# GAN and VAE from an Optimal Transport Point of View

@article{Genevay2017GANAV, title={GAN and VAE from an Optimal Transport Point of View}, author={Aude Genevay and Gabriel Peyr'e and Marco Cuturi}, journal={arXiv: Machine Learning}, year={2017} }

This short article revisits some of the ideas introduced in arXiv:1701.07875 and arXiv:1705.07642 in a simple setup. This sheds some lights on the connexions between Variational Autoencoders (VAE), Generative Adversarial Networks (GAN) and Minimum Kantorovitch Estimators (MKE).

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## 42 Citations

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## References

SHOWING 1-10 OF 13 REFERENCES

From optimal transport to generative modeling: the VEGAN cookbook

- Computer Science
- 2017

It is shown that POT for the 2-Wasserstein distance coincides with the objective heuristically employed in adversarial auto-encoders (AAE) (Makhzani et al., 2016), which provides the first theoretical justification for AAEs known to the authors.

Sinkhorn-AutoDiff: Tractable Wasserstein Learning of Generative Models

- Computer Science
- 2017

This paper presents the first tractable computational method to train large scale generative models using an optimal transport loss, and relies on two key ideas: entropic smoothing, which turns the original OT loss into one that can be computed using Sinkhorn fixed point iterations; and algorithmic (automatic) differentiation of these iterations, which result in a robust and differentiable approximation of the OT loss.

Auto-Encoding Variational Bayes

- Computer ScienceICLR
- 2014

A stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case is introduced.

Wasserstein Training of Restricted Boltzmann Machines

- Computer ScienceNIPS
- 2016

This work proposes a novel approach for Boltzmann machine training which assumes that a meaningful metric between observations is known, and derives a gradient of that distance with respect to the model parameters from the Kullback-Leibler divergence.

Inference in generative models using the Wasserstein distance

- Computer Science, Mathematics
- 2017

This work uses Wasserstein distances between empirical distributions of observed data and empirical distribution of synthetic data drawn from such models to estimate their parameters, and proposes an alternative distance using the Hilbert space-filling curve.

Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling

- Mathematics
- 2015

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.-…

On parameter estimation with the Wasserstein distance

- Mathematics, Computer ScienceInformation and Inference: A Journal of the IMA
- 2019

These results cover the misspecified setting, in which the data-generating process is not assumed to be part of the family of distributions described by the model, and some difficulties arising in the numerical approximation of these estimators are discussed.

Scaling Algorithms for Unbalanced Transport Problems

- Computer Science
- 2017

This article introduces a new class of fast algorithms to approx-imate variational problems involving unbalanced optimal transport, and shows how these methods can be used to solve unbalanced transport, unbalanced gradient flows, and to compute unbalanced barycenters.

Generative Adversarial Nets

- Computer ScienceNIPS
- 2014

We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models: a generative model G that captures the data distribution, and a…