Identifiability and Exact Reconstruction of the Optimal Transport Cost on Finite Spaces

[2410.23146] Identifiability and Exact Reconstruction of the Optimal Transport Cost on Finite Spaces Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2410.23146 (math) [Submitted on 30 Oct 2024 ( v1 ), last revised 6 Aug 2026 (this version, v4)] Title: Identifiability and Exact Reconstruction of the Optimal Transport Cost on Finite Spaces Authors: Alberto González-Sanz , Michel Groppe , Axel Munk View PDF HTML (experimental) Abstract: The goal of optimal transport (OT) is to find optimal assignments or matchings between data sets which minimize the total cost for a given cost function. However, sometimes the cost function is unknown but we have access to (parts of) the solution to the OT problem, e.g.\ the OT plan or the value of the objective function. Recovering the cost from such information is called inverse OT and has become recently of certain interest triggered by novel applications, e.g.\ in social science and economics. This raises the issue under which circumstances such cost is identifiable, i.e., it can be uniquely recovered from other OT quantities. In this work we provide sufficient and necessary conditions for the identifiability of the cost function on finite ground spaces. We find that such conditions correspond to the combinatorial structure of the corresponding linear program and discuss its computational complexity and implications for cost estimation in statistical linear models. Comments: 30 pages, 5 figures Subjects: Optimization and Control (math.OC) MSC classes: 90C08, 52B12, 15A29 (Primary) 62J07, 62F12 (Secondary) Cite as: arXiv:2410.23146 [math.OC] (or arXiv:2410.23146v4 [math.OC] for this version) https://doi.org/10.48550/arXiv.2410.23146 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Discrete Applied Mathematics 394 (2026) 352-371 Related DOI : https://doi.org/10.1016/j.dam.2026.08.012 Focus to learn more DOI(s) linking to related resources Submission history From: Michel Groppe [ view email ] [v1] Wed, 30 Oct 2024 16:03:13 UTC (49 KB) [v2] Mon, 1 Sep 2025 08:26:13 UTC (50 KB) [v3] Wed, 27 May 2026 08:33:20 UTC (60 KB) [v4] Thu, 6 Aug 2026 07:19:53 UTC (61 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2024-10 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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