Asymptotics of Entropy-Regularized Optimal Transport via Chaos Decomposition

11/17/2020
by   Zaid Harchaoui, et al.
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Consider the problem of estimating the optimal coupling (i.e., matching) between N i.i.d. data points sampled from two densities ρ_0 and ρ_1 in ℝ^d. The cost of transport is an arbitrary continuous function that satisfies suitable growth and integrability assumptions. For both computational efficiency and smoothness, often a regularization term using entropy is added to this discrete problem. We introduce a modification of the commonly used discrete entropic regularization (Cuturi '13) such that the optimal coupling for the regularized problem can be thought of as the static Schrödinger bridge with N particles. This paper is on the asymptotic properties of this discrete Schrödinger bridge as N tends to infinity. We show that it converges to the continuum Schrödinger bridge and derive the first two error terms of orders N^-1/2 and N^-1, respectively. This gives us functional CLT, including for the cost of transport, and second order Gaussian chaos limits when the limiting Gaussian variance is zero, extending similar recent results derived for finite state spaces and the quadratic cost. The proofs are based on a novel chaos decomposition of the discrete Schrödinger bridge by polynomial functions of the pair of empirical distributions as a first and second order Taylor approximations in the space of measures. This is achieved by extending the Hoeffding decomposition from the classical theory of U-statistics. The kernels corresponding to the first and second order chaoses are given by Markov operators which have natural interpretations in the Sinkhorn algorithm.

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