Quantitative Uniqueness of Kantorovich Potentials

Abstract: This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal measure's support is rectifiably connected (path-connected by rectifiable paths), the optimal dual potentials are unique up to a constant. This represents the first uniqueness result applicable even when both marginal measures are concentrated on lower-dimensional subsets of the ambient space, and also applies in cases where optimal potentials are nowhere differentiable on the supports of the marginals. On the quantitative side, we control the diameter of the set of optimal dual potentials by the Hausdorff distance between the support of one of the marginal measures and a regular connected set. In this way, we quantify the extent to which optimisers are almost unique when the support of one marginal measure is almost connected. This is a consequence of a novel characterisation of the set of optimal dual potentials as the intersection of an explicit family of half-spaces.

• The paper introduces a qualitative uniqueness theorem demonstrating that if one marginal has rectifiably connected support, then the dual Kantorovich potentials are unique up to a constant.
• The paper develops a quantitative framework by bounding the L^p-diameter of Kantorovich potentials using the Hausdorff distance, yielding sharp stability estimates in various regimes.
• It characterizes the set of optimal potentials as an intersection of half-spaces, extending classical uniqueness results to lower-dimensional and empirical settings.

Quantitative Uniqueness of Kantorovich Potentials: Analytical Summary

Motivation and Problem Statement

The paper "Quantitative Uniqueness of Kantorovich Potentials" (2603.29595) addresses the fundamental problem of uniqueness in the dual formulation of optimal transport. It considers two central questions:

• Under what conditions are Kantorovich potentials unique up to a constant (qualitative uniqueness)?
• Under what conditions are Kantorovich potentials nearly unique, i.e., can one quantify the diameter of the set of dual optimizers (quantitative uniqueness)?

Both questions are crucial for the stability and well-posedness of the dual optimal transport problem, especially when marginal measures are supported on lower-dimensional or disconnected sets.

Main Qualitative Uniqueness Results

The paper delivers a significant qualitative uniqueness theorem: if one marginal measure has rectifiably connected support, then the dual potentials are unique up to a constant. This result applies even when both marginals are singular or concentrated on lower-dimensional subsets, and even if the optimal potentials are nowhere differentiable on the support.

A generalization yields a graph-based condition: decomposing supports into rectifiably connected components, dual uniqueness holds if the bipartite incidence graph (coupling nonzero mass between components) is connected.

These theorems extend previous dual uniqueness results by weakening the “full-dimensional closure of connected open set” assumptions. Notably, the former covers rectifiable connectivity, which includes connected submanifolds but not arbitrary path-connected sets.

Quantitative Uniqueness: Diameter Bounds

The paper introduces a novel quantitative framework for uniqueness: it defines the $L^p(\rho)$-diameter of the set of Kantorovich potentials modulo constants, measuring the oscillation between optimizers.

A central result bounds this diameter by the Hausdorff distance between the support of one marginal and a connected set satisfying a regularity arc property (specifically, connectivity by uniformly bounded curvature arcs). This yields sharp estimates for cases where the marginal is “almost connected.”

For $\mathcal{C}^{1,\alpha}$ costs and compact marginals, the $L^\infty$ diameter is controlled by a power of the Hausdorff distance:

$\diam_{L^\infty}(\Phi_c(\rho, \mu)) \leq C(1+C_\Omega)^2 \left( d_H^{1-1/(\alpha+1)} + d_H \right),$

where $d_H$ is the Hausdorff distance between the support and a connected reference set.

Specific results address grid-approximated measures and empirical measures from i.i.d. samples, yielding diameter bounds proportional to discretization scale or sampling radius, and establishing optimal rates in those regimes.

Characterization via Half-Spaces

A major technical innovation is the explicit characterization of the optimal set $\Phi_c(\rho, \mu)$ as an intersection of a family of half-spaces, determined by admissibility intervals derived from the cost function and the support of optimal couplings. This structure generalizes discrete LP theory into the infinite-dimensional context and underpins both qualitative and quantitative results.

Related Work and Context

The paper situates its results by comparing to classical qualitative uniqueness results (Santambrogio/Villani), recent combinatorial graph-based approaches in the discrete case, and prior partial results for empirical measures. It notes that most prior quantitative stability results assume both primal and dual uniqueness, and that its quantitative results remove this requirement.

For regularized variants of optimal transport (entropic, quadratic), dual uniqueness holds without connectivity requirements due to strict concavity.

Implications, Applications, and Open Problems

The theoretical results have direct implications for the stability analysis of dual optimal transport and for algorithms relying on uniqueness in empirical or grid-based settings. The explicit diameter bounds inform approximation schemes, sampling, and robustness analysis.

Several open questions remain, particularly around optimal exponents in diameter bounds, the necessity of certain regularity assumptions, and extensions to unbounded target measures.

Future work may further refine quantitative stability for singular marginals, explore sharper bounds in empirical contexts, and bridge these results with computational optimal transport frameworks.

Conclusion

This paper establishes both qualitative and quantitative uniqueness results for Kantorovich potentials, extending classical uniqueness theory to lower-dimensional, disconnected, and empirically sampled marginals. The explicit diameter bounds and half-space intersection characterization provide new analytical tools and inform practical applications in numerical and empirical optimal transport. The results refine the foundational understanding of dual uniqueness and pave the way for further advances in quantitative stability and robust optimal transport analysis.