--- title: Stability of Optimal Transport Maps url: https://www.emergentmind.com/topics/stability-of-optimal-transport-maps type: topic --- # Stability of Optimal Transport Maps Stability of optimal transport maps concerns the quantitative and qualitative dependence of the optimal transport map—typically denoted $T_\mu$—on perturbations of the input probability measures, most often the target measure $\mu$, with the source measure fixed. This property governs the robustness of optimal transport—the minimizer of the Monge problem for a given cost function—against variations of the marginals. Stability results are essential both for the analysis of the nonlinear PDEs underpinning optimal transport and for applications in computational geometry, statistics, numerical analysis, and data science, as they provide guarantees on the behavior of the transport map under data perturbation or discretization. ## 1. Quantitative and Qualitative Stability: Definitions and Fundamental Results A central question is to determine, for fixed source measure $p$ (often absolutely continuous), how sensitively the optimal transport map $T_\mu$ varies in response to variations in the target measure $\mu$, as measured in Wasserstein distance (or similar). The prototypical form of a quantitative stability bound is \[ \|T_\mu - T_\nu\|_{L^2(p)} \leq C \, W_p(\mu, \nu)^\alpha, \] where $C > 0$ and $\alpha > 0$ are constants, and $W_p$ is the $p$-Wasserstein distance. Many works establish such a bound under regularity assumptions on the marginals and/or domain. For instance, when the source density is bounded away from zero and infinity on a bounded convex domain, and the cost is quadratic, it follows that the mapping $\mu \mapsto T_\mu$ is locally Hölder continuous with a positive exponent [2103.05934, 1910.05954, 2411.04908]: \[ \|T_\mu - T_\nu\|_{L^2(p)} \leq C\, W_1(\mu, \nu)^\gamma. \] The value of $\gamma$ varies with the regularity and geometry: for Lebesgue measure on a convex set, dimension-free exponents such as $\gamma = 2/15$ or $1/6$ have been proven [1910.05954, 2411.04908]. The foundational work [1202.5561] establishes strong $W^{2,1}_{\mathrm{loc}}$ stability of Alexandrov solutions to the Monge-Ampère equation under strong $L^1_{\mathrm{loc}}$ convergence of the right-hand side, which implies strong $W^{1,1}_{\mathrm{loc}}$ stability for the corresponding optimal transport maps. This ensures not only pointwise, but strong Sobolev convergence. It is also established that the optimal transport map is unstable in certain settings: for source densities blowing up superpolynomially at boundary points, no power-type stability inequality can hold [2510.13265]. Even for bounded densities supported on domains with intricate geometries (e.g., configurations close to loss of uniqueness), stability can break down or the best possible modulus is much lower than $1$. ## 2. Mechanisms Underpinning Stability and Instability Stability is enforced by various structural and analytic properties: - **Source Density Regularity:** If the source density $p$ is uniformly bounded above and below on its support (which is convex or satisfies a John condition), stability is typically secured. Quantitative exponents depend sensitively on this lower and upper bound. - **Domain Geometry:** Convexity or satisfying the John domain condition is critical for patching local variance inequalities into global stability. Counterexamples (e.g., room-and-passage domains) demonstrate the loss of any Hölder modulus in non-John domains [2411.04908]. - **Uniqueness of Optimal Plans:** Instability arises not only from unbounded densities but also near configurations with non-uniqueness of optimal transport plans. Small perturbations in the target measure can then induce large, even discontinuous, changes in the map [2510.13265]. In two dimensions, as shown in classical examples [1710.03708], nonconvex or merely Lipschitz perturbations of domains break stability of the identity map. - **Modulus of Stability:** For regular data, $\alpha = 1/2$ (i.e., square root behavior) is optimal, as seen in sharp lower bounds [1803.00785, 1910.05954]. For rougher data or higher cost exponents, exponents can degrade (e.g., $\alpha = 1/3$ if the optimal transport potential is just convex and Lipschitz [2401.01088]). ## 3. Methodological Frameworks Various analytical methods underpin stability estimates: - **Monge–Ampère Techniques:** In convex, regular regimes, the PDE viewpoint (Monge–Ampère equation) allows sharp $W^{2,1}_{\mathrm{loc}}$ or $L^2$-norm comparisons of potentials and their gradients [1202.5561, 1803.00785]. - **Variance Inequalities and Gluing:** Recent works [2411.04908, 2504.05412] employ local variance inequalities for potentials on small convex pieces; Whitney decompositions and Boman chain conditions are used to glue these into global stability bounds. Spectral and graph-theoretic methods also enable local-to-global patching, especially on nonconvex sets. - **Entropy-Regularized Approaches:** Entropic regularization (entropic Brenier maps) allows leveraging the smoothness of regularized functionals to prove quantitative stability with near-optimal exponents and to bridge to the unregularized setting, improving previously exponential-in-regularization constants [2404.02855, 2504.05412]. - **Strong c-Concavity:** For general costs, ensuring strong c-concavity of Kantorovich potentials (through second derivative or MTW-type conditions) yields stability via explicit $L^2$ estimates controlling changes in the map by