No-collision Transportation Maps

Abstract: Transportation maps between probability measures are critical objects in numerous areas of mathematics and applications such as PDE, fluid mechanics, geometry, machine learning, computer science, and economics. Given a pair of source and target measures, one searches for a map that has suitable properties and transports the source measure to the target one. Here, we study maps that possess the \textit{no-collision} property; that is, particles simultaneously traveling from sources to targets in a unit time with uniform velocities do not collide. These maps are particularly relevant for applications in swarm control problems. We characterize these no-collision maps in terms of \textit{half-space preserving} property and establish a direct connection between these maps and \textit{binary-space-partitioning (BSP) tree} structures. Based on this characterization, we provide explicit BSP algorithms, of cost $O(n \log n)$, to construct no-collision maps. Moreover, interpreting these maps as approximations of optimal transportation maps, we find that they succeed in computing nearly optimal maps for $q$-Wasserstein metric ($q=1,2$). In some cases, our maps yield costs that are just a few percent off from being optimal.