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Vivien Seguy

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JMLR Journal 2021 Journal Article

POT: Python Optimal Transport

  • Rémi Flamary
  • Nicolas Courty
  • Alexandre Gramfort
  • Mokhtar Z. Alaya
  • Aurélie Boisbunon
  • Stanislas Chambon
  • Laetitia Chapel
  • Adrien Corenflos

Optimal transport has recently been reintroduced to the machine learning community thanks in part to novel efficient optimization procedures allowing for medium to large scale applications. We propose a Python toolbox that implements several key optimal transport ideas for the machine learning community. The toolbox contains implementations of a number of founding works of OT for machine learning such as Sinkhorn algorithm and Wasserstein barycenters, but also provides generic solvers that can be used for conducting novel fundamental research. This toolbox, named POT for Python Optimal Transport, is open source with an MIT license. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2021. ( edit, beta )

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NeurIPS Conference 2015 Conference Paper

Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric

  • Vivien Seguy
  • Marco Cuturi

We consider in this work the space of probability measures $P(X)$ on a Hilbert space $X$ endowed with the 2-Wasserstein metric. Given a finite family of probability measures in $P(X)$, we propose an iterative approach to compute geodesic principal components that summarize efficiently that dataset. The 2-Wasserstein metric provides $P(X)$ with a Riemannian structure and associated concepts (Fr\'echet mean, geodesics, tangent vectors) which prove crucial to follow the intuitive approach laid out by standard principal component analysis. To make our approach feasible, we propose to use an alternative parameterization of geodesics proposed by \citet[\S 9. 2]{ambrosio2006gradient}. These \textit{generalized} geodesics are parameterized with two velocity fields defined on the support of the Wasserstein mean of the data, each pointing towards an ending point of the generalized geodesic. The resulting optimization problem of finding principal components is solved by adapting a projected gradient descend method. Experiment results show the ability of the computed principal components to capture axes of variability on histograms and probability measures data.

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