Empirical Optimal Transport between Different Measures Adapts to Lower Complexity

Abstract

The empirical optimal transport (OT) cost between two probability measures from random data is a fundamental quantity in transport based data analysis. In this work, we derive novel guarantees for its convergence rate when the involved measures are different, possibly supported on different spaces. Our central observation is that the statistical performance of the empirical OT cost is determined by the less complex measure, a phenomenon we refer to as lower complexity adaptation of empirical OT. For instance, under Lipschitz ground costs, we find that the empirical OT cost based on n observations converges at least with rate n −1 d to the population quantity if one of the two measures is concentrated on a d-dimensional manifold, while the other can be arbitrary. For semi-concave ground costs, we show that the upper bound for the rate improves to n −2 d . Similarly, our theory establishes the general convergence rate n −1 2 for semi-discrete OT. All of these results are valid in the two-sample case as well, meaning that the convergence rate is still governed by the simpler of the two measures. On a conceptual level, our findings therefore suggest that the curse of dimensionality only affects the estimation of the OT cost when both measures exhibit a high intrinsic dimension. Our proofs are based on the dual formulation of OT as a maximization over a suitable function class F c and the observation that the c-transform of F c under bounded costs has the same uniform metric entropy as F c itself.

Unbalanced Optimal Transport, from Theory to Numerics

“…The rate can be refined by replacing the dimension d of the ambient space (e.g. X = R d ) by the intrinsic dimension (such as the Hausdorff dimension) of the input measures [177], and by the lowest intrinsic dimension of the two considered inputs [84]. Considering densities smooth to the order p reduces the complexity to O(N −p/d ) [178], but the estimators (α N , β N ) rely on wavelet decomposition or kernel estimation, which complicates the computation of OT(α N , β N ).…”

Unbalanced Kantorovich-Rubinstein distance and barycenter for finitely supported measures: A statistical perspective

“…This can be used to estimate OT and its plan. In general, this yields a statistically efficient estimator [Dudley, 1969, Ajtai et al, 1984, Sommerfeld et al, 2019, Weed and Bach, 2019, Hütter and Rigollet, 2021, Manole et al, 2021, Hundrieser et al, 2022 see, however, Niles- Weed and Berthet [2022] for certain improvements. For OT barycenters based on empirical measures significantly less is known, though recently some progress has been made in the context of finitely supported measures [Heinemann et al, 2022b].…”

Statistical and Structural Aspects of Unbalanced Optimal Transport Barycenters

“…2 In the naive upper bound based on the triangle inequality the potentially slower rate of the two measures dominates the speed of convergence. However, there are scenarios where the opposite holds true [Hundrieser et al, 2022]. This phenomenon is known as lower complexity adaptation.…”

“…Another interesting question is whether considering the dual formulation of the (p, C)barycenter problem directly and then using tools from empirical process theory would allow to recover a phenomenon similar to the lower complexity adaptation [Hundrieser et al, 2022]. The constants in empirical deviation bounds in this chapter depend on the average of the constants arising from J individual estimation problem.…”