The Riemannian geometry of Sinkhorn divergences
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912593280499712 |
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| author | Lavenant, Hugo Luckhardt, Jonas Mordant, Gilles Schmitzer, Bernhard Tamanini, Luca |
| author_facet | Lavenant, Hugo Luckhardt, Jonas Mordant, Gilles Schmitzer, Bernhard Tamanini, Luca |
| contents | We propose a new metric between probability measures on a compact metric space that mirrors the Riemannian manifold-like structure of quadratic optimal transport but includes entropic regularization. Its metric tensor is given by the Hessian of the Sinkhorn divergence, a debiased variant of entropic optimal transport. We precisely identify the tangent space it induces, which turns out to be related to a Reproducing Kernel Hilbert Space (RKHS). As usual in Riemannian geometry, the distance is built by looking for shortest paths. We prove that our distance is geodesic, metrizes the weak-star topology, and is equivalent to a RKHS norm. Still it retains the geometric flavor of optimal transport: as a paradigmatic example, translations are geodesics for the quadratic cost on $\mathbb{R}^d$. We also show two negative results on the Sinkhorn divergence that may be of independent interest: that it is not jointly convex, and that its square root is not a distance because it fails to satisfy the triangle inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Riemannian geometry of Sinkhorn divergences Lavenant, Hugo Luckhardt, Jonas Mordant, Gilles Schmitzer, Bernhard Tamanini, Luca Optimization and Control Metric Geometry 49Q22 (Primary) 51F99 (Secondary) We propose a new metric between probability measures on a compact metric space that mirrors the Riemannian manifold-like structure of quadratic optimal transport but includes entropic regularization. Its metric tensor is given by the Hessian of the Sinkhorn divergence, a debiased variant of entropic optimal transport. We precisely identify the tangent space it induces, which turns out to be related to a Reproducing Kernel Hilbert Space (RKHS). As usual in Riemannian geometry, the distance is built by looking for shortest paths. We prove that our distance is geodesic, metrizes the weak-star topology, and is equivalent to a RKHS norm. Still it retains the geometric flavor of optimal transport: as a paradigmatic example, translations are geodesics for the quadratic cost on $\mathbb{R}^d$. We also show two negative results on the Sinkhorn divergence that may be of independent interest: that it is not jointly convex, and that its square root is not a distance because it fails to satisfy the triangle inequality. |
| title | The Riemannian geometry of Sinkhorn divergences |
| topic | Optimization and Control Metric Geometry 49Q22 (Primary) 51F99 (Secondary) |
| url | https://arxiv.org/abs/2405.04987 |