Abstract
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.
Highlights
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
It has been widely recognized that a significant gain in statistical inference can be achieved by the use of non-Euclidean distances to better capture the geometry of the sets to which the data truly belong
This leads to statistical inference problems from multiple points clouds such as those displayed in Figure 2 that can be modeled as discrete probability measures supported on Rd
Summary
In many fields of interest (e.g. in signal and image processing or bio-informatics), one records data in the form of high-dimensional vectors or matrices. The development of this technology leads to datasets made of multiple measurements (e.g. up to 18) of millions of individuals cells from different subjects This leads to statistical inference problems from multiple points clouds such as those displayed in Figure 2 that can be modeled as discrete probability measures supported on Rd. Techniques based on optimal transport for data science have recently received an increasing interest in mathematical and computational statistics [8,10,11,12,13,14,15,28,29,44,45,47,49,51,54,58,62,66,67], machine learning [4, 6,22,23,32,36,37,38,40,55,57], image processing and computer vision [7,17,24,30,31,41,52,59,60] or computational biology [56]. Throughout the paper, we discuss some research perspectives and open problems related to the statistical aspects of barycenters and GPCA in the Wasserstein space
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