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.