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The empirical optimal transport (OT) cost between two probability measuresfrom random data is a fundamental quantity in transport based data analysis. Inthis work, we derive novel guarantees for its convergence rate when theinvolved measures are different, possibly supported on different spaces. Ourcentral observation is that the statistical performance of the empirical OTcost is determined by the less complex measure, a phenomenon we refer to aslower complexity adaptation of empirical OT. For instance, under Lipschitzground costs, we find that the empirical OT cost based on $n$ observationsconverges at least with rate $n^{-1/d}$ to the population quantity if one ofthe two measures is concentrated on a $d$-dimensional manifold, while the othercan be arbitrary. For semi-concave ground costs, we show that the upper boundfor the rate improves to $n^{-2/d}$. Similarly, our theory establishes thegeneral convergence rate $n^{-1/2}$ for semi-discrete OT. All of these resultsare valid in the two-sample case as well, meaning that the convergence rate isstill governed by the simpler of the two measures. On a conceptual level, ourfindings therefore suggest that the curse of dimensionality only affects theestimation of the OT cost when both measures exhibit a high intrinsicdimension. Our proofs are based on the dual formulation of OT as a maximizationover a suitable function class $\mathcal{F}_c$ and the observation that the$c$-transform of $\mathcal{F}_c$ under bounded costs has the same uniformmetric entropy as $\mathcal{F}_c$ itself.

Empirical Optimal Transport between Different Measures Adapts to Lower Complexity