Closed Forms for Gaussian Kullback--Leibler Unbalanced Optimal Transport without Coupling Entropy
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910188691259392 |
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| author | Yang, Jiaping Zhang, Yunxin |
| author_facet | Yang, Jiaping Zhang, Yunxin |
| contents | We obtain an explicit solution for the static Kullback--Leibler (KL) unbalanced optimal transport problem between finite non-degenerate Gaussian measures with quadratic cost, two independent positive marginal relaxation parameters, and no entropy penalty on the coupling. The minimizer is a scaled Wasserstein coupling between two adjusted Gaussian marginals and is supported on an affine graph; in entropic Gaussian unbalanced transport, by contrast, the optimal plan is non-degenerate on the product space. The covariance map is the unique positive definite solution of a Riccati equation and admits a principal-square-root representation. Compared with the known equal-penalty Gaussian Hellinger--Kantorovich endpoint, the result treats the asymmetric two-sided Kullback--Leibler relaxation and gives the modified marginals, joint minimizer, value, and a direct quadratic KL-dual certificate. The large-relaxation limit recovers the Gaussian Wasserstein cost for equal masses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02497 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Closed Forms for Gaussian Kullback--Leibler Unbalanced Optimal Transport without Coupling Entropy Yang, Jiaping Zhang, Yunxin Optimization and Control We obtain an explicit solution for the static Kullback--Leibler (KL) unbalanced optimal transport problem between finite non-degenerate Gaussian measures with quadratic cost, two independent positive marginal relaxation parameters, and no entropy penalty on the coupling. The minimizer is a scaled Wasserstein coupling between two adjusted Gaussian marginals and is supported on an affine graph; in entropic Gaussian unbalanced transport, by contrast, the optimal plan is non-degenerate on the product space. The covariance map is the unique positive definite solution of a Riccati equation and admits a principal-square-root representation. Compared with the known equal-penalty Gaussian Hellinger--Kantorovich endpoint, the result treats the asymmetric two-sided Kullback--Leibler relaxation and gives the modified marginals, joint minimizer, value, and a direct quadratic KL-dual certificate. The large-relaxation limit recovers the Gaussian Wasserstein cost for equal masses. |
| title | Closed Forms for Gaussian Kullback--Leibler Unbalanced Optimal Transport without Coupling Entropy |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2605.02497 |