Entropy-Kantorovich potentials
• Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimation