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Title: A Greedy Approach to Max-Sliced Wasserstein GANs

Paper content:
Generative Adversarial Networks have made data generation possible in various use cases, but in case of complex, high-dimensional distributions it can be difficult to train them, because of convergence problems and the appearance of mode collapse.
Sliced Wasserstein GANs and especially the application of the Max-Sliced Wasserstein distance made it possible to approximate Wasserstein distance during training in an efficient and stable way and helped ease convergence problems of these architectures.
This method transforms sample assignment and distance calculation into sorting the one-dimensional projection of the samples, which results a sufficient approximation of the high-dimensional Wasserstein distance.
In this paper we will demonstrate that the approximation of the Wasserstein distance by sorting the samples is not always the optimal approach and the greedy assignment of the real and fake samples can result faster convergence and better approximation of the original distribution.
Generative Adversarial Networks (GANs) were first introduced in Goodfellow et al. (2014) , where instead of the application of a mathematically well-established loss function an other differentiable neural network, a discriminator was applied to approximate the distance between two distributions.
These methods are popularly applied in data generation and has significantly improved the modelling capabilities of neural networks.
It was demonstrated in various use cases that these approaches can approximate complex high-dimensional distributions in practice Karras et al. (2017) , Yu et al. (2017) , Brock et al. (2018) .
Apart from the theoretical advantage of GANs and applying a discriminator network instead of a distance metric (e.g. 1 or 2 loss), modelling high-dimensional distributions with GANs often proves to be problematic in practice.
The two most common problems are mode collapse, where the generator gets stuck in a state where only a small portion of the whole distribution is modeled and convergence problems, where either the generator or the discriminator solves his task almost perfectly, providing low or no gradients for training for the other network.
Convergence problems were improved, by introducing the Wasserstein distance Gulrajani et al. (2017) , which instead of a point-wise distance calculation (e.g. cross-entropy or 1 distance) calculates a minimal transportation distance (earth mover's distance) between the two distributions.
The approximation and calculation of the Wasserstein distance is complex and difficult in highdimensions, since in case of a large sample size calculation and minimization of the transport becomes exponentially complex, also distance can have various magnitudes in the different dimensions.
In Deshpande et al. (2018) it was demonstrated that high-dimensional distributions can be approximated by using a high number of one dimensional projections.
For a selected projection the minimal transport between the one dimensional samples can be calculated by sorting both the real and the fake samples and assigning them according to their sorted indices correspondingly.
As an additional advantage, it was also demonstrated in Deshpande et al. (2018) that instead of the regular mini-max game of adversarial training, the distribution of the real samples could be approximated directly by the generator only, omitting the discriminator and turning training into a simple and more stable minimization problem.
The theory of this novel method is well described and it was demonstrated that it works in practice, but unfortunately for complex, high-dimensional distributions a large amount of projections are needed.
In Deshpande et al. (2019) it was demonstrated how the high number of random projections could be substituted by a single continuously optimized plane.
The parameters of this projection are optimized in an adversarial manner selecting the "worst" projection, which maximizes separation between the real and fake samples using a surrogate function.
This modification brought the regular adversarial training back and created a mini-max game again, where the generator creates samples which resemble well to the original distribution according to the selected plane and the discriminator tries to find a projection, which separates the real and fake samples from each other.
The essence of Sliced Wasserstein distances is how they provide a method to calculate minimal transportation between the projected samples in one-dimension with ease, which approximates the Wasserstein distance in the original high-dimension.
In theory this approach is sound and works well in practise.
It was proven in Kolouri et al. (2019) that the sliced Wasserstein distance satisfies the properties of non-negativity, identity of indiscernibles, symmetry, and triangle inequality, this way forming a true metric.
However it approximates high-dimensional distributions well, we would like to demonstrate in this paper that the assignment of real and fake samples by sorting them in one dimension also has its flaws and a greedy assignment approach can perform better on commonly applied datasets.
We would also argue regarding the application of the Wasserstein distance.
We will demonstrate that in many cases various assignments can result the same minimal transportation during training and calculation of the Wasserstein distance with sorting can alter the distribution of perfectly modeled samples even when only a single sample differs from the approximated distribution.
In this paper we have introduced greedy sample assignment for Max-Sliced Wasserstein GANs.
We have shown that using one-dimensional samples, in many cases multiple assignments can result optimal transportation and in most cases sorting changes all the samples, meanwhile those parts of the distribution which are at a "good" position should not generate error.
We proposed greedy assignment as a possible solution, where samples will be assigned to their most similar counterparts.
We have also introduced how the combination of the two methods can be applied resulting a hybrid approach in which it can automatically selected -based on the difference of the two measures -which assignment will be used.
We have demonstrated on simple toy datasets that greedy assignment performs better than sorting the samples and we have evaluated both the greedy and the hybrid methods on commonly investigated datasets (MNIST and CelebA).
With all datasets the greedy assignment resulted lower KullbackLeibler divergence and higher correlation than the traditional approach.
We have used the Max-Sliced Wasserstein distance for the base of our comparison, since this is the most recent version, which also results the best performance, but all the approaches can be exploited in case of regular Sliced Wasserstein distances as well.
Also our approach changes the distance calculation only and it can be applied together with various other improved techniques and architectures which are used in GAN training.

assistant: text: We apply a greedy assignment on the projected samples instead of sorting to approximate Wasserstein distance