Statistical and Structural Aspects of Unbalanced Optimal Transport Barycenters

Abstract

find the OT map by picking the most cost-efficient pairwise assignment between the points. However, when considering part b), this approach fails. Here, one measure still has eight support points with mass 1/8, but the other one has seven support points with mass 1/7, respectively. It is immediately clear that in this scenario, the set of transport maps is empty. Though, of course it is still possible to transport mass between the two measures. The key ingredient which is missing is the ability to split mass, i.e. for a given location x ∈ supp(µ 1 ), to be allowed to send parts of its mass to different locations in supp(µ 2 ). This does not provide an OT map, however, it leads to finding an Intuitively, the curve (ν t ) t∈[0,1] describes a locally shortest path between µ 1 and µ 2 with respect to the geometry of the metric space (P(Y), W p ). A specific time point ν t can be understood as a W p -interpolation of µ 1 and µ 2 with weight 1 − t for µ 1 and t for µ 2 . Recall the example OT plan from Figure 1.2(b). Following the trajectory of the corresponding geodesic (see Figure 1.3(b)), it can be seen that the ellipse is being squeezed together, while it simultaneously moves from bottom-left to the top-right.This can be seen easily by noting that since ν is a geodesic it holds W p p (µ 1 , ν 0.5 ) + W p p (µ 2 , ν 0.5 ) = 2 1−p W p p (µ 1 , µ 2 ) (1.7)