Section: New Results

Unbalanced Optimal Transport

G. Carlier, F-X. Vialard, B. Schmitzer, L. Chizat Classical optimal transport theory and algorithms assume that the input measures are normalized, i.e. that their total mass is 1. This is an important limitation for many problems in imaging sciences and machine learning, where input data are typically not normalized, and where one should enables local creation or destruction of mass. Handling such “unbalanced” transportation problem is also relevant for applications in biological modeling, for instance to take into account cellular growth through optimal transport gradient flows.

Recently, several researchers of MOKAPLAN made important progress on this problem, by deriving a general framework extending optimal transport to this “unbalanced” setting. In [38] we derived a dynamic optimal transport formulation that enables a source term in the initial formulation of Benamou and Brenier  [55] . We proved that it defines a distance on positive measures, enjoy many important properties (dual formulation) and can be computed using fast first order convex optimization methods. We then provided in [39] an even larger class of “unbalanced” optimal transport optimization problems, that are obtained via a static formulation, and show that one can recovers the dynamic formulation in some specific cases. Similar models were derived independently and at the same time by two other international teams  [143] , [137] , which shows the timeliness of our research. We believe these new theoretical and numerical findings will have a strong impact on the developpement of optimal transport methods in imaging sciences and machine learning.