Perturbed Optimal Transport Advances

Updated 3 July 2026

Perturbed optimal transport is a framework that extends classical optimal transport by incorporating small data and cost perturbations to analyze model stability and robustness.
The theory provides explicit convergence rates and stability bounds for transport maps and potentials under varied perturbations, ensuring reliable quantitative assessments.
Applications span robust risk management, computational finance, and dynamic systems, where structured perturbations enhance both reliability and computational efficiency.

Perturbed optimal transport refers to a range of frameworks in which classical optimal transport theory is extended or analyzed under small or structured perturbations of the input data, cost structure, or model constraints. These perturbations may be due to contamination of marginals, distributional uncertainty, regularization, dynamic system interactions, or relaxation of the mass conservation requirement. The resulting theory encompasses stability and breakdown analysis, regularity results for perturbed costs, robustness and minimax statistical estimation, and connections to robust and unbalanced transport models. Perturbed optimal transport thus provides a rigorous quantitative foundation for analyzing the sensitivity, robustness, and computational tractability of transport-based models in high-dimensional probability, statistics, and applied mathematics.

1. Breakdown and Robustness of Optimal Transport Maps

A central direction in perturbed optimal transport is the study of the breakdown point of the Monge map under contamination of the target measure. For strictly convex, coercive costs $c(x, y) = h(x-y)$ satisfying Gangbo–McCann structural conditions, the breakdown point of the transport map $T_{Q \to P}$ at $u$ is defined as the smallest fraction $\epsilon$ such that there exists a contaminant $\mu$ with $P_{\epsilon, \mu} = (1-\epsilon)P + \epsilon\mu$ for which the transport image of $u$ becomes unbounded: $BP(\mathbf T_{Q \to P}(u), P) = \inf\{ \epsilon \in [0,1]: \sup_\mu \sup_{v \in \mathbf T_{Q \to P_{\epsilon, \mu}}(u)} \|v\| = \infty \}.$ The sharp result is that this breakdown point equals the Tukey halfspace depth of $u$ with respect to $Q$ ,

$T_{Q \to P}$ 0

independent of the particular strictly convex cost $T_{Q \to P}$ 1. This establishes a cost-independent characterization of the robustness of transport-based quantiles (Gonzalez-Sanz et al., 16 Mar 2026). The proof combines geometric and convex analysis, leveraging duality and extremal contamination directions constructed using the convex conjugate of $T_{Q \to P}$ 2.

2. Stability and Instability of Transport under Perturbation

Quantitative stability of optimal transport maps and potentials under perturbations is key to understanding the sensitivity of transport-based procedures. For power costs $T_{Q \to P}$ 3, if the source $T_{Q \to P}$ 4 is log-concave, compactly supported, and bounded above/below, then the Kantorovich potentials satisfy

$T_{Q \to P}$ 5

with $T_{Q \to P}$ 6 ( $T_{Q \to P}$ 7) or $T_{Q \to P}$ 8 ( $T_{Q \to P}$ 9); the transport maps themselves obey

$u$ 0

with $u$ 1 ( $u$ 2) or $u$ 3 ( $u$ 4) (Mischler et al., 2024). These results yield explicit rates for how map and potential estimates respond to perturbations of the target.

However, sharp counterexamples demonstrate that uniform Hölder stability may fail catastrophically if the source density $u$ 5 blows up at isolated singular points or if the support approaches configurations with nonunique optimal plans. In particular, for sources with superpolynomial concentration near the boundary, no universal stability exponent exists; and near loss of map uniqueness, small $u$ 6 differences in the target measure can induce large $u$ 7 deviations in the transport map (Letrouit, 15 Oct 2025).

3. Regularity Theory for Perturbed and Regularized Costs

Regularity of optimal transport solutions under perturbed cost functionals is a major research theme. For entropic optimal transport problems with cost

$u$ 8

it is established that if the minimizer is a local quasi-minimizer for the quadratic problem (after suitable affine rescalings) and long trajectories are exponentially penalized, then $u$ 9 Morrey–Campanato regularity propagates down to the entropic length $\epsilon$ 0, with precise scaling in $\epsilon$ 1 and control at all intermediate scales (Gvalani et al., 13 Jan 2025). This framework extends to any cost functional that is a sufficiently controlled perturbation of the quadratic cost.

