{"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/families-of-costs-with-zero-and-nonnegative","title":"Families of costs with zero and nonnegative MTW tensor in optimal transport","arxiv_id":"2401.00953","date":"2024-01-01","proceeding":null,"authors":["Du Nguyen"],"abstract":"We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on $\\mathbb{R}^n$ with a cost function of form $\\mathsf{c}(x, y) = \\mathsf{u}(x^{\\mathfrak{t}}y)$, where $\\mathsf{u}$ is a scalar function with inverse $\\mathsf{s}$, $x^{\\ft}y$ is a nondegenerate bilinear pairing of vectors $x, y$ belonging to an open subset of $\\mathbb{R}^n$. The condition that the MTW-tensor vanishes on null vectors under the Kim-McCann metric is a fourth-order nonlinear ODE, which could be reduced to a linear ODE of the form $\\mathsf{s}^{(2)} - S\\mathsf{s}^{(1)} + P\\mathsf{s} = 0$ with constant coefficients $P$ and $S$. The resulting inverse functions include {\\it Lambert} and {\\it generalized inverse hyperbolic\\slash trigonometric} functions. The square Euclidean metric and $\\log$-type costs are equivalent to instances of these solutions. The optimal map for the family is also explicit. For cost functions of a similar form on a hyperboloid model of the hyperbolic space and unit sphere, we also express this tensor in terms of algebraic expressions in derivatives of $\\mathsf{s}$ using the Gauss-Codazzi equation, obtaining new families of strictly regular costs for these manifolds, including new families of {\\it power function costs}. We analyze the $\\sinh$-type hyperbolic cost, providing examples of $\\mathsf{c}$-convex functions and divergence.","url_abs":"https://arxiv.org/abs/2401.00953v1","url_pdf":"https://arxiv.org/pdf/2401.00953v1.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":"abstracts"},"code_links":[{"paper_slug":"families-of-costs-with-zero-and-nonnegative","repo_url":"https://github.com/dnguyend/regularmtw","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":null}],"tasks":[{"task_slug":"form","task_name":"Form"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}