I found this explanation on Quora simple and helpful for understanding why the Wasserstein distance should improve GAN performance.

https://www.quora.com/What-is-new-with-Wasserstein-GAN

Here’s the Tensorflow implementation on github that explanation came from:

Copied & pasted with relevant sentences bolded so you don’t even have to leave the forums

A pretty interesting paper that takes on the problem of stability in GANs and interpretability of the loss function during training. GANs essentially are models that try to learn the distribution of real data by minimizing f-divergence (difference in probabilty distribution) by generating adversarial data. The convergence in min max objective of the originally proposed GAN can be interpreted as minimizing the Jensen Shannon (JS) divergence. In this paper, the authors point out the shortcomings in such metrics when the support of the two distributions being compared do not overlap and propose using the earth movers/wasserstein distance as an alternative to JS. The parallel lines example provides a nice intuition to the differences in the f-divergence metrices. Note that when the f-divergence is discrete as in JS, KL we might face problems in learning models with gradients as the divergence loss is not differentiable everywhere.

Theorem 1 proposed in the paper is probably the key takeaway for anyone wondering why wasserstein distance might help in training GANS. The theorem basically states that a distribution mapping function (critic) that is continuous with respect to its parameters and locally lipschitz has a continuous and almost everywhere differentiable wasserstein distance.

**A continuous and almost everywhere differentiable metric would mean we can strongly train the discriminator before doing an update to the generator which in turn would receive improved reliable gradients to train from the discriminator. With the earlier formulations of GAN such training was not possible since training discriminator strongly would lead to vanishing gradients.**

Given that neural networks are generally continuous w.r.t to its parameters, the thing to make sure is the critic being Lipschitz. By clipping the weight parameters in the critic, we prevent the model from saturating while the growth is made almost linear. This would mean the gradients of the function is bounded by the slope of this linearity becoming Lipschitz bound.