Empirical optimal transport potentials: fast rates and a functional central limit theorem

What the math gives to ML

The paper isolates a useful asymmetry in quadratic optimal transport: under strong convexity and regularity, a potential can be controlled in an invariant L1 norm by a weak discrepancy between the distributions induced by its gradient, whereas direct transport-map error is governed by the slower Wasserstein rate. The normalization by a median removes the otherwise unavoidable additive-constant ambiguity. This suggests training convex neural potentials through weak pushforward matching while explicitly enforcing strong convexity and median normalization, rather than relying only on noisy pointwise map regression or W2 objectives. The strongest initial test is an input-convex neural network used as an OT map generator and evaluated for potential stability, sample efficiency, and objective convergence.