Langevin Diffusion Approximation to Same Marginal Schrödinger Bridge
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918425878593536 |
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| author | Agarwal, Medha Harchaoui, Zaid Mulcahy, Garrett Pal, Soumik |
| author_facet | Agarwal, Medha Harchaoui, Zaid Mulcahy, Garrett Pal, Soumik |
| contents | We introduce a novel approximation to the same marginal Schrödinger bridge using the Langevin diffusion. As $\varepsilon \downarrow 0$, it is known that the barycentric projection (also known as the entropic Brenier map) of the Schrödinger bridge converges to the Brenier map, which is the identity. Our diffusion approximation is leveraged to show that, under suitable assumptions, the difference between the two is $\varepsilon$ times the gradient of the marginal log density (i.e., the score function), in $\mathbf{L}^2$. More generally, we show that the family of Markov operators, indexed by $\varepsilon > 0$, derived from integrating test functions against the conditional density of the static Schrödinger bridge at temperature $\varepsilon$, admits a derivative at $\varepsilon=0$ given by the generator of the Langevin semigroup. Hence, these operators satisfy an approximate semigroup property at low temperatures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Langevin Diffusion Approximation to Same Marginal Schrödinger Bridge Agarwal, Medha Harchaoui, Zaid Mulcahy, Garrett Pal, Soumik Probability Machine Learning 49N99, 49Q22, 60J60 We introduce a novel approximation to the same marginal Schrödinger bridge using the Langevin diffusion. As $\varepsilon \downarrow 0$, it is known that the barycentric projection (also known as the entropic Brenier map) of the Schrödinger bridge converges to the Brenier map, which is the identity. Our diffusion approximation is leveraged to show that, under suitable assumptions, the difference between the two is $\varepsilon$ times the gradient of the marginal log density (i.e., the score function), in $\mathbf{L}^2$. More generally, we show that the family of Markov operators, indexed by $\varepsilon > 0$, derived from integrating test functions against the conditional density of the static Schrödinger bridge at temperature $\varepsilon$, admits a derivative at $\varepsilon=0$ given by the generator of the Langevin semigroup. Hence, these operators satisfy an approximate semigroup property at low temperatures. |
| title | Langevin Diffusion Approximation to Same Marginal Schrödinger Bridge |
| topic | Probability Machine Learning 49N99, 49Q22, 60J60 |
| url | https://arxiv.org/abs/2505.07647 |