wasserstein distance computation

The massive use of Machine Learning (ML) tools in industry comes with critical challenges, such as the lack of explainable models and the use of black-box algorithms. We address this issue by applying Optimal Transport theory in the analysis of responses of ML models to variations in the distribution of input variables. We find the closest distribution, in the Wasserstein sense, that satisfies a given constraintt and examine its impact on model behavior. Furthermore, we establish convergence results for this projected distribution and demonstrate our approach using examples and real-world datasets in both regression and classification settings.

To address the sensitivity of the Wasserstein barycenter to outlier distributions, this paper proposes the Wasserstein median—defined as the Fréchet median under the 2-Wasserstein distance—as a robust alternative. Methodologically, we develop a generic iterative algorithmic framework built upon existing Wasserstein barycenter solvers and provide a rigorous proof of its convergence. Theoretically, we establish, for the first time, the existence, strong consistency, and outlier-robustness of the Wasserstein median. Empirically, experiments on synthetic and real-world data—including single-cell gene expression profiles and image distributions—demonstrate substantial improvements in robustness: the median reduces sensitivity to outliers by 40%–65% compared to the barycenter, while preserving interpretability and computational tractability. This work introduces a new paradigm and practical tool for robust summarization of collections of probability distributions.

Robust comparison of high-dimensional positive measures in the presence of outliers and noise remains challenging. Method: This paper proposes the Sliced Unbalanced Optimal Transport (SUOT) framework, introducing two novel sliced unbalanced OT loss functions. The method employs a Frank–Wolfe-type optimization algorithm, reducing computational complexity to linear time while enhancing stability in high dimensions and robustness to outliers. Contribution/Results: We establish the topological structure, statistical consistency, and convergence properties of the proposed losses. SUOT unifies and generalizes both sliced OT and unbalanced OT—two dominant paradigms in modern optimal transport. Extensive experiments on synthetic and real-world datasets demonstrate that SUOT outperforms standard OT and its variants in both computational efficiency and robustness, particularly under contamination and high-dimensional settings.

This paper addresses the conceptual and methodological divide between φ-divergence-based (likelihood-ratio-centric) and Wasserstein-based (outcome-space-centric) paradigms in distributionally robust optimization (DRO). To bridge this gap, we propose the first unified DRO framework. Methodologically, we integrate optimal transport theory with conditional moment constraints to construct a novel DRO model capable of simultaneously perturbing both likelihood ratios and outcome distributions. Via Lagrangian duality analysis, we derive a computationally tractable closed-form dual reformulation, whose equivalent problem is solvable in polynomial time. Theoretical contributions include: (i) establishing a rigorous strong duality theorem under conditional moment constraints; and (ii) introducing a new modeling paradigm for optimal transport that explicitly incorporates such constraints. Empirical results demonstrate that the proposed framework significantly enhances generalization and robustness under distributional shifts.

This work addresses the topological information loss and degraded geometric awareness inherent in the Sliced Wasserstein (SW) distance due to its reliance on linear projections. To resolve this, we propose the Tree-Sliced Wasserstein distance with Splitting Maps (TSW-SL)—the first extension of SW to metric tree structures. Our core innovation lies in introducing a splitting map and a tree-domain Radon transform, rigorously proving their invertibility and metric properties while preserving the closed-form optimal transport solution. Theoretically, this generalizes both the Radon transform and optimal transport frameworks to non-Euclidean, hierarchical domains. Algorithmically, TSW-SL enables efficient gradient flow optimization via differentiable tree projections. Experiments demonstrate that TSW-SL consistently outperforms SW and its variants in gradient flow simulation, image style transfer, and generative modeling—achieving superior geometric robustness without sacrificing computational efficiency.

This work addresses the computational challenges of evaluating Wasserstein distances in large-scale non-Euclidean spaces by proposing a projection-based approach grounded in 1-Lipschitz observables. The method pushes forward probability measures onto the real line, computes their one-dimensional Wasserstein distances, and constructs a hierarchy of pseudometrics over nested subspaces to approximate the original distance. This hierarchical framework balances accuracy and efficiency through tunable parameters and establishes a theoretical link between the metric covering dimension of the support set and the order required for unique measure recovery—providing an analogue of the Cramér–Wold theorem in non-Euclidean settings. Theoretical analysis confirms that measures can be uniquely recovered at specific hierarchy levels, and numerical experiments on finite discrete grids demonstrate the method’s effectiveness and practicality.

This study addresses the problem of efficiently learning an unknown probability distribution from limited samples and provides non-asymptotic, computable error bounds in the Wasserstein distance. To this end, the authors propose a novel framework that integrates optimal transport theory, concentration inequalities, and mixed-integer linear programming. By leveraging intelligent clustering to optimize the selection of support points, the method constructs a tractable optimization problem that depends only on the size of the support set of the empirical distribution. This approach achieves, for the first time, high-confidence Wasserstein error bounds without requiring prior knowledge of the true distribution. Experimental results demonstrate that the method significantly reduces the support set size across multiple benchmarks while yielding substantially tighter error bounds.

This study investigates whether empirical subgradients of sample-based optimal transport objectives converge to the subdifferential of the population objective, thereby ensuring that subgradient methods consistently approximate population stationary points. By leveraging subdifferential analysis and graphical convergence theory, the work establishes—for the first time—the graphical convergence of empirical subgradients within the optimal transport framework. It further reveals the critical role of parametric smoothness in balancing statistical consistency and optimization stability, showing that nonsmooth settings may induce derivative instability even with large samples. The theoretical findings are validated in applications including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems, demonstrating that standard subgradient methods indeed converge consistently to population stationary points.

This work addresses the computational inefficiency of exact empirical Wasserstein median computation, which traditionally relies on nested optimization. The authors propose a fixed-weight, free-support direct solver that eliminates inner loops by solving an exact optimal transport subproblem, performing barycentric projection, and updating support points at each iteration. A key innovation is the use of an inverse-distance weighted averaging strategy to update support points, which yields a tight majorization-minimization (MM) surrogate function. This construction guarantees monotonic descent of the objective, preserves the convex hull of the support set, and ensures finite-time convergence. Experiments demonstrate that the method achieves objective values comparable to the nested Weiszfeld algorithm with significantly fewer optimal transport calls and exhibits superior robustness over Wasserstein barycenters in contaminated data, posterior aggregation, and image prototyping tasks.

This study addresses the problem of probability density estimation under prescribed expectation constraints. The authors propose an optimization framework based on the Wasserstein distance, which minimizes the discrepancy between the estimated density and a prior distribution while incorporating a regularization term to suppress artifacts in the target measure. The key innovation lies in the novel integration of optimal transport theory with expectation-constrained density estimation, alongside the development of an annealing algorithm capable of handling nonsmooth constraints. Experimental results on both synthetic data and real-world financial applications demonstrate the method’s effectiveness: the resulting density estimates rigorously satisfy the imposed constraints and exhibit strong structural plausibility.