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Kantorovich Strikes Back! Wasserstein GANs are not Optimal Transport?
Alexander Korotin · Alexander Kolesov · Evgeny Burnaev


Wasserstein Generative Adversarial Networks (WGANs) are the popular generative models built on the theory of Optimal Transport (OT) and the Kantorovich duality. Despite the success of WGANs, it is still unclear how well the underlying OT dual solvers approximate the OT cost (Wasserstein-1 distance, W1) and the OT gradient needed to update the generator. In this paper, we address these questions. We construct 1-Lipschitz functions and use them to build ray monotone transport plans. This strategy yields pairs of continuous benchmark distributions with the analytically known OT plan, OT cost and OT gradient in high-dimensional spaces such as spaces of images. We thoroughly evaluate popular WGAN dual form solvers (gradient penalty, spectral normalization, entropic regularization, etc.) using these benchmark pairs. Even though these solvers perform well in WGANs, none of them faithfully compute W1 in high dimensions. Nevertheless, many provide a meaningful approximation of the OT gradient. These observations suggest that these solvers should not be treated as good estimators of W1 but to some extent they indeed can be used in variational problems requiring the minimization of W1.

Author Information

Alexander Korotin (Skolkovo Institute of Science and Technology)
Alexander Kolesov (The Skolkovo Institute of Science and Technology)
Evgeny Burnaev (Skoltech)

Evgeny Burnaev obtained his MSc in Applied Physics and Mathematics from the Moscow Institute of Physics and Technology in 2006. After successfully defending his PhD thesis in Foundations of Computer Science at the Institute for Information Transmission Problem RAS (IITP RAS) in 2008, Evgeny stayed with the Institute as a head of IITP Data Analysis and Predictive Modeling Lab. Since 2007 Evgeny Burnaev carried out a number of successful industrial projects with Airbus, SAFT, IHI, and Sahara Force India Formula 1 team among others. The corresponding data analysis algorithms, developed by Evgeny Burnaev and his scientific group, formed a core of the algorithmic software library for metamodeling and optimization. Thanks to the developed functionality, engineers can construct fast mathematical approximations to long running computer codes (realizing physical models) based on available data and perform design space exploration for trade-off studies. The software library passed the final Technology Readiness Level 6 certification in Airbus. According to Airbus experts, application of the library “provides the reduction of up to 10% of lead time and cost in several areas of the aircraft design process”. Nowadays a spin-off company Datadvance develops a Software platform for Design Space Exploration with GUI based on this algorithmic core. Since 2016 Evgeny Burnaev works as Associate Professor of Skoltech and manages his research group for Advanced Data Analytics in Science and Engineering For his scientific achievements in the year 2017 Evgeny Burnaev was honored with the Moscow Government Prize for Young Scientists in the category for the Transmission, Storage, Processing and Protection of Information for leading the project “The development of methods for predictive analytics for processing industrial, biomedical and financial data.”

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