scholarly journals Characterization of barycenters in the Wasserstein space by averaging optimal transport maps

2018 ◽  
Vol 22 ◽  
pp. 35-57 ◽  
Author(s):  
Jérémie Bigot ◽  
Thierry Klein
Keyword(s):  
Optimal Transport ◽  
Probability Measures ◽  
Reference Measure ◽  
Precise Characterization ◽  
Fixed Reference ◽  
Random Probability ◽  
Wasserstein Space ◽  
Transport Maps ◽  
Random Probability Measures

This paper is concerned by the study of barycenters for random probability measures in the Wasserstein space. Using a duality argument, we give a precise characterization of the population barycenter for various parametric classes of random probability measures with compact support. In particular, we make a connection between averaging in the Wasserstein space as introduced in Agueh and Carlier [SIAM J. Math. Anal. 43 (2011) 904–924], and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach in statistics for the analysis of deformable models in signal and image processing. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered.

scholarly journals Statistical data analysis in the Wasserstein space

2020 ◽  
Vol 68 ◽  
pp. 1-19
Author(s):  
Jérémie Bigot
Keyword(s):  
Optimal Transport ◽  
Point Clouds ◽  
Data Set ◽  
Inference Problems ◽  
Research Perspectives ◽  
Random Probability ◽  
Wasserstein Space ◽  
Geometric Variation ◽  
Random Probability Measures ◽  
Wasserstein Distances

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.

scholarly journals On the multimodality of random probability measures

2007 ◽  
Vol 2 (1) ◽  
pp. 213-219 ◽  
Author(s):  
George Kokolakis ◽  
George Kouvaras
Keyword(s):  
Probability Measures ◽  
Random Probability ◽  
Random Probability Measures

scholarly journals A GEOMETRIC STUDY OF WASSERSTEIN SPACES: HADAMARD SPACES

2012 ◽  
Vol 04 (04) ◽  
pp. 515-542 ◽  
Author(s):  
JÉRÔME BERTRAND ◽  
BENOÎT KLOECKNER
Keyword(s):  
Large Scale ◽  
Euclidean Plane ◽  
Optimal Transport ◽  
Probability Measures ◽  
Hadamard Space ◽  
Geodesic Boundary ◽  
Wasserstein Space ◽  
Scale Properties ◽  
Positively Curved ◽  
Geometric Study

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space [Formula: see text]. In this paper we investigate the geometry of [Formula: see text] when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, [Formula: see text] is not non-positively curved, our results show that [Formula: see text] have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for [Formula: see text] that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in [Formula: see text].

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