Issue |
ESAIM: ProcS
Volume 68, 2020
Journées MAS 2018 - Sampling and Processes
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Page(s) | 1 - 19 | |
DOI | https://doi.org/10.1051/proc/202068001 | |
Published online | 10 June 2020 |
Statistical data analysis in the Wasserstein space*
Institut de Mathématiques de Bordeaux et CNRS (UMR 5251), Université de Bordeaux
This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on the use of Wasserstein distances and tools from optimal transport to analyse such data. In particular, we highlight the benefits of using the notions of barycenter and geodesic PCA in the Wasserstein space for the purpose of learning the principal modes of geometric variation in a dataset. In this setting, we discuss existing works and we present some research perspectives related to the emerging field of statistical optimal transport.
Résumé
Cet article est motivé par des problèmes d’inférence statistique à partir d’un ensemble de données dont les éléments peuvent être modélisés comme des mesures de probabilités tels que des histogrammes ou des nuages de points. Il est proposé de donner un aperçu des contributions récentes en statistique sur l’utilisation des distances de Wasserstein et des outils issus du transport optimal pour l’analyse de telles données. Nous insistons en particulier sur l’intérêt des notions de barycentre et d’ACP géodésique dans l’espace de Wasserstein dans le but d’apprendre les principaux modes de variation géométrique de ce type de données. Dans ce contexte, nous discutons des travaux existants et présentons quelques perspectives de recherche dans le domaine émergent du transport optimal pour la statistique.
Jérémie Bigot is a member of Institut Universitaire de France (IUF), and this work has been carried out with financial support from the IUF. I would like to thank Hervé Cardot and Pierre Calka for proposing to write this review, as well as the two anonymous referees for their comments and suggestions of improvement.
© EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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