Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure (2103.05934v2)

Abstract: This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. Assuming that the source density $\rho$ is bounded from above and below on a compact convex set, we prove that the map $\mu$ $\rightarrow$ T$\mu$ is bi-H{\"o}lder continuous on large families of probability measures, such as the set of probability measures whose moment of order p > d is bounded by some constant. These stability estimates show that the linearized optimal transport metric W2,$\rho$($\mu$, $\nu$) = T$\mu$ -- T$\nu$ L 2 ($\rho$,R d) is bi-H{\"o}lder equivalent to the 2-Wasserstein distance on such sets, justifiying its use in applications.

• The paper establishes that optimal transport maps exhibit bi-Hölder continuity for measures with finite moments above the space dimension.
• It introduces a Linearized Optimal Transport framework linking the 2-Wasserstein distance to OT map distances, which aids statistical computations.
• The study demonstrates robust quantitative stability of OT maps, ensuring their reliability for computational applications in imaging and data modeling.

Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure

Introduction

The paper focuses on the paper of the quantitative stability of optimal transport (OT) maps, particularly when varying the target measure. The OT problem, given two probability measures $\rho$ and $\mu$, involves finding a map that transports one measure into the other with minimal cost, typically modeled as the quadratic cost. The paper addresses the issue of stability of these maps when the target measure $\mu$ changes, with specific interest in applications that require these maps' continuity properties.

Theoretical Foundations

The discussion begins with Brenier's theorem, which states that when the source measure $\rho$ is absolutely continuous, the optimal transport map is unique and can be described as the gradient of a convex function. The stability of these OT maps is essential in many applications since it influences the robustness of computations involving these maps.

The paper introduces various mathematical tools to paper this stability. Central to the analysis is the continuity of the map $\mu \mapsto T_\mu$ in different probability measure spaces and under specific conditions on the moments of the measures involved.

Main Contributions

Bi-Hölder Continuity

The authors establish that the transport map exhibits a bi-Hölder continuous behavior across families of probability measures. Specifically, for measures with bounded moments of order $p > d$, where $d$ is the dimension, the map $\mu \mapsto T_\mu$ is bi-Hölder. This results in an equivalent linearization of the transport distance in certain spaces, justifying the use of these mappings in computational scenarios.

Linearized Transport Framework

The paper formalizes the Linearized Optimal Transport (LOT) framework, linking the $2$-Wasserstein distance with the respective OT map distances. This framework simplifies statistical computation for probability measures by leveraging the underlying geometry of the transport maps. It creates a Hilbert space embedding for these maps, enabling the direct application of standard statistical tools.

Quantitative Stability Results

The robustness of the transport maps is shown even under varying conditions for the regularity and boundedness of the involved measures. The measures are evaluated with respect to their finite moments and support, affecting the continuity properties of the associated OT maps and potentials.

Practical Implications

The quantitative stability insights direct applications in various fields such as image processing, pattern recognition, and data-driven modeling. These findings assist in ensuring that the transport maps used in such applications stay reliable under perturbations of the target measure.

The framework developed is particularly relevant for computational applications where large datasets and the need for efficient algorithms align. The LOT approach further supports experimental tasks in machine learning where the structure of data distributions is managed via transport maps.

Conclusion

The paper provides extensive theoretical elaborations that reinforce the practical utility of OT maps in real-world data scenarios. By confirming the bi-Hölder continuity of these mappings in certain measure spaces, this research enhances understanding of OT maps' stability properties. This contributes broadly to fields that rely on the stability of data transformations, such as biometric recognition and other high-dimensional data processing tasks. The derived results hold promise for continued exploration and refinement in OT-related applications in computational mathematics and engineering fields.