Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916275187351552 |
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| author | Gonzalez-Sanz, Alberto Loubes, Jean-Michel Niles-Weed, Jonathan |
| author_facet | Gonzalez-Sanz, Alberto Loubes, Jean-Michel Niles-Weed, Jonathan |
| contents | This work deals with the asymptotic distribution of both potentials and couplings of entropic regularized optimal transport for compactly supported probabilities in $\R^d$. We first provide the central limit theorem of the Sinkhorn potentials -- the solutions of the dual problem -- as a Gaussian process in $\Cs$. Then we obtain the weak limits of the couplings -- the solutions of the primal problem -- evaluated on integrable functions, proving a conjecture of \cite{ChaosDecom}. In both cases, their limit is a real Gaussian random variable. Finally we consider the weak limit of the entropic Sinkhorn divergence under both assumptions $H_0:\ {\rm P}={\rm Q}$ or $H_1:\ {\rm P}\neq{\rm Q}$. Under $H_0$ the limit is a quadratic form applied to a Gaussian process in a Sobolev space, while under $H_1$, the limit is Gaussian. We provide also a different characterisation of the limit under $H_0$ in terms of an infinite sum of an i.i.d. sequence of standard Gaussian random variables. Such results enable statistical inference based on entropic regularized optimal transport. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_07427 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences Gonzalez-Sanz, Alberto Loubes, Jean-Michel Niles-Weed, Jonathan Probability Statistics Theory This work deals with the asymptotic distribution of both potentials and couplings of entropic regularized optimal transport for compactly supported probabilities in $\R^d$. We first provide the central limit theorem of the Sinkhorn potentials -- the solutions of the dual problem -- as a Gaussian process in $\Cs$. Then we obtain the weak limits of the couplings -- the solutions of the primal problem -- evaluated on integrable functions, proving a conjecture of \cite{ChaosDecom}. In both cases, their limit is a real Gaussian random variable. Finally we consider the weak limit of the entropic Sinkhorn divergence under both assumptions $H_0:\ {\rm P}={\rm Q}$ or $H_1:\ {\rm P}\neq{\rm Q}$. Under $H_0$ the limit is a quadratic form applied to a Gaussian process in a Sobolev space, while under $H_1$, the limit is Gaussian. We provide also a different characterisation of the limit under $H_0$ in terms of an infinite sum of an i.i.d. sequence of standard Gaussian random variables. Such results enable statistical inference based on entropic regularized optimal transport. |
| title | Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2207.07427 |