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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.21464 |
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| _version_ | 1866912789069561856 |
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| author | Yun, Ho Zemel, Yoav |
| author_facet | Yun, Ho Zemel, Yoav |
| contents | We explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space. In this general setting, the optimal transport map is formally the subgradient of a convex function that is infinite almost everywhere, rendering conventional duality-based variational methods ineffective. We overcome this analytical barrier by exploiting a constructive operator-theoretic approach. Our central result proves that the Kantorovich problem for any pair of Gaussian measures reduces to a Monge problem; that is, an optimal transport map exists in at least one direction between two measures. This reduction allows for a complete characterization with explicit formulas for all optimal (potentially unbounded) Monge transport map and Kantorovich couplings, as well as establishing their uniqueness. Furthermore, we provide a full description of the convex set of geodesics between degenerate measures, revealing a rich geometric structure where the classical McCann interpolants arise as the extreme points. We apply these findings to construct transport maps for Gaussian processes and introduce a novel framework for Wasserstein barycenters based on random Green's operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21464 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gaussian Optimal Transport Beyond Brenier's Theorem Yun, Ho Zemel, Yoav Functional Analysis 49Q22 (Primary) 60B11, 47B25, 53C22, 46N30 (Secondary) We explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space. In this general setting, the optimal transport map is formally the subgradient of a convex function that is infinite almost everywhere, rendering conventional duality-based variational methods ineffective. We overcome this analytical barrier by exploiting a constructive operator-theoretic approach. Our central result proves that the Kantorovich problem for any pair of Gaussian measures reduces to a Monge problem; that is, an optimal transport map exists in at least one direction between two measures. This reduction allows for a complete characterization with explicit formulas for all optimal (potentially unbounded) Monge transport map and Kantorovich couplings, as well as establishing their uniqueness. Furthermore, we provide a full description of the convex set of geodesics between degenerate measures, revealing a rich geometric structure where the classical McCann interpolants arise as the extreme points. We apply these findings to construct transport maps for Gaussian processes and introduce a novel framework for Wasserstein barycenters based on random Green's operators. |
| title | Gaussian Optimal Transport Beyond Brenier's Theorem |
| topic | Functional Analysis 49Q22 (Primary) 60B11, 47B25, 53C22, 46N30 (Secondary) |
| url | https://arxiv.org/abs/2512.21464 |