Abstract: Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by these distances on the space of probability measures reveals interesting differences. In particular, we can establish surprising approximate global convergence guarantees for the 1-Wasserstein distance, even when the parametric generator has a nonconvex parametrization.

@incollection{bottou-geometry-2018,
  author = {Bottou, L{\'e}on and Arjovsky, Martin and Lopez-Paz, David and Oquab, Maxime},
  title = {Geometrical Insights for Implicit Generative Modeling},
  booktitle = {Braverman Readings in Machine Learning: Key Ideas from Inception to Current State},
  editor = {Lev Rozonoer, Boris Mirkin, Ilya Muchnik},
  series = {LNAI Vol. 11100},
  publisher = {Springer},
  year = {2018},
  pages = {229--268},
  url = {http://leon.bottou.org/papers/bottou-geometry-2018},
}