GT Transport optimal - EDP - Machine learning
Properties of Discrete Sliced Wasserstein Losses
June 2024
Intervenant : Eloi Tanguy
Institution : MAP5 (U. Paris-Cité)
Heure : 15h00 - 15h50
Lieu : 3L8

 The Sliced Wasserstein (SW) distance has become a common alternative to the Wasserstein distance for the comparison of probability measures. Widespread applications include image processing, domain adaptation and generative modelling, where it is typical to optimise some parameters in order to minimise SW, which in practice serves as a loss function between discrete probability measures. These optimisation problems all bear the same sub-problem, which is minimising the SW distance between two uniform discrete measures with the same amount of points as a function of the support (i.e. a matrix of data points) of one of the measures. We study the regularity and optimisation properties of this energy, as well as its Monte-Carlo approximation (estimating the expectation in SW using projection samples), as well as asymptotical and non-asymptotical statistical properties of the Monte-Carlo approximation. Finally, we show that in a certain sense, Stochastic Gradient Descent methods minimising these energies converge towards (Clarke) critical points, with an extension to Generative Neural Network training.

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