This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research.
Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.
Optimal transportation.- The Wasserstein space.- Fréchet means in the
Wasserstein space.- Phase variation and Fréchet means.- Construction of
Fréchet means and multicouplings.
Prof. Victor M. Panaretos holds the Chair of Mathematical Statistics at Institute of Mathematics of the EPFL. He completed his undergraduate studies in Mathematics and Statistics in Athens and Dublin. He then received his PhD in 2007 from the University of California at Berkeley, for which he was awarded the Erich Lehmann Award. He is an Elected Member of the International Statistical Institute, a Fellow of the Institute for Mathematical Statistics, the recipient of an ERC Starting Grant Award, and was the 2019 Bernoulli Society Forum Lecturer. He serves on the editorial boards of the Annals of Statistics, Biometrika, and the Journal of the American Statistical Association. Dr. Yoav Zemel graduated with a PhD in Mathematics at the EPFL in 2017 and is currently a postdoctoral researcher in the Statistical Laboratory at the University of Cambridge. He completed his undergraduate studies in Mathematics and Economics at the Hebrew University of Jerusalem, and earned an MSc in Applied Mathematics at the EPFL. He is the recipient of the EPFL excellence scholarship, as well as the Hebrew University Amirim scholarship, Rectors Prize and Deans Prize.
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