metric changes in the target [2207.11042]. - **Statistical and Numerical Discretization:** In semi- or fully-discrete settings, stability bounds control approximation error, establishing $L^2$ estimates of the order $h^{1/2}$ for point cloud grid size $h$ when solving for discrete OT maps [1803.00785, 2004.05299]. ## 4. Extensions to General Costs and Geometric Settings Progress has been made in extending stability theory: - **Power Cost Functions:** For $c(x, y) = |x - y|^p$ with $p > 1$, explicit stability bounds for both Kantorovich potentials and optimal maps have been established under log-concavity and bounded support, with exponents varying as $1-1/p$ for $1 < p < 2$ and $1/2$ for $p \geq 2$ in the potential, and exponents for the map derived via interpolation inequalities and regularity of the cost [2407.19337]. - **Riemannian Manifolds:** On manifolds equipped with the squared Riemannian distance cost, entropy-regularized and integral-geometric arguments yield stability bounds—e.g., \[ \|\varphi_\mu - \varphi_\nu\|_{L^2(p)} \leq C\, W_1(\mu, \nu)^{1/2}, \qquad \int \operatorname{dist}(T_\mu(x), T_\nu(x))^2\, dp(x) \leq C W_1(\mu,\nu)^{1/6} \] for suitable $p$, $C$ depending on geometric parameters [2504.05412]. - **Sphere and Non-Euclidean Spaces:** Studies of optimal transport on the sphere yield Hölder stability results with exponents as low as $1/9$ in semi-discrete settings [2501.08492]. The differential geometry of the cost and the structure of Laguerre cells become essential in such cases. - **Unbalanced Optimal Transport:** The extension to unbalanced transport (where marginals are only approximately fitted) reveals a localization of stability between Sobolev seminorms and full $H^1$-norms depending on the entropy regularization and complexity of the potential class [2203.09143]. ## 5. Implications for Computational, Statistical, and Applied Settings Stability results have direct implications: - **Numerical Approximations:** Quantitative error bounds (e.g., $O(h^{1/2})$) ensure consistency of discrete approximations of the transport map against measurement or sampling error [1803.00785, 2004.05299]. These inform mesh design and algorithmic accuracy standards in computational geometry and physics. - **Statistical Estimation and Inference:** Bounds that link transport map error to approximation of the measures in Wasserstein distance allow minimax rate analysis and control of plug-in estimators. Fast rates can be obtained under regularity/smoothness assumptions, and central limit theorems for plug-in estimators of $W_2^2$ are available in smooth regimes [2107.12364, 2502.12326, 2303.10155]. - **Linearization and Machine Learning:** The linearized optimal transport metric $W_{2, \rho}(\mu, \nu) = \|T_\mu - T_\nu\|_{L^2(\rho)}$ provides an isomorphic embedding up to Hölder equivalence, justifying its use in regression, clustering, or learning on spaces of measures [1910.05954, 2103.05934, 2411.04908]. - **Uncertainty Quantification:** Explicit stability moduli inform the robustness of transport-based statistical tests, the derivation of confidence bands for empirical maps, and Bayesian contraction rates for regression in non-Euclidean settings, including on the sphere [2501.08492]. ## 6. Limitations, Counterexamples, and Open Problems Despite many positive results, several scenarios remain delicate or unresolved: - **Instabilities from Irregular Densities and Geometry:** For source measures with density blowing up superpolynomially at isolated boundary points, or for carefully constructed domains, no modulus of continuity (even arbitrarily weak) for the map in terms of the Wasserstein distance holds [2510.13265]. - **Near Non-Uniqueness Regions:** When the optimal transport problem is near a region with multiple minimizers, even for uniformly bounded source densities, the mapping $\mu \mapsto T_\mu$ is unstable. Quantitative lower bounds indicate that, as one approaches this regime, the best permissible modulus of continuity degrades substantially. - **Boundary of Stability:** The precise characterization of exponents and their optimality in all cases remains an open topic. While $1/2$ is optimal in uniformly convex, regular regimes, strong instability can force exponents arbitrarily low or eliminate any stability altogether. - **Extension to Broader Geometric Classes:** While approaches using gluing and entropy-regularization apply to John domains and Riemannian manifolds, the nature of quantitative stability on more general (e.g., non-smooth or singular spaces) is not fully understood. - **Statistical and Numerical Ramifications:** There remain open questions about devising robust statistical estimators and numerical algorithms capable of adapting to or overcoming settings where stability fails, as well as the design of regularization schemes to mitigate such instability. ## 7. Summary Table: Stability Moduli for OT Maps | Setting | Source Density Condition | Stability Modulus Exponent ($\alpha$) | Reference | |-----------------------------------------------------------|------------------------------|----------------------------------------|---------------------| | Bounded, convex support, $c(x,y)=|x-y|^2$ | $p$ bounded above/below | $\geq 1/2$ | [2411.04908] | | Convex, $C^{1,1}$ domains, $C^\infty$ densities | $p$, $\mu$ smooth | $1/2$ (optimal) | [1803.00785] | | Compact, convex source, sphere, semi-discrete setting | $p$ uniform on sphere | $1/9$ | [2501.08492] | | Power cost $c(x,y)=|x-y|^p$, $p>1$ | $p$ log-concave, bounded | $1-1/p$ ($1