Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences
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This work deals with the asymptotic distribution of both potentials and couplings of entropic regularized optimal transport for compactly supported probabilities in ^d. We first provide the central limit theorem of the Sinkhorn potentials – the solutions of the dual problem – as a Gaussian process in . Then we obtain the weak limits of the couplings – the solutions of the primal problem – evaluated on integrable functions, proving a conjecture of <cit.>. 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: P= Q or H_1: P≠ 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.
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