For Monge–Ampère type parabolic flows associated with a cost $\epsilon$ 2 that is a small $\epsilon$ 3 perturbation of a base cost $\epsilon$ 4 satisfying the (weak) Ma–Trudinger–Wang (MTW) condition, global-in-time solutions exist and converge to the OT potential for $\epsilon$ 5, with all regularity constants and bounds uniform in the size of the perturbation (Abedin et al., 2021).

4. Structured Perturbed Transport Models

Several frameworks systematically design or interpret perturbed optimal transport. In semi-discrete OT, perturbing the cost function via random utility shocks or ambiguous disturbances leads to a family of dual-regularized (smoothed) problems. The corresponding robustified dual is equivalent to a regularized primal with divergence-type penalties (e.g., entropic, $\epsilon$ 6, Tsallis). These are directly linked to discrete choice models in economics, and stochastic gradient methods exploiting dual smoothing achieve fast convergence rates in the presence of perturbations (Taskesen et al., 2021).

In the context of nonconservative or unbalanced transport, the perturbation is built into the feasible set or cost via mass-change factors or marginal penalties. The nonconservative OT problem with a transport-dependent efficiency factor $\epsilon$ 7 generalizes both balanced and standard unbalanced OT, allowing for explicit perturbative analysis when $\epsilon$ 8 (Kováčová et al., 1 Oct 2025). Unbalanced OT with quadratic cost and KL marginal penalties identifies the optimal Monge-type object as a transport–growth pair $\epsilon$ 9; minimax rates for their estimation are achieved under sound perturbation control and value-based stability reductions (Ponnoprat et al., 9 May 2026).

5. Applications and High-Dimensional Risk Scenarios

Perturbed optimal transport finds direct application in computational finance and robust risk management. A prominent example is the linear-response framework for martingale entropic joint calibration of SPX and VIX smiles. Here, Fisher-information-based linearization provides fast, dimension-reduced risk estimates that remain accurate relative to full recalibration. This method leverages the strict convexity and smoothness of the entropic dual and analytic properties of solution paths under market data perturbations (Che et al., 11 Mar 2026). In robust multi-marginal OT for risk aggregation, stability results guarantee continuous dependence of the worst-case risk value on the marginals and the cost/output function, with explicit Lipschitz bounds in Wasserstein distance in the one-dimensional case (Ennaji et al., 2022).

6. Dynamical and Graph-Based Perturbed Transport

The set-oriented, graph-based framework for optimal perturbations in nonlinear systems analyzes the minimal discrete-time interventions required to drive the state distribution between prescribed measures under interleaved dynamics and Monge–Kantorovich-type map-based corrections. At each perturbation, a discrete OT problem is solved, and the full multi-step problem is cast as a convex optimization over pseudo-time evolution on a graph. The spatially localized and temporally adaptive structure of the perturbations is governed by the natural transport pathways of the underlying dynamical system (Grover et al., 2016).

Summary Table: Core Themes in Perturbed Optimal Transport

Theme	Mathematical Focus	Representative Papers
Breakdown and robustness	Breakdown point, Tukey depth, contaminations	(Gonzalez-Sanz et al., 16 Mar 2026)
Quantitative stability/instability	Hölder/Lipschitz bounds, pathologies	(Mischler et al., 2024, Letrouit, 15 Oct 2025)
Regularity for perturbed costs	$\mu$ 0-regularity, parabolic flows	(Gvalani et al., 13 Jan 2025, Abedin et al., 2021)
Structured (robust, unbalanced) OT	Regularization, mass-loss, KL penalties	(Taskesen et al., 2021, Kováčová et al., 1 Oct 2025, Ponnoprat et al., 9 May 2026)
Computational and risk applications	Fisher linearization, multi-marginal risk	(Che et al., 11 Mar 2026, Ennaji et al., 2022)
Dynamical systems and graphs	Graph flow, dynamic OT under perturbations	(Grover et al., 2016)

Perturbed optimal transport thus unifies a rich spectrum of analytical, statistical, and computational phenomena. The theory provides sharp characterizations of robustness and instability, rate-optimal estimation methods, stability criteria, and efficient computational strategies, with applications spanning statistics, mathematical finance, nonlinear dynamics, and robust optimization.