Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space

[2507.01709] Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Statistics Theory arXiv:2507.01709 (math) [Submitted on 2 Jul 2025 ( v1 ), last revised 9 Apr 2026 (this version, v2)] Title: Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space Authors: Paul Freulon , Nikitas Georgakis , Victor Panaretos View a PDF of the paper titled Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space, by Paul Freulon and 1 other authors View PDF HTML (experimental) Abstract: The Optimal Transport (OT) problem with squared Euclidean cost consists in finding a coupling between two input measures that maximizes correlation. Consequently, the optimal coupling is often singular with respect to the Lebesgue measure. Regularizing the OT problem with an entropy term yields an approximation called entropic optimal transport. Entropic penalties steer the induced coupling toward a reference measure with desired properties. For instance, when seeking a diffuse coupling, the most popular reference measures are the Lebesgue measure and the product of the two input measures. In this work, we study the case where the reference coupling is not a product, focussing on the Gaussian case as a core paradigm. We establish a reduction of such a regularised OT problem to a matrix optimization problem, enabling us to provide a complete description of the solution, both in terms of the primal variable and the dual variables. Beyond its intrinsic interest, allowing non-product references is essential in dynamic statistical settings. As a key motivation, we address the reconstruction of trajectory dynamics from finitely many time marginals where, unlike product references, Gaussian process references produce transitions that assemble into a coherent continuous-time process. Comments: 39 pages, 5 figures Subjects: Statistics Theory (math.ST) ; Machine Learning (stat.ML) MSC classes: 62H99 ACM classes: G.3 Cite as: arXiv:2507.01709 [math.ST] (or arXiv:2507.01709v2 [math.ST] for this version) https://doi.org/10.48550/arXiv.2507.01709 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Paul Freulon [ view email ] [v1] Wed, 2 Jul 2025 13:40:21 UTC (260 KB) [v2] Thu, 9 Apr 2026 10:20:44 UTC (1,049 KB) Full-text links: Access Paper: View a PDF of the paper titled Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space, by Paul Freulon and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.ST < prev | next > new | recent | 2025-07 Change to browse by: math stat stat.ML stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from

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