Appendix 1 Methods details

This mapping is done through what's referred to as a transport plan. Pi,jCi,j γH(P) subject to P Rt t+,PT1t = 1t,P1t = 1t, (6) where Pi,j is the transport plan and Ci,j is the ground metric that measures the distance between point i in the source and j in the target. A common expression for the ground metric is to use either the L2 distance for the Mahalanobis metric which can be defined in terms of a matrix L as Ci,j = L(ui) L(vj) 22. The final term introduces some regularization into our objective by adding a entropy term on the transport plan. This will induce some smoothness and wiggle room in the solution of our objective.