Tight stability bounds for entropic Brenier maps
This paper derives tight stability bounds for entropic Brenier maps, optimizing Lipschitz constants under support constraints, with applications to semi-discrete optimal transport.
Key Findings
Methodology
The authors analyze the sensitivity of entropic Brenier maps by examining derivatives of Brenier potentials and leveraging transportation inequalities. They introduce the concept of tilt-stability for conditional entropic couplings, bounding the covariance and matrix norms via the potentials' smoothness. Combining these tools, they derive explicit Lipschitz constants depending on support bounds and potential regularity. The approach involves decomposing the transport maps into primal-dual structures, controlling the perturbations through entropy and covariance bounds, and establishing linear stability estimates under support constraints.
Key Results
- For measures supported within a radius R, the stability bound ∥ Tμ - Tν ∥_{L^2( ho)} \le (1 + 2 R^2 / \epsilon) W_2(\mu, u) is established, significantly improving previous exponential bounds, and shown to be tight via constructed examples.
- Under the assumption that the entropic Brenier potential is Λ-Lipschitz, the stability bound improves to ∥ Tμ - Tν ∥_{L^2( ho)} \le (1 + 2 \sqrt{R \Lambda} / \epsilon) W_2, providing a regularization-independent estimate in the smooth case.
- In the semi-discrete setting, the unregularized Brenier map's stability satisfies ∥ Tμ^0 - Tν^0 ∥_{L^2( ho)} \le W_2^{1/3}(\mu, u), filling a gap in the literature and demonstrating near-optimal bounds under regularity conditions.
Significance
These results advance the theoretical understanding of entropic optimal transport's stability, crucial for scalable algorithms and statistical estimation. By tightening Lipschitz bounds under support constraints, the work enables more robust and precise transfer maps in high-dimensional data analysis. The insights into the zero-regularization limit bridge the gap between regularized and classical optimal transport, fostering developments in machine learning, data science, and computational mathematics.
Technical Contribution
The paper introduces a novel combination of tilt-stability concepts for conditional entropic couplings with transportation inequalities, leading to explicit Lipschitz bounds that are tight under support constraints. It extends stability analysis beyond smooth potentials, covering non-smooth and semi-discrete cases, and provides near-optimal bounds for unregularized Brenier maps. The methodology unifies geometric, probabilistic, and variational tools, offering a comprehensive framework for stability analysis in entropy-regularized optimal transport.
Novelty
This is the first work to establish linear stability bounds for entropic Brenier maps supported on bounded regions, surpassing prior exponential bounds. The integration of tilt-stability with transportation inequalities to derive explicit Lipschitz constants under minimal assumptions marks a significant methodological breakthrough. The extension to semi-discrete and unregularized settings broadens the applicability, making this a foundational contribution to the theory of regularized optimal transport.
Limitations
- Results rely on support boundedness or strong smoothness assumptions of potentials; in cases with unbounded support or singular measures, bounds may not hold or be loose.
- High-dimensional settings pose challenges due to potential degeneracy of the potentials' smoothness, limiting practical applicability without further assumptions.
- Computational aspects, such as estimating the constants or implementing algorithms at scale, remain non-trivial and require further development.
Future Work
Future research will focus on relaxing support and smoothness conditions, extending stability bounds to unbounded or singular measures. Developing scalable algorithms that incorporate these bounds, and exploring the behavior in non-Euclidean spaces or with non-convex costs, are promising directions. Additionally, integrating these theoretical insights into deep learning frameworks for robust transfer learning and domain adaptation is a key long-term goal.
AI Executive Summary
This study addresses a fundamental challenge in optimal transport: understanding how small changes in target measures affect the associated transport maps, especially under entropy regularization. Prior work established exponential bounds on the stability of entropic Brenier maps, limiting their practical utility. The authors introduce a novel framework that leverages the smoothness of Brenier potentials, transportation inequalities, and tilt-stability concepts to derive sharp, linear Lipschitz bounds. In particular, they prove that when the measures are supported within a bounded region, the stability constant can be tightened to a linear function of the Wasserstein distance, with explicit dependence on the support radius R and regularization parameter ε. This represents a substantial improvement over previous exponential bounds and is shown to be tight via explicit examples. Further, under smoothness assumptions on the potentials, the bounds become independent of ε, matching the stability properties of classical Brenier maps. Extending these results, the paper also establishes near-optimal stability bounds for semi-discrete unregularized optimal transport maps, filling a notable gap in the literature. These theoretical advances have significant implications for computational optimal transport, statistical estimation, and machine learning applications, enabling more robust and scalable algorithms. The work opens avenues for future exploration into unbounded measures, high-dimensional regimes, and non-convex cost functions, promising to deepen our understanding of the stability and approximation properties of optimal transport solutions.
Deep Dive
Abstract
Entropic Brenier maps are regularized analogues of Brenier maps (optimal transport maps) which converge to Brenier maps as the regularization parameter shrinks. In this work, we prove quantitative stability bounds between entropic Brenier maps under variations of the target measure. In particular, when all measures have bounded support, we establish the optimal Lipschitz constant for the mapping from probability measures to entropic Brenier maps. This provides an exponential improvement to a result of Carlier, Chizat, and Laborde (2024). As an application, we prove near-optimal bounds for the stability of semi-discrete \emph{unregularized} Brenier maps for a family of discrete target measures.