Robust Entropic Optimal Transport (E-ROBOT)

• E-ROBOT is a framework that blends robust optimal transport with entropic regularization to enhance stability and computational efficiency.
• It leverages Schrödinger bridge theory and regularized Sinkhorn divergences to achieve dimension-free sample complexity in high-dimensional contaminated data.
• E-ROBOT finds applications in statistical inference, image processing, and gradient flows, offering robust performance against outliers.

Entropic-regularized Robust Optimal Transport (E-ROBOT) encompasses a class of methods and theoretical results that combine the classical robustness principles of robust optimal transport (ROBOT) with the computational, statistical, and regularizing advantages of entropic optimal transport and Schrödinger bridge approaches. E-ROBOT is characterized by the introduction of two key parameters: a regularization strength $\varepsilon$ (entropic penalty) and a robustness parameter $\lambda$ that shapes a robust ground cost. The framework provides dimension-free sample complexity, stable algorithmic properties, and strong applicability to high-dimensional and contaminated data regimes (Vecchia et al., 15 Sep 2025).

1. Theoretical Foundations and Mathematical Formulation

E-ROBOT is formulated on the basis of the Schrödinger bridge problem, which seeks a coupling between probability measures that minimizes a combined cost involving transport and entropy. The central object is the robust Sinkhorn divergence, $\overline{W}_{\varepsilon,\lambda}$, defined between probability measures $\mu$ and $\nu$ as: $\overline{W}_{\varepsilon,\lambda}(\mu,\nu)$ where the entropic regularization parameter $\varepsilon$ controls the smoothness of the optimal transport plan, and the robustness parameter $\lambda$ governs the sensitivity of the transport cost to outliers.

The negentropy term for $\mu$ is expressed as: $$\frac{1}{\varepsilon} F_{\varepsilon, \lambda}(\mu) + \frac{1}{2} = \inf_{\xi \in \mathcal{P}(X)} \left\{ \int \ln \left(\frac{d\mu}{d\xi}\right) d\mu + \frac{1}{2} \iint k_{\varepsilon,\lambda}(x, y) d\xi(x) d\xi(y) \right\}$$ where

$$k_{\varepsilon,\lambda}(x, y) = \exp\left( -\frac{c_{\lambda}(x,y)}{\varepsilon} \right)$$

and $c_{\lambda}(x,y)$ is a robust ground cost parameterized by $\lambda$, typically satisfying symmetry and boundedness constraints for computational stability and theoretical regularity.

The dual formulation involves optimal potentials $(\varphi^*, \psi^*)$ satisfying certain regularity properties (Lipschitz and boundedness), derived from the robust cost structure, and fixed-point relations inherited from Schrödinger system theory.

2. Dimension-Free Sample Complexity

A principal result of E-ROBOT is the establishment of dimension-free sample complexity for the robust Sinkhorn divergence: $$\overline{W}_{\varepsilon,\lambda}(\mu_n, \nu_n) - \overline{W}_{\varepsilon,\lambda}(\mu, \nu) = \mathcal{O}(n^{-1/2})$$ as shown in Theorem 9 and Corollary 10 of (Vecchia et al., 15 Sep 2025). Here,

• $n$ is the sample size,
• $\mu_n$, $\nu_n$ are empirical measures drawn from $\mu$, $\nu$,
• the convergence rate does not deteriorate with the ambient dimension.

This property distinguishes E-ROBOT from standard robust OT (ROBOT) and classical (unregularized) OT where rates typically degrade with dimension. The convergence analysis leverages the regularity of dual Schödinger potentials associated with entropic regularization, enforced by the interplay of $\varepsilon$ and the smoothness of $c_\lambda$.

3. Robustness and the Role of the Robust Kernel

The robustness parameter $\lambda$ modulates the ground cost $c_\lambda(x, y)$ in the kernel $k_{\varepsilon,\lambda}$, introducing a tunable resistance to outliers. For instance, $c_\lambda$ may implement truncation (e.g., capped or Huberized costs), Laplace-type tails, or other robust loss structures. The exponential scaling in $k_{\varepsilon,\lambda}$ ensures that large deviations in $c_\lambda$ are exponentially suppressed, and the robust divergence de-emphasizes pairs $(x, y)$ with large pairwise distance.

