apply optimal transport

Existing optimal transport (OT) mapping estimation theory heavily relies on Brenier’s theorem—which requires quadratic cost and absolutely continuous source distributions—rendering it inadequate for stochastic OT mappings with mass splitting, commonly encountered in real-world settings involving singular, discrete, or corrupted source/target distributions. Method: We propose a novel metric to quantify the quality of stochastic OT mappings and develop the first universal, robust, finite-sample optimal risk bound framework. Our approach integrates generalization error analysis, adversarially robust statistical learning, parameterized stochastic mapping modeling, and regularized empirical risk minimization. Contribution/Results: We derive near-optimal finite-sample risk bounds under minimal distributional assumptions. Experiments demonstrate substantial improvements in transport accuracy over conventional OT methods in challenging non-absolutely-continuous and corrupted-data regimes where standard approaches fail.

This work addresses the optimal transport barycenter problem by proposing a density-aware pairwise distance metric. Methodologically, it formulates a Lagrangian functional incorporating the underlying data distribution as a prior, where the cost between two points is defined as the minimal-action path that avoids low-probability regions. The path is parameterized via Chebyshev polynomials, and endpoint adversarial constraints are introduced to enforce exact matching of source and target distributions. The resulting distance is differentiable, fully data-driven, and explicitly encodes both intrinsic geometric structure and probability density—departing from conventional Euclidean or fixed-cost assumptions. Empirically, on synthetic data, the approach significantly improves barycenter estimation accuracy and robustness in clustering and distribution alignment tasks. It establishes a novel paradigm for density-sensitive optimal transport modeling.

Bridging the gap between optimal transport (OT) theory and modern machine learning practice—particularly in the theoretical modeling and evaluation of generative models. Method: We systematically construct a unified, ML-oriented OT framework encompassing the Monge–Kantorovich formulation, Brenier’s theorem, dual and dynamic representations, the Bures metric, and Wasserstein gradient flows. For the first time, we deeply integrate the full mathematical OT machinery into generative adversarial networks (GANs), diffusion models, Transformer token dynamics, and neural network training gradient flows. Our approach unifies linear programming, semi-discrete solvers, entropic regularization, and dynamic OT modeling to yield geometrically interpretable distribution matching. Contribution/Results: The framework enhances theoretical rigor and algorithmic stability in generative model design, providing principled geometric insights and enabling consistent analysis across diverse generative paradigms.

This work addresses the problem of efficiently learning Wasserstein geodesics and their associated optimal transport velocity fields directly from samples to model the dynamic transport process between a source and a target distribution. Building upon the dynamical formulation of optimal transport, the constrained optimization problem is reformulated as a minimax game, enabling joint approximation of the geodesic, optimal map, and full velocity field using deep neural networks. The proposed method constitutes the first purely sample-driven neural solver capable of handling general cost functions—including the quadratic cost—without requiring explicit density estimation. Experiments on both synthetic and real-world datasets demonstrate that the approach accurately reconstructs geodesics and velocity fields and enables direct sampling from the target distribution, thereby validating its effectiveness and broad applicability.

This work investigates the statistical limits and sample complexity lower bounds for estimating any valid transport map without relying on optimal transport (OT). Formalizing the estimation task within a minimax framework, the study integrates tools from optimal transport theory, minimax analysis, and stability-based hypothesis testing to establish, for the first time, rigorous statistical lower bounds for non-optimal yet valid transport maps. The results demonstrate that under standard stability conditions, the statistical difficulty of estimating an arbitrary valid transport map is comparable to that of estimating the OT map itself. However, when the stability assumption fails, alternative valid maps can substantially improve estimation accuracy, outperforming the OT-based approach.

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 lack of statistical guarantees and the neglect of mass variation in existing Monge-type estimators for unbalanced optimal transport. The authors reformulate the objective as a transport–growth pair and develop estimators tailored to both general and smooth density settings, leveraging optimal transport plans and kernel methods, respectively. By introducing a value stability reduction technique and integrating a quadratic-cost model with KL marginal penalization, kernel density estimation, and minimax lower bound analysis, they establish—for the first time—that the proposed estimators achieve minimax-optimal convergence rates across multiple regimes. This provides a rigorous statistical foundation for Monge-type estimation in unbalanced optimal transport.

This work addresses the challenge of low-rank optimal transport (OT), which, despite its ability to reveal latent data structures and enhance statistical stability, is notoriously non-convex and NP-hard. The authors propose the first reduction of this problem to a clustering task, introducing a novel "transport clustering" algorithm: it first computes a full-rank OT solution to obtain correspondences and then clusters these to construct a low-rank transport plan. The method achieves a constant-factor approximation in polynomial time and provides theoretical approximation guarantees under negative-type metrics and kernel-based costs. Empirical evaluations demonstrate that the algorithm significantly outperforms existing low-rank OT solvers on both synthetic and large-scale high-dimensional datasets, offering a favorable combination of computational efficiency and theoretical rigor.