{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/no-collision-transportation-maps","title":"No-collision Transportation Maps","arxiv_id":"1912.02317","date":"2019-12-05","proceeding":null,"authors":["Levon Nurbekyan","Alexander Iannantuono","Adam M. Oberman"],"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.","url_abs":"https://arxiv.org/abs/1912.02317v1","url_pdf":"https://arxiv.org/pdf/1912.02317v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"no-collision-transportation-maps","repo_url":"https://github.com/armeehn/no-collision-transportation-maps","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}