The Schrödinger bridge duality gives rise to robust dual potentials $(\varphi^*, \psi^*)$ whose Lipschitz continuity and boundedness (Proposition 3 in (Vecchia et al., 15 Sep 2025)) ensure that convergence properties hold uniformly in the presence of data corruption or contamination.

4. Applications in Statistics and Machine Learning

E-ROBOT has strong utility in several applied domains, specifically::

• Goodness-of-Fit and Two-Sample Testing: The robust Sinkhorn divergence provides a robust, convex discrepancy measure. Because its concentration bounds are dimension-free, such tests maintain statistical power in high dimensions and under contamination.
• Barycenter Computation for Corrupted Shapes: The robust entropic barycenter is less biased by outlier points in the constituent shapes. The robust kernel $k_{\varepsilon,\lambda}$ downweights anomalous alignments, leading to stable barycenter shapes even in adversarial contamination.
• Gradient Flows: E-ROBOT induces gradient flows with respect to the robust Sinkhorn divergence. Unlike flows generated by certain MMDs (where kernel tails suppress gradients far from data), the robust entropic flow provides nonvanishing gradients even in outlying regions, facilitating effective convergence toward the target distribution.
• Image Color Transfer: In computer vision, E-ROBOT yields improved color transfer mappings by mitigating adverse effects of atypical (outlier) color values, leading to more visually consistent results even when input images are subject to noise or artifacts.

This is supported by a theory that establishes uniform convergence bounds and the existence of robust dual potentials under general contamination models.

5. Computational Implementation

E-ROBOT is implemented using variants of the Sinkhorn algorithm and Schrödinger bridge solvers, often requiring only small modifications to existing entropic OT codes:

• The cost matrix is constructed from the robust cost $c_\lambda$.
• The kernel $k_{\varepsilon,\lambda}$ is used to define the scaling variables and update rules in Sinkhorn-like iterations (which may be stabilized in the log-domain).
• Dual potentials are updated until convergence to machine precision or until the error matches the theoretical statistical tolerance (proportional to $n^{-1/2}$).
• For practical purposes, log-domain stabilization methods are essential to avoid numerical overflow, especially for small $\varepsilon$ or data with large dynamic range in $c_\lambda$.

This yields efficient and numerically stable estimators suitable for integration into large-scale pipelines for modern machine learning and statistical data analysis.

6. Connections, Impact, and Research Directions

E-ROBOT provides a bridge between robust statistics, classic OT, and entropic regularization methods, leveraging the strengths of each paradigm. The dimension-free sample complexity is especially impactful for contemporary statistical machine learning, where data scales and data contamination are both common.

Future research directions suggested by (Vecchia et al., 15 Sep 2025) include:

• Extensions to richer families of robust costs (beyond truncated or Laplace-like $c_\lambda$), potentially modeling heteroskedastic noise or domain-specific outlier patterns.
• More efficient and scalable algorithms, such as stochastic or GPU-parallelized Sinkhorn solvers for robust costs, to further improve applicability in large data settings.
• Integration into deep learning, using E-ROBOT divergences as loss functions for robust generative modeling or adversarial domain adaptation.
• Extensions to settings involving model selection, hypothesis testing, or nonparametric inference, particularly under heavy-tailed or contaminated data.
• Theoretical refinement of regularity and stability properties for yet more general cost structures and data irregularities.

Table: Key Parameters and Their Roles

Parameter	Role in E-ROBOT	Effect
$\varepsilon$	Entropic smoothing; regularization strength	Controls smoothness/statistical complexity; smaller $\varepsilon$ tightens approximation to classical OT
$\lambda$	Robustness of cost function	Downweights influence of outliers; dictates strength of robustness
$k_{\varepsilon,\lambda}$	Robust kernel for Sinkhorn updates	Encodes both entropic smoothing and robustness

Conclusion

E-ROBOT synthesizes robust optimal transport methods with entropic regularization to achieve dimension-free sample complexity, robust performance in the presence of outliers, and algorithmic efficiency via scalable entropic OT solvers. Through its Schrödinger bridge foundations, robust dual potentials, and smooth divergence structure, E-ROBOT enables robust statistical inference, barycenter computation, gradient flows, and image processing in high dimensions and under contamination. Its computational implementation is compatible with existing OT solvers, and ongoing research continues to expand its theoretical and practical scope (Vecchia et al., 15 Sep 2025).