Sharpness of the exponent in general stability bounds for nonsmooth OT maps

Determine whether the exponent 1/3 in the one-sample quantitative stability inequality for optimal transport maps—namely, the bound ∥T̂ − T₀∥_{L²(P)}² ≲ W₂^{1/3}(Q̂, Q) under mild regularity on the source distribution P and no smoothness assumptions on T₀—is optimal. Specifically, ascertain the best possible exponent α in ∥T̂ − T₀∥_{L²(P)}² ≲ W₂^{α}(Q̂, Q) within the assumptions of the stated stability framework, by deriving matching lower bounds or improved upper bounds.

Background

A general quantitative stability bound shows that, without smoothness assumptions on the optimal transport map T₀ and under mild regularity on the source distribution P, the L²(P) error between the perturbed map T̂ and T₀ scales with the Wasserstein perturbation W₂(Q̂, Q) to the power 1/3. This provides dimension-dependent risk bounds for OT map estimation in low-regularity regimes but may be conservative.

Clarifying the optimal exponent in this stability relationship would sharpen risk bounds for nonsmooth OT maps and guide minimax rate analyses when smoothness or strong convexity assumptions fail. Establishing tightness would also inform the design and evaluation of estimators in low-regularity settings.

References

Although the sharpness of this exponent remains an open question, Theorem~\ref{thm:stability_general} already leads to upper bounds on the minimax estimation risk of nonsmooth optimal transport maps in the one-sample setting.

— Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives  (2506.19025 - Balakrishnan et al., 23 Jun 2025) in Section 4.6 (Beyond Smooth OT Maps)

The example given in Proposition \ref{prop: sharp exponent stability} is for fixed source, perturbing only the target. Thus we do not rule out the possibility of a better exponent under perturbations of the source, with fixed target.

— Quantitative Stability, Coercivity and Uniqueness of Optimal Transport Plans  (2609.35574 - Ford, 28 Sep 2026) in Introduction, immediately following the statement of the bi-marginal plan stability theorem

It is likely that Theorem \ref{t:p>2} generalizes to optimal transport with respect to $p$-cost on $Rd$ for $p>1$: one would obtain for $q>p$ and $R>0$ a quantitative stability estimate, uniform over $\mathcal{M}_{q,R}$, and with dimension-free exponent (depending only on $p,q$). But proving this would probably require first generalizing to $p$-costs, which is not yet done.

— Embedding large subsets of Wasserstein spaces into Banach spaces  (2610.12029 - Letrouit et al., 8 Oct 2026) in Remark \ref{r:qcostnotdone}, in the proof section for the linearized optimal transport embedding