A Unifying Approach to Distributional Limits for Empirical Optimal Transport
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We provide a unifying approach to central limit type theorems for empirical optimal transport (OT). In general, the limit distributions are characterized as suprema of Gaussian processes. We explicitly characterize when the limit distribution is centered normal or degenerates to a Dirac measure. Moreover, in contrast to recent contributions on distributional limit laws for empirical OT on Euclidean spaces which require centering around its expectation, the distributional limits obtained here are centered around the population quantity, which is well-suited for statistical applications. At the heart of our theory is Kantorovich duality representing OT as a supremum over a function class ℱ_c for an underlying sufficiently regular cost function c. In this regard, OT is considered as a functional defined on ℓ^∞(ℱ_c) the Banach space of bounded functionals from ℱ_c to ℝ and equipped with uniform norm. We prove the OT functional to be Hadamard directional differentiable and conclude distributional convergence via a functional delta method that necessitates weak convergence of an underlying empirical process in ℓ^∞(ℱ_c). The latter can be dealt with empirical process theory and requires ℱ_c to be a Donsker class. We give sufficient conditions depending on the dimension of the ground space, the underlying cost function and the probability measures under consideration to guarantee the Donsker property. Overall, our approach reveals a noteworthy trade-off inherent in central limit theorems for empirical OT: Kantorovich duality requires ℱ_c to be sufficiently rich, while the empirical processes only converges weakly if ℱ_c is not too complex.
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