Smooth Optimal Transport Estimation beyond Dimensionality

Definitions and mathematical statement

For a general definition of transport cost, let $(X,\mathcal A)$ and $(Y,\mathcal B)$ be measurable spaces, let $\mu$ and $\nu$ be probability measures on them, and let $c:X\times Y\to[0,\infty]$ be measurable. Their set of couplings and associated optimal transport value are \[ \Pi(\mu,\nu)=\{\pi\text{ a probability measure on }X\times Y:\pi(A\times Y)=\mu(A),\ \pi(X\times B)=\nu(B)\text{ for all }A\in\mathcal A,\ B\in\mathcal B\}, \qquad \operatorname{OT}_c(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int_{X\times Y}c(x,y)\,\mathrm d\pi(x,y). \] This is a general definitional framework; the supplied record does not select these spaces or this class of costs as its intended setting. An optimal coupling is a coupling attaining the infimum, when such a coupling exists. The record does not specify whether the quantity to be estimated is the transport value, an optimal coupling, or an optimal transport map.

Source question. Can strong smoothness assumptions on the probability distributions be used to overcome the curse of dimensionality in statistical estimation of optimal transport?

Here the curse of dimensionality refers to deterioration of estimation performance as the dimension increases. A precise quantified conjecture requires specification of the dimension-dependent distribution classes, the meaning and strength of smoothness, the transport cost, the estimand, the observation model, and the estimation loss. None of these is fixed in the supplied statement. In particular, a convergence exponent independent of dimension and a bound whose constants are also independent of dimension are distinct possible targets. The supplied record does not determine which uniformity or quantifiers are intended, so no particular rate bound or universal assertion is imposed here.

The supplied dated assessment labels the problem partially solved: it reports dimension-free statistical and computational rates under sufficient smoothness, while describing a general removal of dimensional dependence and dimension-robust constants as unresolved. This assessment is supplied curation, not an independently verified mathematical conclusion.

Short English statement

The question asks whether sufficiently smooth probability distributions allow optimal transport to be estimated without the usual deterioration in high dimension. The source leaves the transport setting, smoothness class, statistical model, loss, and meaning of dimension independence unspecified. Its supplied partially solved label and literature assessment require independent review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004631$, title Smooth Optimal Transport Estimation beyond Dimensionality; source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year $2021$ and names no individual proposer.
• Supplied report citation: Applications of Optimal Transportation in the Natural Sciences (2022), Oberwolfach Reports, pp. 507--587, DOI 10.4171/owr/2021/10. https://doi.org/10.4171/owr/2021/10. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: "Vacher et al., dimension-free smooth OT." https://proceedings.mlr.press/v134/vacher21a.html.
• Supplied literature reference: "Sparse regularized OT." https://arxiv.org/abs/2505.04721.
• Supplied literature reference: "Oberwolfach report." https://doi.org/10.4171/owr/2021/10.
• Literature checked at: 2026-08-22. This is the exact date of supplied curation, not independent verification. Its assessment reports known dimension-free statistical and computational rates for sufficiently smooth optimal transport estimation, and describes general removal of the curse of dimensionality and dimension-robust constants as unresolved. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review for self-containment and preservation of the mathematical target. That supplied review does not resolve the unspecified mathematical setting identified here.