Robustness Aspects of Optimal Transportation

E. Ronchetti

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Research Center for Statistics and Geneva School of Economics and Management, University of Geneva, Geneva, Switzerland [Elvezio.Ronchetti@unige.ch]

Keywords: Contamination – Optimal transportation map – Penalization – Wasserstein distance

Optimal transportation (OT) goes back to the seminal work by Monge in 1781, who considered the problem of finding the optimal way to move given piles of sand to fill up given holes of the same total volume. Monge’s problem was revisited by Kantorovich in 1942, in relation to the economic problem of optimal allocation of resources. Nowadays OT is a flourishing area of research and applications in many fields, including mathematics, statistics, economics, computer vision, imaging, and machine learning. After a short introduction into the basic concepts of OT, we give an overview of the stability issues related to the use of these techniques by providing a discussion of several results available in the literature. Several open problems still remain in order to develop a theory of robust OT.