Stability and Oracle Inequalities for Optimal Transport Maps between General Distributions

Optimal transport (OT) provides a powerful framework for comparing and transforming probability distributions, with wide applications in generative modeling, AI4Science and statistical inference. However, existing estimation theory typically requires stringent smoothness conditions on the underlying Brenier potentials and assumes bounded distribution supports, limiting practical applicability. In this paper, we introduce a unified theoretical framework for semi-dual OT map estimation that relaxes both of these restrictions. Building on sieved convex conjugate, our framework has two key contributions: (i) a new map stability bounds that holds without any second-order regularity assumptions on the true Brenier potentials, and (ii) an oracle inequality that cleanly decomposes the estimation error into statistical error, sieved bias, and approximation error. Specifically, our approximation error is measured in the L 1 norm rather than Sobolev norm in the existing results, aligning more naturally with classical approximation theory. Leveraging these tools, we provide statistical error of semi-dual estimators with mild and verifiable conditions on the true OT map. Moreover, we establish the first theoretical guarantee for deep neural network OT map estimator between general distributions, with Tanh network function class as an example.

OT map is both ↵-Hölder smooth and strongly convex. These assumptions exclude many practical OT applications.

In parallel, Gunsilius [25] introduced an alternative analysis using the Poincaré inequality, a fundamental tool in PDEs and functional analysis. While still assuming compact and convex supports, this approach removed the need for Hölder smoothness or strong convexity of the Brenier potential. Building on this, [21] extended the theory to sub-exponential µ, under (↵, a)-convexity and ( , a)smoothness conditions on the Brenier potential. More recently, [20] relaxed the Poincaré inequality assumption to accommodate heavy-tailed µ and ⌫, and eliminated the (↵, a)-convexity condition via a new sieved OT estimator, characterizing how tail thickness of µ and ⌫ influences convergence.

We also note recent advances in plug-in type OT estimators, which directly compute the OT map between estimated distributions [10,17,34,37,5]. In parallel with these theoretical advances, significant efforts have focused on improving the computational efficiency of OT, including entropic regularization [15,1,22,11,19] and sum-of-squares formulations [35,41]. Meanwhile, neuralnetwork-based OT estimators have advanced rapidly, powering large-scale applications such as image generation and translation [33,30,4,31,12].

Despite these advances, a few core regularity assumptions on the true Brenier potential ' 0 and distributions still limit the applicability of dual-type estimators. To clarify these limitations, we decompose the overall estimation error into two components: statistical error and approximation error. The statistical error measures the discrepancy between the estimator and the best possible function in F, reflecting the randomness from finite samples. The approximation error captures the gap between the true Brenier potential and its best approximation within F, since we do not assume that the true Brenier potential ' 0 lies in F. Existing analyses of each error component often rely on restrictive smoothness or convexity assumptions, which we discuss in more below.